Subgroups — C₂×#22573986 #31764150

Order 17152.

12982 conjugacy classes of subgroups.

The subgroup lattice has 596 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	2	yes	maximal, complemented
2	#22573986	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16864068	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16864065	C₂	1	2	yes	maximal, complemented
5	C₂×D₄×D₂₆₈	C₂	1	3	yes	maximal, complemented
6	C₂×#16864031	C₂	1	3	yes	maximal, complemented
7	C₂×#16864030	C₂	1	6	yes	maximal, complemented
8	C₂×#16863962	C₂	1	6	yes	maximal, complemented
9	C₂×#16863943	C₂	1	3	yes	maximal, complemented
10	C₂×#127×D₆₇	C₂	1	3	yes	maximal, complemented
11	C₂×C₄×#11761629	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#550×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#14524089	—	2	32	no	—
14	C₂×#14524089	C₂²	1	32	yes	complemented
15	C₂²×#11761629	C₂²	1	1	yes	characteristic
16	C₂²×#11761629	C₂²	1	2	yes	complemented
17	C₂²×#11761628	C₂²	1	6	yes	complemented
18	C₂³×D₂₆₈	C₂²	1	6	yes	complemented
19	C₂²×D₄×D₆₇	C₂²	1	6	yes	complemented
20	C₂²×D₄×D₆₇	C₂²	1	6	yes	complemented
21	C₂²×D₄×D₆₇	C₂²	1	6	yes	complemented
22	C₂²×D₄×D₆₇	C₂²	1	6	yes	complemented
23	#16864068	C₂²	1	16	yes	complemented
24	#16864065	C₂²	1	32	yes	complemented
25	D₄×D₂₆₈	C₂²	1	48	yes	complemented
26	#16864031	C₂²	1	48	yes	complemented
27	#16864030	C₂²	1	96	yes	complemented
28	#16863962	C₂²	1	96	yes	complemented
29	#16863943	C₂²	1	48	yes	complemented
30	C₂×#14524042	C₂²	1	1	yes	characteristic, complemented
31	C₂×#14524040	C₂²	1	3	yes	complemented
32	C₂×#14524039	C₂²	1	6	yes	complemented
33	C₂×#14524039	C₂²	1	3	yes	complemented
34	C₂×#14524039	C₂²	1	6	yes	complemented
35	#127×D₆₇	C₂²	1	48	yes	complemented
36	C₂×#14524029	C₂²	1	6	yes	complemented
37	C₂×#14524018	C₂²	1	6	yes	complemented
38	C₂×#14524018	C₂²	1	3	yes	complemented
39	C₂×#14524017	C₂²	1	6	yes	complemented
40	C₂×#14524016	C₂²	1	6	yes	complemented
41	C₂×#14524014	C₂²	1	3	yes	complemented
42	C₂×#14524014	C₂²	1	6	yes	complemented
43	C₂×#14524013	C₂²	1	12	yes	complemented
44	C₂×#14524011	C₂²	1	12	yes	complemented
45	C₄×#11761629	C₂²	1	16	yes	complemented
46	C₂×Q₈×Dic₆₇	C₂²	1	1	yes	characteristic, complemented
47	C₂×C₄×D₂₆₈	C₂²	1	3	yes	complemented
48	C₂×C₄×D₂₆₈	C₂²	1	3	yes	complemented
49	C₂×#14524005	C₂²	1	3	yes	complemented
50	C₂×C₄×C₂²⋊D₆₇	C₂²	1	6	yes	complemented
51	C₂×#14524001	C₂²	1	3	yes	complemented
52	C₂×C₄²×D₆₇	C₂²	1	3	yes	complemented
53	C₂²×#41×C₆₇	C₂²	1	2	yes	complemented
54	#550×C₆₇	C₂²	1	16	yes	complemented
55	C₂×#127×C₆₇	C₂²	1	3	yes	complemented
56	C₂×#121×C₆₇	C₂²	1	6	yes	complemented
57	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
58	C₂×C₄×D₄×C₆₇	C₂²	1	3	yes	complemented
59	C₂⁴×D₆₇	C₂³	1	12	yes	—
60	#14524089	—	2	256	no	—
61	C₂×#11761629	—	2	16	no	—
62	C₂×#11761629	D₄	1	16	yes	complemented
63	C₂×#11761629	C₂³	1	8	yes	—
64	C₂×#11761629	C₂³	1	16	yes	complemented
65	C₂×#11761628	—	2	48	no	—
66	C₂×#11761628	C₂³	1	48	yes	complemented
67	C₂²×Dic₁₃₄	C₂³	1	2	yes	—
68	C₂²×D₂₆₈	—	2	48	no	—
69	C₂²×D₂₆₈	C₂³	1	3	yes	—
70	C₂²×D₂₆₈	C₂³	1	6	yes	—
71	C₂²×D₂₆₈	C₂³	1	6	yes	—
72	C₂²×D₂₆₈	C₂³	1	6	yes	—
73	C₂²×D₂₆₈	C₂³	1	3	yes	—
74	C₂²×D₂₆₈	C₂³	1	48	yes	complemented
75	C₂²×C₂²⋊D₆₇	C₂³	1	6	yes	—
76	C₂²×C₂²⋊D₆₇	C₂³	1	6	yes	—
77	C₂²×C₂²⋊D₆₇	C₂³	1	6	yes	—
78	C₂²×C₂²⋊D₆₇	C₂³	1	12	yes	—
79	C₂×D₄×D₆₇	—	2	48	no	—
80	C₂×D₄×D₆₇	—	2	48	no	—
81	C₂×D₄×D₆₇	—	4	24	no	—
82	C₂×D₄×D₆₇	—	4	24	no	—
83	C₂×D₄×D₆₇	C₂³	1	48	yes	complemented
84	C₂×D₄×D₆₇	C₂³	1	48	yes	complemented
85	C₂×D₄×D₆₇	C₂³	1	48	yes	complemented
86	C₂×D₄×D₆₇	C₂³	1	48	yes	complemented
87	C₂²×C₄×D₆₇	C₂³	1	3	yes	—
88	C₂²×C₄×D₆₇	C₂³	1	3	yes	—
89	C₂²×C₄×D₆₇	C₂³	1	6	yes	—
90	C₂²×C₄×D₆₇	C₂³	1	3	yes	—
91	#14524042	C₂³	1	8	yes	complemented
92	#14524040	C₂³	1	24	yes	complemented
93	#14524039	C₂³	1	48	yes	complemented
94	#14524039	C₂³	1	24	yes	complemented
95	#14524039	C₂³	1	48	yes	complemented
96	#14524029	C₂³	1	48	yes	complemented
97	#14524018	C₂³	1	48	yes	complemented
98	#14524018	C₂³	1	24	yes	complemented
99	#14524017	C₂³	1	48	yes	complemented
100	#14524016	C₂³	1	48	yes	complemented
101	#14524014	C₂³	1	24	yes	complemented
102	#14524014	C₂³	1	48	yes	complemented
103	#14524013	C₂³	1	96	yes	complemented
104	#14524011	C₂³	1	96	yes	complemented
105	Q₈×Dic₆₇	C₂³	1	8	yes	complemented
106	C₄×D₂₆₈	C₂³	1	24	yes	complemented
107	C₄×D₂₆₈	C₂³	1	24	yes	complemented
108	#14524005	C₂³	1	24	yes	complemented
109	C₄×C₂²⋊D₆₇	C₂³	1	48	yes	complemented
110	#14524001	C₂³	1	24	yes	complemented
111	C₂×C₄⋊Dic₆₇	C₂³	1	3	yes	—
112	C₂×C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
113	C₂×Dic₆₇⋊C₄	C₂³	1	2	yes	—
114	C₂×C₂²⋊Dic₆₇	C₂³	1	6	yes	—
115	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	—
116	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	—
117	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	—
118	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
119	C₂×C₄×Dic₆₇	C₂³	1	3	yes	—
120	C₂×C₄×Dic₆₇	C₂³	1	3	yes	—
121	C₄²×D₆₇	C₂³	1	24	yes	complemented
122	C₂×#41×C₆₇	—	2	16	no	—
123	C₂×#41×C₆₇	C₂³	1	16	yes	complemented
124	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
125	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
126	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
127	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
128	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
129	#127×C₆₇	C₂³	1	24	yes	complemented
130	#121×C₆₇	C₂³	1	48	yes	complemented
131	C₄×Q₈×C₆₇	C₂³	1	8	yes	complemented
132	C₄×D₄×C₆₇	C₂³	1	24	yes	complemented
133	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
134	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
135	C₂×C₂²⋊C₄×C₆₇	C₂³	1	6	yes	—
136	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
137	C₂³×D₆₇	—	2	48	no	—
138	C₂³×D₆₇	—	4	24	no	—
139	C₂³×D₆₇	—	4	24	no	—
140	C₂³×D₆₇	C₂⁴	1	6	yes	—
141	C₂³×D₆₇	C₂⁴	1	3	yes	—
142	C₂³×D₆₇	C₂⁴	1	48	yes	—
143	#11761629	—	2	64	no	—
144	#11761629	C₂×D₄	1	64	yes	complemented
145	#11761628	—	2	192	no	—
146	C₂×Dic₁₃₄	—	2	8	no	—
147	C₂×Dic₁₃₄	C₂⁴	1	8	yes	—
148	C₂×D₂₆₈	—	2	24	no	—
149	C₂×D₂₆₈	—	2	24	no	—
150	C₂×D₂₆₈	—	2	12	no	—
151	C₂×D₂₆₈	—	2	192	no	—
152	C₂×D₂₆₈	—	4	12	no	—
153	C₂×D₂₆₈	C₂×D₄	1	24	yes	complemented
154	C₂×D₂₆₈	C₂⁴	1	12	yes	—
155	C₂×D₂₆₈	C₂⁴	1	24	yes	—
156	C₂×D₂₆₈	C₂⁴	1	24	yes	—
157	C₂×D₂₆₈	C₂⁴	1	24	yes	—
158	C₂×D₂₆₈	C₂⁴	1	12	yes	—
159	C₂×C₂²⋊D₆₇	—	2	24	no	—
160	C₂×C₂²⋊D₆₇	—	2	24	no	—
161	C₂×C₂²⋊D₆₇	—	2	24	no	—
162	C₂×C₂²⋊D₆₇	—	4	24	no	—
163	C₂×C₂²⋊D₆₇	C₂⁴	1	24	yes	—
164	C₂×C₂²⋊D₆₇	C₂⁴	1	24	yes	—
165	C₂×C₂²⋊D₆₇	C₂⁴	1	24	yes	—
166	C₂×C₂²⋊D₆₇	C₂⁴	1	48	yes	—
167	D₄×D₆₇	—	2	192	no	—
168	D₄×D₆₇	—	2	192	no	—
169	D₄×D₆₇	—	4	96	no	—
170	D₄×D₆₇	—	4	96	no	—
171	C₂²×Dic₆₇	C₂⁴	1	3	yes	—
172	C₂²×Dic₆₇	C₂⁴	1	2	yes	—
173	C₂²×Dic₆₇	C₂⁴	1	1	yes	characteristic
174	C₂²×Dic₆₇	C₂⁴	1	1	yes	characteristic
175	C₂×C₄×D₆₇	—	2	24	no	—
176	C₂×C₄×D₆₇	—	2	12	no	—
177	C₂×C₄×D₆₇	—	2	12	no	—
178	C₂×C₄×D₆₇	C₂×D₄	1	24	yes	complemented
179	C₂×C₄×D₆₇	C₂⁴	1	12	yes	—
180	C₂×C₄×D₆₇	C₂⁴	1	12	yes	—
181	C₂×C₄×D₆₇	C₂⁴	1	24	yes	—
182	C₂×C₄×D₆₇	C₂⁴	1	12	yes	—
183	C₄⋊Dic₆₇	C₂⁴	1	12	yes	—
184	C₄⋊Dic₆₇	C₂⁴	1	4	yes	—
185	Dic₆₇⋊C₄	C₂⁴	1	8	yes	—
186	C₂²⋊Dic₆₇	C₂⁴	1	24	yes	—
187	(C₂×D₆₇)⋊C₄	C₂⁴	1	24	yes	—
188	(C₂×D₆₇)⋊C₄	C₂⁴	1	24	yes	—
189	(C₂×D₆₇)⋊C₄	C₂⁴	1	24	yes	—
190	C₄×Dic₆₇	C₂⁴	1	4	yes	—
191	C₄×Dic₆₇	C₂⁴	1	12	yes	—
192	C₄×Dic₆₇	C₂⁴	1	12	yes	—
193	C₂⁴×C₆₇	C₂⁴	1	6	yes	—
194	#41×C₆₇	—	2	64	no	—
195	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	complemented
196	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
197	C₂×D₄×C₆₇	—	2	24	no	—
198	C₂×D₄×C₆₇	—	2	12	no	—
199	C₂×D₄×C₆₇	—	4	12	no	—
200	C₂×D₄×C₆₇	C₂⁴	1	12	yes	—
201	C₂×D₄×C₆₇	C₂⁴	1	24	yes	—
202	C₂×D₄×C₆₇	C₂⁴	1	24	yes	—
203	C₂²×C₄×C₆₇	—	2	24	no	—
204	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
205	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
206	C₂²×C₄×C₆₇	C₂⁴	1	24	yes	—
207	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
208	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
209	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
210	C₄⋊C₄×C₆₇	C₂⁴	1	12	yes	—
211	C₂²⋊C₄×C₆₇	C₂⁴	1	24	yes	—
212	C₄²×C₆₇	C₂⁴	1	12	yes	—
213	C₂³×C₆₇	—	2	12	no	—
214	C₂³×C₆₇	—	2	12	no	—
215	C₂³×C₆₇	—	2	12	no	—
216	C₂³×C₆₇	C₂⁵	1	12	yes	—
217	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
218	C₂²×D₆₇	—	2	6	no	—
219	C₂²×D₆₇	—	2	6	no	—
220	C₂²×D₆₇	—	2	12	no	—
221	C₂²×D₆₇	—	2	12	no	—
222	C₂²×D₆₇	—	2	12	no	—
223	C₂²×D₆₇	—	2	96	no	—
224	C₂²×D₆₇	—	4	48	no	—
225	C₂²×D₆₇	—	4	48	no	—
226	C₂²×D₆₇	—	8	96	no	—
227	C₂²×D₆₇	C₂²×D₄	1	12	yes	—
228	C₂²×D₆₇	C₂⁵	1	12	yes	—
229	C₂²×D₆₇	C₂⁵	1	6	yes	—
230	Q₈×C₆₇	C₂²×D₄	1	16	yes	complemented
231	D₄×C₆₇	—	2	48	no	—
232	D₄×C₆₇	—	2	24	no	—
233	D₄×C₆₇	—	4	24	no	—
234	C₂×C₄×C₆₇	—	2	4	no	—
235	C₂×C₄×C₆₇	—	2	6	no	—
236	C₂×C₄×C₆₇	—	2	48	no	—
237	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
238	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	complemented
239	C₂×C₄×C₆₇	C₂⁵	1	6	yes	—
240	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
241	C₂×C₄×C₆₇	C₂⁵	1	6	yes	—
242	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
243	C₂²⋊D₆₇	—	2	48	no	—
244	C₂²⋊D₆₇	—	2	48	no	—
245	C₂²⋊D₆₇	—	2	48	no	—
246	C₂²⋊D₆₇	—	4	48	no	—
247	C₂×Dic₆₇	—	2	4	no	—
248	C₂×Dic₆₇	—	2	6	no	—
249	C₂×Dic₆₇	C₂²×D₄	1	4	yes	—
250	C₂×Dic₆₇	H(5,𝔽₂)	1	4	yes	complemented
251	C₂×Dic₆₇	C₂⁵	1	6	yes	—
252	C₂×Dic₆₇	C₂⁵	1	4	yes	—
253	C₂×Dic₆₇	C₂⁵	1	2	yes	—
254	C₂×Dic₆₇	C₂⁵	1	2	yes	—
255	D₂₆₈	—	2	48	no	—
256	D₂₆₈	—	2	48	no	—
257	D₂₆₈	—	2	24	no	—
258	D₂₆₈	—	4	24	no	—
259	D₂₆₈	C₂²×D₄	1	48	yes	complemented
260	C₄×D₆₇	—	2	48	no	—
261	C₄×D₆₇	—	2	24	no	—
262	C₄×D₆₇	—	2	24	no	—
263	C₄×D₆₇	C₂²×D₄	1	48	yes	complemented
264	Dic₁₃₄	—	2	16	no	—
265	C₂²×C₆₇	—	2	12	no	—
266	C₂²×C₆₇	—	2	12	no	—
267	C₂²×C₆₇	—	2	12	no	—
268	C₂²×C₆₇	—	4	24	no	—
269	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
270	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
271	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
272	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
273	C₂×D₆₇	—	2	6	no	—
274	C₂×D₆₇	—	2	6	no	—
275	C₂×D₆₇	—	2	24	no	—
276	C₂×D₆₇	—	2	12	no	—
277	C₂×D₆₇	—	2	12	no	—
278	C₂×D₆₇	—	2	12	no	—
279	C₂×D₆₇	—	4	24	no	—
280	C₂×D₆₇	—	8	96	no	—
281	C₂×D₆₇	C₂³×D₄	1	12	yes	—
282	C₄×C₆₇	—	2	4	no	—
283	C₄×C₆₇	—	2	6	no	—
284	C₄×C₆₇	C₂³×D₄	1	12	yes	—
285	C₄×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	complemented
286	Dic₆₇	—	2	4	no	—
287	Dic₆₇	—	2	6	no	—
288	Dic₆₇	C₂³×D₄	1	4	yes	—
289	Dic₆₇	C₂×H(5,𝔽₂)	1	4	yes	complemented
290	C₂×#3498	—	67	1	no	maximal, Sylow
291	C₂×C₆₇	—	4	12	no	—
292	C₂×C₆₇	#3498	1	4	yes	complemented
293	C₂×C₆₇	C₂⁴×D₄	1	1	yes	characteristic
294	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
295	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
296	D₆₇	—	2	12	no	—
297	D₆₇	—	4	12	no	—
298	C₂²×H(5,𝔽₂)	—	67	2	no	—
299	#3498	—	67	32	no	—
300	C₂×#550	—	67	2	no	—
301	C₂×#550	—	67	1	no	—
302	C₂×#550	D₆₇	1	1	yes	characteristic, complemented
303	C₂×D₄²	—	67	3	no	—
304	C₂×D₄²	—	67	6	no	—
305	C₂×D₄²	—	67	3	no	—
306	C₂×#535	—	67	6	no	—
307	C₂×#532	—	67	3	no	—
308	C₂²×#127	—	67	3	no	—
309	C₂×C₄×#41	—	67	1	no	—
310	C₆₇	C₂×#3498	1	1	yes	characteristic, complemented, Sylow
311	C₂×H(5,𝔽₂)	—	67	32	no	—
312	C₂×H(5,𝔽₂)	—	134	32	no	—
313	C₂²×#41	—	67	1	no	—
314	C₂²×#41	—	67	2	no	—
315	C₂²×#41	—	67	6	no	—
316	C₂²×#41	C₂×D₆₇	1	2	yes	complemented
317	C₂³×D₄	—	67	6	no	—
318	C₂³×D₄	—	67	6	no	—
319	C₂³×D₄	—	67	6	no	—
320	C₂³×D₄	—	67	6	no	—
321	C₂³×D₄	—	67	6	no	—
322	#550	—	67	32	no	—
323	#550	—	67	16	no	—
324	#550	C₂×D₆₇	1	16	yes	complemented
325	D₄²	—	67	48	no	—
326	D₄²	—	67	96	no	—
327	D₄²	—	67	48	no	—
328	#535	—	67	96	no	—
329	#532	—	67	48	no	—
330	C₂×C₄⋊Q₈	—	67	1	no	—
331	C₂×#127	—	67	3	no	—
332	C₂×#127	—	67	6	no	—
333	C₂×#127	—	67	6	no	—
334	C₂×#127	—	67	48	no	—
335	C₂×#127	—	67	3	no	—
336	C₂×#127	C₂×D₆₇	1	3	yes	complemented
337	C₂×#124	—	67	6	no	—
338	C₂×#121	—	67	3	no	—
339	C₂×#121	—	67	6	no	—
340	C₂×#121	—	67	6	no	—
341	C₂×#121	—	67	3	no	—
342	C₂×#121	—	67	6	no	—
343	C₂×#121	—	67	6	no	—
344	C₂×#121	C₂×D₆₇	1	6	yes	complemented
345	C₂×#120	—	67	12	no	—
346	C₂×#120	—	67	12	no	—
347	C₄×#41	—	67	16	no	—
348	C₂×C₄×Q₈	—	67	1	no	—
349	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic, complemented
350	C₂×C₄×D₄	—	67	6	no	—
351	C₂×C₄×D₄	—	67	3	no	—
352	C₂×C₄×D₄	—	67	3	no	—
353	C₂×C₄×D₄	—	67	3	no	—
354	C₂×C₄×D₄	C₂×D₆₇	1	3	yes	complemented
355	C₂×#117	—	67	3	no	—
356	C₂²×C₄²	—	67	3	no	—
357	C₂⁵	—	67	12	no	—
358	H(5,𝔽₂)	—	134	256	no	—
359	C₂×#41	—	2	16	no	—
360	C₂×#41	—	67	16	no	—
361	C₂×#41	—	67	8	no	—
362	C₂×#41	—	67	16	no	—
363	C₂×#41	—	67	48	no	—
364	C₂×#41	—	134	48	no	—
365	C₂×#41	—	134	16	no	—
366	C₂×#41	C₂²×D₆₇	1	16	yes	complemented
367	C₂²×Q₈	—	67	2	no	—
368	C₂²×Q₈	C₂²×D₆₇	1	1	yes	characteristic
369	C₂²×D₄	—	268	24	no	—
370	C₂²×D₄	—	268	24	no	—
371	C₂²×D₄	—	67	12	no	—
372	C₂²×D₄	—	67	48	no	—
373	C₂²×D₄	—	67	48	no	—
374	C₂²×D₄	—	67	6	no	—
375	C₂²×D₄	—	67	6	no	—
376	C₂²×D₄	—	67	6	no	—
377	C₂²×D₄	—	67	48	no	—
378	C₂²×D₄	—	67	48	no	—
379	C₂²×D₄	—	67	6	no	—
380	C₂²×D₄	—	67	3	no	—
381	C₂²×D₄	—	67	3	no	—
382	C₂²×D₄	—	67	6	no	—
383	C₂²×D₄	—	67	6	no	—
384	C₂²×D₄	—	67	48	no	—
385	C₂²×D₄	—	134	48	no	—
386	C₂²×D₄	—	134	48	no	—
387	C₂²×D₄	—	134	48	no	—
388	C₂²×D₄	C₂²×D₆₇	1	3	yes	—
389	C₂²×D₄	C₂²×D₆₇	1	6	yes	—
390	C₂²×D₄	C₂²×D₆₇	1	6	yes	—
391	C₂³×C₄	—	67	3	no	—
392	C₂³×C₄	—	67	3	no	—
393	C₂³×C₄	—	67	3	no	—
394	C₂³×C₄	—	67	6	no	—
395	C₂³×C₄	C₂²×D₆₇	1	6	yes	—
396	C₄⋊Q₈	—	67	8	no	—
397	#127	—	67	24	no	—
398	#127	—	67	48	no	—
399	#127	—	67	48	no	—
400	#127	—	67	24	no	—
401	#127	C₂²×D₆₇	1	24	yes	complemented
402	#124	—	67	48	no	—
403	#121	—	67	24	no	—
404	#121	—	67	48	no	—
405	#121	—	67	48	no	—
406	#121	—	67	24	no	—
407	#121	—	67	48	no	—
408	#121	—	67	48	no	—
409	#121	C₂²×D₆₇	1	48	yes	complemented
410	#120	—	67	96	no	—
411	#120	—	67	96	no	—
412	C₄×Q₈	—	67	8	no	—
413	C₄×Q₈	C₂²×D₆₇	1	8	yes	complemented
414	C₄×D₄	—	67	48	no	—
415	C₄×D₄	—	67	24	no	—
416	C₄×D₄	—	67	24	no	—
417	C₄×D₄	—	67	24	no	—
418	C₄×D₄	C₂²×D₆₇	1	24	yes	complemented
419	#117	—	67	24	no	—
420	C₂×C₄⋊C₄	—	67	3	no	—
421	C₂×C₄⋊C₄	—	67	2	no	—
422	C₂×C₄⋊C₄	—	67	1	no	—
423	C₂×C₄⋊C₄	C₂²×D₆₇	1	3	yes	—
424	C₂×C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
425	C₂×C₂²⋊C₄	—	67	6	no	—
426	C₂×C₂²⋊C₄	—	67	6	no	—
427	C₂×C₂²⋊C₄	—	67	6	no	—
428	C₂×C₂²⋊C₄	—	67	6	no	—
429	C₂×C₂²⋊C₄	C₂²×D₆₇	1	6	yes	—
430	C₂×C₄²	—	67	3	no	—
431	C₂×C₄²	—	67	3	no	—
432	C₂×C₄²	—	67	1	no	—
433	C₂×C₄²	—	67	24	no	—
434	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
435	C₂⁴	—	268	24	no	—
436	C₂⁴	—	268	24	no	—
437	C₂⁴	—	67	48	no	—
438	C₂⁴	—	67	3	no	—
439	C₂⁴	—	67	6	no	—
440	C₂⁴	—	134	48	no	—
441	C₂⁴	C₂³×D₆₇	1	6	yes	—
442	#41	—	2	64	no	—
443	#41	—	67	64	no	—
444	#41	—	134	192	no	—
445	#41	—	134	64	no	—
446	C₂×Q₈	—	67	8	no	—
447	C₂×Q₈	—	134	8	no	—
448	C₂×Q₈	D₄×D₆₇	1	8	yes	complemented
449	C₂×Q₈	C₂³×D₆₇	1	4	yes	—
450	C₂×D₄	—	2	24	no	—
451	C₂×D₄	—	2	12	no	—
452	C₂×D₄	—	4	12	no	—
453	C₂×D₄	—	268	12	no	—
454	C₂×D₄	—	268	96	no	—
455	C₂×D₄	—	268	96	no	—
456	C₂×D₄	—	268	24	no	—
457	C₂×D₄	—	67	48	no	—
458	C₂×D₄	—	67	24	no	—
459	C₂×D₄	—	67	24	no	—
460	C₂×D₄	—	67	24	no	—
461	C₂×D₄	—	67	24	no	—
462	C₂×D₄	—	67	24	no	—
463	C₂×D₄	—	67	12	no	—
464	C₂×D₄	—	67	12	no	—
465	C₂×D₄	—	67	24	no	—
466	C₂×D₄	—	67	24	no	—
467	C₂×D₄	—	134	192	no	—
468	C₂×D₄	—	134	24	no	—
469	C₂×D₄	—	134	192	no	—
470	C₂×D₄	—	134	24	no	—
471	C₂×D₄	—	134	24	no	—
472	C₂×D₄	—	134	192	no	—
473	C₂×D₄	—	134	24	no	—
474	C₂×D₄	—	134	24	no	—
475	C₂×D₄	—	134	12	no	—
476	C₂×D₄	C₂³×D₆₇	1	12	yes	—
477	C₂×D₄	C₂³×D₆₇	1	24	yes	—
478	C₂×D₄	C₂³×D₆₇	1	24	yes	—
479	C₂²×C₄	—	2	24	no	—
480	C₂²×C₄	—	67	1	no	—
481	C₂²×C₄	—	67	1	no	—
482	C₂²×C₄	—	67	24	no	—
483	C₂²×C₄	—	67	12	no	—
484	C₂²×C₄	—	67	3	no	—
485	C₂²×C₄	—	67	2	no	—
486	C₂²×C₄	—	67	12	no	—
487	C₂²×C₄	—	67	12	no	—
488	C₂²×C₄	—	67	24	no	—
489	C₂²×C₄	—	134	12	no	—
490	C₂²×C₄	—	134	24	no	—
491	C₂²×C₄	—	134	12	no	—
492	C₂²×C₄	C₂³×D₆₇	1	24	yes	—
493	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
494	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
495	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
496	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
497	C₄⋊C₄	—	67	12	no	—
498	C₄⋊C₄	—	67	8	no	—
499	C₄⋊C₄	—	67	4	no	—
500	C₄⋊C₄	C₂³×D₆₇	1	12	yes	—
501	C₄⋊C₄	C₂³×D₆₇	1	4	yes	—
502	C₂²⋊C₄	—	67	24	no	—
503	C₂²⋊C₄	—	67	24	no	—
504	C₂²⋊C₄	—	67	24	no	—
505	C₂²⋊C₄	—	67	24	no	—
506	C₂²⋊C₄	C₂³×D₆₇	1	24	yes	—
507	C₄²	—	67	4	no	—
508	C₄²	—	67	12	no	—
509	C₄²	—	67	12	no	—
510	C₄²	C₂³×D₆₇	1	12	yes	—
511	C₂³	—	2	12	no	—
512	C₂³	—	2	12	no	—
513	C₂³	—	2	12	no	—
514	C₂³	—	268	48	no	—
515	C₂³	—	268	48	no	—
516	C₂³	—	536	96	no	—
517	C₂³	—	67	12	no	—
518	C₂³	—	67	6	no	—
519	C₂³	—	67	12	no	—
520	C₂³	—	134	96	no	—
521	C₂³	—	134	12	no	—
522	C₂³	—	134	12	no	—
523	C₂³	—	134	12	no	—
524	C₂³	—	134	6	no	—
525	C₂³	—	134	6	no	—
526	C₂³	C₂⁴×D₆₇	1	12	yes	—
527	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
528	Q₈	—	134	16	no	—
529	Q₈	C₂×D₄×D₆₇	1	16	yes	complemented
530	D₄	—	2	48	no	—
531	D₄	—	2	24	no	—
532	D₄	—	4	24	no	—
533	D₄	—	268	24	no	—
534	D₄	—	268	48	no	—
535	D₄	—	67	48	no	—
536	D₄	—	134	48	no	—
537	D₄	—	134	48	no	—
538	D₄	—	134	48	no	—
539	D₄	—	134	48	no	—
540	D₄	—	134	48	no	—
541	D₄	—	134	24	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	6	no	—
544	C₂×C₄	—	2	48	no	—
545	C₂×C₄	—	67	4	no	—
546	C₂×C₄	—	67	4	no	—
547	C₂×C₄	—	67	2	no	—
548	C₂×C₄	—	67	2	no	—
549	C₂×C₄	—	67	48	no	—
550	C₂×C₄	—	67	6	no	—
551	C₂×C₄	—	67	4	no	—
552	C₂×C₄	—	134	24	no	—
553	C₂×C₄	—	134	4	no	—
554	C₂×C₄	—	134	6	no	—
555	C₂×C₄	—	134	48	no	—
556	C₂×C₄	—	134	24	no	—
557	C₂×C₄	C₂×D₄×D₆₇	1	12	yes	—
558	C₂×C₄	#14524089	1	4	yes	complemented
559	C₂×C₄	C₂⁴×D₆₇	1	6	yes	—
560	C₂×C₄	C₂⁴×D₆₇	1	4	yes	—
561	C₂×C₄	C₂⁴×D₆₇	1	6	yes	—
562	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
563	C₂²	—	2	12	no	—
564	C₂²	—	2	12	no	—
565	C₂²	—	2	12	no	—
566	C₂²	—	4	24	no	—
567	C₂²	—	268	24	no	—
568	C₂²	—	536	96	no	—
569	C₂²	—	67	12	no	—
570	C₂²	—	134	12	no	—
571	C₂²	—	134	12	no	—
572	C₂²	—	134	12	no	—
573	C₂²	—	134	6	no	—
574	C₂²	—	134	24	no	—
575	C₂²	—	134	6	no	—
576	C₂²	C₂²×D₄×D₆₇	1	2	yes	—
577	C₂²	H(5,𝔽₂)×D₆₇	1	2	yes	—
578	C₂²	C₂×#14524089	1	2	yes	—
579	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
580	C₄	—	2	4	no	—
581	C₄	—	2	6	no	—
582	C₄	—	67	4	no	—
583	C₄	—	67	4	no	—
584	C₄	—	134	4	no	—
585	C₄	—	134	6	no	—
586	C₄	C₂²×D₄×D₆₇	1	12	yes	—
587	C₄	C₂×#14524089	1	4	yes	complemented
588	C₂	—	4	12	no	—
589	C₂	—	268	12	no	—
590	C₂	—	134	12	no	—
591	C₂	#22573986	1	4	yes	direct product factor
592	C₂	C₂³×D₄×D₆₇	1	1	yes	characteristic
593	C₂	C₂×H(5,𝔽₂)×D₆₇	1	1	yes	characteristic
594	C₂	C₂²×#14524089	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂×#550×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁵
hypercenter	C₂×#550	D₆₇
hypocenter	C₆₇	C₂×#3498
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×#550

Complements

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

N	H ≅ G/N	type
1. C₂²×#14524089	C₂	semidirect
2. #22573986	C₂	direct (N × H)
3. C₂×#16864068	C₂	semidirect
4. C₂×#16864065	C₂	semidirect
5. C₂×D₄×D₂₆₈	C₂	semidirect
6. C₂×#16864031	C₂	semidirect
7. C₂×#16864030	C₂	semidirect
8. C₂×#16863962	C₂	semidirect
9. C₂×#16863943	C₂	semidirect
10. C₂×#127×D₆₇	C₂	semidirect
11. C₂×C₄×#11761629	C₂	semidirect
12. C₂×#550×C₆₇	C₂	semidirect
14. C₂×#14524089	C₂²	semidirect
16. C₂²×#11761629	C₂²	semidirect
17. C₂²×#11761628	C₂²	semidirect
18. C₂³×D₂₆₈	C₂²	semidirect
19. C₂²×D₄×D₆₇	C₂²	semidirect
20. C₂²×D₄×D₆₇	C₂²	semidirect
21. C₂²×D₄×D₆₇	C₂²	semidirect
22. C₂²×D₄×D₆₇	C₂²	semidirect
23. #16864068	C₂²	semidirect
24. #16864065	C₂²	semidirect
25. D₄×D₂₆₈	C₂²	semidirect
26. #16864031	C₂²	semidirect
27. #16864030	C₂²	semidirect
28. #16863962	C₂²	semidirect
29. #16863943	C₂²	semidirect
30. C₂×#14524042	C₂²	semidirect
31. C₂×#14524040	C₂²	semidirect
32. C₂×#14524039	C₂²	semidirect
33. C₂×#14524039	C₂²	semidirect
34. C₂×#14524039	C₂²	semidirect
35. #127×D₆₇	C₂²	semidirect
36. C₂×#14524029	C₂²	semidirect
37. C₂×#14524018	C₂²	semidirect
38. C₂×#14524018	C₂²	semidirect
39. C₂×#14524017	C₂²	semidirect
40. C₂×#14524016	C₂²	semidirect
41. C₂×#14524014	C₂²	semidirect
42. C₂×#14524014	C₂²	semidirect
43. C₂×#14524013	C₂²	semidirect
44. C₂×#14524011	C₂²	semidirect
45. C₄×#11761629	C₂²	semidirect
46. C₂×Q₈×Dic₆₇	C₂²	semidirect
47. C₂×C₄×D₂₆₈	C₂²	semidirect
48. C₂×C₄×D₂₆₈	C₂²	semidirect
49. C₂×#14524005	C₂²	semidirect
50. C₂×C₄×C₂²⋊D₆₇	C₂²	semidirect
51. C₂×#14524001	C₂²	semidirect
52. C₂×C₄²×D₆₇	C₂²	semidirect
53. C₂²×#41×C₆₇	C₂²	semidirect
54. #550×C₆₇	C₂²	semidirect
55. C₂×#127×C₆₇	C₂²	semidirect
56. C₂×#121×C₆₇	C₂²	semidirect
57. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
58. C₂×C₄×D₄×C₆₇	C₂²	semidirect
62. C₂×#11761629	D₄	semidirect
64. C₂×#11761629	C₂³	semidirect
66. C₂×#11761628	C₂³	semidirect
74. C₂²×D₂₆₈	C₂³	semidirect
83. C₂×D₄×D₆₇	C₂³	semidirect
84. C₂×D₄×D₆₇	C₂³	semidirect
85. C₂×D₄×D₆₇	C₂³	semidirect
86. C₂×D₄×D₆₇	C₂³	semidirect
91. #14524042	C₂³	semidirect
92. #14524040	C₂³	semidirect
93. #14524039	C₂³	semidirect
94. #14524039	C₂³	semidirect
95. #14524039	C₂³	semidirect
96. #14524029	C₂³	semidirect
97. #14524018	C₂³	semidirect
98. #14524018	C₂³	semidirect
99. #14524017	C₂³	semidirect
100. #14524016	C₂³	semidirect
101. #14524014	C₂³	semidirect
102. #14524014	C₂³	semidirect
103. #14524013	C₂³	semidirect
104. #14524011	C₂³	semidirect
105. Q₈×Dic₆₇	C₂³	semidirect
106. C₄×D₂₆₈	C₂³	semidirect
107. C₄×D₂₆₈	C₂³	semidirect
108. #14524005	C₂³	semidirect
109. C₄×C₂²⋊D₆₇	C₂³	semidirect
110. #14524001	C₂³	semidirect
121. C₄²×D₆₇	C₂³	semidirect
123. C₂×#41×C₆₇	C₂³	semidirect
129. #127×C₆₇	C₂³	semidirect
130. #121×C₆₇	C₂³	semidirect
131. C₄×Q₈×C₆₇	C₂³	semidirect
132. C₄×D₄×C₆₇	C₂³	semidirect
144. #11761629	C₂×D₄	semidirect
153. C₂×D₂₆₈	C₂×D₄	semidirect
178. C₂×C₄×D₆₇	C₂×D₄	semidirect
195. C₂×Q₈×C₆₇	C₂×D₄	semidirect
230. Q₈×C₆₇	C₂²×D₄	semidirect
238. C₂×C₄×C₆₇	H(5,𝔽₂)	semidirect
250. C₂×Dic₆₇	H(5,𝔽₂)	semidirect
259. D₂₆₈	C₂²×D₄	semidirect
263. C₄×D₆₇	C₂²×D₄	semidirect
285. C₄×C₆₇	C₂×H(5,𝔽₂)	semidirect
289. Dic₆₇	C₂×H(5,𝔽₂)	semidirect
292. C₂×C₆₇	#3498	semidirect
302. C₂×#550	D₆₇	semidirect
310. C₆₇	C₂×#3498	semidirect
316. C₂²×#41	C₂×D₆₇	semidirect
324. #550	C₂×D₆₇	semidirect
336. C₂×#127	C₂×D₆₇	semidirect
344. C₂×#121	C₂×D₆₇	semidirect
349. C₂×C₄×Q₈	C₂×D₆₇	semidirect
354. C₂×C₄×D₄	C₂×D₆₇	semidirect
366. C₂×#41	C₂²×D₆₇	semidirect
401. #127	C₂²×D₆₇	semidirect
409. #121	C₂²×D₆₇	semidirect
413. C₄×Q₈	C₂²×D₆₇	semidirect
418. C₄×D₄	C₂²×D₆₇	semidirect
448. C₂×Q₈	D₄×D₆₇	semidirect
529. Q₈	C₂×D₄×D₆₇	semidirect
558. C₂×C₄	#14524089	semidirect
587. C₄	C₂×#14524089	semidirect
591. C₂	#22573986	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
#22573986	2
C₂×#16864068	2
C₂×#16864065	2
C₂×D₄×D₂₆₈	2
C₂×#16864031	2
C₂×#16864030	2
C₂×#16863962	2
C₂×#16863943	2
C₂×#127×D₆₇	2
C₂×C₄×#11761629	2
C₂×#550×C₆₇	2
C₂×#3498	67

Subgroups — C2×#22573986 #31764150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22573986 #31764150