Subgroups — C₂×#22573878 #31764144

Order 17152.

12366 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524089	C₂	1	1	yes	maximal, characteristic, complemented
2	#22573878	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16864065	C₂	1	2	yes	maximal, complemented
4	C₂×#16864057	C₂	1	2	yes	maximal, complemented
5	C₂×#16864034	C₂	1	6	yes	maximal, complemented
6	C₂×D₄×C₂²⋊D₆₇	C₂	1	6	yes	maximal, complemented
7	C₂×#16863974	C₂	1	3	yes	maximal, complemented
8	C₂×#16863960	C₂	1	3	yes	maximal, complemented
9	C₂×#16863907	C₂	1	3	yes	maximal, complemented
10	C₂²×#14524017	C₂	1	3	yes	maximal, complemented
11	C₂×#16863879	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#14524089	—	2	16	no	—
14	C₂×#14524089	C₂²	1	16	yes	complemented
15	C₂²×#11761629	C₂²	1	2	yes	complemented
16	C₂²×#11761628	C₂²	1	3	yes	complemented
17	C₂³×D₂₆₈	C₂²	1	3	yes	complemented
18	C₂³×C₂²⋊D₆₇	C₂²	1	6	yes	complemented
19	C₂²×D₄×D₆₇	C₂²	1	6	yes	complemented
20	#16864065	C₂²	1	32	yes	complemented
21	#16864057	C₂²	1	32	yes	complemented
22	#16864034	C₂²	1	96	yes	complemented
23	D₄×C₂²⋊D₆₇	C₂²	1	96	yes	complemented
24	#16863974	C₂²	1	48	yes	complemented
25	#16863960	C₂²	1	48	yes	complemented
26	C₂×#14524039	C₂²	1	6	yes	complemented
27	C₂×#14524032	C₂²	1	6	yes	complemented
28	#16863907	C₂²	1	48	yes	complemented
29	C₂×#14524026	C₂²	1	6	yes	complemented
30	C₂×#14524022	C₂²	1	1	yes	characteristic, complemented
31	C₂×#14524022	C₂²	1	3	yes	complemented
32	C₂×#14524019	C₂²	1	6	yes	complemented
33	C₂×#14524017	C₂²	1	6	yes	complemented
34	C₂×#14524017	C₂²	1	6	yes	complemented
35	C₂×#14524017	C₂²	1	3	yes	complemented
36	C₂×#14524017	C₂²	1	6	yes	complemented
37	C₂×#14524017	C₂²	1	3	yes	complemented
38	C₂×#14524017	C₂²	1	48	yes	complemented
39	C₂×#14524015	C₂²	1	12	yes	complemented
40	C₂×#14524014	C₂²	1	6	yes	complemented
41	C₂×#14524012	C₂²	1	6	yes	complemented
42	C₂×#14524011	C₂²	1	12	yes	complemented
43	#16863879	C₂²	1	16	yes	complemented
44	C₂×Q₈×Dic₆₇	C₂²	1	2	yes	complemented
45	C₂×D₄×Dic₆₇	C₂²	1	6	yes	complemented
46	C₂×C₄×C₂²⋊D₆₇	C₂²	1	6	yes	complemented
47	C₂×C₄×C₂²⋊D₆₇	C₂²	1	6	yes	complemented
48	C₂×#14524002	C₂²	1	3	yes	complemented
49	C₂²×C₄⋊Dic₆₇	C₂²	1	3	yes	complemented
50	C₂²×(C₂×D₆₇)⋊C₄	C₂²	1	6	yes	complemented
51	C₂²×#41×C₆₇	—	2	16	no	—
52	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
53	C₂²×#41×C₆₇	C₂²	1	3	yes	complemented
54	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic, complemented
55	C₂²×#41×C₆₇	C₂²	1	3	yes	complemented
56	C₂²×#41×C₆₇	C₂²	1	16	yes	complemented
57	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
58	C₂³×D₄×C₆₇	C₂²	1	3	yes	complemented
59	C₂⁴×C₄×C₆₇	C₂²	1	3	yes	complemented
60	C₂⁴×D₆₇	C₂³	1	6	yes	—
61	#14524089	—	2	128	no	—
62	C₂×#11761629	—	2	16	no	—
63	C₂×#11761629	C₂³	1	16	yes	complemented
64	C₂×#11761628	—	2	24	no	—
65	C₂×#11761628	C₂³	1	24	yes	complemented
66	C₂²×Dic₁₃₄	C₂³	1	1	yes	characteristic
67	C₂²×D₂₆₈	—	2	24	no	—
68	C₂²×D₂₆₈	C₂³	1	3	yes	—
69	C₂²×D₂₆₈	C₂³	1	6	yes	—
70	C₂²×D₂₆₈	C₂³	1	24	yes	complemented
71	C₂²×C₂²⋊D₆₇	—	4	24	no	—
72	C₂²×C₂²⋊D₆₇	C₂³	1	6	yes	—
73	C₂²×C₂²⋊D₆₇	C₂³	1	6	yes	—
74	C₂²×C₂²⋊D₆₇	C₂³	1	6	yes	—
75	C₂²×C₂²⋊D₆₇	C₂³	1	12	yes	—
76	C₂²×C₂²⋊D₆₇	C₂³	1	48	yes	complemented
77	C₂×D₄×D₆₇	—	2	48	no	—
78	C₂×D₄×D₆₇	C₂³	1	48	yes	complemented
79	C₂³×Dic₆₇	C₂³	1	6	yes	—
80	C₂²×C₄×D₆₇	C₂³	1	6	yes	—
81	#14524039	C₂³	1	48	yes	complemented
82	#14524032	C₂³	1	48	yes	complemented
83	#14524026	C₂³	1	48	yes	complemented
84	#14524022	C₂³	1	8	yes	complemented
85	#14524022	C₂³	1	24	yes	complemented
86	#14524019	C₂³	1	48	yes	complemented
87	#14524017	C₂³	1	48	yes	complemented
88	#14524017	C₂³	1	48	yes	complemented
89	#14524017	C₂³	1	24	yes	complemented
90	#14524017	C₂³	1	48	yes	complemented
91	#14524017	C₂³	1	24	yes	complemented
92	#14524015	C₂³	1	96	yes	complemented
93	#14524014	C₂³	1	48	yes	complemented
94	#14524012	C₂³	1	48	yes	complemented
95	#14524011	C₂³	1	96	yes	complemented
96	Q₈×Dic₆₇	C₂³	1	16	yes	complemented
97	D₄×Dic₆₇	C₂³	1	48	yes	complemented
98	C₄×C₂²⋊D₆₇	C₂³	1	48	yes	complemented
99	C₄×C₂²⋊D₆₇	C₂³	1	48	yes	complemented
100	#14524002	C₂³	1	24	yes	complemented
101	C₂×C₄⋊Dic₆₇	C₂³	1	3	yes	—
102	C₂×C₄⋊Dic₆₇	C₂³	1	6	yes	—
103	C₂×C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊Dic₆₇	C₂³	1	24	yes	complemented
105	C₂×Dic₆₇⋊C₄	C₂³	1	2	yes	—
106	C₂×Dic₆₇⋊C₄	C₂³	1	6	yes	—
107	C₂×C₂²⋊Dic₆₇	C₂³	1	6	yes	—
108	C₂×C₂²⋊Dic₆₇	C₂³	1	2	yes	—
109	C₂×C₂²⋊Dic₆₇	C₂³	1	6	yes	—
110	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	—
111	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	12	yes	—
112	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	—
113	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	48	yes	complemented
114	C₂×C₄×Dic₆₇	C₂³	1	6	yes	—
115	C₂⁵×C₆₇	C₂³	1	3	yes	—
116	C₂×#41×C₆₇	—	2	128	no	—
117	C₂×#41×C₆₇	—	2	8	no	—
118	C₂×#41×C₆₇	—	2	24	no	—
119	C₂×#41×C₆₇	—	2	24	no	—
120	C₂×#41×C₆₇	D₄	1	16	yes	complemented
121	C₂×#41×C₆₇	C₂³	1	8	yes	—
122	C₂×#41×C₆₇	C₂³	1	24	yes	complemented
123	C₂×#41×C₆₇	C₂³	1	8	yes	complemented
124	C₂×#41×C₆₇	C₂³	1	24	yes	complemented
125	C₂²×Q₈×C₆₇	—	2	8	no	—
126	C₂²×Q₈×C₆₇	C₂³	1	3	yes	—
127	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
128	C₂²×Q₈×C₆₇	C₂³	1	8	yes	complemented
129	C₂²×D₄×C₆₇	—	2	24	no	—
130	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
131	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
132	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
133	C₂²×D₄×C₆₇	C₂³	1	24	yes	complemented
134	C₂³×C₄×C₆₇	—	2	24	no	—
135	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
136	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
137	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
138	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
139	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
140	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
141	C₂³×C₄×C₆₇	C₂³	1	24	yes	complemented
142	C₂³×D₆₇	—	2	24	no	—
143	C₂³×D₆₇	—	4	12	no	—
144	C₂³×D₆₇	—	4	12	no	—
145	C₂³×D₆₇	C₂⁴	1	6	yes	—
146	C₂³×D₆₇	C₂⁴	1	24	yes	—
147	#11761629	—	2	64	no	—
148	#11761628	—	2	96	no	—
149	C₂×Dic₁₃₄	—	2	4	no	—
150	C₂×Dic₁₃₄	C₂⁴	1	4	yes	—
151	C₂×D₂₆₈	—	2	24	no	—
152	C₂×D₂₆₈	—	2	12	no	—
153	C₂×D₂₆₈	—	2	96	no	—
154	C₂×D₂₆₈	C₂⁴	1	12	yes	—
155	C₂×D₂₆₈	C₂⁴	1	24	yes	—
156	C₂×C₂²⋊D₆₇	—	2	24	no	—
157	C₂×C₂²⋊D₆₇	—	2	24	no	—
158	C₂×C₂²⋊D₆₇	—	4	24	no	—
159	C₂×C₂²⋊D₆₇	—	4	12	no	—
160	C₂×C₂²⋊D₆₇	—	4	96	no	—
161	C₂×C₂²⋊D₆₇	C₂⁴	1	24	yes	—
162	C₂×C₂²⋊D₆₇	C₂⁴	1	24	yes	—
163	C₂×C₂²⋊D₆₇	C₂⁴	1	24	yes	—
164	C₂×C₂²⋊D₆₇	C₂⁴	1	48	yes	—
165	D₄×D₆₇	—	2	192	no	—
166	C₂²×Dic₆₇	—	4	12	no	—
167	C₂²×Dic₆₇	C₂⁴	1	2	yes	—
168	C₂²×Dic₆₇	C₂⁴	1	6	yes	—
169	C₂²×Dic₆₇	C₂⁴	1	2	yes	—
170	C₂²×Dic₆₇	C₂⁴	1	24	yes	—
171	C₂×C₄×D₆₇	—	2	24	no	—
172	C₂×C₄×D₆₇	C₂⁴	1	24	yes	—
173	C₄⋊Dic₆₇	C₂⁴	1	12	yes	—
174	C₄⋊Dic₆₇	C₂⁴	1	24	yes	—
175	C₄⋊Dic₆₇	C₂⁴	1	4	yes	—
176	Dic₆₇⋊C₄	C₂⁴	1	8	yes	—
177	Dic₆₇⋊C₄	C₂⁴	1	24	yes	—
178	C₂²⋊Dic₆₇	C₂⁴	1	24	yes	—
179	C₂²⋊Dic₆₇	C₂⁴	1	8	yes	—
180	C₂²⋊Dic₆₇	C₂⁴	1	24	yes	—
181	(C₂×D₆₇)⋊C₄	C₂⁴	1	24	yes	—
182	(C₂×D₆₇)⋊C₄	C₂⁴	1	48	yes	—
183	(C₂×D₆₇)⋊C₄	C₂⁴	1	24	yes	—
184	C₄×Dic₆₇	C₂⁴	1	24	yes	—
185	C₂⁴×C₆₇	—	2	12	no	—
186	C₂⁴×C₆₇	—	2	12	no	—
187	C₂⁴×C₆₇	—	4	6	no	—
188	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
189	C₂⁴×C₆₇	C₂⁴	1	12	yes	—
190	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
191	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
192	#41×C₆₇	—	2	32	no	—
193	#41×C₆₇	—	2	96	no	—
194	#41×C₆₇	—	2	96	no	—
195	#41×C₆₇	C₂×D₄	1	64	yes	complemented
196	C₂×Q₈×C₆₇	—	2	32	no	—
197	C₂×Q₈×C₆₇	—	2	12	no	—
198	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	complemented
199	C₂×Q₈×C₆₇	C₂⁴	1	12	yes	—
200	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
201	C₂×D₄×C₆₇	—	2	24	no	—
202	C₂×D₄×C₆₇	—	2	96	no	—
203	C₂×D₄×C₆₇	—	2	12	no	—
204	C₂×D₄×C₆₇	C₂×D₄	1	24	yes	complemented
205	C₂×D₄×C₆₇	C₂⁴	1	24	yes	—
206	C₂×D₄×C₆₇	C₂⁴	1	12	yes	—
207	C₂×D₄×C₆₇	C₂⁴	1	12	yes	—
208	C₂²×C₄×C₆₇	—	2	12	no	—
209	C₂²×C₄×C₆₇	—	2	4	no	—
210	C₂²×C₄×C₆₇	—	2	12	no	—
211	C₂²×C₄×C₆₇	—	2	12	no	—
212	C₂²×C₄×C₆₇	—	2	12	no	—
213	C₂²×C₄×C₆₇	—	2	96	no	—
214	C₂²×C₄×C₆₇	C₂×D₄	1	24	yes	complemented
215	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
216	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
217	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
218	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
219	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
220	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
221	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
222	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
223	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
224	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
225	C₂³×C₆₇	—	2	2	no	—
226	C₂³×C₆₇	—	2	6	no	—
227	C₂³×C₆₇	—	2	2	no	—
228	C₂³×C₆₇	—	2	6	no	—
229	C₂³×C₆₇	—	2	24	no	—
230	C₂³×C₆₇	—	2	6	no	—
231	C₂³×C₆₇	—	2	6	no	—
232	C₂³×C₆₇	—	2	24	no	—
233	C₂³×C₆₇	—	2	6	no	—
234	C₂³×C₆₇	—	4	12	no	—
235	C₂³×C₆₇	—	4	48	no	—
236	C₂³×C₆₇	C₂²×D₄	1	12	yes	—
237	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	complemented
238	C₂³×C₆₇	C₂⁵	1	6	yes	—
239	C₂³×C₆₇	C₂⁵	1	6	yes	—
240	C₂³×C₆₇	C₂⁵	1	2	yes	—
241	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
242	C₂²×D₆₇	—	2	12	no	—
243	C₂²×D₆₇	—	2	12	no	—
244	C₂²×D₆₇	—	2	12	no	—
245	C₂²×D₆₇	—	2	48	no	—
246	C₂²×D₆₇	—	4	24	no	—
247	C₂²×D₆₇	—	4	24	no	—
248	C₂²×D₆₇	—	8	48	no	—
249	C₂²×D₆₇	C₂⁵	1	12	yes	—
250	Q₈×C₆₇	—	2	24	no	—
251	Q₈×C₆₇	C₂²×D₄	1	16	yes	complemented
252	D₄×C₆₇	—	2	48	no	—
253	D₄×C₆₇	—	2	24	no	—
254	D₄×C₆₇	C₂²×D₄	1	48	yes	complemented
255	C₂×C₄×C₆₇	—	2	2	no	—
256	C₂×C₄×C₆₇	—	2	6	no	—
257	C₂×C₄×C₆₇	—	2	24	no	—
258	C₂×C₄×C₆₇	—	2	8	no	—
259	C₂×C₄×C₆₇	—	2	24	no	—
260	C₂×C₄×C₆₇	—	2	24	no	—
261	C₂×C₄×C₆₇	—	2	24	no	—
262	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
263	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
264	C₂×C₄×C₆₇	C₂²×D₄	1	48	yes	complemented
265	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
266	C₂×C₄×C₆₇	C₂⁵	1	6	yes	—
267	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
268	C₂×C₄×C₆₇	C₂⁵	1	6	yes	—
269	C₂²⋊D₆₇	—	2	48	no	—
270	C₂²⋊D₆₇	—	2	48	no	—
271	C₂²⋊D₆₇	—	4	48	no	—
272	C₂²⋊D₆₇	—	4	24	no	—
273	C₂×Dic₆₇	—	2	4	no	—
274	C₂×Dic₆₇	—	2	12	no	—
275	C₂×Dic₆₇	—	2	4	no	—
276	C₂×Dic₆₇	—	4	24	no	—
277	C₂×Dic₆₇	C₂⁵	1	4	yes	—
278	C₂×Dic₆₇	C₂⁵	1	12	yes	—
279	C₂×Dic₆₇	C₂⁵	1	4	yes	—
280	D₂₆₈	—	2	48	no	—
281	D₂₆₈	—	2	24	no	—
282	C₄×D₆₇	—	2	48	no	—
283	Dic₁₃₄	—	2	8	no	—
284	C₂²×C₆₇	—	2	2	no	—
285	C₂²×C₆₇	—	2	6	no	—
286	C₂²×C₆₇	—	2	6	no	—
287	C₂²×C₆₇	—	2	2	no	—
288	C₂²×C₆₇	—	2	24	no	—
289	C₂²×C₆₇	—	2	8	no	—
290	C₂²×C₆₇	—	2	6	no	—
291	C₂²×C₆₇	—	2	6	no	—
292	C₂²×C₆₇	—	2	6	no	—
293	C₂²×C₆₇	—	4	48	no	—
294	C₂²×C₆₇	—	4	12	no	—
295	C₂²×C₆₇	C₂³×D₄	1	12	yes	—
296	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
297	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
298	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	complemented
299	C₂²×C₆₇	#585	1	2	yes	—
300	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
301	C₂×D₆₇	—	2	12	no	—
302	C₂×D₆₇	—	2	12	no	—
303	C₂×D₆₇	—	2	12	no	—
304	C₂×D₆₇	—	4	24	no	—
305	C₂×D₆₇	—	8	48	no	—
306	C₄×C₆₇	—	2	2	no	—
307	C₄×C₆₇	—	2	6	no	—
308	C₄×C₆₇	C₂³×D₄	1	4	yes	—
309	C₄×C₆₇	C₂³×D₄	1	12	yes	—
310	Dic₆₇	—	2	4	no	—
311	Dic₆₇	—	2	12	no	—
312	Dic₆₇	—	2	4	no	—
313	C₂×#3488	—	67	1	no	maximal, Sylow
314	C₂×C₆₇	—	2	12	no	—
315	C₂×C₆₇	—	2	4	no	—
316	C₂×C₆₇	—	4	6	no	—
317	C₂×C₆₇	#3488	1	4	yes	complemented
318	C₂×C₆₇	C₂⁴×D₄	1	1	yes	characteristic
319	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
320	C₂×C₆₇	C₂×#585	1	1	yes	characteristic
321	D₆₇	—	4	12	no	—
322	C₂²×H(5,𝔽₂)	—	67	1	no	—
323	C₂³×#41	D₆₇	1	1	yes	characteristic, complemented
324	#3488	—	67	32	no	—
325	C₂×#550	—	67	2	no	—
326	C₂×#548	—	67	2	no	—
327	C₂×#546	—	67	6	no	—
328	C₂×D₄²	—	67	6	no	—
329	C₂×#537	—	67	3	no	—
330	C₂×#535	—	67	3	no	—
331	C₂×#525	—	67	3	no	—
332	C₂²×#121	—	67	3	no	—
333	C₂×#520	—	67	1	no	—
334	C₆₇	C₂×#3488	1	1	yes	characteristic, complemented, Sylow
335	C₂×H(5,𝔽₂)	—	67	16	no	—
336	C₂×H(5,𝔽₂)	—	134	16	no	—
337	C₂²×#41	—	2	16	no	—
338	C₂²×#41	—	67	2	no	—
339	C₂²×#41	—	67	3	no	—
340	C₂²×#41	C₂×D₆₇	1	1	yes	characteristic
341	C₂²×#41	C₂×D₆₇	1	3	yes	complemented
342	C₂²×#41	C₂×D₆₇	1	3	yes	complemented
343	C₂²×#41	C₂×D₆₇	1	1	yes	characteristic, complemented
344	C₂²×#41	C₂×D₆₇	1	16	yes	complemented
345	C₂³×Q₈	C₂×D₆₇	1	1	yes	characteristic, complemented
346	C₂³×D₄	—	67	6	no	—
347	C₂³×D₄	—	67	6	no	—
348	C₂³×D₄	—	67	3	no	—
349	C₂³×D₄	C₂×D₆₇	1	3	yes	complemented
350	C₂⁴×C₄	C₂×D₆₇	1	3	yes	complemented
351	#550	—	67	32	no	—
352	#548	—	67	32	no	—
353	#546	—	67	96	no	—
354	D₄²	—	67	96	no	—
355	#537	—	67	48	no	—
356	#535	—	67	48	no	—
357	C₂×#127	—	67	6	no	—
358	C₂×#124	—	67	6	no	—
359	#525	—	67	48	no	—
360	C₂×#123	—	67	6	no	—
361	C₂×C₂²⋊Q₈	—	67	6	no	—
362	C₂×C₂²⋊Q₈	—	67	1	no	—
363	C₂×C₂²⋊Q₈	—	67	3	no	—
364	C₂×#121	—	67	12	no	—
365	C₂×#121	—	67	6	no	—
366	C₂×#121	—	67	3	no	—
367	C₂×#121	—	67	6	no	—
368	C₂×#121	—	67	6	no	—
369	C₂×#121	—	67	6	no	—
370	C₂×#121	—	67	3	no	—
371	C₂×#121	—	67	48	no	—
372	C₂×#120	—	67	6	no	—
373	C₂×#120	—	67	12	no	—
374	#520	—	67	16	no	—
375	C₂×C₄×Q₈	—	67	2	no	—
376	C₂×C₄×D₄	—	67	6	no	—
377	C₂×C₄×D₄	—	67	6	no	—
378	C₂×C₄×D₄	—	67	6	no	—
379	C₂×#117	—	67	3	no	—
380	C₂²×C₄⋊C₄	—	67	3	no	—
381	C₂²×C₂²⋊C₄	—	67	6	no	—
382	C₂⁵	—	67	6	no	—
383	C₂⁵	C₂²×D₆₇	1	3	yes	—
384	H(5,𝔽₂)	—	134	128	no	—
385	C₂×#41	—	2	128	no	—
386	C₂×#41	—	2	8	no	—
387	C₂×#41	—	2	24	no	—
388	C₂×#41	—	2	24	no	—
389	C₂×#41	—	67	16	no	—
390	C₂×#41	—	67	24	no	—
391	C₂×#41	—	134	16	no	—
392	C₂×#41	—	134	24	no	—
393	C₂×#41	C₂²⋊D₆₇	1	16	yes	complemented
394	C₂×#41	C₂²×D₆₇	1	8	yes	—
395	C₂×#41	C₂²×D₆₇	1	24	yes	complemented
396	C₂×#41	C₂²×D₆₇	1	24	yes	complemented
397	C₂×#41	C₂²×D₆₇	1	8	yes	complemented
398	C₂²×Q₈	—	2	8	no	—
399	C₂²×Q₈	—	67	1	no	—
400	C₂²×Q₈	C₂²×D₆₇	1	3	yes	—
401	C₂²×Q₈	C₂²×D₆₇	1	1	yes	characteristic
402	C₂²×Q₈	C₂²×D₆₇	1	8	yes	complemented
403	C₂²×D₄	—	2	24	no	—
404	C₂²×D₄	—	268	24	no	—
405	C₂²×D₄	—	67	6	no	—
406	C₂²×D₄	—	67	12	no	—
407	C₂²×D₄	—	67	48	no	—
408	C₂²×D₄	—	67	6	no	—
409	C₂²×D₄	—	67	6	no	—
410	C₂²×D₄	—	67	48	no	—
411	C₂²×D₄	—	67	3	no	—
412	C₂²×D₄	—	67	6	no	—
413	C₂²×D₄	—	67	24	no	—
414	C₂²×D₄	—	134	48	no	—
415	C₂²×D₄	—	134	24	no	—
416	C₂²×D₄	C₂²×D₆₇	1	6	yes	—
417	C₂²×D₄	C₂²×D₆₇	1	3	yes	—
418	C₂²×D₄	C₂²×D₆₇	1	3	yes	—
419	C₂²×D₄	C₂²×D₆₇	1	24	yes	complemented
420	C₂³×C₄	—	2	24	no	—
421	C₂³×C₄	—	67	6	no	—
422	C₂³×C₄	—	67	6	no	—
423	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
424	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
425	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
426	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
427	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
428	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
429	C₂³×C₄	C₂²×D₆₇	1	24	yes	complemented
430	#127	—	67	48	no	—
431	#124	—	67	48	no	—
432	#123	—	67	48	no	—
433	C₂²⋊Q₈	—	67	48	no	—
434	C₂²⋊Q₈	—	67	8	no	—
435	C₂²⋊Q₈	—	67	24	no	—
436	#121	—	67	96	no	—
437	#121	—	67	48	no	—
438	#121	—	67	24	no	—
439	#121	—	67	48	no	—
440	#121	—	67	48	no	—
441	#121	—	67	48	no	—
442	#121	—	67	24	no	—
443	#120	—	67	48	no	—
444	#120	—	67	96	no	—
445	C₄×Q₈	—	67	16	no	—
446	C₄×D₄	—	67	48	no	—
447	C₄×D₄	—	67	48	no	—
448	C₄×D₄	—	67	48	no	—
449	#117	—	67	24	no	—
450	C₂×C₄⋊C₄	—	67	3	no	—
451	C₂×C₄⋊C₄	—	67	6	no	—
452	C₂×C₄⋊C₄	—	67	24	no	—
453	C₂×C₄⋊C₄	—	67	1	no	—
454	C₂×C₄⋊C₄	—	67	2	no	—
455	C₂×C₄⋊C₄	—	67	6	no	—
456	C₂×C₂²⋊C₄	—	67	6	no	—
457	C₂×C₂²⋊C₄	—	67	2	no	—
458	C₂×C₂²⋊C₄	—	67	6	no	—
459	C₂×C₂²⋊C₄	—	67	6	no	—
460	C₂×C₂²⋊C₄	—	67	12	no	—
461	C₂×C₂²⋊C₄	—	67	48	no	—
462	C₂×C₂²⋊C₄	—	67	6	no	—
463	C₂×C₄²	—	67	6	no	—
464	C₂⁴	—	2	12	no	—
465	C₂⁴	—	2	12	no	—
466	C₂⁴	—	4	6	no	—
467	C₂⁴	—	268	12	no	—
468	C₂⁴	—	268	12	no	—
469	C₂⁴	—	67	24	no	—
470	C₂⁴	—	67	6	no	—
471	C₂⁴	—	134	24	no	—
472	C₂⁴	C₂³×D₆₇	1	3	yes	—
473	C₂⁴	C₂³×D₆₇	1	12	yes	—
474	C₂⁴	C₂³×D₆₇	1	3	yes	—
475	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
476	#41	—	2	32	no	—
477	#41	—	2	96	no	—
478	#41	—	2	96	no	—
479	#41	—	134	64	no	—
480	#41	—	134	96	no	—
481	#41	C₂×C₂²⋊D₆₇	1	64	yes	complemented
482	C₂×Q₈	—	2	32	no	—
483	C₂×Q₈	—	2	12	no	—
484	C₂×Q₈	—	67	4	no	—
485	C₂×Q₈	—	134	4	no	—
486	C₂×Q₈	C₂×C₂²⋊D₆₇	1	8	yes	complemented
487	C₂×Q₈	C₂³×D₆₇	1	12	yes	—
488	C₂×Q₈	C₂³×D₆₇	1	4	yes	—
489	C₂×D₄	—	2	24	no	—
490	C₂×D₄	—	2	96	no	—
491	C₂×D₄	—	2	12	no	—
492	C₂×D₄	—	268	96	no	—
493	C₂×D₄	—	268	24	no	—
494	C₂×D₄	—	268	12	no	—
495	C₂×D₄	—	67	24	no	—
496	C₂×D₄	—	67	48	no	—
497	C₂×D₄	—	67	24	no	—
498	C₂×D₄	—	67	24	no	—
499	C₂×D₄	—	67	12	no	—
500	C₂×D₄	—	67	24	no	—
501	C₂×D₄	—	134	24	no	—
502	C₂×D₄	—	134	24	no	—
503	C₂×D₄	—	134	192	no	—
504	C₂×D₄	—	134	12	no	—
505	C₂×D₄	—	134	24	no	—
506	C₂×D₄	—	134	96	no	—
507	C₂×D₄	C₂×C₂²⋊D₆₇	1	24	yes	complemented
508	C₂×D₄	C₂³×D₆₇	1	24	yes	—
509	C₂×D₄	C₂³×D₆₇	1	12	yes	—
510	C₂×D₄	C₂³×D₆₇	1	12	yes	—
511	C₂²×C₄	—	2	12	no	—
512	C₂²×C₄	—	2	4	no	—
513	C₂²×C₄	—	2	12	no	—
514	C₂²×C₄	—	2	12	no	—
515	C₂²×C₄	—	2	12	no	—
516	C₂²×C₄	—	2	96	no	—
517	C₂²×C₄	—	268	12	no	—
518	C₂²×C₄	—	67	24	no	—
519	C₂²×C₄	—	67	2	no	—
520	C₂²×C₄	—	67	6	no	—
521	C₂²×C₄	—	67	2	no	—
522	C₂²×C₄	—	67	24	no	—
523	C₂²×C₄	—	134	24	no	—
524	C₂²×C₄	C₂×C₂²⋊D₆₇	1	24	yes	complemented
525	C₂²×C₄	C₂³×D₆₇	1	12	yes	—
526	C₂²×C₄	C₂³×D₆₇	1	12	yes	—
527	C₂²×C₄	C₂³×D₆₇	1	12	yes	—
528	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
529	C₂²×C₄	C₂³×D₆₇	1	12	yes	—
530	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
531	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
532	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
533	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
534	C₂²×C₄	C₂³×D₆₇	1	12	yes	—
535	C₄⋊C₄	—	67	12	no	—
536	C₄⋊C₄	—	67	24	no	—
537	C₄⋊C₄	—	67	4	no	—
538	C₄⋊C₄	—	67	8	no	—
539	C₄⋊C₄	—	67	24	no	—
540	C₂²⋊C₄	—	67	24	no	—
541	C₂²⋊C₄	—	67	8	no	—
542	C₂²⋊C₄	—	67	24	no	—
543	C₂²⋊C₄	—	67	24	no	—
544	C₂²⋊C₄	—	67	48	no	—
545	C₂²⋊C₄	—	67	24	no	—
546	C₄²	—	67	24	no	—
547	C₂³	—	2	2	no	—
548	C₂³	—	2	6	no	—
549	C₂³	—	2	2	no	—
550	C₂³	—	2	6	no	—
551	C₂³	—	2	24	no	—
552	C₂³	—	2	6	no	—
553	C₂³	—	2	6	no	—
554	C₂³	—	2	24	no	—
555	C₂³	—	2	6	no	—
556	C₂³	—	4	12	no	—
557	C₂³	—	4	48	no	—
558	C₂³	—	268	24	no	—
559	C₂³	—	268	24	no	—
560	C₂³	—	536	48	no	—
561	C₂³	—	67	12	no	—
562	C₂³	—	134	12	no	—
563	C₂³	—	134	12	no	—
564	C₂³	—	134	12	no	—
565	C₂³	—	134	48	no	—
566	C₂³	C₂²×C₂²⋊D₆₇	1	12	yes	—
567	C₂³	#14524089	1	4	yes	complemented
568	C₂³	C₂⁴×D₆₇	1	6	yes	—
569	C₂³	C₂⁴×D₆₇	1	6	yes	—
570	C₂³	C₂⁴×D₆₇	1	2	yes	—
571	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
572	Q₈	—	2	24	no	—
573	Q₈	—	134	8	no	—
574	Q₈	C₂²×C₂²⋊D₆₇	1	16	yes	complemented
575	D₄	—	2	48	no	—
576	D₄	—	2	24	no	—
577	D₄	—	268	48	no	—
578	D₄	—	268	24	no	—
579	D₄	—	134	48	no	—
580	D₄	—	134	48	no	—
581	D₄	—	134	24	no	—
582	D₄	—	134	48	no	—
583	D₄	C₂²×C₂²⋊D₆₇	1	48	yes	complemented
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	6	no	—
586	C₂×C₄	—	2	24	no	—
587	C₂×C₄	—	2	8	no	—
588	C₂×C₄	—	2	24	no	—
589	C₂×C₄	—	2	24	no	—
590	C₂×C₄	—	2	24	no	—
591	C₂×C₄	—	268	24	no	—
592	C₂×C₄	—	67	4	no	—
593	C₂×C₄	—	67	12	no	—
594	C₂×C₄	—	67	4	no	—
595	C₂×C₄	—	134	4	no	—
596	C₂×C₄	—	134	12	no	—
597	C₂×C₄	—	134	4	no	—
598	C₂×C₄	—	134	48	no	—
599	C₂×C₄	C₂²×C₂²⋊D₆₇	1	4	yes	—
600	C₂×C₄	C₂²×C₂²⋊D₆₇	1	12	yes	—
601	C₂×C₄	C₂²×C₂²⋊D₆₇	1	48	yes	complemented
602	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
603	C₂×C₄	C₂⁴×D₆₇	1	6	yes	—
604	C₂×C₄	C₂⁴×D₆₇	1	6	yes	—
605	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
606	C₂²	—	2	8	no	—
607	C₂²	—	2	24	no	—
608	C₂²	—	2	2	no	—
609	C₂²	—	2	6	no	—
610	C₂²	—	2	2	no	—
611	C₂²	—	2	6	no	—
612	C₂²	—	2	6	no	—
613	C₂²	—	2	6	no	—
614	C₂²	—	2	6	no	—
615	C₂²	—	4	48	no	—
616	C₂²	—	4	12	no	—
617	C₂²	—	268	24	no	—
618	C₂²	—	536	48	no	—
619	C₂²	—	134	12	no	—
620	C₂²	—	134	12	no	—
621	C₂²	—	134	12	no	—
622	C₂²	C₂³×C₂²⋊D₆₇	1	12	yes	—
623	C₂²	C₂³×C₂²⋊D₆₇	1	2	yes	—
624	C₂²	C₂×#14524089	1	2	yes	—
625	C₂²	C₂×#14524089	1	4	yes	complemented
626	C₂²	#16864232	1	2	yes	—
627	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
628	C₄	—	2	2	no	—
629	C₄	—	2	6	no	—
630	C₄	—	134	4	no	—
631	C₄	—	134	12	no	—
632	C₄	—	134	4	no	—
633	C₄	C₂³×C₂²⋊D₆₇	1	4	yes	—
634	C₄	C₂³×C₂²⋊D₆₇	1	12	yes	—
635	C₂	—	2	4	no	—
636	C₂	—	2	12	no	—
637	C₂	—	4	6	no	—
638	C₂	—	268	12	no	—
639	C₂	#22573878	1	4	yes	direct product factor
640	C₂	C₂⁴×C₂²⋊D₆₇	1	1	yes	characteristic
641	C₂	C₂²×#14524089	1	1	yes	characteristic
642	C₂	C₂×#16864232	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂³×#41×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁵
hypercenter	C₂³×#41	D₆₇
hypocenter	C₆₇	C₂×#3488
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×#41

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×#14524089	C₂	semidirect
2. #22573878	C₂	direct (N × H)
3. C₂×#16864065	C₂	semidirect
4. C₂×#16864057	C₂	semidirect
5. C₂×#16864034	C₂	semidirect
6. C₂×D₄×C₂²⋊D₆₇	C₂	semidirect
7. C₂×#16863974	C₂	semidirect
8. C₂×#16863960	C₂	semidirect
9. C₂×#16863907	C₂	semidirect
10. C₂²×#14524017	C₂	semidirect
11. C₂×#16863879	C₂	semidirect
12. C₂³×#41×C₆₇	C₂	semidirect
14. C₂×#14524089	C₂²	semidirect
15. C₂²×#11761629	C₂²	semidirect
16. C₂²×#11761628	C₂²	semidirect
17. C₂³×D₂₆₈	C₂²	semidirect
18. C₂³×C₂²⋊D₆₇	C₂²	semidirect
19. C₂²×D₄×D₆₇	C₂²	semidirect
20. #16864065	C₂²	semidirect
21. #16864057	C₂²	semidirect
22. #16864034	C₂²	semidirect
23. D₄×C₂²⋊D₆₇	C₂²	semidirect
24. #16863974	C₂²	semidirect
25. #16863960	C₂²	semidirect
26. C₂×#14524039	C₂²	semidirect
27. C₂×#14524032	C₂²	semidirect
28. #16863907	C₂²	semidirect
29. C₂×#14524026	C₂²	semidirect
30. C₂×#14524022	C₂²	semidirect
31. C₂×#14524022	C₂²	semidirect
32. C₂×#14524019	C₂²	semidirect
33. C₂×#14524017	C₂²	semidirect
34. C₂×#14524017	C₂²	semidirect
35. C₂×#14524017	C₂²	semidirect
36. C₂×#14524017	C₂²	semidirect
37. C₂×#14524017	C₂²	semidirect
38. C₂×#14524017	C₂²	semidirect
39. C₂×#14524015	C₂²	semidirect
40. C₂×#14524014	C₂²	semidirect
41. C₂×#14524012	C₂²	semidirect
42. C₂×#14524011	C₂²	semidirect
43. #16863879	C₂²	semidirect
44. C₂×Q₈×Dic₆₇	C₂²	semidirect
45. C₂×D₄×Dic₆₇	C₂²	semidirect
46. C₂×C₄×C₂²⋊D₆₇	C₂²	semidirect
47. C₂×C₄×C₂²⋊D₆₇	C₂²	semidirect
48. C₂×#14524002	C₂²	semidirect
49. C₂²×C₄⋊Dic₆₇	C₂²	semidirect
50. C₂²×(C₂×D₆₇)⋊C₄	C₂²	semidirect
53. C₂²×#41×C₆₇	C₂²	semidirect
54. C₂²×#41×C₆₇	C₂²	semidirect
55. C₂²×#41×C₆₇	C₂²	semidirect
56. C₂²×#41×C₆₇	C₂²	semidirect
57. C₂³×Q₈×C₆₇	C₂²	semidirect
58. C₂³×D₄×C₆₇	C₂²	semidirect
59. C₂⁴×C₄×C₆₇	C₂²	semidirect
63. C₂×#11761629	C₂³	semidirect
65. C₂×#11761628	C₂³	semidirect
70. C₂²×D₂₆₈	C₂³	semidirect
76. C₂²×C₂²⋊D₆₇	C₂³	semidirect
78. C₂×D₄×D₆₇	C₂³	semidirect
81. #14524039	C₂³	semidirect
82. #14524032	C₂³	semidirect
83. #14524026	C₂³	semidirect
84. #14524022	C₂³	semidirect
85. #14524022	C₂³	semidirect
86. #14524019	C₂³	semidirect
87. #14524017	C₂³	semidirect
88. #14524017	C₂³	semidirect
89. #14524017	C₂³	semidirect
90. #14524017	C₂³	semidirect
91. #14524017	C₂³	semidirect
92. #14524015	C₂³	semidirect
93. #14524014	C₂³	semidirect
94. #14524012	C₂³	semidirect
95. #14524011	C₂³	semidirect
96. Q₈×Dic₆₇	C₂³	semidirect
97. D₄×Dic₆₇	C₂³	semidirect
98. C₄×C₂²⋊D₆₇	C₂³	semidirect
99. C₄×C₂²⋊D₆₇	C₂³	semidirect
100. #14524002	C₂³	semidirect
104. C₂×C₄⋊Dic₆₇	C₂³	semidirect
113. C₂×(C₂×D₆₇)⋊C₄	C₂³	semidirect
120. C₂×#41×C₆₇	D₄	semidirect
122. C₂×#41×C₆₇	C₂³	semidirect
123. C₂×#41×C₆₇	C₂³	semidirect
124. C₂×#41×C₆₇	C₂³	semidirect
128. C₂²×Q₈×C₆₇	C₂³	semidirect
133. C₂²×D₄×C₆₇	C₂³	semidirect
141. C₂³×C₄×C₆₇	C₂³	semidirect
195. #41×C₆₇	C₂×D₄	semidirect
198. C₂×Q₈×C₆₇	C₂×D₄	semidirect
204. C₂×D₄×C₆₇	C₂×D₄	semidirect
214. C₂²×C₄×C₆₇	C₂×D₄	semidirect
237. C₂³×C₆₇	H(5,𝔽₂)	semidirect
251. Q₈×C₆₇	C₂²×D₄	semidirect
254. D₄×C₆₇	C₂²×D₄	semidirect
264. C₂×C₄×C₆₇	C₂²×D₄	semidirect
298. C₂²×C₆₇	C₂×H(5,𝔽₂)	semidirect
317. C₂×C₆₇	#3488	semidirect
323. C₂³×#41	D₆₇	semidirect
334. C₆₇	C₂×#3488	semidirect
341. C₂²×#41	C₂×D₆₇	semidirect
342. C₂²×#41	C₂×D₆₇	semidirect
343. C₂²×#41	C₂×D₆₇	semidirect
344. C₂²×#41	C₂×D₆₇	semidirect
345. C₂³×Q₈	C₂×D₆₇	semidirect
349. C₂³×D₄	C₂×D₆₇	semidirect
350. C₂⁴×C₄	C₂×D₆₇	semidirect
393. C₂×#41	C₂²⋊D₆₇	semidirect
395. C₂×#41	C₂²×D₆₇	semidirect
396. C₂×#41	C₂²×D₆₇	semidirect
397. C₂×#41	C₂²×D₆₇	semidirect
402. C₂²×Q₈	C₂²×D₆₇	semidirect
419. C₂²×D₄	C₂²×D₆₇	semidirect
429. C₂³×C₄	C₂²×D₆₇	semidirect
481. #41	C₂×C₂²⋊D₆₇	semidirect
486. C₂×Q₈	C₂×C₂²⋊D₆₇	semidirect
507. C₂×D₄	C₂×C₂²⋊D₆₇	semidirect
524. C₂²×C₄	C₂×C₂²⋊D₆₇	semidirect
567. C₂³	#14524089	semidirect
574. Q₈	C₂²×C₂²⋊D₆₇	semidirect
583. D₄	C₂²×C₂²⋊D₆₇	semidirect
601. C₂×C₄	C₂²×C₂²⋊D₆₇	semidirect
625. C₂²	C₂×#14524089	semidirect
639. C₂	#22573878	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#14524089	2
#22573878	2
C₂×#16864065	2
C₂×#16864057	2
C₂×#16864034	2
C₂×D₄×C₂²⋊D₆₇	2
C₂×#16863974	2
C₂×#16863960	2
C₂×#16863907	2
C₂²×#14524017	2
C₂×#16863879	2
C₂³×#41×C₆₇	2
C₂×#3488	67

Subgroups — C2×#22573878 #31764144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22573878 #31764144