Subgroups — #30764783

Order 17152.

842 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573614	C₂	1	1	yes	maximal, characteristic
2	#22559948	C₂	1	1	yes	maximal, characteristic
3	#22559947	C₂	1	2	yes	maximal
4	#22559905	C₂	1	2	yes	maximal
5	C₆₇×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	#16864027	—	2	2	no	—
8	#16864024	—	2	4	no	—
9	#16864022	C₄	1	4	yes	—
10	#16863953	—	2	2	no	—
11	#16863949	—	2	4	no	—
12	C₂×#14524002	C₂²	1	2	yes	—
13	#16863165	—	2	2	no	—
14	#16863164	—	2	4	no	—
15	#16863150	—	2	4	no	—
16	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
19	#336×C₆₇	—	2	2	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761612	Q₈	1	4	yes	complemented
26	C₂²×C₆₇⋊C₈	—	2	2	no	—
27	#14524043	—	2	4	no	—
28	#14524042	—	2	8	no	—
29	#14524035	—	2	4	no	—
30	#14524034	—	2	8	no	—
31	#14524022	—	2	8	no	—
32	#14524020	—	2	16	no	—
33	C₄×Dic₁₃₄	—	2	8	no	—
34	#14524002	—	2	2	no	—
35	#14524002	—	2	2	no	—
36	#14524002	—	2	2	no	—
37	#14524002	—	2	2	no	—
38	C₂×C₄⋊Dic₆₇	D₄	1	4	yes	—
39	C₂×C₂²⋊Dic₆₇	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₆₇	D₄	1	4	yes	—
41	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₆₇	—	2	1	no	—
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	—	2	2	no	—
45	C₂×M₁₆×C₆₇	Q₈	1	2	yes	complemented
46	C₂²×C₈×C₆₇	—	2	1	no	—
47	C₂²×C₈×C₆₇	—	2	2	no	—
48	C₂²×C₈×C₆₇	—	2	1	no	—
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	—	2	1	no	—
51	C₂²×C₈×C₆₇	Q₈	1	2	yes	complemented
52	#117×C₆₇	—	2	2	no	—
53	#117×C₆₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₆₇	D₄	1	2	yes	—
57	C₂×Dic₁₃₄	—	2	8	no	—
58	C₂²×Dic₆₇	C₂²⋊C₄	1	8	yes	—
59	#11761612	—	2	4	no	—
60	#11761612	—	2	4	no	—
61	#11761612	—	4	2	no	—
62	#11761612	—	4	2	no	—
63	C₂×C₆₇⋊C₈	—	2	4	no	—
64	C₂×C₆₇⋊C₈	—	4	2	no	—
65	C₂×C₆₇⋊C₈	—	4	2	no	—
66	C₄⋊Dic₆₇	—	2	8	no	—
67	C₄⋊Dic₆₇	—	2	4	no	—
68	C₄⋊Dic₆₇	—	2	4	no	—
69	Dic₆₇⋊C₄	—	2	16	no	—
70	C₂²⋊Dic₆₇	—	2	4	no	—
71	C₂²⋊Dic₆₇	—	2	4	no	—
72	C₄×Dic₆₇	—	2	4	no	—
73	C₄×Dic₆₇	—	2	4	no	—
74	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	#41	1	2	yes	—
84	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
86	M₁₆×C₆₇	—	2	4	no	—
87	M₁₆×C₆₇	—	4	2	no	—
88	C₂×C₈×C₆₇	—	2	4	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	4	1	no	—
92	C₂×C₈×C₆₇	—	4	1	no	—
93	C₂×C₈×C₆₇	—	4	2	no	—
94	C₂×C₈×C₆₇	—	4	1	no	—
95	C₂×C₈×C₆₇	—	4	1	no	—
96	C₄⋊C₄×C₆₇	—	2	4	no	—
97	C₂²⋊C₄×C₆₇	—	2	4	no	—
98	C₄²×C₆₇	—	2	2	no	—
99	C₄²×C₆₇	—	2	2	no	—
100	C₂³×C₆₇	—	4	2	no	—
101	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
104	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
105	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	4	4	no	—
108	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#120	1	2	yes	—
111	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₆₇	#124	1	2	yes	—
119	C₂×C₄×C₆₇	#124	1	2	yes	—
120	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
126	C₂×Dic₆₇	—	2	8	no	—
127	C₂×Dic₆₇	—	4	8	no	—
128	Dic₁₃₄	—	8	8	no	—
129	C₈×C₆₇	—	4	2	no	—
130	C₈×C₆₇	—	4	2	no	—
131	C₆₇⋊C₈	—	4	4	no	—
132	C₂²×C₆₇	—	2	4	no	—
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	—	4	2	no	—
135	C₂²×C₆₇	—	4	1	no	—
136	C₂²×C₆₇	—	4	1	no	—
137	C₂²×C₆₇	#379	1	1	yes	characteristic
138	C₂²×C₆₇	#391	1	2	yes	—
139	C₂²×C₆₇	#391	1	2	yes	—
140	C₂²×C₆₇	#391	1	1	yes	characteristic
141	C₂²×C₆₇	#391	1	1	yes	characteristic
142	C₄×C₆₇	—	4	4	no	—
143	C₄×C₆₇	#393	1	1	yes	characteristic
144	C₄×C₆₇	#393	1	1	yes	characteristic
145	C₄×C₆₇	#393	1	2	yes	—
146	C₄×C₆₇	#395	1	1	yes	characteristic
147	C₄×C₆₇	#395	1	1	yes	characteristic
148	C₄×C₆₇	#395	1	2	yes	—
149	Dic₆₇	—	4	8	no	—
150	#11391	—	67	1	no	maximal, Sylow
151	C₂×C₆₇	—	2	2	no	—
152	C₂×C₆₇	—	2	2	no	—
153	C₂×C₆₇	—	2	1	no	—
154	C₂×C₆₇	—	2	1	no	—
155	C₂×C₆₇	#1446	1	1	yes	characteristic
156	C₂×C₆₇	#1791	1	2	yes	—
157	#3458	—	67	1	no	—
158	#1742	—	67	2	no	—
159	#1742	—	67	1	no	—
160	#1729	—	67	2	no	—
161	#1729	D₆₇	1	1	yes	characteristic
162	C₆₇	#11391	1	1	yes	characteristic, complemented, Sylow
163	C₂²×M₁₆	—	67	2	no	—
164	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
165	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
166	#544	—	134	4	no	—
167	#544	—	134	2	no	—
168	#543	—	67	4	no	—
169	#533	—	134	4	no	—
170	#533	—	134	2	no	—
171	C₂×#117	—	67	2	no	—
172	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
173	#343	—	134	4	no	—
174	#343	—	134	2	no	—
175	#336	—	2	2	no	—
176	#336	—	134	4	no	—
177	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	67	4	no	—
182	C₂×M₁₆	—	134	2	no	—
183	C₂×M₁₆	—	134	2	no	—
184	C₂×M₁₆	—	134	2	no	—
185	C₂×M₁₆	—	134	2	no	—
186	C₂×M₁₆	—	134	2	no	—
187	C₂×M₁₆	Dic₁₃₄	1	2	yes	complemented
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	134	2	no	—
194	C₂²×C₈	Dic₁₃₄	1	2	yes	complemented
195	C₄⋊Q₈	—	134	4	no	—
196	C₄⋊Q₈	—	134	8	no	—
197	#125	—	134	4	no	—
198	#125	—	134	8	no	—
199	C₂²⋊Q₈	—	134	16	no	—
200	C₂²⋊Q₈	—	134	8	no	—
201	C₄×Q₈	—	134	8	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	—	134	2	no	—
205	#117	—	134	2	no	—
206	#117	—	134	2	no	—
207	#117	—	134	2	no	—
208	C₂×C₄⋊C₄	—	67	4	no	—
209	C₂×C₄⋊C₄	D₂₆₈	1	2	yes	—
210	C₂×C₂²⋊C₄	—	67	4	no	—
211	C₂×C₂²⋊C₄	C₄×D₆₇	1	2	yes	—
212	C₂×C₄²	—	67	4	no	—
213	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
214	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
215	C₂×Q₈	—	134	8	no	—
216	C₂²×C₄	—	67	8	no	—
217	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	4	yes	—
218	C₂²×C₄	D₄×D₆₇	1	2	yes	—
219	C₂²×C₄	D₄×D₆₇	1	2	yes	—
220	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
222	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
223	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
224	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
225	C₂²×C₄	#11761627	1	2	yes	—
226	C₂²×C₄	#11761628	1	1	yes	characteristic
227	C₂²×C₄	#11761628	1	1	yes	characteristic
228	M₁₆	—	2	4	no	—
229	M₁₆	—	4	2	no	—
230	M₁₆	—	268	2	no	—
231	M₁₆	—	268	2	no	—
232	M₁₆	—	134	4	no	—
233	M₁₆	—	134	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	268	2	no	—
243	C₂×C₈	—	268	2	no	—
244	C₂×C₈	—	134	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	134	16	no	—
247	C₄⋊C₄	—	134	8	no	—
248	C₄⋊C₄	—	134	4	no	—
249	C₄⋊C₄	—	134	4	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	134	4	no	—
252	C₂²⋊C₄	—	134	4	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	134	4	no	—
256	C₄²	—	134	4	no	—
257	C₂³	—	4	2	no	—
258	C₂³	C₂²⋊C₄×D₆₇	1	2	yes	—
259	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
260	C₂³	#14524008	1	2	yes	—
261	C₂³	C₄×Dic₁₃₄	1	1	yes	characteristic
262	C₂³	C₄×Dic₁₃₄	1	1	yes	characteristic
263	Q₈	—	536	8	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	268	8	no	—
267	C₂×C₄	—	134	8	no	—
268	C₂×C₄	#14524011	1	1	yes	characteristic
269	C₂×C₄	#14524011	1	1	yes	characteristic
270	C₂×C₄	#14524013	1	2	yes	—
271	C₂×C₄	#14524020	1	2	yes	—
272	C₂×C₄	#14524020	1	2	yes	—
273	C₂×C₄	#14524021	1	2	yes	—
274	C₂×C₄	#14524021	1	2	yes	—
275	C₂×C₄	#14524022	1	1	yes	characteristic
276	C₂×C₄	#14524022	1	1	yes	characteristic
277	C₂×C₄	#14524023	1	2	yes	—
278	C₂×C₄	#14524029	1	2	yes	—
279	C₂×C₄	#14524029	1	2	yes	—
280	C₂×C₄	#14524030	1	1	yes	characteristic
281	C₂×C₄	#14524031	1	1	yes	characteristic
282	C₂×C₄	#14524041	1	1	yes	characteristic
283	C₂×C₄	#14524042	1	2	yes	—
284	C₂×C₄	#14524042	1	2	yes	—
285	C₂×C₄	#14524043	1	1	yes	characteristic
286	C₈	—	4	2	no	—
287	C₈	—	4	2	no	—
288	C₈	—	268	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	#16863251	1	1	yes	characteristic
295	C₂²	#16863300	1	2	yes	—
296	C₂²	#16863300	1	2	yes	—
297	C₂²	#16863301	1	1	yes	characteristic
298	C₂²	#16863302	1	1	yes	characteristic
299	C₄	—	4	4	no	—
300	C₄	—	268	8	no	—
301	C₄	#16863308	1	1	yes	characteristic
302	C₄	#16863308	1	1	yes	characteristic
303	C₄	#16863309	1	2	yes	—
304	C₄	#16863316	1	1	yes	characteristic
305	C₄	#16863316	1	1	yes	characteristic
306	C₄	#16863317	1	2	yes	—
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	#22558479	1	1	yes	characteristic
312	C₂	#22560159	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524011
commutator, prime radical	C₂⁴×C₆₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₆₇×#1729	C₂
socle	C₂²×C₆₇	#379
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	#1729	D₆₇
hypocenter	C₆₇	#11391
LCS	G ▷ C₂⁴×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
25. C₂×#11761612	Q₈	semidirect
45. C₂×M₁₆×C₆₇	Q₈	semidirect
51. C₂²×C₈×C₆₇	Q₈	semidirect
162. C₆₇	#11391	semidirect
187. C₂×M₁₆	Dic₁₃₄	semidirect
194. C₂²×C₈	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22573614	2
#22559948	2
#22559947	2
#22559905	2
C₆₇×#1729	2
#11391	67