Subgroups — #30875831

Order 17152.

616 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561150	C₂	1	2	yes	maximal
2	#22560775	C₂	1	2	yes	maximal
3	#22560768	C₂	1	2	yes	maximal
4	#22560354	C₂	1	2	yes	maximal
5	#22560326	C₂	1	1	yes	maximal, characteristic
6	#22560084	C₂	1	2	yes	maximal
7	#22560062	C₂	1	1	yes	maximal, characteristic
8	#22559961	C₂	1	2	yes	maximal
9	C₆₇×#2295	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×#14523954	C₂²	1	2	yes	—
13	C₂×#14523954	C₂²	1	2	yes	—
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
15	#16863151	C₂²	1	1	yes	characteristic
16	#16863151	C₂²	1	2	yes	—
17	#16863151	C₂²	1	2	yes	—
18	#16863151	C₂²	1	2	yes	—
19	#16863151	C₂²	1	1	yes	characteristic
20	#16863151	C₂²	1	2	yes	—
21	#16863151	C₂²	1	2	yes	—
22	#16863151	C₂²	1	2	yes	—
23	#16863151	C₂²	1	1	yes	characteristic
24	#16863151	C₂²	1	1	yes	characteristic
25	#16863151	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
27	#391×C₆₇	C₄	1	1	yes	characteristic
28	#391×C₆₇	C₄	1	1	yes	characteristic
29	#389×C₆₇	C₄	1	1	yes	characteristic
30	#389×C₆₇	C₄	1	1	yes	characteristic
31	#387×C₆₇	C₄	1	2	yes	—
32	#387×C₆₇	C₄	1	2	yes	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
35	#376×C₆₇	C₂²	1	2	yes	—
36	C₄³×C₆₇	C₂²	1	1	yes	characteristic
37	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
38	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
39	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
40	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
41	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
42	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
43	#14523954	—	2	4	no	—
44	#14523954	—	2	4	no	—
45	C₄×C₆₇⋊C₈	—	2	4	no	—
46	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₄×Q₈×C₆₇	—	2	2	no	—
48	C₄×Q₈×C₆₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
60	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
61	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	#95×C₆₇	C₂×C₄	1	2	yes	—
65	#95×C₆₇	C₂×C₄	1	2	yes	—
66	#95×C₆₇	C₂×C₄	1	2	yes	—
67	#95×C₆₇	C₂×C₄	1	2	yes	—
68	#95×C₆₇	C₂×C₄	1	2	yes	—
69	#95×C₆₇	C₂×C₄	1	2	yes	—
70	#95×C₆₇	C₂×C₄	1	2	yes	—
71	C₂×C₆₇⋊C₈	—	2	2	no	—
72	C₂×C₆₇⋊C₈	—	2	2	no	—
73	C₂×C₆₇⋊C₈	—	2	2	no	—
74	C₂×C₆₇⋊C₈	—	2	2	no	—
75	C₂×C₆₇⋊C₈	—	2	2	no	—
76	C₂×C₆₇⋊C₈	—	2	2	no	—
77	C₂×C₆₇⋊C₈	—	2	2	no	—
78	C₂×C₆₇⋊C₈	—	2	2	no	—
79	C₂×C₆₇⋊C₈	—	2	2	no	—
80	C₂×C₆₇⋊C₈	—	2	2	no	—
81	C₂×C₆₇⋊C₈	—	2	2	no	—
82	C₂×C₆₇⋊C₈	#41	1	4	yes	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	4	1	no	—
85	C₂×Q₈×C₆₇	—	4	1	no	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	—	2	4	no	—
99	C₄⋊C₄×C₆₇	—	2	4	no	—
100	C₄⋊C₄×C₆₇	—	4	1	no	—
101	C₄⋊C₄×C₆₇	—	4	1	no	—
102	C₄⋊C₄×C₆₇	—	4	2	no	—
103	C₄²×C₆₇	—	2	2	no	—
104	C₄²×C₆₇	—	2	2	no	—
105	C₄²×C₆₇	—	2	2	no	—
106	C₄²×C₆₇	—	2	2	no	—
107	C₄²×C₆₇	—	2	2	no	—
108	C₄²×C₆₇	—	2	2	no	—
109	C₄²×C₆₇	—	2	2	no	—
110	C₄²×C₆₇	—	4	1	no	—
111	C₄²×C₆₇	—	4	1	no	—
112	C₄²×C₆₇	—	4	1	no	—
113	C₄²×C₆₇	#41	1	2	yes	—
114	C₄²×C₆₇	#41	1	2	yes	—
115	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
116	Q₈×C₆₇	—	4	4	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	1	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	1	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	1	no	—
137	C₂×C₄×C₆₇	—	2	1	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	#117	1	2	yes	—
145	C₂×C₄×C₆₇	#117	1	2	yes	—
146	C₂×C₄×C₆₇	#131	1	2	yes	—
147	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
148	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
151	C₆₇⋊C₈	—	2	4	no	—
152	C₆₇⋊C₈	—	4	2	no	—
153	C₆₇⋊C₈	—	4	2	no	—
154	C₆₇⋊C₈	—	4	2	no	—
155	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
156	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
157	C₂²×C₆₇	#519	1	1	yes	characteristic
158	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
159	C₂²×C₆₇	#568	1	1	yes	characteristic
160	C₂²×C₆₇	#568	1	1	yes	characteristic
161	C₂²×C₆₇	#568	1	1	yes	characteristic
162	C₄×C₆₇	—	2	2	no	—
163	C₄×C₆₇	—	2	2	no	—
164	C₄×C₆₇	—	2	2	no	—
165	C₄×C₆₇	—	4	2	no	—
166	C₄×C₆₇	—	4	2	no	—
167	C₄×C₆₇	—	4	1	no	—
168	C₄×C₆₇	—	4	2	no	—
169	C₄×C₆₇	—	4	2	no	—
170	C₄×C₆₇	—	4	1	no	—
171	C₄×C₆₇	—	4	1	no	—
172	C₄×C₆₇	#552	1	2	yes	—
173	C₄×C₆₇	#556	1	2	yes	—
174	#22831	—	67	1	no	maximal, Sylow
175	C₂×C₆₇	#2356	1	1	yes	characteristic
176	C₂×C₆₇	#2921	1	1	yes	characteristic
177	C₂×C₆₇	#2925	1	1	yes	characteristic
178	C₂×C₆₇	#2967	1	1	yes	characteristic
179	C₂×C₆₇	#2968	1	1	yes	characteristic
180	C₂×C₆₇	#2985	1	1	yes	characteristic
181	C₂×C₆₇	#2987	1	1	yes	characteristic
182	#2295	D₆₇	1	1	yes	characteristic
183	#1985	—	67	2	no	—
184	#1917	—	67	2	no	—
185	#1916	—	67	2	no	—
186	#1840	—	67	2	no	—
187	#1831	—	67	1	no	—
188	#1771	—	67	2	no	—
189	#1767	—	67	1	no	—
190	#1746	—	67	2	no	—
191	C₆₇	#22831	1	1	yes	characteristic, complemented, Sylow
192	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
193	C₂×C₄⋊C₈	—	67	2	no	—
194	C₂×C₄⋊C₈	—	67	2	no	—
195	C₂×#97	—	67	2	no	—
196	C₂×#97	—	67	2	no	—
197	C₂×C₄×C₈	—	67	2	no	—
198	#391	Dic₆₇	1	1	yes	characteristic
199	#391	Dic₆₇	1	1	yes	characteristic
200	#389	Dic₆₇	1	1	yes	characteristic
201	#389	Dic₆₇	1	1	yes	characteristic
202	#387	Dic₆₇	1	2	yes	—
203	#387	Dic₆₇	1	2	yes	—
204	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
205	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
206	#376	C₂×D₆₇	1	2	yes	—
207	C₄³	C₂×D₆₇	1	1	yes	characteristic
208	#336	—	67	2	no	—
209	#336	—	67	1	no	—
210	#336	—	67	1	no	—
211	#336	—	67	1	no	—
212	#336	—	67	2	no	—
213	#336	—	67	2	no	—
214	#336	—	67	2	no	—
215	#336	—	67	2	no	—
216	#336	—	67	2	no	—
217	#336	—	67	2	no	—
218	#336	—	67	1	no	—
219	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
220	C₂²×C₈	—	67	2	no	—
221	C₂²×C₈	—	67	2	no	—
222	C₂²×C₈	—	67	2	no	—
223	C₂²×C₈	—	67	2	no	—
224	C₄×Q₈	—	2	2	no	—
225	C₄×Q₈	—	2	2	no	—
226	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
227	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
228	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
229	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
230	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
231	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
232	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
233	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
234	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
235	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
236	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
237	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
239	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
240	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
241	C₄⋊C₈	—	134	4	no	—
242	C₄⋊C₈	—	134	4	no	—
243	#97	—	134	4	no	—
244	#97	—	134	4	no	—
245	C₄×C₈	—	134	4	no	—
246	#95	C₂×Dic₆₇	1	2	yes	—
247	#95	C₂×Dic₆₇	1	2	yes	—
248	#95	C₂×Dic₆₇	1	2	yes	—
249	#95	C₂×Dic₆₇	1	2	yes	—
250	#95	C₂×Dic₆₇	1	2	yes	—
251	#95	C₂×Dic₆₇	1	2	yes	—
252	#95	C₂×Dic₆₇	1	2	yes	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
257	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
258	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
259	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
260	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
261	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
262	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
263	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
264	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
265	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
266	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
267	C₂×C₈	—	67	4	no	—
268	C₂×C₈	—	134	2	no	—
269	C₂×C₈	—	134	2	no	—
270	C₂×C₈	—	134	2	no	—
271	C₂×C₈	—	134	2	no	—
272	C₂×C₈	—	134	2	no	—
273	C₂×C₈	—	134	2	no	—
274	C₂×C₈	—	134	2	no	—
275	C₂×C₈	—	134	2	no	—
276	C₂×C₈	—	134	2	no	—
277	C₂×C₈	—	134	2	no	—
278	C₂×C₈	—	134	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	4	no	—
281	C₄⋊C₄	—	2	4	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	4	1	no	—
293	C₄²	—	4	1	no	—
294	C₄²	—	4	1	no	—
295	C₄²	#11761628	1	2	yes	—
296	C₄²	#11761628	1	2	yes	—
297	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
298	Q₈	—	4	4	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	#14524002	1	2	yes	—
327	C₂×C₄	#14524002	1	2	yes	—
328	C₂×C₄	#14524050	1	2	yes	—
329	C₂×C₄	#41×D₆₇	1	2	yes	—
330	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
331	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
332	C₂×C₄	#14524091	1	2	yes	—
333	C₈	—	268	2	no	—
334	C₈	—	268	2	no	—
335	C₈	—	268	2	no	—
336	C₈	—	134	4	no	—
337	C₂²	C₂×#14524002	1	1	yes	characteristic
338	C₂²	#41×Dic₆₇	1	1	yes	characteristic
339	C₂²	#16863874	1	1	yes	characteristic
340	C₂²	C₂×#14524050	1	1	yes	characteristic
341	C₂²	#16864163	1	1	yes	characteristic
342	C₂²	#16864163	1	1	yes	characteristic
343	C₂²	#16864163	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	2	no	—
346	C₄	—	2	2	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	2	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	2	no	—
354	C₄	#16864087	1	2	yes	—
355	C₄	#16864118	1	2	yes	—
356	C₂	#22563174	1	1	yes	characteristic
357	C₂	#22568847	1	1	yes	characteristic
358	C₂	#22568875	1	1	yes	characteristic
359	C₂	#22569196	1	1	yes	characteristic
360	C₂	#22569205	1	1	yes	characteristic
361	C₂	#22569304	1	1	yes	characteristic
362	C₂	#22569315	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#2295	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2295	D₆₇
hypocenter	C₆₇	#22831
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
191. C₆₇	#22831	semidirect

Maximal subgroups

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

subgroup	index
#22561150	2
#22560775	2
#22560768	2
#22560354	2
#22560326	2
#22560084	2
#22560062	2
#22559961	2
C₆₇×#2295	2
#22831	67