Subgroups — C₂×#22574086 #31764192

Order 17152.

8622 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	#22574086	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16864131	C₂	1	4	yes	maximal, complemented
3	C₂×#16864068	C₂	1	4	yes	maximal, complemented
4	C₂×#16864037	C₂	1	8	yes	maximal, complemented
5	C₂×#16864002	C₂	1	4	yes	maximal, complemented
6	C₂×#16863965	C₂	1	2	yes	maximal, complemented
7	C₂×#16863947	C₂	1	4	yes	maximal, complemented
8	C₂²×#14524040	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16863870	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×#11761628	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#11761629	C₂²	1	8	yes	complemented
13	C₂²×#11761628	C₂²	1	1	yes	characteristic
14	C₂³×D₂₆₈	C₂²	1	2	yes	—
15	C₂³×D₂₆₈	C₂²	1	4	yes	—
16	#16864131	C₂²	1	64	yes	complemented
17	#16864068	C₂²	1	64	yes	complemented
18	#16864037	C₂²	1	128	yes	complemented
19	#16864002	C₂²	1	64	yes	complemented
20	#16863965	C₂²	1	32	yes	complemented
21	#16863947	C₂²	1	64	yes	complemented
22	C₂×#14524040	C₂²	1	2	yes	—
23	C₂×#14524040	C₂²	1	4	yes	—
24	C₂×#14524040	C₂²	1	2	yes	—
25	C₂×#14524040	C₂²	1	16	yes	complemented
26	C₂×#14524040	C₂²	1	4	yes	complemented
27	C₂×#14524032	C₂²	1	8	yes	complemented
28	C₂×#14524031	C₂²	1	4	yes	complemented
29	C₂×#14524028	C₂²	1	16	yes	complemented
30	C₂×#14524018	C₂²	1	16	yes	complemented
31	C₂×#14524018	C₂²	1	8	yes	complemented
32	C₂×#14524018	C₂²	1	8	yes	complemented
33	C₂×#14524013	C₂²	1	8	yes	—
34	#16863870	C₂²	1	32	yes	complemented
35	C₄×#11761628	C₂²	1	16	yes	complemented
36	C₂×C₄×Dic₁₃₄	C₂²	1	2	yes	complemented
37	C₂×C₄×D₂₆₈	C₂²	1	2	yes	complemented
38	C₂×C₄×D₂₆₈	C₂²	1	8	yes	complemented
39	C₂×#14524005	C₂²	1	8	yes	complemented
40	C₂×#14524005	C₂²	1	8	yes	complemented
41	C₂×C₄×C₂²⋊D₆₇	C₂²	1	4	yes	—
42	C₂×#14524002	C₂²	1	1	yes	characteristic
43	C₂×#14524000	C₂²	1	2	yes	complemented
44	C₂×C₂²⋊C₄×D₆₇	C₂²	1	8	yes	—
45	C₂×C₄²×D₆₇	C₂²	1	2	yes	complemented
46	#533×C₆₇	C₂²	1	16	yes	complemented
47	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
48	C₂×#125×C₆₇	C₂²	1	2	yes	complemented
49	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
50	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	complemented
51	C₂×#117×C₆₇	C₂²	1	2	yes	—
52	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
53	C₂⁴×D₆₇	C₂³	1	4	yes	—
54	C₂×#11761629	—	4	32	no	—
55	C₂×#11761629	C₂³	1	64	yes	complemented
56	C₂×#11761628	—	2	8	no	—
57	C₂×#11761628	C₂³	1	8	yes	—
58	C₂²×Dic₁₃₄	C₂³	1	1	yes	characteristic
59	C₂²×D₂₆₈	—	2	16	no	—
60	C₂²×D₂₆₈	—	4	16	no	—
61	C₂²×D₂₆₈	C₂³	1	4	yes	—
62	C₂²×D₂₆₈	C₂³	1	8	yes	—
63	C₂²×D₂₆₈	C₂³	1	4	yes	—
64	C₂²×D₂₆₈	C₂³	1	1	yes	characteristic
65	C₂²×D₂₆₈	C₂³	1	16	yes	—
66	C₂²×D₂₆₈	C₂³	1	32	yes	—
67	C₂²×D₂₆₈	C₂³	1	16	yes	—
68	C₂²×D₂₆₈	C₂³	1	8	yes	—
69	C₂²×C₂²⋊D₆₇	C₂³	1	2	yes	—
70	C₂²×C₄×D₆₇	C₂³	1	2	yes	—
71	C₂²×C₄×D₆₇	C₂³	1	4	yes	—
72	C₂²×C₄×D₆₇	C₂³	1	16	yes	—
73	#14524040	C₂³	1	16	yes	—
74	#14524040	C₂³	1	32	yes	—
75	#14524040	C₂³	1	16	yes	—
76	#14524040	C₂³	1	32	yes	complemented
77	#14524032	C₂³	1	64	yes	complemented
78	#14524031	C₂³	1	32	yes	complemented
79	#14524028	C₂³	1	128	yes	complemented
80	#14524018	C₂³	1	128	yes	complemented
81	#14524018	C₂³	1	64	yes	complemented
82	#14524018	C₂³	1	64	yes	complemented
83	#14524013	C₂³	1	64	yes	—
84	C₄×Dic₁₃₄	C₂³	1	16	yes	complemented
85	C₄×D₂₆₈	C₂³	1	16	yes	complemented
86	C₄×D₂₆₈	C₂³	1	64	yes	complemented
87	#14524005	C₂³	1	64	yes	complemented
88	#14524005	C₂³	1	64	yes	complemented
89	C₄×C₂²⋊D₆₇	C₂³	1	32	yes	—
90	#14524002	C₂³	1	8	yes	—
91	#14524000	C₂³	1	16	yes	complemented
92	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
93	C₂×Dic₆₇⋊C₄	C₂³	1	4	yes	—
94	C₂×C₂²⋊Dic₆₇	C₂³	1	2	yes	—
95	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	16	yes	—
96	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
97	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	16	yes	—
98	C₂²⋊C₄×D₆₇	C₂³	1	64	yes	—
99	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
100	C₂×C₄×Dic₆₇	C₂³	1	4	yes	—
101	C₄²×D₆₇	C₂³	1	16	yes	complemented
102	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
103	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
104	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
105	C₄⋊Q₈×C₆₇	C₂³	1	16	yes	complemented
106	#125×C₆₇	C₂³	1	16	yes	complemented
107	C₂²⋊Q₈×C₆₇	C₂³	1	32	yes	—
108	C₄×Q₈×C₆₇	C₂³	1	32	yes	complemented
109	#117×C₆₇	C₂³	1	16	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
114	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
115	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
118	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
119	C₂³×D₆₇	—	2	16	no	—
120	C₂³×D₆₇	—	4	16	no	—
121	C₂³×D₆₇	C₂⁴	1	8	yes	—
122	C₂³×D₆₇	C₂⁴	1	2	yes	—
123	C₂³×D₆₇	C₂⁴	1	16	yes	—
124	#11761629	—	4	128	no	—
125	#11761628	—	2	32	no	—
126	C₂×Dic₁₃₄	—	2	4	no	—
127	C₂×Dic₁₃₄	C₂⁴	1	4	yes	—
128	C₂×D₂₆₈	—	2	16	no	—
129	C₂×D₂₆₈	—	2	16	no	—
130	C₂×D₂₆₈	—	2	4	no	—
131	C₂×D₂₆₈	—	2	64	no	—
132	C₂×D₂₆₈	—	2	32	no	—
133	C₂×D₂₆₈	—	4	64	no	—
134	C₂×D₂₆₈	—	4	16	no	—
135	C₂×D₂₆₈	—	4	32	no	—
136	C₂×D₂₆₈	C₂⁴	1	16	yes	—
137	C₂×D₂₆₈	C₂⁴	1	32	yes	—
138	C₂×D₂₆₈	C₂⁴	1	16	yes	—
139	C₂×D₂₆₈	C₂⁴	1	4	yes	—
140	C₂×D₂₆₈	C₂⁴	1	64	yes	—
141	C₂×D₂₆₈	C₂⁴	1	32	yes	—
142	C₂×C₂²⋊D₆₇	—	2	8	no	—
143	C₂×C₂²⋊D₆₇	C₂⁴	1	8	yes	—
144	C₂²×Dic₆₇	C₂⁴	1	2	yes	—
145	C₂²×Dic₆₇	C₂⁴	1	4	yes	—
146	C₂×C₄×D₆₇	—	2	8	no	—
147	C₂×C₄×D₆₇	—	2	16	no	—
148	C₂×C₄×D₆₇	—	4	32	no	—
149	C₂×C₄×D₆₇	C₂⁴	1	8	yes	—
150	C₂×C₄×D₆₇	C₂⁴	1	16	yes	—
151	C₂×C₄×D₆₇	C₂⁴	1	64	yes	—
152	C₄⋊Dic₆₇	C₂⁴	1	8	yes	—
153	Dic₆₇⋊C₄	C₂⁴	1	16	yes	—
154	C₂²⋊Dic₆₇	C₂⁴	1	8	yes	—
155	(C₂×D₆₇)⋊C₄	C₂⁴	1	64	yes	—
156	(C₂×D₆₇)⋊C₄	C₂⁴	1	16	yes	—
157	(C₂×D₆₇)⋊C₄	C₂⁴	1	64	yes	—
158	C₄×Dic₆₇	C₂⁴	1	8	yes	—
159	C₄×Dic₆₇	C₂⁴	1	16	yes	—
160	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
161	C₂×Q₈×C₆₇	—	2	16	no	—
162	C₂×Q₈×C₆₇	C₂⁴	1	16	yes	—
163	C₂²×C₄×C₆₇	—	2	8	no	—
164	C₂²×C₄×C₆₇	#41	1	8	yes	complemented
165	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
168	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
172	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
173	C₄⋊C₄×C₆₇	C₂⁴	1	32	yes	—
174	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
175	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
176	C₄²×C₆₇	C₂⁴	1	8	yes	—
177	C₄²×C₆₇	C₂⁴	1	4	yes	—
178	C₄²×C₆₇	C₂⁴	1	4	yes	—
179	C₄²×C₆₇	C₂⁴	1	16	yes	—
180	C₂³×C₆₇	—	2	4	no	—
181	C₂³×C₆₇	C₂×#41	1	4	yes	—
182	C₂³×C₆₇	C₂⁵	1	2	yes	—
183	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
184	C₂²×D₆₇	—	2	4	no	—
185	C₂²×D₆₇	—	2	16	no	—
186	C₂²×D₆₇	—	2	8	no	—
187	C₂²×D₆₇	—	2	32	no	—
188	C₂²×D₆₇	—	2	32	no	—
189	C₂²×D₆₇	—	4	32	no	—
190	C₂²×D₆₇	—	8	32	no	—
191	C₂²×D₆₇	C₂⁵	1	16	yes	—
192	C₂²×D₆₇	C₂⁵	1	4	yes	—
193	Q₈×C₆₇	—	2	32	no	—
194	C₂×C₄×C₆₇	—	2	4	no	—
195	C₂×C₄×C₆₇	—	2	16	no	—
196	C₂×C₄×C₆₇	—	2	16	no	—
197	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
198	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
199	C₂×C₄×C₆₇	C₂×#41	1	16	yes	complemented
200	C₂×C₄×C₆₇	H(5,𝔽₂)	1	8	yes	—
201	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
202	C₂×C₄×C₆₇	C₂⁵	1	16	yes	—
203	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
204	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
205	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
206	C₂²⋊D₆₇	—	2	16	no	—
207	C₂×Dic₆₇	—	2	4	no	—
208	C₂×Dic₆₇	—	2	8	no	—
209	C₂×Dic₆₇	C₂⁵	1	4	yes	—
210	C₂×Dic₆₇	C₂⁵	1	8	yes	—
211	D₂₆₈	—	2	32	no	—
212	D₂₆₈	—	2	32	no	—
213	D₂₆₈	—	2	8	no	—
214	D₂₆₈	—	2	64	no	—
215	D₂₆₈	—	4	32	no	—
216	D₂₆₈	—	4	64	no	—
217	C₄×D₆₇	—	2	16	no	—
218	C₄×D₆₇	—	2	32	no	—
219	C₄×D₆₇	—	4	64	no	—
220	Dic₁₃₄	—	2	8	no	—
221	C₂²×C₆₇	—	2	4	no	—
222	C₂²×C₆₇	—	2	8	no	—
223	C₂²×C₆₇	C₂²×#41	1	4	yes	—
224	C₂²×C₆₇	C₂²×#41	1	2	yes	—
225	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
226	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
227	C₂×D₆₇	—	2	4	no	—
228	C₂×D₆₇	—	2	16	no	—
229	C₂×D₆₇	—	2	8	no	—
230	C₂×D₆₇	—	2	32	no	—
231	C₂×D₆₇	—	4	8	no	—
232	C₂×D₆₇	—	4	32	no	—
233	C₂×D₆₇	—	8	32	no	—
234	C₄×C₆₇	—	2	16	no	—
235	C₄×C₆₇	—	2	4	no	—
236	C₄×C₆₇	C₂²×#41	1	4	yes	—
237	C₄×C₆₇	C₂²×#41	1	4	yes	—
238	C₄×C₆₇	C₂×H(5,𝔽₂)	1	8	yes	—
239	Dic₆₇	—	2	4	no	—
240	Dic₆₇	—	2	8	no	—
241	C₂×#3508	—	67	1	no	maximal, Sylow
242	C₂×C₆₇	—	2	4	no	—
243	C₂×C₆₇	#3508	1	4	yes	complemented
244	C₂×C₆₇	C₂³×#41	1	1	yes	characteristic
245	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	2	yes	—
246	D₆₇	—	4	4	no	—
247	D₆₇	—	4	16	no	—
248	#3508	—	67	32	no	—
249	C₂×#559	—	67	4	no	—
250	C₂×#550	—	67	4	no	—
251	C₂×#546	—	67	8	no	—
252	C₂×#540	—	67	4	no	—
253	C₂×#535	—	67	2	no	—
254	C₂×#533	D₆₇	1	1	yes	characteristic, complemented
255	C₂×#532	—	67	4	no	—
256	C₂²×#127	—	67	1	no	—
257	C₂×#518	—	67	2	no	—
258	C₂×C₄×#41	—	67	1	no	—
259	C₆₇	C₂×#3508	1	1	yes	characteristic, complemented, Sylow
260	C₂²×#41	—	67	8	no	—
261	C₂²×#41	—	67	1	no	—
262	C₂³×D₄	—	67	4	no	—
263	C₂³×D₄	—	67	2	no	—
264	#559	—	67	64	no	—
265	#550	—	67	64	no	—
266	#546	—	67	128	no	—
267	#540	—	67	64	no	—
268	#535	—	67	32	no	—
269	#533	C₂×D₆₇	1	16	yes	complemented
270	#532	—	67	64	no	—
271	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	complemented
272	C₂×#127	—	67	2	no	—
273	C₂×#127	—	67	2	no	—
274	C₂×#127	—	67	4	no	—
275	C₂×#127	—	67	16	no	—
276	C₂×#127	—	67	4	no	—
277	C₂×#125	C₂×D₆₇	1	2	yes	complemented
278	C₂×#124	—	67	8	no	—
279	C₂×#124	—	67	4	no	—
280	C₂×#123	—	67	16	no	—
281	C₂×C₂²⋊Q₈	C₂×D₆₇	1	4	yes	—
282	C₂×#121	—	67	8	no	—
283	C₂×#121	—	67	8	no	—
284	C₂×#121	—	67	16	no	—
285	C₂×#120	—	67	8	no	—
286	#518	—	67	32	no	—
287	C₄×#41	—	67	16	no	—
288	C₂×C₄×Q₈	—	67	2	no	—
289	C₂×C₄×Q₈	C₂×D₆₇	1	4	yes	complemented
290	C₂×C₄×D₄	—	67	4	no	—
291	C₂×C₄×D₄	—	67	8	no	—
292	C₂×C₄×D₄	—	67	8	no	—
293	C₂×C₄×D₄	—	67	2	no	—
294	C₂×C₄×D₄	—	67	8	no	—
295	C₂×#117	—	67	1	no	—
296	C₂×#117	—	67	2	no	—
297	C₂×#117	C₂×D₆₇	1	2	yes	—
298	C₂²×C₂²⋊C₄	—	67	8	no	—
299	C₂²×C₄²	—	67	2	no	—
300	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
301	C₂⁵	—	67	4	no	—
302	C₂×#41	—	268	32	no	—
303	C₂×#41	—	67	64	no	—
304	C₂×#41	—	67	8	no	—
305	C₂×#41	—	134	8	no	—
306	C₂²×Q₈	—	67	1	no	—
307	C₂²×Q₈	C₂²×D₆₇	1	4	yes	—
308	C₂²×D₄	—	268	16	no	—
309	C₂²×D₄	—	67	2	no	—
310	C₂²×D₄	—	67	16	no	—
311	C₂²×D₄	—	67	8	no	—
312	C₂²×D₄	—	67	32	no	—
313	C₂²×D₄	—	67	4	no	—
314	C₂²×D₄	—	67	4	no	—
315	C₂²×D₄	—	67	1	no	—
316	C₂²×D₄	—	67	8	no	—
317	C₂²×D₄	—	67	16	no	—
318	C₂²×D₄	—	134	16	no	—
319	C₂³×C₄	—	67	16	no	—
320	C₂³×C₄	—	67	2	no	—
321	C₂³×C₄	—	67	4	no	—
322	C₂³×C₄	C₂²×D₆₇	1	2	yes	—
323	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
324	C₄⋊Q₈	C₂²×D₆₇	1	16	yes	complemented
325	#127	—	67	16	no	—
326	#127	—	67	16	no	—
327	#127	—	67	32	no	—
328	#127	—	67	32	no	—
329	#125	C₂²×D₆₇	1	16	yes	complemented
330	#124	—	67	64	no	—
331	#124	—	67	32	no	—
332	#123	—	67	128	no	—
333	C₂²⋊Q₈	C₂²×D₆₇	1	32	yes	—
334	#121	—	67	64	no	—
335	#121	—	67	64	no	—
336	#121	—	67	128	no	—
337	#120	—	67	64	no	—
338	C₄×Q₈	—	67	16	no	—
339	C₄×Q₈	C₂²×D₆₇	1	32	yes	complemented
340	C₄×D₄	—	67	32	no	—
341	C₄×D₄	—	67	64	no	—
342	C₄×D₄	—	67	64	no	—
343	C₄×D₄	—	67	16	no	—
344	C₄×D₄	—	67	64	no	—
345	#117	—	67	8	no	—
346	#117	—	67	16	no	—
347	#117	C₂²×D₆₇	1	16	yes	—
348	C₂×C₄⋊C₄	—	67	2	no	—
349	C₂×C₄⋊C₄	—	67	4	no	—
350	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
351	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
352	C₂×C₄⋊C₄	C₂²×D₆₇	1	8	yes	—
353	C₂×C₂²⋊C₄	—	67	2	no	—
354	C₂×C₂²⋊C₄	—	67	16	no	—
355	C₂×C₂²⋊C₄	—	67	4	no	—
356	C₂×C₂²⋊C₄	—	67	16	no	—
357	C₂×C₂²⋊C₄	—	67	64	no	—
358	C₂×C₂²⋊C₄	C₂²×D₆₇	1	4	yes	—
359	C₂×C₄²	—	67	2	no	—
360	C₂×C₄²	—	67	4	no	—
361	C₂×C₄²	—	67	16	no	—
362	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
363	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
364	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
365	C₂×C₄²	C₂²×D₆₇	1	8	yes	—
366	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
367	C₂⁴	—	268	16	no	—
368	C₂⁴	—	67	16	no	—
369	C₂⁴	—	67	8	no	—
370	C₂⁴	—	67	2	no	—
371	C₂⁴	—	134	16	no	—
372	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
373	#41	—	268	128	no	—
374	#41	—	134	32	no	—
375	C₂×Q₈	—	2	16	no	—
376	C₂×Q₈	—	67	4	no	—
377	C₂×Q₈	—	134	4	no	—
378	C₂×Q₈	C₂³×D₆₇	1	16	yes	—
379	C₂×D₄	—	268	16	no	—
380	C₂×D₄	—	268	32	no	—
381	C₂×D₄	—	268	64	no	—
382	C₂×D₄	—	67	8	no	—
383	C₂×D₄	—	67	64	no	—
384	C₂×D₄	—	67	32	no	—
385	C₂×D₄	—	67	16	no	—
386	C₂×D₄	—	67	16	no	—
387	C₂×D₄	—	67	4	no	—
388	C₂×D₄	—	67	32	no	—
389	C₂×D₄	—	134	8	no	—
390	C₂×D₄	—	134	16	no	—
391	C₂×D₄	—	134	16	no	—
392	C₂×D₄	—	134	4	no	—
393	C₂×D₄	—	134	64	no	—
394	C₂×D₄	—	134	32	no	—
395	C₂²×C₄	—	2	8	no	—
396	C₂²×C₄	—	268	32	no	—
397	C₂²×C₄	—	67	64	no	—
398	C₂²×C₄	—	67	2	no	—
399	C₂²×C₄	—	67	4	no	—
400	C₂²×C₄	—	67	8	no	—
401	C₂²×C₄	—	67	16	no	—
402	C₂²×C₄	—	134	8	no	—
403	C₂²×C₄	—	134	16	no	—
404	C₂²×C₄	#11761629	1	8	yes	complemented
405	C₂²×C₄	C₂³×D₆₇	1	8	yes	—
406	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
407	C₂²×C₄	C₂³×D₆₇	1	8	yes	—
408	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
409	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
410	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
411	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
412	C₄⋊C₄	—	67	8	no	—
413	C₄⋊C₄	—	67	16	no	—
414	C₄⋊C₄	C₂³×D₆₇	1	16	yes	—
415	C₄⋊C₄	C₂³×D₆₇	1	16	yes	—
416	C₄⋊C₄	C₂³×D₆₇	1	32	yes	—
417	C₂²⋊C₄	—	67	8	no	—
418	C₂²⋊C₄	—	67	64	no	—
419	C₂²⋊C₄	—	67	16	no	—
420	C₂²⋊C₄	—	67	64	no	—
421	C₂²⋊C₄	C₂³×D₆₇	1	16	yes	—
422	C₄²	—	67	8	no	—
423	C₄²	—	67	16	no	—
424	C₄²	C₂³×D₆₇	1	4	yes	—
425	C₄²	C₂³×D₆₇	1	4	yes	—
426	C₄²	C₂³×D₆₇	1	8	yes	—
427	C₄²	C₂³×D₆₇	1	16	yes	—
428	C₂³	—	2	4	no	—
429	C₂³	—	268	32	no	—
430	C₂³	—	536	32	no	—
431	C₂³	—	67	16	no	—
432	C₂³	—	67	4	no	—
433	C₂³	—	134	4	no	—
434	C₂³	—	134	16	no	—
435	C₂³	—	134	32	no	—
436	C₂³	—	134	8	no	—
437	C₂³	—	134	32	no	—
438	C₂³	C₂×#11761629	1	4	yes	—
439	C₂³	C₂⁴×D₆₇	1	2	yes	—
440	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
441	Q₈	—	2	32	no	—
442	Q₈	—	134	8	no	—
443	D₄	—	268	32	no	—
444	D₄	—	268	64	no	—
445	D₄	—	134	16	no	—
446	D₄	—	134	32	no	—
447	D₄	—	134	32	no	—
448	D₄	—	134	8	no	—
449	D₄	—	134	64	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	16	no	—
452	C₂×C₄	—	2	16	no	—
453	C₂×C₄	—	268	64	no	—
454	C₂×C₄	—	67	4	no	—
455	C₂×C₄	—	67	8	no	—
456	C₂×C₄	—	134	4	no	—
457	C₂×C₄	—	134	8	no	—
458	C₂×C₄	—	134	16	no	—
459	C₂×C₄	—	134	32	no	—
460	C₂×C₄	C₂×#11761629	1	4	yes	—
461	C₂×C₄	C₂×#11761629	1	4	yes	—
462	C₂×C₄	C₂×#11761629	1	16	yes	complemented
463	C₂×C₄	#14524089	1	8	yes	—
464	C₂×C₄	C₂⁴×D₆₇	1	4	yes	—
465	C₂×C₄	C₂⁴×D₆₇	1	16	yes	—
466	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
467	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
468	C₂×C₄	C₂⁴×D₆₇	1	4	yes	—
469	C₂²	—	2	8	no	—
470	C₂²	—	2	4	no	—
471	C₂²	—	268	32	no	—
472	C₂²	—	268	8	no	—
473	C₂²	—	536	32	no	—
474	C₂²	—	134	4	no	—
475	C₂²	—	134	16	no	—
476	C₂²	—	134	32	no	—
477	C₂²	—	134	8	no	—
478	C₂²	C₂²×#11761629	1	4	yes	—
479	C₂²	C₂²×#11761629	1	2	yes	—
480	C₂²	C₂×#14524089	1	4	yes	—
481	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
482	C₄	—	2	4	no	—
483	C₄	—	2	16	no	—
484	C₄	—	134	4	no	—
485	C₄	—	134	8	no	—
486	C₄	C₂²×#11761629	1	4	yes	—
487	C₄	C₂²×#11761629	1	4	yes	—
488	C₄	C₂×#14524089	1	8	yes	—
489	C₂	—	2	4	no	—
490	C₂	—	268	16	no	—
491	C₂	—	268	4	no	—
492	C₂	#22574086	1	4	yes	direct product factor
493	C₂	C₂³×#11761629	1	1	yes	characteristic
494	C₂	C₂²×#14524089	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂×#533×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁵
hypercenter	C₂×#533	D₆₇
hypocenter	C₆₇	C₂×#3508
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
1. #22574086	C₂	direct (N × H)
2. C₂×#16864131	C₂	semidirect
3. C₂×#16864068	C₂	semidirect
4. C₂×#16864037	C₂	semidirect
5. C₂×#16864002	C₂	semidirect
6. C₂×#16863965	C₂	semidirect
7. C₂×#16863947	C₂	semidirect
8. C₂²×#14524040	C₂	semidirect
9. C₂×#16863870	C₂	semidirect
10. C₂×C₄×#11761628	C₂	semidirect
11. C₂×#533×C₆₇	C₂	semidirect
12. C₂²×#11761629	C₂²	semidirect
16. #16864131	C₂²	semidirect
17. #16864068	C₂²	semidirect
18. #16864037	C₂²	semidirect
19. #16864002	C₂²	semidirect
20. #16863965	C₂²	semidirect
21. #16863947	C₂²	semidirect
25. C₂×#14524040	C₂²	semidirect
26. C₂×#14524040	C₂²	semidirect
27. C₂×#14524032	C₂²	semidirect
28. C₂×#14524031	C₂²	semidirect
29. C₂×#14524028	C₂²	semidirect
30. C₂×#14524018	C₂²	semidirect
31. C₂×#14524018	C₂²	semidirect
32. C₂×#14524018	C₂²	semidirect
34. #16863870	C₂²	semidirect
35. C₄×#11761628	C₂²	semidirect
36. C₂×C₄×Dic₁₃₄	C₂²	semidirect
37. C₂×C₄×D₂₆₈	C₂²	semidirect
38. C₂×C₄×D₂₆₈	C₂²	semidirect
39. C₂×#14524005	C₂²	semidirect
40. C₂×#14524005	C₂²	semidirect
43. C₂×#14524000	C₂²	semidirect
45. C₂×C₄²×D₆₇	C₂²	semidirect
46. #533×C₆₇	C₂²	semidirect
47. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
48. C₂×#125×C₆₇	C₂²	semidirect
50. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
55. C₂×#11761629	C₂³	semidirect
76. #14524040	C₂³	semidirect
77. #14524032	C₂³	semidirect
78. #14524031	C₂³	semidirect
79. #14524028	C₂³	semidirect
80. #14524018	C₂³	semidirect
81. #14524018	C₂³	semidirect
82. #14524018	C₂³	semidirect
84. C₄×Dic₁₃₄	C₂³	semidirect
85. C₄×D₂₆₈	C₂³	semidirect
86. C₄×D₂₆₈	C₂³	semidirect
87. #14524005	C₂³	semidirect
88. #14524005	C₂³	semidirect
91. #14524000	C₂³	semidirect
101. C₄²×D₆₇	C₂³	semidirect
105. C₄⋊Q₈×C₆₇	C₂³	semidirect
106. #125×C₆₇	C₂³	semidirect
108. C₄×Q₈×C₆₇	C₂³	semidirect
164. C₂²×C₄×C₆₇	#41	semidirect
199. C₂×C₄×C₆₇	C₂×#41	semidirect
243. C₂×C₆₇	#3508	semidirect
254. C₂×#533	D₆₇	semidirect
259. C₆₇	C₂×#3508	semidirect
269. #533	C₂×D₆₇	semidirect
271. C₂×C₄⋊Q₈	C₂×D₆₇	semidirect
277. C₂×#125	C₂×D₆₇	semidirect
289. C₂×C₄×Q₈	C₂×D₆₇	semidirect
324. C₄⋊Q₈	C₂²×D₆₇	semidirect
329. #125	C₂²×D₆₇	semidirect
339. C₄×Q₈	C₂²×D₆₇	semidirect
404. C₂²×C₄	#11761629	semidirect
462. C₂×C₄	C₂×#11761629	semidirect
492. C₂	#22574086	direct (N × H)

Maximal subgroups

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

subgroup	index
#22574086	2
C₂×#16864131	2
C₂×#16864068	2
C₂×#16864037	2
C₂×#16864002	2
C₂×#16863965	2
C₂×#16863947	2
C₂²×#14524040	2
C₂×#16863870	2
C₂×C₄×#11761628	2
C₂×#533×C₆₇	2
C₂×#3508	67

Subgroups — C2×#22574086 #31764192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22574086 #31764192