Subgroups — #34110287

Order 20096.

430 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757823	C₂	1	1	yes	maximal, characteristic, complemented
2	#23757597	C₂	1	2	yes	maximal, complemented
3	#23757558	C₂	1	1	yes	maximal, characteristic, complemented
4	#23757553	C₂	1	1	yes	maximal, characteristic, complemented
5	#23757473	C₂	1	1	yes	maximal, characteristic, complemented
6	#23757468	C₂	1	1	yes	maximal, characteristic, complemented
7	#23757418	C₂	1	2	yes	maximal, complemented
8	#23757383	C₂	1	2	yes	maximal, complemented
9	#23757371	C₂	1	2	yes	maximal, complemented
10	#23757330	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14578954	—	2	1	no	—
13	C₂×#14578941	C₂²	1	2	yes	—
14	C₂×#14578939	C₂²	1	2	yes	—
15	#19097771	C₂²	1	1	yes	characteristic
16	#19097768	C₂²	1	1	yes	characteristic
17	#19097759	—	2	1	no	—
18	#19097745	—	2	1	no	—
19	#19097745	C₂²	1	2	yes	—
20	C₄×D₆₂₈	—	2	1	no	—
21	C₄×D₆₂₈	C₂²	1	2	yes	—
22	#19097704	C₂²	1	2	yes	—
23	#19097702	C₂²	1	2	yes	—
24	C₄⋊#11929434	C₂²	1	1	yes	characteristic
25	C₄⋊#11929434	C₂²	1	1	yes	characteristic
26	#19097695	C₂²	1	2	yes	—
27	#19097692	C₂²	1	2	yes	—
28	#19097691	C₂²	1	2	yes	—
29	#19097691	C₂²	1	2	yes	—
30	C₂²⋊#11929434	C₂²	1	2	yes	—
31	#19097682	C₂²	1	2	yes	—
32	C₂×#41×C₁₅₇	C₂²	1	1	yes	characteristic
33	C₄×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
35	C₄×D₄×C₁₅₇	—	2	1	no	—
36	C₄×D₄×C₁₅₇	—	2	1	no	—
37	C₄×D₄×C₁₅₇	C₂²	1	1	yes	characteristic
38	C₄×D₄×C₁₅₇	C₂²	1	1	yes	characteristic
39	#117×C₁₅₇	—	2	1	no	—
40	#117×C₁₅₇	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₁₅₇	—	2	1	no	—
42	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
43	#14578954	—	4	4	no	—
44	C₂×Dic₃₁₄	—	2	1	no	—
45	C₂×D₆₂₈	—	2	1	no	—
46	C₂×D₆₂₈	—	2	1	no	—
47	C₂×D₆₂₈	C₂³	1	2	yes	—
48	C₂×C₂²⋊D₁₅₇	—	2	2	no	—
49	C₂×C₄×D₁₅₇	—	2	2	no	—
50	#14578941	—	2	4	no	—
51	#14578939	—	2	4	no	—
52	C₂×#11929434	C₂³	1	2	yes	—
53	C₂×#11929434	C₂³	1	2	yes	—
54	C₄⋊Dic₁₅₇	—	2	1	no	—
55	C₄⋊Dic₁₅₇	C₂³	1	2	yes	—
56	#14578931	—	2	2	no	—
57	#14578931	—	2	2	no	—
58	#41×C₁₅₇	—	2	2	no	—
59	#41×C₁₅₇	—	2	2	no	—
60	C₂×Q₈×C₁₅₇	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₅₇	—	2	1	no	—
62	C₂×D₄×C₁₅₇	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₅₇	—	2	1	no	—
64	C₂²×C₄×C₁₅₇	—	2	1	no	—
65	C₂²×C₄×C₁₅₇	—	2	1	no	—
66	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₁₅₇	—	2	1	no	—
70	C₄⋊C₄×C₁₅₇	—	2	1	no	—
71	C₄⋊C₄×C₁₅₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₅₇	C₂³	1	1	yes	characteristic
73	C₂²⋊C₄×C₁₅₇	—	2	1	no	—
74	C₂²⋊C₄×C₁₅₇	—	2	1	no	—
75	C₂²⋊C₄×C₁₅₇	—	2	1	no	—
76	C₄²×C₁₅₇	—	2	1	no	—
77	C₄²×C₁₅₇	—	2	1	no	—
78	C₄²×C₁₅₇	—	2	1	no	—
79	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
80	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
81	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
82	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
83	C₂²×D₁₅₇	—	2	2	no	—
84	Dic₃₁₄	—	4	2	no	—
85	D₆₂₈	—	2	2	no	—
86	D₆₂₈	—	2	2	no	—
87	D₆₂₈	—	4	2	no	—
88	D₆₂₈	—	4	2	no	—
89	C₂²⋊D₁₅₇	—	4	4	no	—
90	C₂×Dic₁₅₇	—	2	2	no	—
91	C₄×D₁₅₇	—	4	4	no	—
92	#11929434	—	2	2	no	—
93	#11929434	—	2	2	no	—
94	C₂³×C₁₅₇	—	2	1	no	—
95	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
96	Q₈×C₁₅₇	—	2	1	no	—
97	Q₈×C₁₅₇	#41	1	2	yes	complemented
98	D₄×C₁₅₇	—	2	4	no	—
99	D₄×C₁₅₇	—	2	1	no	—
100	D₄×C₁₅₇	#41	1	2	yes	complemented
101	C₂×C₄×C₁₅₇	—	2	4	no	—
102	C₂×C₄×C₁₅₇	—	2	1	no	—
103	C₂×C₄×C₁₅₇	—	2	2	no	—
104	C₂×C₄×C₁₅₇	—	2	1	no	—
105	C₂×C₄×C₁₅₇	—	2	1	no	—
106	C₂×C₄×C₁₅₇	—	2	1	no	—
107	C₂×C₄×C₁₅₇	—	2	1	no	—
108	C₂×C₄×C₁₅₇	—	4	1	no	—
109	C₂×C₄×C₁₅₇	—	4	2	no	—
110	C₂×C₄×C₁₅₇	—	4	2	no	—
111	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
117	C₂²×C₁₅₇	—	2	1	no	—
118	C₂²×C₁₅₇	—	2	2	no	—
119	C₂²×C₁₅₇	—	2	1	no	—
120	C₂²×C₁₅₇	—	2	1	no	—
121	C₂²×C₁₅₇	—	4	1	no	—
122	C₂²×C₁₅₇	—	4	1	no	—
123	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×D₁₅₇	—	4	2	no	—
125	C₂×D₁₅₇	—	4	2	no	—
126	C₂×D₁₅₇	—	4	2	no	—
127	C₄×C₁₅₇	—	2	1	no	—
128	C₄×C₁₅₇	—	2	2	no	—
129	C₄×C₁₅₇	—	2	1	no	—
130	C₄×C₁₅₇	—	2	1	no	—
131	C₄×C₁₅₇	—	2	1	no	—
132	C₄×C₁₅₇	—	4	1	no	—
133	C₄×C₁₅₇	—	4	1	no	—
134	C₄×C₁₅₇	AGL(1,ℤ/8)	1	2	yes	—
135	C₄×C₁₅₇	C₂×#41	1	1	yes	characteristic
136	C₄×C₁₅₇	C₂×#41	1	1	yes	characteristic
137	Dic₁₅₇	—	4	2	no	—
138	C₂×C₁₅₇	—	4	1	no	—
139	C₂×C₁₅₇	—	4	2	no	—
140	C₂×C₁₅₇	#525	1	1	yes	characteristic
141	C₂×C₁₅₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₁₅₇	#575	1	1	yes	characteristic
143	D₁₅₇	—	8	2	no	—
144	C₁₅₇	#3115	1	1	yes	characteristic, complemented, Sylow
145	#3115	—	157	1	no	maximal, Sylow
146	#532	—	157	1	no	—
147	C₄×#41	D₁₅₇	1	1	yes	characteristic, complemented
148	#482	—	157	2	no	—
149	#475	—	157	1	no	—
150	#474	—	157	1	no	—
151	#460	—	157	1	no	—
152	#459	—	157	1	no	—
153	#449	—	157	2	no	—
154	#442	—	157	2	no	—
155	#440	—	157	2	no	—
156	#432	—	157	1	no	—
157	C₂×#41	—	314	1	no	—
158	C₂×#41	C₂×D₁₅₇	1	1	yes	characteristic
159	C₂×SD₁₆	—	157	2	no	—
160	C₂×D₈	—	157	2	no	—
161	C₄⋊Q₈	—	157	1	no	—
162	#127	—	157	1	no	—
163	#124	—	314	1	no	—
164	#121	—	314	1	no	—
165	#121	—	157	2	no	—
166	C₄×Q₈	C₂×D₁₅₇	1	1	yes	characteristic
167	C₄×Q₈	C₂×D₁₅₇	1	1	yes	characteristic
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	—	314	1	no	—
171	C₄×D₄	—	157	2	no	—
172	C₄×D₄	C₂×D₁₅₇	1	1	yes	characteristic
173	C₄×D₄	C₂×D₁₅₇	1	1	yes	characteristic
174	#117	—	2	1	no	—
175	#117	C₂×D₁₅₇	1	1	yes	characteristic
176	C₂×C₄²	—	2	1	no	—
177	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
178	#107	—	157	2	no	—
179	#106	—	157	2	no	—
180	C₄⋊C₈	—	157	1	no	—
181	C₄⋊C₈	—	157	1	no	—
182	#103	—	157	2	no	—
183	#102	—	157	2	no	—
184	#102	—	157	2	no	—
185	#102	—	157	2	no	—
186	C₂²⋊C₈	—	157	2	no	—
187	#97	—	157	2	no	—
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	#41	—	628	4	no	—
191	C₂×Q₈	—	314	1	no	—
192	C₂×Q₈	C₂²×D₁₅₇	1	1	yes	characteristic
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	—	314	2	no	—
195	C₂×D₄	—	314	1	no	—
196	C₂×D₄	—	314	1	no	—
197	C₂×D₄	—	157	2	no	—
198	C₂×D₄	C₂²×D₁₅₇	1	1	yes	characteristic
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	314	2	no	—
203	C₂²×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
206	SD₁₆	—	314	4	no	—
207	D₈	—	314	4	no	—
208	C₂×C₈	—	157	2	no	—
209	C₂×C₈	—	157	2	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	314	1	no	—
213	C₄⋊C₄	—	157	2	no	—
214	C₄⋊C₄	C₂²×D₁₅₇	1	1	yes	characteristic
215	C₄⋊C₄	C₂²×D₁₅₇	1	1	yes	characteristic
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	314	2	no	—
220	C₂²⋊C₄	—	314	2	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
225	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
226	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
227	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
228	C₂³	—	2	1	no	—
229	C₂³	—	314	2	no	—
230	C₂³	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
231	Q₈	—	2	1	no	—
232	Q₈	—	628	2	no	—
233	Q₈	#14578954	1	2	yes	complemented
234	D₄	—	2	4	no	—
235	D₄	—	2	1	no	—
236	D₄	—	314	2	no	—
237	D₄	—	314	2	no	—
238	D₄	—	628	2	no	—
239	D₄	—	628	4	no	—
240	D₄	—	628	2	no	—
241	D₄	#14578954	1	2	yes	complemented
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	314	2	no	—
253	C₂×C₄	—	628	4	no	—
254	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
255	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
259	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
260	C₈	—	314	2	no	—
261	C₈	—	314	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	628	2	no	—
269	C₂²	—	628	2	no	—
270	C₂²	—	628	2	no	—
271	C₂²	C₂²×C₂²⋊D₁₅₇	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	628	2	no	—
280	C₄	#19097796	1	2	yes	—
281	C₄	C₂×#14578954	1	1	yes	characteristic
282	C₄	C₂×#14578954	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	4	2	no	—
285	C₂	—	1256	2	no	—
286	C₂	#23757784	1	1	yes	characteristic
287	C₂	C₂×#19097796	1	1	yes	characteristic
288	C₂	#23758071	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₅₇
commutator, cosocle	C₂×C₄×C₁₅₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	C₄×#41×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂²×D₄
hypercenter	C₄×#41	D₁₅₇
hypocenter	C₁₅₇	#3115
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #23757823	C₂	semidirect
2. #23757597	C₂	semidirect
3. #23757558	C₂	semidirect
4. #23757553	C₂	semidirect
5. #23757473	C₂	semidirect
6. #23757468	C₂	semidirect
7. #23757418	C₂	semidirect
8. #23757383	C₂	semidirect
9. #23757371	C₂	semidirect
10. #23757330	C₂	semidirect
11. C₄×#41×C₁₅₇	C₂	semidirect
97. Q₈×C₁₅₇	#41	semidirect
100. D₄×C₁₅₇	#41	semidirect
144. C₁₅₇	#3115	semidirect
147. C₄×#41	D₁₅₇	semidirect
233. Q₈	#14578954	semidirect
241. D₄	#14578954	semidirect

Maximal subgroups

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

subgroup	index
#23757823	2
#23757597	2
#23757558	2
#23757553	2
#23757473	2
#23757468	2
#23757418	2
#23757383	2
#23757371	2
#23757330	2
C₄×#41×C₁₅₇	2
#3115	157