Subgroups — #30843769

Order 17152.

858 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22571332	C₂	1	2	yes	maximal, complemented
2	#22570788	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863801	C₂	1	1	yes	maximal, characteristic, complemented
4	#22570668	C₂	1	2	yes	maximal, complemented
5	#22559880	C₂	1	4	yes	maximal, complemented
6	#22559827	C₂	1	2	yes	maximal, complemented
7	#22559801	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524076	—	2	4	no	—
10	#16863946	C₂²	1	2	yes	—
11	C₂×#14524043	C₂²	1	2	yes	—
12	#16863801	—	2	2	no	—
13	#16863801	C₂²	1	1	yes	characteristic
14	#16863801	C₂²	1	2	yes	—
15	#16863801	C₂²	1	4	yes	—
16	#16863801	C₂²	1	2	yes	—
17	#16863790	—	2	4	no	—
18	#16863776	C₂²	1	4	yes	—
19	#16863773	C₂²	1	1	yes	characteristic
20	#16863769	C₂²	1	4	yes	—
21	#16863743	C₂²	1	2	yes	—
22	#16863743	C₂²	1	4	yes	—
23	#16863710	—	2	4	no	—
24	#16863677	—	2	4	no	—
25	#16863632	—	2	2	no	—
26	C₄×D₅₃₆	—	2	2	no	—
27	C₂×#14523977	—	2	2	no	—
28	#431×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#322×C₆₇	C₂²	1	2	yes	—
32	C₈²×C₆₇	C₂²	1	2	yes	—
33	C₂×#11761628	—	2	2	no	—
34	C₂²×Dic₁₃₄	—	2	2	no	—
35	#14524076	—	4	16	no	—
36	C₂×Dic₂₆₈	—	2	8	no	—
37	C₂×#11761617	—	2	8	no	—
38	C₂×D₅₃₆	—	2	2	no	—
39	#14524043	—	2	2	no	—
40	#14524043	—	2	2	no	—
41	#14524043	—	2	2	no	—
42	#14524043	C₂³	1	4	yes	—
43	#14524043	C₂³	1	2	yes	—
44	#14524040	C₂³	1	2	yes	—
45	#14524031	—	2	2	no	—
46	#14524017	—	2	2	no	—
47	#14524017	—	2	2	no	—
48	C₄×D₂₆₈	—	2	2	no	—
49	C₄×D₂₆₈	—	2	2	no	—
50	C₂×C₄⋊Dic₆₇	—	2	2	no	—
51	C₂×C₄⋊Dic₆₇	—	2	2	no	—
52	#14523977	—	2	4	no	—
53	#14523977	—	2	4	no	—
54	#14523977	—	2	2	no	—
55	#14523977	—	2	8	no	—
56	#14523965	—	2	4	no	—
57	#14523965	—	2	8	no	—
58	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
59	C₂²×C₈×C₆₇	D₄	1	2	yes	—
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
62	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
63	#97×C₆₇	D₄	1	2	yes	—
64	C₄×C₈×C₆₇	D₄	1	2	yes	—
65	C₄×C₈×C₆₇	C₂³	1	2	yes	—
66	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₆₇	C₂³	1	2	yes	—
68	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	#11761628	—	4	8	no	—
70	C₂×Dic₁₃₄	—	2	2	no	—
71	C₂×Dic₁₃₄	—	2	4	no	—
72	C₂×Dic₁₃₄	—	4	8	no	—
73	C₂×Dic₁₃₄	—	4	2	no	—
74	C₂×D₂₆₈	—	2	2	no	—
75	C₂×D₂₆₈	—	2	2	no	—
76	C₂×D₂₆₈	—	2	2	no	—
77	C₂×C₂²⋊D₆₇	—	4	2	no	—
78	C₂²×Dic₆₇	—	4	2	no	—
79	C₂×C₄×D₆₇	—	4	2	no	—
80	Dic₂₆₈	—	4	16	no	—
81	#11761617	—	4	16	no	—
82	D₅₃₆	—	4	4	no	—
83	C₄⋊Dic₆₇	—	2	4	no	—
84	C₄⋊Dic₆₇	—	2	2	no	—
85	C₄⋊Dic₆₇	—	2	4	no	—
86	C₄⋊Dic₆₇	—	2	2	no	—
87	C₄⋊Dic₆₇	—	4	2	no	—
88	C₄⋊Dic₆₇	—	4	2	no	—
89	(C₂×D₆₇)⋊C₄	—	4	2	no	—
90	(C₂×D₆₇)⋊C₄	—	4	2	no	—
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
94	M₁₆×C₆₇	D₈	1	8	yes	complemented
95	C₂×C₈×C₆₇	—	2	4	no	—
96	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
97	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
99	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
100	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
104	C₂³×C₆₇	#127	1	1	yes	characteristic
105	C₂²×D₆₇	—	4	2	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
112	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
115	C₂²⋊D₆₇	—	8	4	no	—
116	C₂×Dic₆₇	—	4	2	no	—
117	C₂×Dic₆₇	—	4	4	no	—
118	C₂×Dic₆₇	—	8	4	no	—
119	D₂₆₈	—	4	4	no	—
120	D₂₆₈	—	4	4	no	—
121	D₂₆₈	—	4	4	no	—
122	C₄×D₆₇	—	8	4	no	—
123	Dic₁₃₄	—	4	8	no	—
124	Dic₁₃₄	—	4	4	no	—
125	Dic₁₃₄	—	8	4	no	—
126	C₈×C₆₇	—	2	2	no	—
127	C₈×C₆₇	—	2	4	no	—
128	C₈×C₆₇	C₂×D₈	1	8	yes	—
129	C₈×C₆₇	#137	1	4	yes	complemented
130	C₂²×C₆₇	—	2	1	no	—
131	C₂²×C₆₇	—	2	1	no	—
132	C₂²×C₆₇	#493	1	2	yes	—
133	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
134	C₂×D₆₇	—	8	2	no	—
135	C₂×D₆₇	—	8	2	no	—
136	C₂×D₆₇	—	8	2	no	—
137	C₄×C₆₇	—	2	1	no	—
138	C₄×C₆₇	—	2	1	no	—
139	C₄×C₆₇	#493	1	2	yes	—
140	C₄×C₆₇	C₂²×D₈	1	2	yes	—
141	C₄×C₆₇	C₂×#137	1	2	yes	—
142	C₄×C₆₇	#578	1	2	yes	—
143	Dic₆₇	—	8	4	no	—
144	Dic₆₇	—	8	2	no	—
145	#19630	—	67	1	no	maximal, Sylow
146	C₂×C₆₇	—	2	2	no	—
147	C₂×C₆₇	C₂×#493	1	1	yes	characteristic
148	C₂×C₆₇	#3152	1	1	yes	characteristic
149	C₂×C₆₇	#3156	1	1	yes	characteristic
150	D₆₇	—	16	2	no	—
151	#3226	—	67	2	no	—
152	#3169	—	67	1	no	—
153	C₂×#500	—	67	1	no	—
154	#3152	—	67	2	no	—
155	#1722	—	67	4	no	—
156	#1713	—	67	2	no	—
157	#1706	—	67	2	no	—
158	C₈×M₁₆	D₆₇	1	1	yes	characteristic, complemented
159	C₆₇	#19630	1	1	yes	characteristic, complemented, Sylow
160	C₂×#137	—	134	4	no	—
161	#532	—	67	2	no	—
162	C₂×C₄⋊Q₈	—	67	2	no	—
163	#500	—	67	2	no	—
164	#500	—	67	2	no	—
165	#500	—	67	1	no	—
166	#500	—	67	4	no	—
167	#500	—	134	2	no	—
168	#497	—	134	4	no	—
169	#494	—	67	4	no	—
170	#493	—	67	1	no	—
171	#492	—	67	4	no	—
172	#486	—	67	2	no	—
173	#486	—	67	4	no	—
174	#480	—	134	4	no	—
175	#474	—	134	4	no	—
176	#466	—	134	2	no	—
177	C₄×D₈	—	134	2	no	—
178	#431	C₂×D₆₇	1	1	yes	characteristic
179	C₂×#107	—	134	2	no	—
180	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
182	#322	C₂×D₆₇	1	2	yes	—
183	C₈²	C₂×D₆₇	1	2	yes	—
184	C₂×#41	—	134	2	no	—
185	C₂²×Q₈	—	134	2	no	—
186	#137	—	268	16	no	—
187	C₂×Q₁₆	—	134	8	no	—
188	C₂×SD₁₆	—	134	8	no	—
189	C₂×D₈	—	134	2	no	—
190	C₂×M₁₆	D₂₆₈	1	2	yes	—
191	C₂²×C₈	D₂₆₈	1	2	yes	—
192	C₄⋊Q₈	—	67	4	no	—
193	C₄⋊Q₈	—	67	2	no	—
194	C₄⋊Q₈	—	134	2	no	—
195	C₄⋊Q₈	—	134	2	no	—
196	C₄⋊Q₈	—	134	2	no	—
197	#127	—	67	2	no	—
198	#124	—	134	2	no	—
199	#121	—	134	2	no	—
200	#121	—	134	2	no	—
201	C₄×D₄	—	134	2	no	—
202	C₄×D₄	—	134	2	no	—
203	C₂×C₄⋊C₄	—	134	2	no	—
204	C₂×C₄⋊C₄	—	134	2	no	—
205	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
206	#107	—	134	4	no	—
207	#107	—	134	4	no	—
208	#107	—	134	2	no	—
209	#107	—	134	8	no	—
210	C₄⋊C₈	D₂₆₈	1	2	yes	—
211	#102	—	134	4	no	—
212	#102	—	134	8	no	—
213	C₂²⋊C₈	D₂₆₈	1	2	yes	—
214	#97	D₂₆₈	1	2	yes	—
215	C₄×C₈	D₂₆₈	1	2	yes	—
216	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₆₇	1	2	yes	—
218	C₄×C₈	C₂²×D₆₇	1	2	yes	—
219	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
220	#41	—	268	8	no	—
221	C₂×Q₈	—	268	2	no	—
222	C₂×Q₈	—	268	8	no	—
223	C₂×Q₈	—	134	2	no	—
224	C₂×Q₈	—	134	4	no	—
225	C₂×D₄	—	268	2	no	—
226	C₂×D₄	—	134	2	no	—
227	C₂×D₄	—	134	2	no	—
228	C₂×D₄	—	134	2	no	—
229	C₂²×C₄	—	268	2	no	—
230	C₂²×C₄	—	268	2	no	—
231	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
234	Q₁₆	—	268	16	no	—
235	SD₁₆	—	268	16	no	—
236	D₈	—	268	4	no	—
237	M₁₆	D₅₃₆	1	8	yes	complemented
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
240	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
241	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
242	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
243	C₄⋊C₄	—	268	2	no	—
244	C₄⋊C₄	—	268	2	no	—
245	C₄⋊C₄	—	134	4	no	—
246	C₄⋊C₄	—	134	2	no	—
247	C₄⋊C₄	—	134	4	no	—
248	C₄⋊C₄	—	134	2	no	—
249	C₂²⋊C₄	—	268	2	no	—
250	C₂²⋊C₄	—	268	2	no	—
251	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
252	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
253	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
254	C₄²	C₂³×D₆₇	1	1	yes	characteristic
255	C₂³	—	268	2	no	—
256	C₂³	#14524040	1	1	yes	characteristic
257	Q₈	—	268	8	no	—
258	Q₈	—	268	4	no	—
259	Q₈	—	536	4	no	—
260	D₄	—	268	4	no	—
261	D₄	—	268	4	no	—
262	D₄	—	268	4	no	—
263	D₄	—	536	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	268	2	no	—
267	C₂×C₄	—	268	4	no	—
268	C₂×C₄	—	536	4	no	—
269	C₂×C₄	—	536	4	no	—
270	C₂×C₄	#14524040	1	1	yes	characteristic
271	C₂×C₄	#14524040	1	1	yes	characteristic
272	C₂×C₄	#14524040	1	1	yes	characteristic
273	C₂×C₄	C₂×D₅₃₆	1	4	yes	—
274	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	4	no	—
279	C₈	C₂×D₅₃₆	1	8	yes	—
280	C₈	#14524076	1	4	yes	complemented
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	536	2	no	—
284	C₂²	—	536	2	no	—
285	C₂²	—	536	2	no	—
286	C₂²	#16863773	1	2	yes	—
287	C₂²	C₂×#14524040	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	536	2	no	—
291	C₄	—	536	4	no	—
292	C₄	#16863773	1	2	yes	—
293	C₄	C₂²×D₅₃₆	1	2	yes	—
294	C₄	C₂×#14524076	1	2	yes	—
295	C₄	#16864213	1	2	yes	—
296	C₂	—	2	2	no	—
297	C₂	—	1072	2	no	—
298	C₂	C₂×#16863773	1	1	yes	characteristic
299	C₂	#22570668	1	1	yes	characteristic
300	C₂	#22570703	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14524040
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₈×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#127
prime radical	#22570788	C₂
hypercenter	C₈×M₁₆	D₆₇
hypocenter	C₆₇	#19630
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. #22571332	C₂	semidirect
3. C₂×#16863801	C₂	semidirect
4. #22570668	C₂	semidirect
5. #22559880	C₂	semidirect
6. #22559827	C₂	semidirect
7. #22559801	C₂	semidirect
8. C₈×M₁₆×C₆₇	C₂	semidirect
94. M₁₆×C₆₇	D₈	semidirect
129. C₈×C₆₇	#137	semidirect
158. C₈×M₁₆	D₆₇	semidirect
159. C₆₇	#19630	semidirect
237. M₁₆	D₅₃₆	semidirect
280. C₈	#14524076	semidirect

Maximal subgroups

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

subgroup	index
#22571332	2
#22570788	2
C₂×#16863801	2
#22570668	2
#22559880	2
#22559827	2
#22559801	2
C₈×M₁₆×C₆₇	2
#19630	67