Subgroups — C₂×#22573983 #31764151

Order 17152.

12982 conjugacy classes of subgroups.

The subgroup lattice has 496 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	#22573983	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16864065	C₂	1	4	yes	maximal, complemented
4	C₂×D₄×C₂²⋊D₆₇	C₂	1	12	yes	maximal, complemented
5	C₂×#16863963	C₂	1	3	yes	maximal, complemented
6	C₂×#16863960	C₂	1	3	yes	maximal, complemented
7	C₂×#16863942	C₂	1	3	yes	maximal, complemented
8	C₂²×#14524039	C₂	1	3	yes	maximal, complemented
9	C₂×#41×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×H(5,𝔽₂)×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#14524089	—	2	16	no	—
12	C₂×#14524089	C₂²	1	16	yes	complemented
13	C₂²×#11761629	C₂²	1	2	yes	complemented
14	C₂²×#11761628	C₂²	1	3	yes	complemented
15	C₂³×D₂₆₈	C₂²	1	3	yes	complemented
16	C₂³×C₂²⋊D₆₇	C₂²	1	12	yes	complemented
17	C₂²×D₄×D₆₇	C₂²	1	6	yes	complemented
18	#16864065	C₂²	1	64	yes	complemented
19	D₄×C₂²⋊D₆₇	C₂²	1	192	yes	complemented
20	#16863963	C₂²	1	48	yes	complemented
21	#16863960	C₂²	1	48	yes	complemented
22	#16863942	C₂²	1	48	yes	complemented
23	C₂×#14524041	C₂²	1	1	yes	characteristic, complemented
24	C₂×#14524039	C₂²	1	12	yes	complemented
25	C₂×#14524039	C₂²	1	3	yes	complemented
26	C₂×#14524039	C₂²	1	6	yes	complemented
27	C₂×#14524039	C₂²	1	48	yes	complemented
28	C₂×#14524032	C₂²	1	3	yes	complemented
29	C₂×#14524030	C₂²	1	3	yes	complemented
30	C₂×#14524017	C₂²	1	12	yes	complemented
31	C₂×#14524015	C₂²	1	12	yes	complemented
32	C₂×#14524014	C₂²	1	12	yes	complemented
33	C₂×#14524012	C₂²	1	12	yes	complemented
34	C₂×#14524011	C₂²	1	12	yes	complemented
35	#41×Dic₆₇	C₂²	1	16	yes	complemented
36	C₂×Q₈×Dic₆₇	C₂²	1	2	yes	complemented
37	C₂×D₄×Dic₆₇	C₂²	1	6	yes	complemented
38	C₂×C₄×C₂²⋊D₆₇	C₂²	1	12	yes	complemented
39	C₂×#14524002	C₂²	1	3	yes	complemented
40	C₂²×C₄×Dic₆₇	C₂²	1	3	yes	complemented
41	C₂×H(5,𝔽₂)×C₆₇	—	2	16	no	—
42	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	16	yes	complemented
43	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
44	C₂²×#41×C₆₇	C₂²	1	2	yes	complemented
45	C₂²×#41×C₆₇	C₂²	1	3	yes	complemented
46	C₂³×D₄×C₆₇	C₂²	1	6	yes	complemented
47	C₂³×D₄×C₆₇	C₂²	1	3	yes	complemented
48	C₂⁴×D₆₇	C₂³	1	6	yes	—
49	#14524089	—	2	128	no	—
50	C₂×#11761629	—	2	16	no	—
51	C₂×#11761629	C₂³	1	16	yes	complemented
52	C₂×#11761628	—	2	24	no	—
53	C₂×#11761628	C₂³	1	24	yes	complemented
54	C₂²×Dic₁₃₄	C₂³	1	1	yes	characteristic
55	C₂²×D₂₆₈	—	2	24	no	—
56	C₂²×D₂₆₈	C₂³	1	6	yes	—
57	C₂²×D₂₆₈	C₂³	1	3	yes	—
58	C₂²×D₂₆₈	C₂³	1	24	yes	complemented
59	C₂²×C₂²⋊D₆₇	—	4	48	no	—
60	C₂²×C₂²⋊D₆₇	C₂³	1	12	yes	—
61	C₂²×C₂²⋊D₆₇	C₂³	1	6	yes	—
62	C₂²×C₂²⋊D₆₇	C₂³	1	24	yes	—
63	C₂²×C₂²⋊D₆₇	C₂³	1	96	yes	complemented
64	C₂×D₄×D₆₇	—	2	48	no	—
65	C₂×D₄×D₆₇	C₂³	1	48	yes	complemented
66	C₂³×Dic₆₇	C₂³	1	6	yes	—
67	C₂²×C₄×D₆₇	C₂³	1	6	yes	—
68	#14524041	C₂³	1	8	yes	complemented
69	#14524039	C₂³	1	96	yes	complemented
70	#14524039	C₂³	1	24	yes	complemented
71	#14524039	C₂³	1	48	yes	complemented
72	#14524032	C₂³	1	24	yes	complemented
73	#14524030	C₂³	1	24	yes	complemented
74	#14524017	C₂³	1	96	yes	complemented
75	#14524015	C₂³	1	96	yes	complemented
76	#14524014	C₂³	1	96	yes	complemented
77	#14524012	C₂³	1	96	yes	complemented
78	#14524011	C₂³	1	96	yes	complemented
79	Q₈×Dic₆₇	C₂³	1	16	yes	complemented
80	D₄×Dic₆₇	C₂³	1	48	yes	complemented
81	C₄×C₂²⋊D₆₇	C₂³	1	96	yes	complemented
82	#14524002	C₂³	1	24	yes	complemented
83	C₂×C₄⋊Dic₆₇	C₂³	1	6	yes	—
84	C₂×Dic₆₇⋊C₄	C₂³	1	4	yes	—
85	C₂×C₂²⋊Dic₆₇	C₂³	1	6	yes	—
86	C₂×C₂²⋊Dic₆₇	C₂³	1	12	yes	—
87	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	12	yes	—
88	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
89	C₂×C₄×Dic₆₇	C₂³	1	3	yes	—
90	C₂×C₄×Dic₆₇	C₂³	1	6	yes	—
91	C₂×C₄×Dic₆₇	C₂³	1	24	yes	complemented
92	C₂⁵×C₆₇	C₂³	1	6	yes	—
93	H(5,𝔽₂)×C₆₇	—	2	128	no	—
94	C₂×#41×C₆₇	—	2	16	no	—
95	C₂×#41×C₆₇	—	2	24	no	—
96	C₂×#41×C₆₇	D₄	1	16	yes	complemented
97	C₂×#41×C₆₇	C₂³	1	8	yes	—
98	C₂×#41×C₆₇	C₂³	1	16	yes	complemented
99	C₂×#41×C₆₇	C₂³	1	24	yes	complemented
100	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
101	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
102	C₂²×D₄×C₆₇	—	2	48	no	—
103	C₂²×D₄×C₆₇	—	2	24	no	—
104	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
105	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
106	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
107	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
108	C₂²×D₄×C₆₇	C₂³	1	48	yes	complemented
109	C₂²×D₄×C₆₇	C₂³	1	24	yes	complemented
110	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
111	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
112	C₂³×D₆₇	—	2	24	no	—
113	C₂³×D₆₇	—	4	24	no	—
114	C₂³×D₆₇	C₂⁴	1	6	yes	—
115	C₂³×D₆₇	C₂⁴	1	24	yes	—
116	#11761629	—	2	64	no	—
117	#11761628	—	2	96	no	—
118	C₂×Dic₁₃₄	—	2	4	no	—
119	C₂×Dic₁₃₄	C₂⁴	1	4	yes	—
120	C₂×D₂₆₈	—	2	24	no	—
121	C₂×D₂₆₈	—	2	12	no	—
122	C₂×D₂₆₈	—	2	96	no	—
123	C₂×D₂₆₈	C₂⁴	1	24	yes	—
124	C₂×D₂₆₈	C₂⁴	1	12	yes	—
125	C₂×C₂²⋊D₆₇	—	2	24	no	—
126	C₂×C₂²⋊D₆₇	—	2	48	no	—
127	C₂×C₂²⋊D₆₇	—	4	192	no	—
128	C₂×C₂²⋊D₆₇	—	4	48	no	—
129	C₂×C₂²⋊D₆₇	C₂⁴	1	48	yes	—
130	C₂×C₂²⋊D₆₇	C₂⁴	1	24	yes	—
131	C₂×C₂²⋊D₆₇	C₂⁴	1	96	yes	—
132	D₄×D₆₇	—	2	192	no	—
133	C₂²×Dic₆₇	—	2	24	no	—
134	C₂²×Dic₆₇	C₂⁴	1	6	yes	—
135	C₂²×Dic₆₇	C₂⁴	1	24	yes	—
136	C₂²×Dic₆₇	C₂⁴	1	2	yes	—
137	C₂²×Dic₆₇	C₂⁴	1	2	yes	—
138	C₂×C₄×D₆₇	—	2	24	no	—
139	C₂×C₄×D₆₇	C₂⁴	1	24	yes	—
140	C₄⋊Dic₆₇	C₂⁴	1	24	yes	—
141	Dic₆₇⋊C₄	C₂⁴	1	16	yes	—
142	C₂²⋊Dic₆₇	C₂⁴	1	24	yes	—
143	C₂²⋊Dic₆₇	C₂⁴	1	48	yes	—
144	(C₂×D₆₇)⋊C₄	C₂⁴	1	48	yes	—
145	C₄×Dic₆₇	C₂⁴	1	4	yes	—
146	C₄×Dic₆₇	C₂⁴	1	12	yes	—
147	C₄×Dic₆₇	C₂⁴	1	24	yes	—
148	C₂⁴×C₆₇	—	2	24	no	—
149	C₂⁴×C₆₇	—	4	24	no	—
150	C₂⁴×C₆₇	C₂⁴	1	6	yes	—
151	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
152	C₂⁴×C₆₇	C₂⁴	1	24	yes	—
153	#41×C₆₇	—	2	64	no	—
154	#41×C₆₇	—	2	96	no	—
155	#41×C₆₇	C₂×D₄	1	64	yes	complemented
156	C₂×Q₈×C₆₇	—	2	4	no	—
157	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	complemented
158	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
159	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
160	C₂×D₄×C₆₇	—	2	24	no	—
161	C₂×D₄×C₆₇	—	2	192	no	—
162	C₂×D₄×C₆₇	—	2	24	no	—
163	C₂×D₄×C₆₇	—	2	12	no	—
164	C₂×D₄×C₆₇	—	2	96	no	—
165	C₂×D₄×C₆₇	C₂×D₄	1	24	yes	complemented
166	C₂×D₄×C₆₇	C₂⁴	1	24	yes	—
167	C₂×D₄×C₆₇	C₂⁴	1	24	yes	—
168	C₂×D₄×C₆₇	C₂⁴	1	12	yes	—
169	C₂×D₄×C₆₇	C₂⁴	1	12	yes	—
170	C₂²×C₄×C₆₇	—	2	24	no	—
171	C₂²×C₄×C₆₇	C₂×D₄	1	24	yes	complemented
172	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₆₇	C₂⁴	1	24	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
175	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
176	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
177	C₂³×C₆₇	—	2	12	no	—
178	C₂³×C₆₇	—	2	12	no	—
179	C₂³×C₆₇	—	2	24	no	—
180	C₂³×C₆₇	—	2	48	no	—
181	C₂³×C₆₇	—	4	48	no	—
182	C₂³×C₆₇	—	8	48	no	—
183	C₂³×C₆₇	C₂²×D₄	1	12	yes	—
184	C₂³×C₆₇	C₂⁵	1	12	yes	—
185	C₂³×C₆₇	C₂⁵	1	6	yes	—
186	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
187	C₂²×D₆₇	—	2	12	no	—
188	C₂²×D₆₇	—	2	24	no	—
189	C₂²×D₆₇	—	2	48	no	—
190	C₂²×D₆₇	—	4	48	no	—
191	C₂²×D₆₇	—	8	48	no	—
192	C₂²×D₆₇	C₂⁵	1	12	yes	—
193	Q₈×C₆₇	—	2	8	no	—
194	Q₈×C₆₇	C₂²×D₄	1	16	yes	complemented
195	D₄×C₆₇	—	2	48	no	—
196	D₄×C₆₇	—	2	24	no	—
197	D₄×C₆₇	—	2	48	no	—
198	D₄×C₆₇	C₂²×D₄	1	48	yes	complemented
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	2	48	no	—
201	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
202	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
203	C₂×C₄×C₆₇	C₂²×D₄	1	48	yes	complemented
204	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
205	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
206	C₂×C₄×C₆₇	C₂⁵	1	6	yes	—
207	C₂²⋊D₆₇	—	2	48	no	—
208	C₂²⋊D₆₇	—	2	96	no	—
209	C₂²⋊D₆₇	—	4	96	no	—
210	C₂×Dic₆₇	—	2	4	no	—
211	C₂×Dic₆₇	—	2	12	no	—
212	C₂×Dic₆₇	—	2	48	no	—
213	C₂×Dic₆₇	H(5,𝔽₂)	1	8	yes	complemented
214	C₂×Dic₆₇	C₂⁵	1	12	yes	—
215	C₂×Dic₆₇	C₂⁵	1	4	yes	—
216	C₂×Dic₆₇	C₂⁵	1	4	yes	—
217	D₂₆₈	—	2	48	no	—
218	D₂₆₈	—	2	24	no	—
219	C₄×D₆₇	—	2	48	no	—
220	Dic₁₃₄	—	2	8	no	—
221	C₂²×C₆₇	—	2	12	no	—
222	C₂²×C₆₇	—	2	24	no	—
223	C₂²×C₆₇	—	2	12	no	—
224	C₂²×C₆₇	—	2	24	no	—
225	C₂²×C₆₇	—	4	24	no	—
226	C₂²×C₆₇	—	8	48	no	—
227	C₂²×C₆₇	C₂³×D₄	1	12	yes	—
228	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
229	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
230	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
231	C₂×D₆₇	—	2	12	no	—
232	C₂×D₆₇	—	2	24	no	—
233	C₂×D₆₇	—	4	24	no	—
234	C₂×D₆₇	—	8	48	no	—
235	C₄×C₆₇	—	2	4	no	—
236	C₄×C₆₇	C₂³×D₄	1	4	yes	—
237	C₄×C₆₇	C₂³×D₄	1	12	yes	—
238	Dic₆₇	—	2	4	no	—
239	Dic₆₇	—	2	12	no	—
240	Dic₆₇	C₂×H(5,𝔽₂)	1	8	yes	complemented
241	C₂×#3498	—	67	1	no	maximal, Sylow
242	C₂×C₆₇	—	2	12	no	—
243	C₂×C₆₇	—	4	12	no	—
244	C₂×C₆₇	#3498	1	4	yes	complemented
245	C₂×C₆₇	C₂⁴×D₄	1	1	yes	characteristic
246	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	2	yes	—
247	D₆₇	—	4	12	no	—
248	C₂²×H(5,𝔽₂)	—	67	1	no	—
249	C₂²×H(5,𝔽₂)	D₆₇	1	1	yes	characteristic, complemented
250	#3498	—	67	32	no	—
251	C₂×#550	—	67	4	no	—
252	C₂×D₄²	—	67	12	no	—
253	C₂×#535	—	67	3	no	—
254	C₂×#535	—	67	3	no	—
255	C₂×#532	—	67	3	no	—
256	C₂²×#127	—	67	3	no	—
257	C₂×C₄×#41	—	67	1	no	—
258	C₆₇	C₂×#3498	1	1	yes	characteristic, complemented, Sylow
259	C₂×H(5,𝔽₂)	—	2	16	no	—
260	C₂×H(5,𝔽₂)	—	67	16	no	—
261	C₂×H(5,𝔽₂)	—	134	16	no	—
262	C₂×H(5,𝔽₂)	C₂×D₆₇	1	16	yes	complemented
263	C₂²×#41	—	67	2	no	—
264	C₂²×#41	—	67	3	no	—
265	C₂²×#41	C₂×D₆₇	1	1	yes	characteristic
266	C₂²×#41	C₂×D₆₇	1	3	yes	complemented
267	C₂²×#41	C₂×D₆₇	1	2	yes	complemented
268	C₂³×D₄	—	67	12	no	—
269	C₂³×D₄	—	67	6	no	—
270	C₂³×D₄	—	67	3	no	—
271	C₂³×D₄	C₂×D₆₇	1	3	yes	complemented
272	C₂³×D₄	C₂×D₆₇	1	6	yes	complemented
273	#550	—	67	64	no	—
274	D₄²	—	67	192	no	—
275	#535	—	67	48	no	—
276	#535	—	67	48	no	—
277	#532	—	67	48	no	—
278	C₂×C₄⋊Q₈	—	67	1	no	—
279	C₂×#127	—	67	3	no	—
280	C₂×#127	—	67	6	no	—
281	C₂×#127	—	67	12	no	—
282	C₂×#127	—	67	48	no	—
283	C₂×#124	—	67	3	no	—
284	C₂×#124	—	67	3	no	—
285	C₂×#121	—	67	12	no	—
286	C₂×#121	—	67	12	no	—
287	C₂×#121	—	67	12	no	—
288	C₂×#120	—	67	12	no	—
289	C₂×#120	—	67	12	no	—
290	C₄×#41	—	67	16	no	—
291	C₂×C₄×Q₈	—	67	2	no	—
292	C₂×C₄×D₄	—	67	6	no	—
293	C₂×C₄×D₄	—	67	12	no	—
294	C₂×#117	—	67	3	no	—
295	C₂²×C₄²	—	67	3	no	—
296	C₂⁵	—	67	6	no	—
297	C₂⁵	C₂²×D₆₇	1	6	yes	—
298	H(5,𝔽₂)	—	2	128	no	—
299	H(5,𝔽₂)	—	134	128	no	—
300	C₂×#41	—	2	16	no	—
301	C₂×#41	—	2	24	no	—
302	C₂×#41	—	67	16	no	—
303	C₂×#41	—	67	24	no	—
304	C₂×#41	—	134	16	no	—
305	C₂×#41	—	134	24	no	—
306	C₂×#41	C₂²⋊D₆₇	1	16	yes	complemented
307	C₂×#41	C₂²×D₆₇	1	8	yes	—
308	C₂×#41	C₂²×D₆₇	1	24	yes	complemented
309	C₂×#41	C₂²×D₆₇	1	16	yes	complemented
310	C₂²×Q₈	—	67	1	no	—
311	C₂²×Q₈	C₂²×D₆₇	1	1	yes	characteristic
312	C₂²×Q₈	C₂²×D₆₇	1	1	yes	characteristic
313	C₂²×D₄	—	2	48	no	—
314	C₂²×D₄	—	2	24	no	—
315	C₂²×D₄	—	268	48	no	—
316	C₂²×D₄	—	67	24	no	—
317	C₂²×D₄	—	67	96	no	—
318	C₂²×D₄	—	67	12	no	—
319	C₂²×D₄	—	67	6	no	—
320	C₂²×D₄	—	67	48	no	—
321	C₂²×D₄	—	67	6	no	—
322	C₂²×D₄	—	67	3	no	—
323	C₂²×D₄	—	67	24	no	—
324	C₂²×D₄	—	134	48	no	—
325	C₂²×D₄	—	134	24	no	—
326	C₂²×D₄	C₂²×D₆₇	1	6	yes	—
327	C₂²×D₄	C₂²×D₆₇	1	6	yes	—
328	C₂²×D₄	C₂²×D₆₇	1	3	yes	—
329	C₂²×D₄	C₂²×D₆₇	1	3	yes	—
330	C₂²×D₄	C₂²×D₆₇	1	24	yes	complemented
331	C₂²×D₄	C₂²×D₆₇	1	48	yes	complemented
332	C₂³×C₄	—	67	6	no	—
333	C₂³×C₄	—	67	6	no	—
334	C₂³×C₄	C₂²×D₆₇	1	6	yes	—
335	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
336	C₄⋊Q₈	—	67	8	no	—
337	#127	—	67	24	no	—
338	#127	—	67	48	no	—
339	#127	—	67	96	no	—
340	#124	—	67	24	no	—
341	#124	—	67	24	no	—
342	#121	—	67	96	no	—
343	#121	—	67	96	no	—
344	#121	—	67	96	no	—
345	#120	—	67	96	no	—
346	#120	—	67	96	no	—
347	C₄×Q₈	—	67	16	no	—
348	C₄×D₄	—	67	48	no	—
349	C₄×D₄	—	67	96	no	—
350	#117	—	67	24	no	—
351	C₂×C₄⋊C₄	—	67	6	no	—
352	C₂×C₄⋊C₄	—	67	4	no	—
353	C₂×C₂²⋊C₄	—	67	6	no	—
354	C₂×C₂²⋊C₄	—	67	12	no	—
355	C₂×C₂²⋊C₄	—	67	12	no	—
356	C₂×C₄²	—	67	3	no	—
357	C₂×C₄²	—	67	6	no	—
358	C₂×C₄²	—	67	24	no	—
359	C₂×C₄²	—	67	1	no	—
360	C₂⁴	—	2	24	no	—
361	C₂⁴	—	4	24	no	—
362	C₂⁴	—	268	24	no	—
363	C₂⁴	—	67	24	no	—
364	C₂⁴	—	67	6	no	—
365	C₂⁴	—	134	24	no	—
366	C₂⁴	C₂³×D₆₇	1	6	yes	—
367	C₂⁴	C₂³×D₆₇	1	3	yes	—
368	C₂⁴	C₂³×D₆₇	1	24	yes	—
369	#41	—	2	64	no	—
370	#41	—	2	96	no	—
371	#41	—	134	64	no	—
372	#41	—	134	96	no	—
373	#41	C₂×C₂²⋊D₆₇	1	64	yes	complemented
374	C₂×Q₈	—	2	4	no	—
375	C₂×Q₈	—	67	4	no	—
376	C₂×Q₈	—	134	4	no	—
377	C₂×Q₈	C₂×C₂²⋊D₆₇	1	8	yes	complemented
378	C₂×Q₈	C₂³×D₆₇	1	4	yes	—
379	C₂×Q₈	C₂³×D₆₇	1	4	yes	—
380	C₂×D₄	—	2	24	no	—
381	C₂×D₄	—	2	192	no	—
382	C₂×D₄	—	2	24	no	—
383	C₂×D₄	—	2	12	no	—
384	C₂×D₄	—	2	96	no	—
385	C₂×D₄	—	268	48	no	—
386	C₂×D₄	—	268	192	no	—
387	C₂×D₄	—	67	96	no	—
388	C₂×D₄	—	67	48	no	—
389	C₂×D₄	—	67	24	no	—
390	C₂×D₄	—	67	24	no	—
391	C₂×D₄	—	67	12	no	—
392	C₂×D₄	—	134	48	no	—
393	C₂×D₄	—	134	24	no	—
394	C₂×D₄	—	134	192	no	—
395	C₂×D₄	—	134	24	no	—
396	C₂×D₄	—	134	12	no	—
397	C₂×D₄	—	134	96	no	—
398	C₂×D₄	C₂×C₂²⋊D₆₇	1	24	yes	complemented
399	C₂×D₄	C₂³×D₆₇	1	24	yes	—
400	C₂×D₄	C₂³×D₆₇	1	24	yes	—
401	C₂×D₄	C₂³×D₆₇	1	12	yes	—
402	C₂×D₄	C₂³×D₆₇	1	12	yes	—
403	C₂²×C₄	—	2	24	no	—
404	C₂²×C₄	—	67	2	no	—
405	C₂²×C₄	—	67	24	no	—
406	C₂²×C₄	—	67	6	no	—
407	C₂²×C₄	—	67	2	no	—
408	C₂²×C₄	—	67	24	no	—
409	C₂²×C₄	—	134	24	no	—
410	C₂²×C₄	—	134	24	no	—
411	C₂²×C₄	C₂×C₂²⋊D₆₇	1	24	yes	complemented
412	C₂²×C₄	C₂³×D₆₇	1	24	yes	—
413	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
414	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
415	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
416	C₂²×C₄	C₂³×D₆₇	1	12	yes	—
417	C₄⋊C₄	—	67	24	no	—
418	C₄⋊C₄	—	67	16	no	—
419	C₂²⋊C₄	—	67	24	no	—
420	C₂²⋊C₄	—	67	48	no	—
421	C₂²⋊C₄	—	67	48	no	—
422	C₄²	—	67	4	no	—
423	C₄²	—	67	12	no	—
424	C₄²	—	67	24	no	—
425	C₂³	—	2	12	no	—
426	C₂³	—	2	12	no	—
427	C₂³	—	2	24	no	—
428	C₂³	—	2	48	no	—
429	C₂³	—	4	48	no	—
430	C₂³	—	8	48	no	—
431	C₂³	—	268	48	no	—
432	C₂³	—	536	48	no	—
433	C₂³	—	67	12	no	—
434	C₂³	—	134	12	no	—
435	C₂³	—	134	24	no	—
436	C₂³	—	134	48	no	—
437	C₂³	C₂²×C₂²⋊D₆₇	1	12	yes	—
438	C₂³	C₂⁴×D₆₇	1	12	yes	—
439	C₂³	C₂⁴×D₆₇	1	6	yes	—
440	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
441	Q₈	—	2	8	no	—
442	Q₈	—	134	8	no	—
443	Q₈	C₂²×C₂²⋊D₆₇	1	16	yes	complemented
444	D₄	—	2	48	no	—
445	D₄	—	2	48	no	—
446	D₄	—	2	24	no	—
447	D₄	—	268	96	no	—
448	D₄	—	134	96	no	—
449	D₄	—	134	48	no	—
450	D₄	—	134	48	no	—
451	D₄	—	134	24	no	—
452	D₄	C₂²×C₂²⋊D₆₇	1	48	yes	complemented
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	48	no	—
455	C₂×C₄	—	67	8	no	—
456	C₂×C₄	—	67	4	no	—
457	C₂×C₄	—	67	12	no	—
458	C₂×C₄	—	67	4	no	—
459	C₂×C₄	—	134	48	no	—
460	C₂×C₄	—	134	4	no	—
461	C₂×C₄	—	134	12	no	—
462	C₂×C₄	—	134	48	no	—
463	C₂×C₄	C₂²×C₂²⋊D₆₇	1	12	yes	—
464	C₂×C₄	C₂²×C₂²⋊D₆₇	1	4	yes	—
465	C₂×C₄	C₂²×C₂²⋊D₆₇	1	48	yes	complemented
466	C₂×C₄	C₂⁴×D₆₇	1	4	yes	—
467	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
468	C₂×C₄	C₂⁴×D₆₇	1	6	yes	—
469	C₂²	—	2	24	no	—
470	C₂²	—	2	12	no	—
471	C₂²	—	2	12	no	—
472	C₂²	—	2	24	no	—
473	C₂²	—	4	24	no	—
474	C₂²	—	8	48	no	—
475	C₂²	—	268	24	no	—
476	C₂²	—	536	48	no	—
477	C₂²	—	134	12	no	—
478	C₂²	—	134	24	no	—
479	C₂²	C₂³×C₂²⋊D₆₇	1	12	yes	—
480	C₂²	C₂³×C₂²⋊D₆₇	1	2	yes	—
481	C₂²	H(5,𝔽₂)×D₆₇	1	4	yes	—
482	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
483	C₄	—	2	4	no	—
484	C₄	—	67	8	no	—
485	C₄	—	134	4	no	—
486	C₄	—	134	12	no	—
487	C₄	C₂³×C₂²⋊D₆₇	1	12	yes	—
488	C₄	C₂³×C₂²⋊D₆₇	1	4	yes	—
489	C₂	—	2	12	no	—
490	C₂	—	4	12	no	—
491	C₂	—	268	12	no	—
492	C₂	#22573983	1	4	yes	direct product factor
493	C₂	C₂⁴×C₂²⋊D₆₇	1	1	yes	characteristic
494	C₂	C₂×H(5,𝔽₂)×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂²×H(5,𝔽₂)×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁵
hypercenter	C₂²×H(5,𝔽₂)	D₆₇
hypocenter	C₆₇	C₂×#3498
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂²×#14524089	C₂	semidirect
2. #22573983	C₂	direct (N × H)
3. C₂×#16864065	C₂	semidirect
4. C₂×D₄×C₂²⋊D₆₇	C₂	semidirect
5. C₂×#16863963	C₂	semidirect
6. C₂×#16863960	C₂	semidirect
7. C₂×#16863942	C₂	semidirect
8. C₂²×#14524039	C₂	semidirect
9. C₂×#41×Dic₆₇	C₂	semidirect
10. C₂²×H(5,𝔽₂)×C₆₇	C₂	semidirect
12. C₂×#14524089	C₂²	semidirect
13. C₂²×#11761629	C₂²	semidirect
14. C₂²×#11761628	C₂²	semidirect
15. C₂³×D₂₆₈	C₂²	semidirect
16. C₂³×C₂²⋊D₆₇	C₂²	semidirect
17. C₂²×D₄×D₆₇	C₂²	semidirect
18. #16864065	C₂²	semidirect
19. D₄×C₂²⋊D₆₇	C₂²	semidirect
20. #16863963	C₂²	semidirect
21. #16863960	C₂²	semidirect
22. #16863942	C₂²	semidirect
23. C₂×#14524041	C₂²	semidirect
24. C₂×#14524039	C₂²	semidirect
25. C₂×#14524039	C₂²	semidirect
26. C₂×#14524039	C₂²	semidirect
27. C₂×#14524039	C₂²	semidirect
28. C₂×#14524032	C₂²	semidirect
29. C₂×#14524030	C₂²	semidirect
30. C₂×#14524017	C₂²	semidirect
31. C₂×#14524015	C₂²	semidirect
32. C₂×#14524014	C₂²	semidirect
33. C₂×#14524012	C₂²	semidirect
34. C₂×#14524011	C₂²	semidirect
35. #41×Dic₆₇	C₂²	semidirect
36. C₂×Q₈×Dic₆₇	C₂²	semidirect
37. C₂×D₄×Dic₆₇	C₂²	semidirect
38. C₂×C₄×C₂²⋊D₆₇	C₂²	semidirect
39. C₂×#14524002	C₂²	semidirect
40. C₂²×C₄×Dic₆₇	C₂²	semidirect
42. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
44. C₂²×#41×C₆₇	C₂²	semidirect
45. C₂²×#41×C₆₇	C₂²	semidirect
46. C₂³×D₄×C₆₇	C₂²	semidirect
47. C₂³×D₄×C₆₇	C₂²	semidirect
51. C₂×#11761629	C₂³	semidirect
53. C₂×#11761628	C₂³	semidirect
58. C₂²×D₂₆₈	C₂³	semidirect
63. C₂²×C₂²⋊D₆₇	C₂³	semidirect
65. C₂×D₄×D₆₇	C₂³	semidirect
68. #14524041	C₂³	semidirect
69. #14524039	C₂³	semidirect
70. #14524039	C₂³	semidirect
71. #14524039	C₂³	semidirect
72. #14524032	C₂³	semidirect
73. #14524030	C₂³	semidirect
74. #14524017	C₂³	semidirect
75. #14524015	C₂³	semidirect
76. #14524014	C₂³	semidirect
77. #14524012	C₂³	semidirect
78. #14524011	C₂³	semidirect
79. Q₈×Dic₆₇	C₂³	semidirect
80. D₄×Dic₆₇	C₂³	semidirect
81. C₄×C₂²⋊D₆₇	C₂³	semidirect
82. #14524002	C₂³	semidirect
91. C₂×C₄×Dic₆₇	C₂³	semidirect
96. C₂×#41×C₆₇	D₄	semidirect
98. C₂×#41×C₆₇	C₂³	semidirect
99. C₂×#41×C₆₇	C₂³	semidirect
108. C₂²×D₄×C₆₇	C₂³	semidirect
109. C₂²×D₄×C₆₇	C₂³	semidirect
155. #41×C₆₇	C₂×D₄	semidirect
157. C₂×Q₈×C₆₇	C₂×D₄	semidirect
165. C₂×D₄×C₆₇	C₂×D₄	semidirect
171. C₂²×C₄×C₆₇	C₂×D₄	semidirect
194. Q₈×C₆₇	C₂²×D₄	semidirect
198. D₄×C₆₇	C₂²×D₄	semidirect
203. C₂×C₄×C₆₇	C₂²×D₄	semidirect
213. C₂×Dic₆₇	H(5,𝔽₂)	semidirect
240. Dic₆₇	C₂×H(5,𝔽₂)	semidirect
244. C₂×C₆₇	#3498	semidirect
249. C₂²×H(5,𝔽₂)	D₆₇	semidirect
258. C₆₇	C₂×#3498	semidirect
262. C₂×H(5,𝔽₂)	C₂×D₆₇	semidirect
266. C₂²×#41	C₂×D₆₇	semidirect
267. C₂²×#41	C₂×D₆₇	semidirect
271. C₂³×D₄	C₂×D₆₇	semidirect
272. C₂³×D₄	C₂×D₆₇	semidirect
306. C₂×#41	C₂²⋊D₆₇	semidirect
308. C₂×#41	C₂²×D₆₇	semidirect
309. C₂×#41	C₂²×D₆₇	semidirect
330. C₂²×D₄	C₂²×D₆₇	semidirect
331. C₂²×D₄	C₂²×D₆₇	semidirect
373. #41	C₂×C₂²⋊D₆₇	semidirect
377. C₂×Q₈	C₂×C₂²⋊D₆₇	semidirect
398. C₂×D₄	C₂×C₂²⋊D₆₇	semidirect
411. C₂²×C₄	C₂×C₂²⋊D₆₇	semidirect
443. Q₈	C₂²×C₂²⋊D₆₇	semidirect
452. D₄	C₂²×C₂²⋊D₆₇	semidirect
465. C₂×C₄	C₂²×C₂²⋊D₆₇	semidirect
492. C₂	#22573983	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
#22573983	2
C₂×#16864065	2
C₂×D₄×C₂²⋊D₆₇	2
C₂×#16863963	2
C₂×#16863960	2
C₂×#16863942	2
C₂²×#14524039	2
C₂×#41×Dic₆₇	2
C₂²×H(5,𝔽₂)×C₆₇	2
C₂×#3498	67

Subgroups — C2×#22573983 #31764151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22573983 #31764151