Subgroups — #31764367

Order 17152.

11534 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#22573882	C₂	1	8	yes	maximal, complemented
3	#22573871	C₂	1	8	yes	maximal, complemented
4	#22573864	C₂	1	2	yes	maximal, complemented
5	#22573857	C₂	1	2	yes	maximal, complemented
6	#22573849	C₂	1	6	yes	maximal, complemented
7	#22573840	C₂	1	6	yes	maximal, complemented
8	#41×C₂²⋊D₆₇	C₂	1	16	yes	maximal, complemented
9	#22573538	C₂	1	6	yes	maximal, complemented
10	C₂×#16863903	C₂	1	6	yes	maximal, complemented
11	#22573412	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#585×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#11761629	C₂²	1	1	yes	characteristic, complemented
14	C₂²×#11761628	C₂²	1	3	yes	complemented
15	C₂²×#11761627	C₂²	1	3	yes	complemented
16	C₂×#41×D₆₇	—	2	16	no	—
17	C₂×#41×D₆₇	C₂²	1	8	yes	complemented
18	C₂×#41×D₆₇	C₂²	1	8	yes	complemented
19	C₂²×Q₈×D₆₇	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₄×D₆₇	C₂²	1	3	yes	complemented
21	C₂³×C₄×D₆₇	C₂²	1	3	yes	complemented
22	#16864065	C₂²	1	16	yes	complemented
23	Q₈×C₂²⋊D₆₇	C₂²	1	16	yes	complemented
24	#16864057	C₂²	1	16	yes	complemented
25	#16864053	C₂²	1	16	yes	complemented
26	#16864045	C₂²	1	48	yes	complemented
27	#16864040	C₂²	1	48	yes	complemented
28	#16864034	C₂²	1	48	yes	complemented
29	D₄×C₂²⋊D₆₇	C₂²	1	48	yes	complemented
30	#16863974	C₂²	1	24	yes	complemented
31	#16863970	C₂²	1	12	yes	complemented
32	#16863967	C₂²	1	12	yes	complemented
33	#16863963	C₂²	1	12	yes	complemented
34	#16863960	C₂²	1	12	yes	complemented
35	#16863955	C₂²	1	24	yes	complemented
36	#16863942	C₂²	1	24	yes	complemented
37	#16863907	C₂²	1	24	yes	complemented
38	#16863903	C₂²	1	12	yes	complemented
39	#16863903	C₂²	1	4	yes	complemented
40	#16863903	C₂²	1	12	yes	complemented
41	#16863903	C₂²	1	12	yes	complemented
42	#16863903	C₂²	1	24	yes	complemented
43	#16863903	C₂²	1	24	yes	complemented
44	C₂×#14524024	C₂²	1	12	yes	complemented
45	C₂×#14524019	C₂²	1	12	yes	complemented
46	C₂×#14524014	C₂²	1	12	yes	complemented
47	C₂×#14524011	C₂²	1	12	yes	complemented
48	#16863879	C₂²	1	8	yes	complemented
49	#16863874	C₂²	1	2	yes	complemented
50	#16863871	C₂²	1	6	yes	complemented
51	#41×Dic₆₇	C₂²	1	8	yes	complemented
52	C₂×C₄×C₂²⋊D₆₇	C₂²	1	24	yes	complemented
53	C₂×#14524002	C₂²	1	6	yes	complemented
54	#585×C₆₇	—	2	16	no	—
55	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	2	yes	complemented
56	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	4	yes	complemented
57	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	6	yes	complemented
58	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	4	yes	complemented
59	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
60	C₂²×#41×C₆₇	C₂²	1	6	yes	complemented
61	C₂²×#41×C₆₇	C₂²	1	8	yes	complemented
62	C₂⁴×D₆₇	C₂³	1	3	yes	—
63	C₂×#11761629	—	2	8	no	—
64	C₂×#11761629	C₂³	1	8	yes	—
65	C₂×#11761628	—	2	24	no	—
66	C₂×#11761628	C₂³	1	12	yes	—
67	C₂×#11761628	C₂³	1	12	yes	—
68	C₂×#11761627	—	2	24	no	—
69	C₂×#11761627	C₂³	1	24	yes	—
70	#41×D₆₇	—	2	64	no	—
71	#41×D₆₇	—	2	64	no	—
72	C₂²×Dic₁₃₄	C₂³	1	3	yes	—
73	C₂×Q₈×D₆₇	—	2	8	no	—
74	C₂×Q₈×D₆₇	C₂³	1	8	yes	—
75	C₂²×D₂₆₈	C₂³	1	3	yes	—
76	C₂²×C₂²⋊D₆₇	C₂³	1	6	yes	—
77	C₂²×C₂²⋊D₆₇	C₂³	1	24	yes	—
78	C₂×D₄×D₆₇	—	2	24	no	—
79	C₂×D₄×D₆₇	C₂³	1	24	yes	—
80	C₂³×Dic₆₇	C₂³	1	3	yes	—
81	C₂²×C₄×D₆₇	—	2	24	no	—
82	C₂²×C₄×D₆₇	C₂³	1	12	yes	—
83	C₂²×C₄×D₆₇	C₂³	1	12	yes	—
84	C₂²×C₄×D₆₇	C₂³	1	1	yes	characteristic
85	C₂²×C₄×D₆₇	C₂³	1	3	yes	—
86	C₂²×C₄×D₆₇	C₂³	1	3	yes	—
87	C₂²×C₄×D₆₇	C₂³	1	3	yes	—
88	#14524041	C₂³	1	24	yes	complemented
89	#14524039	C₂³	1	24	yes	complemented
90	#14524032	C₂³	1	24	yes	complemented
91	#14524030	C₂³	1	24	yes	complemented
92	#14524026	C₂³	1	48	yes	complemented
93	#14524024	C₂³	1	8	yes	—
94	#14524024	C₂³	1	24	yes	—
95	#14524024	C₂³	1	48	yes	complemented
96	#14524022	C₂³	1	48	yes	complemented
97	#14524019	C₂³	1	24	yes	—
98	#14524019	C₂³	1	8	yes	—
99	#14524019	C₂³	1	24	yes	complemented
100	#14524019	C₂³	1	24	yes	complemented
101	#14524017	C₂³	1	48	yes	complemented
102	#14524015	C₂³	1	96	yes	complemented
103	#14524014	C₂³	1	24	yes	—
104	#14524014	C₂³	1	8	yes	—
105	#14524014	C₂³	1	24	yes	complemented
106	#14524014	C₂³	1	24	yes	complemented
107	#14524012	C₂³	1	48	yes	complemented
108	#14524011	C₂³	1	48	yes	—
109	#14524011	C₂³	1	8	yes	—
110	#14524011	C₂³	1	24	yes	—
111	Q₈×Dic₆₇	C₂³	1	16	yes	complemented
112	D₄×Dic₆₇	C₂³	1	48	yes	complemented
113	C₄×C₂²⋊D₆₇	C₂³	1	48	yes	—
114	C₄×C₂²⋊D₆₇	C₂³	1	16	yes	—
115	C₄×C₂²⋊D₆₇	C₂³	1	48	yes	complemented
116	C₄×C₂²⋊D₆₇	C₂³	1	48	yes	complemented
117	#14524002	C₂³	1	4	yes	—
118	#14524002	C₂³	1	12	yes	—
119	#14524002	C₂³	1	12	yes	complemented
120	#14524002	C₂³	1	12	yes	complemented
121	C₂×C₄⋊Dic₆₇	C₂³	1	12	yes	complemented
122	C₂×Dic₆₇⋊C₄	C₂³	1	24	yes	complemented
123	C₂×C₂²⋊Dic₆₇	C₂³	1	12	yes	—
124	C₂×C₂²⋊Dic₆₇	C₂³	1	24	yes	complemented
125	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	24	yes	—
126	C₂×C₄×Dic₆₇	C₂³	1	12	yes	complemented
127	2₋¹⁺⁴×C₆₇	—	2	32	no	—
128	2₋¹⁺⁴×C₆₇	—	2	16	no	—
129	H(5,𝔽₂)×C₆₇	—	2	32	no	—
130	H(5,𝔽₂)×C₆₇	—	2	48	no	—
131	C₂×#41×C₆₇	—	2	64	no	—
132	C₂×#41×C₆₇	—	2	48	no	—
133	C₂×#41×C₆₇	D₄	1	16	yes	complemented
134	C₂×#41×C₆₇	C₂³	1	4	yes	—
135	C₂×#41×C₆₇	C₂³	1	12	yes	—
136	C₂×#41×C₆₇	C₂³	1	4	yes	—
137	C₂×#41×C₆₇	C₂³	1	4	yes	—
138	C₂×#41×C₆₇	C₂³	1	24	yes	complemented
139	C₂×#41×C₆₇	C₂³	1	8	yes	complemented
140	C₂×#41×C₆₇	C₂³	1	12	yes	complemented
141	C₂×#41×C₆₇	C₂³	1	12	yes	complemented
142	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
143	C₂²×Q₈×C₆₇	C₂³	1	8	yes	complemented
144	C₂²×Q₈×C₆₇	C₂³	1	6	yes	complemented
145	C₂²×D₄×C₆₇	C₂³	1	12	yes	—
146	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
147	C₂²×D₄×C₆₇	C₂³	1	24	yes	complemented
148	C₂²×D₄×C₆₇	C₂³	1	6	yes	complemented
149	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
150	C₂³×C₄×C₆₇	C₂³	1	12	yes	complemented
151	C₂³×D₆₇	—	2	12	no	—
152	C₂³×D₆₇	—	4	12	no	—
153	C₂³×D₆₇	C₂⁴	1	1	yes	characteristic
154	C₂³×D₆₇	C₂⁴	1	12	yes	—
155	C₂³×D₆₇	C₂⁴	1	3	yes	—
156	#11761629	—	2	32	no	—
157	#11761628	—	2	48	no	—
158	#11761628	—	2	48	no	—
159	#11761627	—	2	96	no	—
160	C₂×Dic₁₃₄	—	2	12	no	—
161	C₂×Dic₁₃₄	C₂⁴	1	12	yes	—
162	Q₈×D₆₇	—	2	32	no	—
163	C₂×D₂₆₈	—	2	12	no	—
164	C₂×D₂₆₈	C₂⁴	1	12	yes	—
165	C₂×C₂²⋊D₆₇	—	2	24	no	—
166	C₂×C₂²⋊D₆₇	—	4	48	no	—
167	C₂×C₂²⋊D₆₇	C₂⁴	1	16	yes	—
168	C₂×C₂²⋊D₆₇	C₂⁴	1	24	yes	—
169	C₂×C₂²⋊D₆₇	C₂⁴	1	48	yes	—
170	D₄×D₆₇	—	2	96	no	—
171	C₂²×Dic₆₇	—	2	12	no	—
172	C₂²×Dic₆₇	C₂⁴	1	12	yes	—
173	C₂²×Dic₆₇	C₂⁴	1	1	yes	characteristic
174	C₂²×Dic₆₇	C₂⁴	1	3	yes	—
175	C₂²×Dic₆₇	C₂⁴	1	24	yes	—
176	C₂×C₄×D₆₇	—	2	4	no	—
177	C₂×C₄×D₆₇	—	2	12	no	—
178	C₂×C₄×D₆₇	—	2	48	no	—
179	C₂×C₄×D₆₇	—	2	48	no	—
180	C₂×C₄×D₆₇	—	2	12	no	—
181	C₂×C₄×D₆₇	—	2	12	no	—
182	C₂×C₄×D₆₇	C₂⁴	1	2	yes	—
183	C₂×C₄×D₆₇	C₂⁴	1	12	yes	—
184	C₂×C₄×D₆₇	C₂⁴	1	6	yes	—
185	C₂×C₄×D₆₇	C₂⁴	1	12	yes	—
186	C₂×C₄×D₆₇	C₂⁴	1	2	yes	—
187	C₂×C₄×D₆₇	C₂⁴	1	6	yes	—
188	C₄⋊Dic₆₇	C₂⁴	1	8	yes	—
189	C₄⋊Dic₆₇	C₂⁴	1	24	yes	—
190	Dic₆₇⋊C₄	C₂⁴	1	16	yes	—
191	Dic₆₇⋊C₄	C₂⁴	1	48	yes	—
192	C₂²⋊Dic₆₇	C₂⁴	1	8	yes	—
193	C₂²⋊Dic₆₇	C₂⁴	1	24	yes	—
194	C₂²⋊Dic₆₇	C₂⁴	1	16	yes	—
195	C₂²⋊Dic₆₇	C₂⁴	1	48	yes	—
196	(C₂×D₆₇)⋊C₄	C₂⁴	1	16	yes	—
197	(C₂×D₆₇)⋊C₄	C₂⁴	1	48	yes	—
198	C₄×Dic₆₇	C₂⁴	1	8	yes	—
199	C₄×Dic₆₇	C₂⁴	1	24	yes	—
200	C₂⁴×C₆₇	C₂⁴	1	12	yes	—
201	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
202	#41×C₆₇	—	2	32	no	—
203	#41×C₆₇	—	2	96	no	—
204	#41×C₆₇	—	2	16	no	—
205	#41×C₆₇	—	2	48	no	—
206	#41×C₆₇	—	2	48	no	—
207	#41×C₆₇	—	2	48	no	—
208	#41×C₆₇	C₂×D₄	1	32	yes	complemented
209	#41×C₆₇	C₂×D₄	1	32	yes	complemented
210	C₂×Q₈×C₆₇	—	2	32	no	—
211	C₂×Q₈×C₆₇	—	2	24	no	—
212	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	complemented
213	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
214	C₂×Q₈×C₆₇	C₂⁴	1	12	yes	—
215	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
216	C₂×D₄×C₆₇	—	2	48	no	—
217	C₂×D₄×C₆₇	—	2	96	no	—
218	C₂×D₄×C₆₇	—	2	24	no	—
219	C₂×D₄×C₆₇	C₂×D₄	1	24	yes	complemented
220	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
221	C₂×D₄×C₆₇	C₂⁴	1	24	yes	—
222	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
223	C₂×D₄×C₆₇	C₂⁴	1	12	yes	—
224	C₂×D₄×C₆₇	C₂⁴	1	12	yes	—
225	C₂²×C₄×C₆₇	—	2	48	no	—
226	C₂²×C₄×C₆₇	C₂×D₄	1	24	yes	complemented
227	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
228	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
229	C₂²×C₄×C₆₇	C₂⁴	1	24	yes	—
230	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
231	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
232	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
233	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
234	C₂³×C₆₇	—	2	24	no	—
235	C₂³×C₆₇	—	4	6	no	—
236	C₂³×C₆₇	—	4	24	no	—
237	C₂³×C₆₇	C₂²×D₄	1	12	yes	—
238	C₂³×C₆₇	C₂⁵	1	8	yes	—
239	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
240	C₂³×C₆₇	C₂⁵	1	6	yes	—
241	C₂²×D₆₇	—	2	2	no	—
242	C₂²×D₆₇	—	2	4	no	—
243	C₂²×D₆₇	—	2	24	no	—
244	C₂²×D₆₇	—	2	6	no	—
245	C₂²×D₆₇	—	2	24	no	—
246	C₂²×D₆₇	—	4	12	no	—
247	C₂²×D₆₇	—	4	6	no	—
248	C₂²×D₆₇	—	4	48	no	—
249	C₂²×D₆₇	C₂⁵	1	6	yes	—
250	C₂²×D₆₇	C₂⁵	1	2	yes	—
251	Q₈×C₆₇	—	2	24	no	—
252	Q₈×C₆₇	—	2	8	no	—
253	Q₈×C₆₇	C₂²×D₄	1	16	yes	—
254	D₄×C₆₇	—	2	16	no	—
255	D₄×C₆₇	—	2	48	no	—
256	D₄×C₆₇	—	2	24	no	—
257	D₄×C₆₇	—	2	8	no	—
258	D₄×C₆₇	C₂²×D₄	1	48	yes	—
259	C₂×C₄×C₆₇	—	2	24	no	—
260	C₂×C₄×C₆₇	—	2	16	no	—
261	C₂×C₄×C₆₇	—	2	48	no	—
262	C₂×C₄×C₆₇	C₂²×D₄	1	24	yes	—
263	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
264	C₂×C₄×C₆₇	C₂²×D₄	1	24	yes	—
265	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
266	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
267	C₂×C₄×C₆₇	C₂⁵	1	6	yes	—
268	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
269	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
270	C₂²⋊D₆₇	—	2	48	no	—
271	C₂²⋊D₆₇	—	2	32	no	—
272	C₂²⋊D₆₇	—	4	48	no	—
273	C₂×Dic₆₇	—	2	2	no	—
274	C₂×Dic₆₇	—	2	6	no	—
275	C₂×Dic₆₇	—	2	24	no	—
276	C₂×Dic₆₇	—	4	24	no	—
277	C₂×Dic₆₇	C₂⁵	1	16	yes	—
278	C₂×Dic₆₇	C₂⁵	1	2	yes	—
279	C₂×Dic₆₇	C₂⁵	1	6	yes	—
280	D₂₆₈	—	2	24	no	—
281	C₄×D₆₇	—	2	4	no	—
282	C₄×D₆₇	—	2	4	no	—
283	C₄×D₆₇	—	2	24	no	—
284	C₄×D₆₇	—	2	24	no	—
285	C₄×D₆₇	—	2	12	no	—
286	C₄×D₆₇	—	2	12	no	—
287	Dic₁₃₄	—	2	24	no	—
288	C₂²×C₆₇	—	2	12	no	—
289	C₂²×C₆₇	—	2	2	no	—
290	C₂²×C₆₇	—	2	8	no	—
291	C₂²×C₆₇	—	2	16	no	—
292	C₂²×C₆₇	—	4	12	no	—
293	C₂²×C₆₇	—	8	24	no	—
294	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
295	C₂²×C₆₇	C₂³×D₄	1	12	yes	—
296	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
297	C₂×D₆₇	—	2	8	no	—
298	C₂×D₆₇	—	2	2	no	—
299	C₂×D₆₇	—	2	2	no	—
300	C₂×D₆₇	—	2	6	no	—
301	C₂×D₆₇	—	2	12	no	—
302	C₂×D₆₇	—	4	48	no	—
303	C₂×D₆₇	—	4	12	no	—
304	C₂×D₆₇	#585	1	4	yes	complemented
305	C₄×C₆₇	—	2	8	no	—
306	C₄×C₆₇	C₂³×D₄	1	12	yes	—
307	C₄×C₆₇	C₂³×D₄	1	2	yes	—
308	C₄×C₆₇	C₂³×D₄	1	2	yes	—
309	Dic₆₇	—	2	6	no	—
310	Dic₆₇	—	2	2	no	—
311	Dic₆₇	—	2	16	no	—
312	#62811	—	67	1	no	maximal, Sylow
313	C₂×C₆₇	—	2	2	no	—
314	C₂×C₆₇	—	2	12	no	—
315	C₂×C₆₇	—	4	8	no	—
316	C₂×C₆₇	C₂⁴×D₄	1	1	yes	characteristic
317	C₂×C₆₇	C₂×#585	1	2	yes	—
318	D₆₇	—	2	4	no	—
319	D₆₇	—	4	6	no	—
320	C₂×#585	D₆₇	1	1	yes	characteristic, complemented
321	C₂³×#41	—	67	1	no	—
322	#3489	—	67	8	no	—
323	#3488	—	67	8	no	—
324	#3487	—	67	2	no	—
325	#3486	—	67	2	no	—
326	#3485	—	67	6	no	—
327	#3484	—	67	6	no	—
328	D₄×#41	—	67	16	no	—
329	#3449	—	67	6	no	—
330	C₂×#525	—	67	6	no	—
331	#3428	—	67	1	no	—
332	C₆₇	#62811	1	1	yes	characteristic, complemented, Sylow
333	#585	—	2	16	no	—
334	C₂×2₋¹⁺⁴	C₂×D₆₇	1	2	yes	complemented
335	C₂×2₋¹⁺⁴	C₂×D₆₇	1	4	yes	complemented
336	C₂×H(5,𝔽₂)	C₂×D₆₇	1	6	yes	complemented
337	C₂×H(5,𝔽₂)	C₂×D₆₇	1	4	yes	complemented
338	C₂²×#41	—	67	8	no	—
339	C₂²×#41	—	67	8	no	—
340	C₂²×#41	—	67	1	no	—
341	C₂²×#41	—	67	3	no	—
342	C₂²×#41	—	67	3	no	—
343	C₂²×#41	—	134	16	no	—
344	C₂²×#41	C₂×D₆₇	1	1	yes	characteristic
345	C₂²×#41	C₂×D₆₇	1	6	yes	complemented
346	C₂²×#41	C₂×D₆₇	1	8	yes	complemented
347	C₂³×Q₈	—	67	1	no	—
348	C₂³×D₄	—	67	3	no	—
349	C₂⁴×C₄	—	67	3	no	—
350	#550	—	67	16	no	—
351	D₄×Q₈	—	67	16	no	—
352	#548	—	67	16	no	—
353	#548	—	67	16	no	—
354	#547	—	67	48	no	—
355	#546	—	67	48	no	—
356	#546	—	67	48	no	—
357	D₄²	—	67	48	no	—
358	#537	—	67	24	no	—
359	#536	—	67	12	no	—
360	#536	—	67	12	no	—
361	#535	—	67	12	no	—
362	#535	—	67	12	no	—
363	#534	—	67	24	no	—
364	#532	—	67	24	no	—
365	#525	—	67	4	no	—
366	#525	—	67	12	no	—
367	#525	—	67	12	no	—
368	#525	—	67	12	no	—
369	#525	—	67	24	no	—
370	#525	—	67	24	no	—
371	#525	—	67	24	no	—
372	C₂×#123	—	67	12	no	—
373	C₂×C₂²⋊Q₈	—	67	12	no	—
374	C₂×#121	—	67	12	no	—
375	C₂×#120	—	67	12	no	—
376	#520	—	67	8	no	—
377	#519	—	67	2	no	—
378	#518	—	67	6	no	—
379	C₄×#41	—	67	8	no	—
380	C₂×C₄×D₄	—	67	24	no	—
381	C₂×#117	—	67	6	no	—
382	C₂⁵	—	67	3	no	—
383	2₋¹⁺⁴	—	2	32	no	—
384	2₋¹⁺⁴	—	2	16	no	—
385	H(5,𝔽₂)	—	2	32	no	—
386	H(5,𝔽₂)	—	2	48	no	—
387	C₂×#41	—	2	64	no	—
388	C₂×#41	—	2	48	no	—
389	C₂×#41	—	67	8	no	—
390	C₂×#41	—	67	24	no	—
391	C₂×#41	—	67	12	no	—
392	C₂×#41	—	67	12	no	—
393	C₂×#41	—	134	64	no	—
394	C₂×#41	—	134	64	no	—
395	C₂×#41	—	134	8	no	—
396	C₂×#41	—	134	24	no	—
397	C₂×#41	—	134	24	no	—
398	C₂×#41	C₂²⋊D₆₇	1	16	yes	complemented
399	C₂×#41	C₂²×D₆₇	1	4	yes	—
400	C₂×#41	C₂²×D₆₇	1	12	yes	—
401	C₂×#41	C₂²×D₆₇	1	4	yes	—
402	C₂×#41	C₂²×D₆₇	1	4	yes	—
403	C₂×#41	C₂²×D₆₇	1	12	yes	complemented
404	C₂×#41	C₂²×D₆₇	1	12	yes	complemented
405	C₂×#41	C₂²×D₆₇	1	24	yes	complemented
406	C₂×#41	C₂²×D₆₇	1	8	yes	complemented
407	C₂²×Q₈	—	67	8	no	—
408	C₂²×Q₈	—	67	3	no	—
409	C₂²×Q₈	—	134	8	no	—
410	C₂²×Q₈	C₂²×D₆₇	1	1	yes	characteristic
411	C₂²×Q₈	C₂²×D₆₇	1	6	yes	complemented
412	C₂²×Q₈	C₂²×D₆₇	1	8	yes	complemented
413	C₂²×D₄	—	67	24	no	—
414	C₂²×D₄	—	67	6	no	—
415	C₂²×D₄	—	67	24	no	—
416	C₂²×D₄	—	67	3	no	—
417	C₂²×D₄	—	134	24	no	—
418	C₂²×D₄	C₂²×D₆₇	1	12	yes	—
419	C₂²×D₄	C₂²×D₆₇	1	3	yes	—
420	C₂²×D₄	C₂²×D₆₇	1	6	yes	complemented
421	C₂²×D₄	C₂²×D₆₇	1	24	yes	complemented
422	C₂³×C₄	—	67	3	no	—
423	C₂³×C₄	—	67	12	no	—
424	C₂³×C₄	—	67	12	no	—
425	C₂³×C₄	—	67	1	no	—
426	C₂³×C₄	—	67	3	no	—
427	C₂³×C₄	—	67	3	no	—
428	C₂³×C₄	—	67	3	no	—
429	C₂³×C₄	—	134	24	no	—
430	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
431	C₂³×C₄	C₂²×D₆₇	1	12	yes	complemented
432	C₄⋊Q₈	—	67	24	no	—
433	#127	—	67	24	no	—
434	#124	—	67	24	no	—
435	#124	—	67	24	no	—
436	#123	—	67	8	no	—
437	#123	—	67	24	no	—
438	#123	—	67	48	no	—
439	#123	—	67	48	no	—
440	C₂²⋊Q₈	—	67	8	no	—
441	C₂²⋊Q₈	—	67	24	no	—
442	C₂²⋊Q₈	—	67	24	no	—
443	C₂²⋊Q₈	—	67	24	no	—
444	C₂²⋊Q₈	—	67	48	no	—
445	#121	—	67	8	no	—
446	#121	—	67	24	no	—
447	#121	—	67	24	no	—
448	#121	—	67	24	no	—
449	#121	—	67	96	no	—
450	#121	—	67	48	no	—
451	#120	—	67	48	no	—
452	#120	—	67	8	no	—
453	#120	—	67	24	no	—
454	#120	—	67	48	no	—
455	C₄×Q₈	—	67	16	no	—
456	C₄×D₄	—	67	48	no	—
457	C₄×D₄	—	67	16	no	—
458	C₄×D₄	—	67	48	no	—
459	C₄×D₄	—	67	48	no	—
460	C₄×D₄	—	67	48	no	—
461	#117	—	67	4	no	—
462	#117	—	67	12	no	—
463	#117	—	67	12	no	—
464	#117	—	67	12	no	—
465	C₂×C₄⋊C₄	—	67	12	no	—
466	C₂×C₄⋊C₄	—	67	24	no	—
467	C₂×C₂²⋊C₄	—	67	12	no	—
468	C₂×C₂²⋊C₄	—	67	24	no	—
469	C₂×C₂²⋊C₄	—	67	24	no	—
470	C₂×C₄²	—	67	12	no	—
471	C₂⁴	—	268	12	no	—
472	C₂⁴	—	67	3	no	—
473	C₂⁴	—	67	12	no	—
474	C₂⁴	—	67	1	no	—
475	C₂⁴	—	134	12	no	—
476	C₂⁴	C₂³×D₆₇	1	12	yes	—
477	C₂⁴	C₂³×D₆₇	1	3	yes	—
478	#41	—	2	32	no	—
479	#41	—	2	96	no	—
480	#41	—	2	16	no	—
481	#41	—	2	48	no	—
482	#41	—	2	48	no	—
483	#41	—	2	48	no	—
484	#41	—	134	32	no	—
485	#41	—	134	96	no	—
486	#41	—	134	48	no	—
487	#41	—	134	48	no	—
488	#41	C₂×C₂²⋊D₆₇	1	32	yes	complemented
489	#41	C₂×C₂²⋊D₆₇	1	32	yes	complemented
490	C₂×Q₈	—	2	32	no	—
491	C₂×Q₈	—	2	24	no	—
492	C₂×Q₈	—	67	12	no	—
493	C₂×Q₈	—	134	32	no	—
494	C₂×Q₈	—	134	12	no	—
495	C₂×Q₈	C₂×C₂²⋊D₆₇	1	8	yes	complemented
496	C₂×Q₈	C₂³×D₆₇	1	4	yes	—
497	C₂×Q₈	C₂³×D₆₇	1	12	yes	—
498	C₂×Q₈	C₂³×D₆₇	1	4	yes	—
499	C₂×D₄	—	2	48	no	—
500	C₂×D₄	—	2	96	no	—
501	C₂×D₄	—	2	24	no	—
502	C₂×D₄	—	268	48	no	—
503	C₂×D₄	—	67	48	no	—
504	C₂×D₄	—	67	16	no	—
505	C₂×D₄	—	67	24	no	—
506	C₂×D₄	—	67	12	no	—
507	C₂×D₄	—	134	24	no	—
508	C₂×D₄	—	134	96	no	—
509	C₂×D₄	—	134	12	no	—
510	C₂×D₄	C₂×C₂²⋊D₆₇	1	24	yes	complemented
511	C₂×D₄	C₂³×D₆₇	1	8	yes	—
512	C₂×D₄	C₂³×D₆₇	1	24	yes	—
513	C₂×D₄	C₂³×D₆₇	1	4	yes	—
514	C₂×D₄	C₂³×D₆₇	1	12	yes	—
515	C₂×D₄	C₂³×D₆₇	1	12	yes	—
516	C₂²×C₄	—	2	48	no	—
517	C₂²×C₄	—	67	24	no	—
518	C₂²×C₄	—	67	12	no	—
519	C₂²×C₄	—	67	1	no	—
520	C₂²×C₄	—	67	3	no	—
521	C₂²×C₄	—	67	12	no	—
522	C₂²×C₄	—	67	12	no	—
523	C₂²×C₄	—	67	2	no	—
524	C₂²×C₄	—	67	6	no	—
525	C₂²×C₄	—	67	2	no	—
526	C₂²×C₄	—	67	6	no	—
527	C₂²×C₄	—	134	12	no	—
528	C₂²×C₄	—	134	4	no	—
529	C₂²×C₄	—	134	12	no	—
530	C₂²×C₄	—	134	12	no	—
531	C₂²×C₄	—	134	12	no	—
532	C₂²×C₄	—	134	48	no	—
533	C₂²×C₄	—	134	48	no	—
534	C₂²×C₄	C₂×C₂²⋊D₆₇	1	24	yes	complemented
535	C₂²×C₄	C₂³×D₆₇	1	12	yes	—
536	C₂²×C₄	C₂³×D₆₇	1	24	yes	—
537	C₂²×C₄	C₂³×D₆₇	1	8	yes	—
538	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
539	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
540	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
541	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
542	C₄⋊C₄	—	67	8	no	—
543	C₄⋊C₄	—	67	16	no	—
544	C₄⋊C₄	—	67	24	no	—
545	C₄⋊C₄	—	67	48	no	—
546	C₂²⋊C₄	—	67	8	no	—
547	C₂²⋊C₄	—	67	16	no	—
548	C₂²⋊C₄	—	67	24	no	—
549	C₂²⋊C₄	—	67	48	no	—
550	C₂²⋊C₄	—	67	16	no	—
551	C₂²⋊C₄	—	67	48	no	—
552	C₄²	—	67	24	no	—
553	C₄²	—	67	8	no	—
554	C₂³	—	2	24	no	—
555	C₂³	—	4	6	no	—
556	C₂³	—	4	24	no	—
557	C₂³	—	268	12	no	—
558	C₂³	—	268	6	no	—
559	C₂³	—	268	48	no	—
560	C₂³	—	67	2	no	—
561	C₂³	—	67	6	no	—
562	C₂³	—	134	24	no	—
563	C₂³	—	134	4	no	—
564	C₂³	—	134	6	no	—
565	C₂³	—	134	24	no	—
566	C₂³	—	134	2	no	—
567	C₂³	C₂²×C₂²⋊D₆₇	1	12	yes	—
568	C₂³	C₂⁴×D₆₇	1	8	yes	—
569	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
570	C₂³	C₂⁴×D₆₇	1	6	yes	—
571	Q₈	—	2	24	no	—
572	Q₈	—	2	8	no	—
573	Q₈	—	134	24	no	—
574	Q₈	C₂²×C₂²⋊D₆₇	1	16	yes	—
575	D₄	—	2	48	no	—
576	D₄	—	2	16	no	—
577	D₄	—	2	24	no	—
578	D₄	—	2	8	no	—
579	D₄	—	268	48	no	—
580	D₄	—	134	32	no	—
581	D₄	—	134	48	no	—
582	D₄	—	134	24	no	—
583	D₄	C₂²×C₂²⋊D₆₇	1	48	yes	—
584	C₂×C₄	—	2	24	no	—
585	C₂×C₄	—	2	16	no	—
586	C₂×C₄	—	2	48	no	—
587	C₂×C₄	—	268	24	no	—
588	C₂×C₄	—	67	16	no	—
589	C₂×C₄	—	67	2	no	—
590	C₂×C₄	—	67	6	no	—
591	C₂×C₄	—	134	2	no	—
592	C₂×C₄	—	134	6	no	—
593	C₂×C₄	—	134	24	no	—
594	C₂×C₄	—	134	4	no	—
595	C₂×C₄	—	134	4	no	—
596	C₂×C₄	—	134	24	no	—
597	C₂×C₄	—	134	24	no	—
598	C₂×C₄	—	134	12	no	—
599	C₂×C₄	—	134	12	no	—
600	C₂×C₄	C₂²×C₂²⋊D₆₇	1	24	yes	—
601	C₂×C₄	C₂²×C₂²⋊D₆₇	1	4	yes	—
602	C₂×C₄	C₂²×C₂²⋊D₆₇	1	24	yes	—
603	C₂×C₄	C₂²×C₂²⋊D₆₇	1	12	yes	—
604	C₂×C₄	C₂⁴×D₆₇	1	8	yes	—
605	C₂×C₄	C₂⁴×D₆₇	1	6	yes	—
606	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
607	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
608	C₂²	—	2	12	no	—
609	C₂²	—	2	2	no	—
610	C₂²	—	2	8	no	—
611	C₂²	—	2	16	no	—
612	C₂²	—	4	12	no	—
613	C₂²	—	8	24	no	—
614	C₂²	—	268	12	no	—
615	C₂²	—	268	48	no	—
616	C₂²	—	67	4	no	—
617	C₂²	—	134	8	no	—
618	C₂²	—	134	6	no	—
619	C₂²	—	134	12	no	—
620	C₂²	—	134	2	no	—
621	C₂²	—	134	2	no	—
622	C₂²	C₂³×C₂²⋊D₆₇	1	12	yes	—
623	C₂²	C₂³×C₂²⋊D₆₇	1	2	yes	—
624	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
625	C₄	—	2	8	no	—
626	C₄	—	134	2	no	—
627	C₄	—	134	6	no	—
628	C₄	—	134	16	no	—
629	C₄	C₂³×C₂²⋊D₆₇	1	2	yes	—
630	C₄	C₂³×C₂²⋊D₆₇	1	12	yes	—
631	C₄	C₂³×C₂²⋊D₆₇	1	2	yes	—
632	C₂	—	2	2	no	—
633	C₂	—	2	12	no	—
634	C₂	—	4	8	no	—
635	C₂	—	268	6	no	—
636	C₂	—	134	4	no	—
637	C₂	C₂⁴×C₂²⋊D₆₇	1	1	yes	characteristic
638	C₂	#585×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂⁴×D₆₇
commutator, socle, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂×#585×C₆₇	C₂
hypercenter	C₂×#585	D₆₇
hypocenter	C₆₇	#62811
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#585

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×D₆₇	C₂	semidirect
2. #22573882	C₂	semidirect
3. #22573871	C₂	semidirect
4. #22573864	C₂	semidirect
5. #22573857	C₂	semidirect
6. #22573849	C₂	semidirect
7. #22573840	C₂	semidirect
8. #41×C₂²⋊D₆₇	C₂	semidirect
9. #22573538	C₂	semidirect
10. C₂×#16863903	C₂	semidirect
11. #22573412	C₂	semidirect
12. C₂×#585×C₆₇	C₂	semidirect
13. C₂²×#11761629	C₂²	semidirect
14. C₂²×#11761628	C₂²	semidirect
15. C₂²×#11761627	C₂²	semidirect
17. C₂×#41×D₆₇	C₂²	semidirect
18. C₂×#41×D₆₇	C₂²	semidirect
19. C₂²×Q₈×D₆₇	C₂²	semidirect
20. C₂²×D₄×D₆₇	C₂²	semidirect
21. C₂³×C₄×D₆₇	C₂²	semidirect
22. #16864065	C₂²	semidirect
23. Q₈×C₂²⋊D₆₇	C₂²	semidirect
24. #16864057	C₂²	semidirect
25. #16864053	C₂²	semidirect
26. #16864045	C₂²	semidirect
27. #16864040	C₂²	semidirect
28. #16864034	C₂²	semidirect
29. D₄×C₂²⋊D₆₇	C₂²	semidirect
30. #16863974	C₂²	semidirect
31. #16863970	C₂²	semidirect
32. #16863967	C₂²	semidirect
33. #16863963	C₂²	semidirect
34. #16863960	C₂²	semidirect
35. #16863955	C₂²	semidirect
36. #16863942	C₂²	semidirect
37. #16863907	C₂²	semidirect
38. #16863903	C₂²	semidirect
39. #16863903	C₂²	semidirect
40. #16863903	C₂²	semidirect
41. #16863903	C₂²	semidirect
42. #16863903	C₂²	semidirect
43. #16863903	C₂²	semidirect
44. C₂×#14524024	C₂²	semidirect
45. C₂×#14524019	C₂²	semidirect
46. C₂×#14524014	C₂²	semidirect
47. C₂×#14524011	C₂²	semidirect
48. #16863879	C₂²	semidirect
49. #16863874	C₂²	semidirect
50. #16863871	C₂²	semidirect
51. #41×Dic₆₇	C₂²	semidirect
52. C₂×C₄×C₂²⋊D₆₇	C₂²	semidirect
53. C₂×#14524002	C₂²	semidirect
55. C₂×2₋¹⁺⁴×C₆₇	C₂²	semidirect
56. C₂×2₋¹⁺⁴×C₆₇	C₂²	semidirect
57. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
58. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
60. C₂²×#41×C₆₇	C₂²	semidirect
61. C₂²×#41×C₆₇	C₂²	semidirect
88. #14524041	C₂³	semidirect
89. #14524039	C₂³	semidirect
90. #14524032	C₂³	semidirect
91. #14524030	C₂³	semidirect
92. #14524026	C₂³	semidirect
95. #14524024	C₂³	semidirect
96. #14524022	C₂³	semidirect
99. #14524019	C₂³	semidirect
100. #14524019	C₂³	semidirect
101. #14524017	C₂³	semidirect
102. #14524015	C₂³	semidirect
105. #14524014	C₂³	semidirect
106. #14524014	C₂³	semidirect
107. #14524012	C₂³	semidirect
111. Q₈×Dic₆₇	C₂³	semidirect
112. D₄×Dic₆₇	C₂³	semidirect
115. C₄×C₂²⋊D₆₇	C₂³	semidirect
116. C₄×C₂²⋊D₆₇	C₂³	semidirect
119. #14524002	C₂³	semidirect
120. #14524002	C₂³	semidirect
121. C₂×C₄⋊Dic₆₇	C₂³	semidirect
122. C₂×Dic₆₇⋊C₄	C₂³	semidirect
124. C₂×C₂²⋊Dic₆₇	C₂³	semidirect
126. C₂×C₄×Dic₆₇	C₂³	semidirect
133. C₂×#41×C₆₇	D₄	semidirect
138. C₂×#41×C₆₇	C₂³	semidirect
139. C₂×#41×C₆₇	C₂³	semidirect
140. C₂×#41×C₆₇	C₂³	semidirect
141. C₂×#41×C₆₇	C₂³	semidirect
143. C₂²×Q₈×C₆₇	C₂³	semidirect
144. C₂²×Q₈×C₆₇	C₂³	semidirect
147. C₂²×D₄×C₆₇	C₂³	semidirect
148. C₂²×D₄×C₆₇	C₂³	semidirect
150. C₂³×C₄×C₆₇	C₂³	semidirect
208. #41×C₆₇	C₂×D₄	semidirect
209. #41×C₆₇	C₂×D₄	semidirect
212. C₂×Q₈×C₆₇	C₂×D₄	semidirect
219. C₂×D₄×C₆₇	C₂×D₄	semidirect
226. C₂²×C₄×C₆₇	C₂×D₄	semidirect
304. C₂×D₆₇	#585	semidirect
320. C₂×#585	D₆₇	semidirect
332. C₆₇	#62811	semidirect
334. C₂×2₋¹⁺⁴	C₂×D₆₇	semidirect
335. C₂×2₋¹⁺⁴	C₂×D₆₇	semidirect
336. C₂×H(5,𝔽₂)	C₂×D₆₇	semidirect
337. C₂×H(5,𝔽₂)	C₂×D₆₇	semidirect
345. C₂²×#41	C₂×D₆₇	semidirect
346. C₂²×#41	C₂×D₆₇	semidirect
398. C₂×#41	C₂²⋊D₆₇	semidirect
403. C₂×#41	C₂²×D₆₇	semidirect
404. C₂×#41	C₂²×D₆₇	semidirect
405. C₂×#41	C₂²×D₆₇	semidirect
406. C₂×#41	C₂²×D₆₇	semidirect
411. C₂²×Q₈	C₂²×D₆₇	semidirect
412. C₂²×Q₈	C₂²×D₆₇	semidirect
420. C₂²×D₄	C₂²×D₆₇	semidirect
421. C₂²×D₄	C₂²×D₆₇	semidirect
431. C₂³×C₄	C₂²×D₆₇	semidirect
488. #41	C₂×C₂²⋊D₆₇	semidirect
489. #41	C₂×C₂²⋊D₆₇	semidirect
495. C₂×Q₈	C₂×C₂²⋊D₆₇	semidirect
510. C₂×D₄	C₂×C₂²⋊D₆₇	semidirect
534. C₂²×C₄	C₂×C₂²⋊D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41×D₆₇	2
#22573882	2
#22573871	2
#22573864	2
#22573857	2
#22573849	2
#22573840	2
#41×C₂²⋊D₆₇	2
#22573538	2
C₂×#16863903	2
#22573412	2
C₂×#585×C₆₇	2
#62811	67