Subgroups — #30862303

Order 17152.

902 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569662	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863516	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863514	C₂	1	1	yes	maximal, characteristic, complemented
4	#22568925	C₂	1	1	yes	maximal, characteristic, complemented
5	#22568636	C₂	1	1	yes	maximal, characteristic, complemented
6	#22561126	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560983	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560859	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560855	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560537	C₂	1	1	yes	maximal, characteristic, complemented
11	#22560533	C₂	1	1	yes	maximal, characteristic, complemented
12	#22560228	C₂	1	1	yes	maximal, characteristic, complemented
13	#22560083	C₂	1	1	yes	maximal, characteristic, complemented
14	#22560055	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#1764	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14524065	C₂²	1	1	yes	characteristic
17	C₂×#14524063	C₂²	1	1	yes	characteristic
18	C₂×#14524051	C₂²	1	1	yes	characteristic
19	C₂×#14524049	C₂²	1	1	yes	characteristic
20	C₂×#14524049	C₂²	1	1	yes	characteristic
21	#16863942	C₂²	1	1	yes	characteristic
22	#16863864	C₂²	1	1	yes	characteristic
23	#16863621	C₄	1	1	yes	characteristic, complemented
24	#16863617	C₄	1	1	yes	characteristic, complemented
25	#16863614	C₄	1	1	yes	characteristic, complemented
26	#16863610	C₄	1	1	yes	characteristic, complemented
27	#16863605	C₄	1	1	yes	characteristic, complemented
28	#16863600	C₄	1	1	yes	characteristic, complemented
29	#16863595	C₄	1	1	yes	characteristic, complemented
30	#16863590	C₄	1	1	yes	characteristic, complemented
31	#16863516	—	2	4	no	—
32	#16863514	—	2	4	no	—
33	#16863467	—	2	1	no	—
34	#16863467	—	2	1	no	—
35	#16863465	—	2	1	no	—
36	#16863465	—	2	1	no	—
37	C₂×#14523979	C₂²	1	1	yes	characteristic
38	C₂×#14523978	C₂²	1	1	yes	characteristic
39	#16863419	C₂²	1	1	yes	characteristic
40	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
41	C₂×#14523971	C₂²	1	1	yes	characteristic
42	C₂×#14523971	C₂²	1	1	yes	characteristic
43	C₂×#14523970	C₂²	1	1	yes	characteristic
44	C₂×#14523969	C₂²	1	1	yes	characteristic
45	C₂×#14523969	C₂²	1	1	yes	characteristic
46	#16863390	C₂²	1	1	yes	characteristic
47	#16863364	C₂²	1	1	yes	characteristic
48	#16863363	C₂²	1	1	yes	characteristic
49	#16863353	—	2	1	no	—
50	#16863352	C₂²	1	1	yes	characteristic
51	#16863350	—	2	1	no	—
52	M₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
53	C₄×#11761611	—	2	1	no	—
54	C₂×#14523953	C₂²	1	1	yes	characteristic
55	C₂×#14523953	C₂²	1	1	yes	characteristic
56	#97×D₆₇	—	2	1	no	—
57	C₄×Dic₆₇⋊C₄	C₂²	1	1	yes	characteristic
58	#16863160	C₂²	1	1	yes	characteristic
59	#16863155	C₂²	1	1	yes	characteristic
60	#16863151	C₂²	1	1	yes	characteristic
61	#16863150	C₂²	1	1	yes	characteristic
62	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
63	C₂×#104×C₆₇	C₂²	1	1	yes	characteristic
64	#416×C₆₇	C₂²	1	1	yes	characteristic
65	#405×C₆₇	C₂²	1	1	yes	characteristic
66	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
67	#341×C₆₇	C₂²	1	1	yes	characteristic
68	#339×C₆₇	C₂²	1	1	yes	characteristic
69	C₂×#11761628	C₂³	1	1	yes	characteristic
70	C₂×#11761628	C₂³	1	1	yes	characteristic
71	#14524065	—	2	2	no	—
72	#14524065	—	2	2	no	—
73	#14524063	—	2	2	no	—
74	#14524063	—	2	2	no	—
75	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
76	C₂×#11761618	C₂×C₄	1	1	yes	characteristic
77	C₂×#11761617	C₂×C₄	1	1	yes	characteristic
78	C₂×#11761617	C₂×C₄	1	1	yes	characteristic
79	C₂×#11761616	C₂×C₄	1	1	yes	characteristic
80	C₂×#11761615	C₂×C₄	1	1	yes	characteristic
81	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
82	C₂×#11761613	C₂×C₄	1	1	yes	characteristic
83	#14524051	—	2	2	no	—
84	#14524051	—	2	2	no	—
85	#14524049	—	2	2	no	—
86	#14524049	—	2	2	no	—
87	#14524049	—	2	2	no	—
88	#14524049	D₄	1	4	yes	complemented
89	C₂×#11761612	C₂³	1	1	yes	characteristic
90	C₂×#11761612	C₂³	1	1	yes	characteristic
91	C₂×#11761611	—	2	1	no	—
92	C₂×#11761611	—	2	1	no	—
93	C₂×#11761611	—	2	1	no	—
94	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
95	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
96	C₂×C₈×D₆₇	—	2	1	no	—
97	C₂×C₈×D₆₇	—	2	1	no	—
98	C₂×C₈×D₆₇	—	2	1	no	—
99	#14524041	C₂×C₄	1	1	yes	characteristic
100	#14524039	C₂×C₄	1	1	yes	characteristic
101	#14524032	C₂×C₄	1	1	yes	characteristic
102	#14524030	C₂×C₄	1	1	yes	characteristic
103	#14524015	—	2	1	no	—
104	#14524015	—	2	1	no	—
105	#14524008	C₂×C₄	1	1	yes	characteristic
106	#14524008	C₂×C₄	1	1	yes	characteristic
107	#14524005	C₂×C₄	1	1	yes	characteristic
108	#14524005	C₂×C₄	1	1	yes	characteristic
109	C₄×C₂²⋊D₆₇	—	2	1	no	—
110	C₄×C₂²⋊D₆₇	—	2	1	no	—
111	(C₂²⋊D₆₇)⋊C₄	—	2	1	no	—
112	(C₂²⋊D₆₇)⋊C₄	—	2	1	no	—
113	#14524000	—	2	1	no	—
114	C₂×Dic₆₇⋊C₄	—	2	1	no	—
115	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
116	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
117	C₄²×D₆₇	—	2	1	no	—
118	#14523979	—	2	2	no	—
119	#14523978	—	2	2	no	—
120	Dic₆₇⋊C₈	—	2	1	no	—
121	Dic₆₇⋊C₈	—	2	1	no	—
122	Dic₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
123	Dic₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
124	Dic₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
125	Dic₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
126	#14523971	—	2	2	no	—
127	#14523971	—	2	2	no	—
128	#14523970	—	2	2	no	—
129	#14523969	—	2	2	no	—
130	#14523969	—	2	2	no	—
131	#14523966	C₂×C₄	1	1	yes	characteristic
132	#14523966	C₂×C₄	1	1	yes	characteristic
133	#14523963	C₂×C₄	1	1	yes	characteristic
134	#14523963	C₂×C₄	1	1	yes	characteristic
135	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
136	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
137	(C₂×D₆₇)⋊C₈	—	2	1	no	—
138	(C₂×D₆₇)⋊C₈	—	2	1	no	—
139	#14523954	—	2	1	no	—
140	#14523953	—	2	1	no	—
141	#14523953	—	2	2	no	—
142	#14523953	D₄	1	1	yes	characteristic
143	#14523953	D₄	1	1	yes	characteristic
144	#14523953	D₄	1	1	yes	characteristic
145	#14523953	D₄	1	1	yes	characteristic
146	C₄×C₆₇⋊C₈	—	2	1	no	—
147	C₈×Dic₆₇	—	2	1	no	—
148	C₆₇⋊#95	—	2	1	no	—
149	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
150	#131×C₆₇	—	2	2	no	—
151	#131×C₆₇	—	2	2	no	—
152	C₂×M₁₆×C₆₇	—	2	1	no	—
153	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
154	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
155	C₂²×C₈×C₆₇	—	2	1	no	—
156	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
157	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
158	#117×C₆₇	C₂³	1	1	yes	characteristic
159	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
160	#104×C₆₇	—	2	2	no	—
161	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
162	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
163	#102×C₆₇	C₂×C₄	1	1	yes	characteristic
164	#102×C₆₇	C₂×C₄	1	1	yes	characteristic
165	#97×C₆₇	—	2	1	no	—
166	C₄×C₈×C₆₇	—	2	2	no	—
167	C₄×C₈×C₆₇	—	2	1	no	—
168	#11761628	—	2	1	no	—
169	#11761628	—	2	1	no	—
170	#11761628	—	2	1	no	—
171	#11761628	—	2	1	no	—
172	#11761628	—	2	1	no	—
173	#11761628	—	2	1	no	—
174	#11761628	C₂×D₄	1	2	yes	—
175	#11761628	#41	1	2	yes	—
176	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
177	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
178	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
179	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
180	C₂×C₂²⋊D₆₇	—	2	1	no	—
181	C₂×C₂²⋊D₆₇	—	2	1	no	—
182	C₂×C₂²⋊D₆₇	—	2	1	no	—
183	C₂×C₂²⋊D₆₇	—	2	1	no	—
184	C₂²×Dic₆₇	—	2	1	no	—
185	C₂²×Dic₆₇	#41	1	1	yes	characteristic
186	C₂²×Dic₆₇	#41	1	1	yes	characteristic
187	C₂×C₄×D₆₇	—	2	1	no	—
188	C₂×C₄×D₆₇	—	2	1	no	—
189	C₂×C₄×D₆₇	—	2	1	no	—
190	C₂×C₄×D₆₇	—	2	1	no	—
191	Dic₂₆₈	—	2	2	no	—
192	#11761618	—	2	2	no	—
193	#11761617	—	2	2	no	—
194	#11761617	—	2	2	no	—
195	#11761616	—	2	2	no	—
196	#11761615	—	2	2	no	—
197	D₅₃₆	—	2	2	no	—
198	#11761613	—	2	2	no	—
199	#11761612	—	2	1	no	—
200	#11761612	—	2	1	no	—
201	#11761612	—	2	1	no	—
202	#11761612	C₂×D₄	1	2	yes	—
203	#11761611	—	2	4	no	—
204	#11761611	—	2	4	no	—
205	#11761611	—	4	2	no	—
206	C₂×C₆₇⋊C₈	—	2	1	no	—
207	C₂×C₆₇⋊C₈	—	2	1	no	—
208	C₂×C₆₇⋊C₈	—	2	1	no	—
209	C₂×C₆₇⋊C₈	—	2	1	no	—
210	C₂×C₆₇⋊C₈	—	2	1	no	—
211	C₂×C₆₇⋊C₈	—	2	1	no	—
212	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
213	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
214	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
215	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
216	C₈×D₆₇	—	2	4	no	—
217	C₈×D₆₇	—	2	4	no	—
218	C₈×D₆₇	—	4	2	no	—
219	Dic₆₇⋊C₄	—	2	1	no	—
220	Dic₆₇⋊C₄	—	2	1	no	—
221	Dic₆₇⋊C₄	—	2	1	no	—
222	Dic₆₇⋊C₄	—	2	1	no	—
223	Dic₆₇⋊C₄	—	4	2	no	—
224	C₂²⋊Dic₆₇	—	2	1	no	—
225	C₂²⋊Dic₆₇	—	2	1	no	—
226	(C₂×D₆₇)⋊C₄	—	2	1	no	—
227	(C₂×D₆₇)⋊C₄	—	2	1	no	—
228	(C₂×D₆₇)⋊C₄	—	2	1	no	—
229	(C₂×D₆₇)⋊C₄	—	2	1	no	—
230	C₄×Dic₆₇	—	2	1	no	—
231	C₄×Dic₆₇	—	2	1	no	—
232	C₄×Dic₆₇	—	2	1	no	—
233	C₄×Dic₆₇	—	2	1	no	—
234	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
235	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
236	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
237	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
238	#41×C₆₇	—	2	1	no	—
239	#41×C₆₇	—	2	1	no	—
240	#41×C₆₇	—	2	1	no	—
241	#41×C₆₇	—	2	1	no	—
242	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
243	C₂×D₄×C₆₇	—	2	1	no	—
244	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
245	C₂²×C₄×C₆₇	—	2	1	no	—
246	C₂²×C₄×C₆₇	—	2	1	no	—
247	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
248	M₁₆×C₆₇	—	2	1	no	—
249	M₁₆×C₆₇	—	2	1	no	—
250	M₁₆×C₆₇	—	2	1	no	—
251	M₁₆×C₆₇	—	4	2	no	—
252	M₁₆×C₆₇	#41	1	2	yes	—
253	C₂×C₈×C₆₇	—	2	1	no	—
254	C₂×C₈×C₆₇	—	2	1	no	—
255	C₂×C₈×C₆₇	—	2	1	no	—
256	C₂×C₈×C₆₇	—	2	1	no	—
257	C₂×C₈×C₆₇	—	4	2	no	—
258	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
259	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
260	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
261	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
262	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
263	C₂×C₈×C₆₇	#41	1	2	yes	—
264	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
265	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
266	C₂²⋊C₄×C₆₇	—	2	1	no	—
267	C₄²×C₆₇	—	2	1	no	—
268	C₄²×C₆₇	—	2	1	no	—
269	C₄²×C₆₇	—	2	1	no	—
270	C₂³×C₆₇	—	2	1	no	—
271	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
272	C₂²×D₆₇	—	2	1	no	—
273	C₂²×D₆₇	—	2	1	no	—
274	Q₈×C₆₇	—	2	1	no	—
275	Q₈×C₆₇	—	2	1	no	—
276	D₄×C₆₇	—	2	1	no	—
277	D₄×C₆₇	—	2	1	no	—
278	D₄×C₆₇	—	4	1	no	—
279	D₄×C₆₇	—	4	1	no	—
280	C₂×C₄×C₆₇	—	2	2	no	—
281	C₂×C₄×C₆₇	—	2	1	no	—
282	C₂×C₄×C₆₇	—	2	1	no	—
283	C₂×C₄×C₆₇	—	2	1	no	—
284	C₂×C₄×C₆₇	—	4	1	no	—
285	C₂×C₄×C₆₇	—	4	1	no	—
286	C₂×C₄×C₆₇	—	4	1	no	—
287	C₂×C₄×C₆₇	—	4	1	no	—
288	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
289	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
290	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
291	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
292	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
293	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
294	C₂²⋊D₆₇	—	2	2	no	—
295	C₂²⋊D₆₇	—	2	2	no	—
296	C₂²⋊D₆₇	—	4	1	no	—
297	C₂²⋊D₆₇	—	4	1	no	—
298	C₂²⋊D₆₇	—	8	1	no	—
299	C₂²⋊D₆₇	—	8	1	no	—
300	C₂×Dic₆₇	—	2	1	no	—
301	C₂×Dic₆₇	—	2	1	no	—
302	C₂×Dic₆₇	—	4	1	no	—
303	C₂×Dic₆₇	—	4	1	no	—
304	C₂×Dic₆₇	—	4	1	no	—
305	C₂×Dic₆₇	—	4	1	no	—
306	C₂×Dic₆₇	—	4	1	no	—
307	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
308	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
309	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
310	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
311	D₂₆₈	—	2	1	no	—
312	D₂₆₈	—	2	1	no	—
313	D₂₆₈	—	2	1	no	—
314	D₂₆₈	C₄×D₄	1	2	yes	—
315	C₄×D₆₇	—	2	2	no	—
316	C₄×D₆₇	—	2	2	no	—
317	C₄×D₆₇	—	4	1	no	—
318	C₄×D₆₇	—	4	1	no	—
319	C₄×D₆₇	—	8	1	no	—
320	C₄×D₆₇	—	8	1	no	—
321	Dic₁₃₄	—	2	1	no	—
322	Dic₁₃₄	—	2	1	no	—
323	Dic₁₃₄	—	2	1	no	—
324	Dic₁₃₄	C₄×D₄	1	2	yes	—
325	C₈×C₆₇	—	2	1	no	—
326	C₈×C₆₇	—	2	1	no	—
327	C₈×C₆₇	—	2	2	no	—
328	C₈×C₆₇	—	4	1	no	—
329	C₈×C₆₇	#121	1	2	yes	—
330	C₈×C₆₇	#121	1	2	yes	—
331	C₆₇⋊C₈	—	2	1	no	—
332	C₆₇⋊C₈	—	2	1	no	—
333	C₆₇⋊C₈	—	2	2	no	—
334	C₆₇⋊C₈	—	2	1	no	—
335	C₆₇⋊C₈	—	2	1	no	—
336	C₆₇⋊C₈	—	4	1	no	—
337	C₂²×C₆₇	—	2	1	no	—
338	C₂²×C₆₇	—	2	1	no	—
339	C₂²×C₆₇	—	4	1	no	—
340	C₂²×C₆₇	—	4	1	no	—
341	C₂²×C₆₇	—	4	1	no	—
342	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
343	C₂²×C₆₇	#529	1	1	yes	characteristic
344	C₂²×C₆₇	#532	1	1	yes	characteristic
345	C₂×D₆₇	—	2	2	no	—
346	C₂×D₆₇	—	4	1	no	—
347	C₂×D₆₇	—	4	1	no	—
348	C₂×D₆₇	—	4	1	no	—
349	C₂×D₆₇	—	4	1	no	—
350	C₂×D₆₇	—	4	1	no	—
351	C₄×C₆₇	—	4	2	no	—
352	C₄×C₆₇	—	4	1	no	—
353	C₄×C₆₇	—	4	1	no	—
354	C₄×C₆₇	—	4	1	no	—
355	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
356	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
357	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
358	C₄×C₆₇	#525	1	1	yes	characteristic
359	Dic₆₇	—	2	1	no	—
360	Dic₆₇	—	2	1	no	—
361	Dic₆₇	—	2	1	no	—
362	Dic₆₇	—	2	1	no	—
363	Dic₆₇	—	2	2	no	—
364	Dic₆₇	—	4	1	no	—
365	Dic₆₇	—	8	1	no	—
366	#21519	—	67	1	no	maximal, Sylow
367	C₂×C₆₇	—	2	1	no	—
368	C₂×C₆₇	—	2	1	no	—
369	C₂×C₆₇	—	8	1	no	—
370	C₂×C₆₇	C₄×#121	1	1	yes	characteristic
371	C₂×C₆₇	#2956	1	1	yes	characteristic
372	C₂×C₆₇	#2957	1	1	yes	characteristic
373	D₆₇	—	4	2	no	—
374	D₆₇	—	8	1	no	—
375	#3035	—	67	1	no	—
376	C₂×#444	—	67	1	no	—
377	C₂×#443	—	67	1	no	—
378	#2932	—	67	1	no	—
379	#2875	—	67	1	no	—
380	#1980	—	67	1	no	—
381	#1950	—	67	1	no	—
382	#1929	—	67	1	no	—
383	#1929	—	67	1	no	—
384	#1876	—	67	1	no	—
385	#1876	—	67	1	no	—
386	#1806	—	67	1	no	—
387	#1771	—	67	1	no	—
388	#1765	—	67	1	no	—
389	#1764	D₆₇	1	1	yes	characteristic, complemented
390	C₆₇	#21519	1	1	yes	characteristic, complemented, Sylow
391	C₂×#135	—	67	1	no	—
392	C₂×#135	—	67	1	no	—
393	C₂×#131	—	67	1	no	—
394	C₂×#131	—	67	1	no	—
395	C₂×#131	—	67	1	no	—
396	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
397	#532	—	67	1	no	—
398	C₄×#41	—	67	1	no	—
399	#464	—	67	1	no	—
400	#464	—	67	1	no	—
401	#463	—	67	1	no	—
402	#463	—	67	1	no	—
403	#462	—	67	1	no	—
404	#461	—	67	1	no	—
405	#460	—	67	1	no	—
406	#459	—	67	1	no	—
407	#444	—	134	4	no	—
408	#443	—	134	4	no	—
409	#435	—	134	1	no	—
410	#435	—	134	1	no	—
411	#435	—	134	1	no	—
412	#435	—	134	1	no	—
413	C₂×#108	—	67	1	no	—
414	C₂×#108	—	67	1	no	—
415	#423	—	67	1	no	—
416	C₂×C₄⋊C₈	—	67	1	no	—
417	C₂×#104	—	67	1	no	—
418	C₂×#104	—	67	1	no	—
419	C₂×#104	—	67	1	no	—
420	C₂×#104	—	67	1	no	—
421	C₂×#104	—	67	1	no	—
422	C₂×#104	C₂×D₆₇	1	1	yes	characteristic
423	#416	—	67	1	no	—
424	#416	C₂×D₆₇	1	1	yes	characteristic
425	#408	—	67	1	no	—
426	#408	—	67	1	no	—
427	#405	—	67	1	no	—
428	#405	—	134	1	no	—
429	#405	—	134	1	no	—
430	#405	C₂×D₆₇	1	1	yes	characteristic
431	C₄×M₁₆	—	67	1	no	—
432	C₄×M₁₆	—	134	1	no	—
433	C₂×#97	—	67	1	no	—
434	C₂×#97	—	67	1	no	—
435	C₂×#97	—	134	1	no	—
436	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
437	C₄×C₄⋊C₄	—	67	1	no	—
438	#341	—	67	1	no	—
439	#341	C₂×D₆₇	1	1	yes	characteristic
440	#339	—	67	1	no	—
441	#339	C₂×D₆₇	1	1	yes	characteristic
442	#336	—	67	1	no	—
443	#336	—	67	1	no	—
444	C₂×#41	—	67	1	no	—
445	C₂×#41	—	67	1	no	—
446	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
447	#135	—	134	2	no	—
448	#135	—	134	2	no	—
449	#135	—	134	2	no	—
450	#135	—	134	2	no	—
451	C₂×Q₁₆	—	67	1	no	—
452	C₂×Q₁₆	—	67	1	no	—
453	C₂×SD₁₆	—	67	1	no	—
454	C₂×SD₁₆	—	67	1	no	—
455	C₂×SD₁₆	—	67	1	no	—
456	C₂×SD₁₆	—	67	1	no	—
457	C₂×D₈	—	67	1	no	—
458	C₂×D₈	—	67	1	no	—
459	#131	—	2	2	no	—
460	#131	—	2	2	no	—
461	#131	—	67	4	no	—
462	#131	—	134	2	no	—
463	#131	—	134	2	no	—
464	#131	—	134	2	no	—
465	#131	—	134	2	no	—
466	#131	—	134	2	no	—
467	C₂×M₁₆	—	2	1	no	—
468	C₂×M₁₆	—	67	1	no	—
469	C₂×M₁₆	—	67	1	no	—
470	C₂×M₁₆	—	134	1	no	—
471	C₂×M₁₆	—	134	1	no	—
472	C₂×M₁₆	—	134	1	no	—
473	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
474	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
475	C₂²×C₈	—	2	1	no	—
476	C₂²×C₈	—	67	1	no	—
477	C₂²×C₈	—	67	1	no	—
478	C₂²×C₈	—	134	1	no	—
479	C₂²×C₈	—	134	1	no	—
480	C₂²×C₈	—	134	1	no	—
481	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
482	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
483	C₄⋊Q₈	—	67	1	no	—
484	#127	—	67	1	no	—
485	#124	—	67	1	no	—
486	#124	—	67	1	no	—
487	#121	—	134	1	no	—
488	#121	—	134	1	no	—
489	C₄×Q₈	—	67	1	no	—
490	C₄×Q₈	—	67	1	no	—
491	C₄×D₄	—	67	1	no	—
492	C₄×D₄	—	67	1	no	—
493	C₄×D₄	—	134	1	no	—
494	C₄×D₄	—	134	1	no	—
495	C₄×D₄	—	134	1	no	—
496	C₄×D₄	—	134	1	no	—
497	#117	—	134	1	no	—
498	#117	C₂²×D₆₇	1	1	yes	characteristic
499	C₂×C₄⋊C₄	—	134	1	no	—
500	C₂×C₄²	—	67	1	no	—
501	C₂×C₄²	—	67	1	no	—
502	C₂×C₄²	—	134	1	no	—
503	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
504	#108	—	134	2	no	—
505	#108	—	134	2	no	—
506	C₄⋊C₈	—	67	1	no	—
507	C₄⋊C₈	—	67	1	no	—
508	C₄⋊C₈	—	67	1	no	—
509	C₄⋊C₈	—	67	1	no	—
510	C₄⋊C₈	—	134	1	no	—
511	C₄⋊C₈	—	134	1	no	—
512	#104	—	2	2	no	—
513	#104	—	134	2	no	—
514	#104	—	134	2	no	—
515	#104	—	134	2	no	—
516	#104	—	134	2	no	—
517	#104	—	134	2	no	—
518	#103	—	67	1	no	—
519	#103	—	67	1	no	—
520	#103	C₄×D₆₇	1	1	yes	characteristic
521	#103	C₄×D₆₇	1	1	yes	characteristic
522	#102	—	67	1	no	—
523	#102	—	67	1	no	—
524	#102	C₄×D₆₇	1	1	yes	characteristic
525	#102	C₄×D₆₇	1	1	yes	characteristic
526	C₂²⋊C₈	—	134	1	no	—
527	C₂²⋊C₈	—	134	1	no	—
528	C₂²⋊C₈	—	134	1	no	—
529	C₂²⋊C₈	—	134	1	no	—
530	#97	—	2	1	no	—
531	#97	—	67	1	no	—
532	#97	—	67	1	no	—
533	#97	—	67	1	no	—
534	#97	—	67	1	no	—
535	#97	—	134	1	no	—
536	#97	—	134	1	no	—
537	#97	—	134	2	no	—
538	C₄×C₈	—	2	2	no	—
539	C₄×C₈	—	2	1	no	—
540	C₄×C₈	—	134	1	no	—
541	C₄×C₈	—	134	1	no	—
542	#95	—	134	1	no	—
543	#41	—	2	1	no	—
544	#41	—	2	1	no	—
545	#41	—	2	1	no	—
546	#41	—	2	1	no	—
547	#41	—	67	2	no	—
548	#41	—	67	2	no	—
549	#41	—	134	1	no	—
550	#41	—	134	1	no	—
551	#41	—	134	1	no	—
552	#41	—	134	1	no	—
553	#41	—	134	1	no	—
554	#41	—	134	1	no	—
555	C₂×Q₈	—	67	1	no	—
556	C₂×Q₈	—	67	1	no	—
557	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
558	C₂×D₄	—	2	1	no	—
559	C₂×D₄	—	67	1	no	—
560	C₂×D₄	—	67	1	no	—
561	C₂×D₄	—	134	1	no	—
562	C₂×D₄	—	134	1	no	—
563	C₂×D₄	—	134	1	no	—
564	C₂×D₄	—	134	1	no	—
565	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
566	C₂²×C₄	—	2	1	no	—
567	C₂²×C₄	—	2	1	no	—
568	C₂²×C₄	—	67	1	no	—
569	C₂²×C₄	—	67	1	no	—
570	C₂²×C₄	—	134	1	no	—
571	C₂²×C₄	—	134	1	no	—
572	C₂²×C₄	—	134	1	no	—
573	C₂²×C₄	—	134	1	no	—
574	C₂²×C₄	—	134	1	no	—
575	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
576	Q₁₆	—	134	2	no	—
577	Q₁₆	—	134	2	no	—
578	SD₁₆	—	134	2	no	—
579	SD₁₆	—	134	2	no	—
580	SD₁₆	—	134	2	no	—
581	SD₁₆	—	134	2	no	—
582	D₈	—	134	2	no	—
583	D₈	—	134	2	no	—
584	M₁₆	—	2	1	no	—
585	M₁₆	—	2	1	no	—
586	M₁₆	—	2	1	no	—
587	M₁₆	—	4	2	no	—
588	M₁₆	—	268	2	no	—
589	M₁₆	—	67	2	no	—
590	M₁₆	—	134	4	no	—
591	M₁₆	—	134	1	no	—
592	M₁₆	—	134	1	no	—
593	M₁₆	—	134	1	no	—
594	M₁₆	—	134	4	no	—
595	M₁₆	#11761628	1	2	yes	—
596	C₂×C₈	—	2	1	no	—
597	C₂×C₈	—	2	1	no	—
598	C₂×C₈	—	2	1	no	—
599	C₂×C₈	—	2	1	no	—
600	C₂×C₈	—	4	2	no	—
601	C₂×C₈	—	268	2	no	—
602	C₂×C₈	—	67	1	no	—
603	C₂×C₈	—	67	1	no	—
604	C₂×C₈	—	67	1	no	—
605	C₂×C₈	—	67	1	no	—
606	C₂×C₈	—	134	4	no	—
607	C₂×C₈	—	134	1	no	—
608	C₂×C₈	—	134	1	no	—
609	C₂×C₈	—	134	1	no	—
610	C₂×C₈	—	134	1	no	—
611	C₂×C₈	—	134	1	no	—
612	C₂×C₈	—	134	1	no	—
613	C₂×C₈	—	134	4	no	—
614	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
615	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
616	C₂×C₈	D₄×D₆₇	1	2	yes	—
617	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
618	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
619	C₂×C₈	#11761628	1	2	yes	—
620	C₄⋊C₄	—	268	2	no	—
621	C₄⋊C₄	—	134	1	no	—
622	C₄⋊C₄	—	134	1	no	—
623	C₄⋊C₄	—	134	1	no	—
624	C₄⋊C₄	—	134	1	no	—
625	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
626	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
627	C₂²⋊C₄	—	2	1	no	—
628	C₂²⋊C₄	—	134	1	no	—
629	C₂²⋊C₄	—	134	1	no	—
630	C₂²⋊C₄	—	134	1	no	—
631	C₂²⋊C₄	—	134	1	no	—
632	C₂²⋊C₄	—	134	1	no	—
633	C₂²⋊C₄	—	134	1	no	—
634	C₄²	—	2	1	no	—
635	C₄²	—	2	1	no	—
636	C₄²	—	2	1	no	—
637	C₄²	—	67	1	no	—
638	C₄²	—	67	1	no	—
639	C₄²	—	67	1	no	—
640	C₄²	—	67	1	no	—
641	C₄²	—	134	1	no	—
642	C₄²	—	134	1	no	—
643	C₄²	—	134	1	no	—
644	C₄²	—	134	1	no	—
645	C₂³	—	2	1	no	—
646	C₂³	—	134	1	no	—
647	C₂³	—	134	1	no	—
648	C₂³	#41×D₆₇	1	1	yes	characteristic
649	Q₈	—	2	1	no	—
650	Q₈	—	2	1	no	—
651	Q₈	—	67	2	no	—
652	Q₈	—	134	1	no	—
653	Q₈	—	134	1	no	—
654	Q₈	—	134	1	no	—
655	D₄	—	2	1	no	—
656	D₄	—	2	1	no	—
657	D₄	—	4	1	no	—
658	D₄	—	4	1	no	—
659	D₄	—	268	1	no	—
660	D₄	—	268	1	no	—
661	D₄	—	536	1	no	—
662	D₄	—	536	1	no	—
663	D₄	—	67	2	no	—
664	D₄	—	134	2	no	—
665	D₄	—	134	2	no	—
666	D₄	—	134	1	no	—
667	D₄	—	134	1	no	—
668	D₄	—	134	1	no	—
669	C₂×C₄	—	2	2	no	—
670	C₂×C₄	—	2	1	no	—
671	C₂×C₄	—	2	1	no	—
672	C₂×C₄	—	2	1	no	—
673	C₂×C₄	—	4	1	no	—
674	C₂×C₄	—	4	1	no	—
675	C₂×C₄	—	4	1	no	—
676	C₂×C₄	—	4	1	no	—
677	C₂×C₄	—	268	1	no	—
678	C₂×C₄	—	268	1	no	—
679	C₂×C₄	—	268	1	no	—
680	C₂×C₄	—	268	1	no	—
681	C₂×C₄	—	268	1	no	—
682	C₂×C₄	—	268	1	no	—
683	C₂×C₄	—	268	1	no	—
684	C₂×C₄	—	536	1	no	—
685	C₂×C₄	—	536	1	no	—
686	C₂×C₄	—	67	1	no	—
687	C₂×C₄	—	67	1	no	—
688	C₂×C₄	—	67	1	no	—
689	C₂×C₄	—	67	1	no	—
690	C₂×C₄	—	134	1	no	—
691	C₂×C₄	—	134	1	no	—
692	C₂×C₄	—	134	2	no	—
693	C₂×C₄	—	134	2	no	—
694	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
695	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
696	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
697	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
698	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
699	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
700	C₈	—	2	1	no	—
701	C₈	—	2	1	no	—
702	C₈	—	2	2	no	—
703	C₈	—	4	1	no	—
704	C₈	—	268	1	no	—
705	C₈	—	134	1	no	—
706	C₈	—	134	1	no	—
707	C₈	—	134	2	no	—
708	C₈	—	134	1	no	—
709	C₈	—	134	1	no	—
710	C₈	#14524016	1	2	yes	—
711	C₈	#14524016	1	2	yes	—
712	C₂²	—	2	1	no	—
713	C₂²	—	2	1	no	—
714	C₂²	—	4	1	no	—
715	C₂²	—	4	1	no	—
716	C₂²	—	4	1	no	—
717	C₂²	—	268	1	no	—
718	C₂²	—	268	1	no	—
719	C₂²	—	268	1	no	—
720	C₂²	—	268	1	no	—
721	C₂²	—	268	1	no	—
722	C₂²	—	134	2	no	—
723	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
724	C₂²	#16863924	1	1	yes	characteristic
725	C₂²	#16863943	1	1	yes	characteristic
726	C₄	—	4	2	no	—
727	C₄	—	4	1	no	—
728	C₄	—	4	1	no	—
729	C₄	—	4	1	no	—
730	C₄	—	268	1	no	—
731	C₄	—	536	1	no	—
732	C₄	—	134	1	no	—
733	C₄	—	134	1	no	—
734	C₄	—	134	2	no	—
735	C₄	—	134	1	no	—
736	C₄	—	134	1	no	—
737	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
738	C₄	C₄×#11761628	1	1	yes	characteristic
739	C₄	C₂×#14524016	1	1	yes	characteristic
740	C₄	#16863905	1	1	yes	characteristic
741	C₂	—	2	1	no	—
742	C₂	—	2	1	no	—
743	C₂	—	8	1	no	—
744	C₂	—	268	2	no	—
745	C₂	—	536	1	no	—
746	C₂	C₄×#14524016	1	1	yes	characteristic
747	C₂	#22569114	1	1	yes	characteristic
748	C₂	#22569121	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1764	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22569662	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1764	D₆₇
hypocenter	C₆₇	#21519
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1764

Complements

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

N	H ≅ G/N	type
2. C₂×#16863516	C₂	semidirect
3. C₂×#16863514	C₂	semidirect
4. #22568925	C₂	semidirect
5. #22568636	C₂	semidirect
6. #22561126	C₂	semidirect
7. #22560983	C₂	semidirect
8. #22560859	C₂	semidirect
9. #22560855	C₂	semidirect
10. #22560537	C₂	semidirect
11. #22560533	C₂	semidirect
12. #22560228	C₂	semidirect
13. #22560083	C₂	semidirect
14. #22560055	C₂	semidirect
15. C₆₇×#1764	C₂	semidirect
23. #16863621	C₄	semidirect
24. #16863617	C₄	semidirect
25. #16863614	C₄	semidirect
26. #16863610	C₄	semidirect
27. #16863605	C₄	semidirect
28. #16863600	C₄	semidirect
29. #16863595	C₄	semidirect
30. #16863590	C₄	semidirect
88. #14524049	D₄	semidirect
389. #1764	D₆₇	semidirect
390. C₆₇	#21519	semidirect

Maximal subgroups

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

subgroup	index
#22569662	2
C₂×#16863516	2
C₂×#16863514	2
#22568925	2
#22568636	2
#22561126	2
#22560983	2
#22560859	2
#22560855	2
#22560537	2
#22560533	2
#22560228	2
#22560083	2
#22560055	2
C₆₇×#1764	2
#21519	67