Subgroups — #34109041

Order 20096.

458 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23758070	C₂	1	1	yes	maximal, characteristic
2	#23758038	C₂	1	2	yes	maximal, complemented
3	#23757395	C₂	1	1	yes	maximal, characteristic, complemented
4	#23757394	C₂	1	1	yes	maximal, characteristic, complemented
5	#23757393	C₂	1	2	yes	maximal, complemented
6	#23757391	C₂	1	1	yes	maximal, characteristic, complemented
7	#23757387	C₂	1	2	yes	maximal, complemented
8	#23757320	C₂	1	1	yes	maximal, characteristic, complemented
9	#104×D₁₅₇	C₂	1	2	yes	maximal, complemented
10	C₄×#14578937	C₂	1	1	yes	maximal, characteristic, complemented
11	#443×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₅₇	C₂²	1	2	yes	—
13	#19097803	C₄	1	2	yes	complemented
14	#19097800	C₄	1	2	yes	complemented
15	#19097798	C₄	1	2	yes	complemented
16	#19097795	C₄	1	2	yes	complemented
17	#19097793	C₂²	1	1	yes	characteristic
18	#19097791	C₂²	1	2	yes	—
19	#19097779	—	2	2	no	—
20	#19097779	C₂²	1	2	yes	complemented
21	#19097778	C₂²	1	2	yes	complemented
22	#19097777	—	2	2	no	—
23	#19097777	C₂²	1	2	yes	complemented
24	C₂×#14578937	—	2	2	no	—
25	C₂×#14578937	C₂²	1	1	yes	characteristic
26	C₂×#14578937	C₂²	1	1	yes	characteristic
27	M₁₆×D₁₅₇	—	2	2	no	—
28	M₁₆×D₁₅₇	C₂²	1	2	yes	—
29	C₄²×D₁₅₇	C₂²	1	1	yes	characteristic
30	#19097707	C₂²	1	1	yes	characteristic, complemented
31	#19097706	C₂²	1	2	yes	complemented
32	#19097699	C₂²	1	2	yes	complemented
33	#19097698	C₂²	1	2	yes	complemented
34	#19097697	C₂²	1	2	yes	complemented
35	#19097681	C₂²	1	1	yes	characteristic, complemented
36	#19097681	C₂²	1	1	yes	characteristic, complemented
37	C₄×#11929434	C₂²	1	1	yes	characteristic, complemented
38	#135×C₁₅₇	C₂²	1	1	yes	characteristic
39	#131×C₁₅₇	C₂²	1	2	yes	complemented
40	#108×C₁₅₇	C₂²	1	1	yes	characteristic, complemented
41	#104×C₁₅₇	C₂²	1	2	yes	complemented
42	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic, complemented
43	#14578955	C₂×C₄	1	2	yes	—
44	#14578954	—	2	2	no	—
45	#14578954	C₂³	1	2	yes	—
46	#14578953	—	2	2	no	—
47	#14578953	C₂×C₄	1	2	yes	—
48	Q₈×D₁₅₇	C₂×C₄	1	2	yes	—
49	D₄×D₁₅₇	—	2	2	no	—
50	D₄×D₁₅₇	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₁₅₇	—	2	2	no	—
52	C₂×C₄×D₁₅₇	—	2	1	no	—
53	C₂×C₄×D₁₅₇	C₂³	1	1	yes	characteristic
54	Dic₆₂₈	C₂×C₄	1	1	yes	characteristic, complemented
55	#14578944	C₂×C₄	1	2	yes	complemented
56	#14578943	C₂×C₄	1	2	yes	complemented
57	#14578942	C₂×C₄	1	2	yes	complemented
58	#14578941	C₂×C₄	1	2	yes	complemented
59	D₁₂₅₆	C₂×C₄	1	1	yes	characteristic, complemented
60	#14578939	C₂×C₄	1	2	yes	complemented
61	#14578938	—	2	2	no	—
62	#14578938	C₂³	1	2	yes	—
63	#14578937	—	2	2	no	—
64	#14578937	—	2	2	no	—
65	#14578937	—	2	2	no	—
66	#14578937	C₂×C₄	1	2	yes	—
67	#14578937	C₂×C₄	1	2	yes	—
68	#14578937	D₄	1	2	yes	—
69	#14578937	D₄	1	2	yes	—
70	C₂×#11929434	—	2	2	no	—
71	C₂×#11929434	C₂³	1	1	yes	characteristic
72	C₂×#11929434	C₂³	1	1	yes	characteristic
73	C₈×D₁₅₇	—	2	2	no	—
74	C₈×D₁₅₇	—	2	2	no	—
75	C₈×D₁₅₇	—	2	2	no	—
76	C₄×Dic₁₅₇	—	2	1	no	—
77	C₄×Dic₁₅₇	C₂³	1	1	yes	characteristic
78	#41×C₁₅₇	C₂³	1	2	yes	—
79	Q₁₆×C₁₅₇	C₂×C₄	1	1	yes	characteristic, complemented
80	SD₁₆×C₁₅₇	C₂×C₄	1	2	yes	complemented
81	D₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic, complemented
82	M₁₆×C₁₅₇	—	2	2	no	—
83	M₁₆×C₁₅₇	C₂³	1	2	yes	—
84	C₂×C₈×C₁₅₇	—	2	2	no	—
85	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
87	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
88	C₂²×D₁₅₇	—	2	2	no	—
89	C₂²×D₁₅₇	#41	1	1	yes	characteristic
90	Dic₃₁₄	—	2	2	no	—
91	Dic₃₁₄	C₂²×C₄	1	2	yes	—
92	D₆₂₈	—	2	2	no	—
93	D₆₂₈	C₂²×C₄	1	2	yes	—
94	C₂²⋊D₁₅₇	—	2	2	no	—
95	C₂²⋊D₁₅₇	—	2	2	no	—
96	C₂²⋊D₁₅₇	—	2	2	no	—
97	C₂×Dic₁₅₇	—	2	2	no	—
98	C₂×Dic₁₅₇	—	2	1	no	—
99	C₂×Dic₁₅₇	#41	1	1	yes	characteristic
100	C₄×D₁₅₇	—	2	2	no	—
101	C₄×D₁₅₇	—	2	2	no	—
102	C₄×D₁₅₇	—	2	2	no	—
103	C₄×D₁₅₇	—	4	2	no	—
104	C₄×D₁₅₇	C₂²×C₄	1	2	yes	—
105	C₄×D₁₅₇	C₂×D₄	1	2	yes	—
106	#11929434	—	2	2	no	—
107	#11929434	C₂²×C₄	1	2	yes	—
108	#11929434	C₂×D₄	1	2	yes	—
109	Q₈×C₁₅₇	C₂²×C₄	1	2	yes	—
110	D₄×C₁₅₇	—	2	2	no	—
111	D₄×C₁₅₇	C₂²×C₄	1	2	yes	—
112	C₂×C₄×C₁₅₇	—	2	1	no	—
113	C₂×C₄×C₁₅₇	—	2	2	no	—
114	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
115	C₈×C₁₅₇	—	2	2	no	—
116	C₈×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
117	C₈×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
118	C₈×C₁₅₇	C₂×D₄	1	1	yes	characteristic
119	C₈×C₁₅₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₁₅₇	—	2	2	no	—
121	C₂²×C₁₅₇	C₂×#41	1	1	yes	characteristic
122	C₂×D₁₅₇	—	2	2	no	—
123	C₂×D₁₅₇	—	4	1	no	—
124	C₂×D₁₅₇	—	8	2	no	—
125	C₂×D₁₅₇	C₄×D₄	1	2	yes	—
126	C₄×C₁₅₇	—	2	2	no	—
127	C₄×C₁₅₇	—	2	2	no	—
128	C₄×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
130	Dic₁₅₇	—	2	2	no	—
131	Dic₁₅₇	—	4	1	no	—
132	Dic₁₅₇	C₄×D₄	1	2	yes	—
133	C₂×C₁₅₇	—	2	1	no	—
134	C₂×C₁₅₇	—	4	2	no	—
135	C₂×C₁₅₇	C₂×C₄×D₄	1	1	yes	characteristic
136	D₁₅₇	—	2	2	no	—
137	D₁₅₇	—	4	2	no	—
138	C₁₅₇	#2956	1	1	yes	characteristic, complemented, Sylow
139	#2956	—	157	1	no	maximal, Sylow
140	#575	—	157	1	no	—
141	#568	—	157	2	no	—
142	#444	—	157	1	no	—
143	#444	—	157	2	no	—
144	#444	—	157	1	no	—
145	#443	—	157	2	no	—
146	#443	—	157	1	no	—
147	#443	D₁₅₇	1	1	yes	characteristic, complemented
148	#430	—	157	1	no	—
149	C₂×#104	—	157	2	no	—
150	C₄×M₁₆	—	157	1	no	—
151	C₂×#41	—	157	2	no	—
152	#137	—	157	2	no	—
153	#137	—	157	2	no	—
154	AGL(1,ℤ/8)	—	157	2	no	—
155	AGL(1,ℤ/8)	—	157	2	no	—
156	#135	—	157	2	no	—
157	#135	—	157	1	no	—
158	#135	C₂×D₁₅₇	1	1	yes	characteristic
159	#131	—	314	2	no	—
160	#131	—	314	2	no	—
161	#131	—	157	2	no	—
162	#131	—	157	2	no	—
163	#131	—	157	2	no	—
164	#131	C₂×D₁₅₇	1	2	yes	complemented
165	C₂×M₁₆	—	314	2	no	—
166	C₂×M₁₆	—	314	2	no	—
167	C₂×M₁₆	—	157	2	no	—
168	C₂×M₁₆	—	157	1	no	—
169	C₂×M₁₆	—	157	1	no	—
170	C₂×C₄²	—	157	1	no	—
171	#108	—	157	2	no	—
172	#108	—	157	1	no	—
173	#108	C₂×D₁₅₇	1	1	yes	characteristic, complemented
174	#104	—	157	2	no	—
175	#104	—	157	2	no	—
176	#104	—	157	2	no	—
177	#104	C₂×D₁₅₇	1	2	yes	complemented
178	#97	—	157	1	no	—
179	#97	—	157	1	no	—
180	C₄×C₈	—	157	1	no	—
181	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic, complemented
182	#41	—	314	2	no	—
183	#41	—	314	2	no	—
184	#41	—	157	2	no	—
185	#41	—	157	2	no	—
186	#41	—	157	2	no	—
187	#41	C₂²×D₁₅₇	1	2	yes	—
188	C₂×Q₈	—	157	2	no	—
189	C₂×D₄	—	314	2	no	—
190	C₂×D₄	—	157	2	no	—
191	C₂²×C₄	—	314	2	no	—
192	C₂²×C₄	—	314	1	no	—
193	C₂²×C₄	—	157	1	no	—
194	Q₁₆	—	157	2	no	—
195	Q₁₆	—	157	1	no	—
196	Q₁₆	C₄×D₁₅₇	1	1	yes	characteristic, complemented
197	SD₁₆	—	157	2	no	—
198	SD₁₆	—	157	2	no	—
199	SD₁₆	—	157	2	no	—
200	SD₁₆	C₄×D₁₅₇	1	2	yes	complemented
201	D₈	—	157	2	no	—
202	D₈	—	157	1	no	—
203	D₈	C₄×D₁₅₇	1	1	yes	characteristic, complemented
204	M₁₆	—	2	2	no	—
205	M₁₆	—	314	2	no	—
206	M₁₆	—	314	2	no	—
207	M₁₆	—	314	2	no	—
208	M₁₆	—	314	2	no	—
209	M₁₆	—	157	2	no	—
210	M₁₆	—	157	2	no	—
211	M₁₆	—	157	2	no	—
212	M₁₆	—	157	2	no	—
213	M₁₆	—	157	2	no	—
214	M₁₆	C₂²×D₁₅₇	1	2	yes	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	314	2	no	—
217	C₂×C₈	—	314	2	no	—
218	C₂×C₈	—	314	2	no	—
219	C₂×C₈	—	314	2	no	—
220	C₂×C₈	—	157	1	no	—
221	C₂×C₈	—	157	1	no	—
222	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
224	C₄²	—	314	1	no	—
225	C₄²	—	157	1	no	—
226	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
227	C₂³	—	314	2	no	—
228	C₂³	—	157	1	no	—
229	Q₈	—	314	2	no	—
230	Q₈	—	157	2	no	—
231	Q₈	C₂×C₄×D₁₅₇	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	314	2	no	—
234	D₄	—	314	2	no	—
235	D₄	—	314	2	no	—
236	D₄	—	314	2	no	—
237	D₄	—	157	2	no	—
238	D₄	C₂×C₄×D₁₅₇	1	2	yes	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	314	2	no	—
242	C₂×C₄	—	314	2	no	—
243	C₂×C₄	—	314	2	no	—
244	C₂×C₄	—	314	2	no	—
245	C₂×C₄	—	314	1	no	—
246	C₂×C₄	—	628	2	no	—
247	C₂×C₄	—	157	1	no	—
248	C₂×C₄	—	157	2	no	—
249	C₂×C₄	—	157	2	no	—
250	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	314	2	no	—
253	C₈	—	157	2	no	—
254	C₈	—	157	2	no	—
255	C₈	C₂×C₄×D₁₅₇	1	1	yes	characteristic
256	C₈	C₂×C₄×D₁₅₇	1	1	yes	characteristic
257	C₈	D₄×D₁₅₇	1	1	yes	characteristic
258	C₈	D₄×D₁₅₇	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	314	2	no	—
261	C₂²	—	628	1	no	—
262	C₂²	—	157	2	no	—
263	C₂²	—	1256	2	no	—
264	C₂²	#41×D₁₅₇	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	314	2	no	—
268	C₄	—	628	1	no	—
269	C₄	—	157	2	no	—
270	C₄	C₂²×C₄×D₁₅₇	1	1	yes	characteristic
271	C₄	C₂×D₄×D₁₅₇	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	—	314	2	no	—
275	C₂	—	628	2	no	—
276	C₂	C₄×D₄×D₁₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₅₇
commutator	C₄×C₁₅₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	#443×C₁₅₇	C₂
socle	C₂×C₁₅₇	C₂×C₄×D₄
prime radical	#23758070	C₂
cosocle	C₂×C₄×C₁₅₇	C₂⁴
hypercenter	#443	D₁₅₇
hypocenter	C₁₅₇	#2956
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #23758038	C₂	semidirect
3. #23757395	C₂	semidirect
4. #23757394	C₂	semidirect
5. #23757393	C₂	semidirect
6. #23757391	C₂	semidirect
7. #23757387	C₂	semidirect
8. #23757320	C₂	semidirect
9. #104×D₁₅₇	C₂	semidirect
10. C₄×#14578937	C₂	semidirect
11. #443×C₁₅₇	C₂	semidirect
13. #19097803	C₄	semidirect
14. #19097800	C₄	semidirect
15. #19097798	C₄	semidirect
16. #19097795	C₄	semidirect
20. #19097779	C₂²	semidirect
21. #19097778	C₂²	semidirect
23. #19097777	C₂²	semidirect
30. #19097707	C₂²	semidirect
31. #19097706	C₂²	semidirect
32. #19097699	C₂²	semidirect
33. #19097698	C₂²	semidirect
34. #19097697	C₂²	semidirect
35. #19097681	C₂²	semidirect
36. #19097681	C₂²	semidirect
37. C₄×#11929434	C₂²	semidirect
39. #131×C₁₅₇	C₂²	semidirect
40. #108×C₁₅₇	C₂²	semidirect
41. #104×C₁₅₇	C₂²	semidirect
42. C₄×C₈×C₁₅₇	C₂²	semidirect
54. Dic₆₂₈	C₂×C₄	semidirect
55. #14578944	C₂×C₄	semidirect
56. #14578943	C₂×C₄	semidirect
57. #14578942	C₂×C₄	semidirect
58. #14578941	C₂×C₄	semidirect
59. D₁₂₅₆	C₂×C₄	semidirect
60. #14578939	C₂×C₄	semidirect
79. Q₁₆×C₁₅₇	C₂×C₄	semidirect
80. SD₁₆×C₁₅₇	C₂×C₄	semidirect
81. D₈×C₁₅₇	C₂×C₄	semidirect
138. C₁₅₇	#2956	semidirect
147. #443	D₁₅₇	semidirect
164. #131	C₂×D₁₅₇	semidirect
173. #108	C₂×D₁₅₇	semidirect
177. #104	C₂×D₁₅₇	semidirect
181. C₄×C₈	C₂×D₁₅₇	semidirect
196. Q₁₆	C₄×D₁₅₇	semidirect
200. SD₁₆	C₄×D₁₅₇	semidirect
203. D₈	C₄×D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23758070	2
#23758038	2
#23757395	2
#23757394	2
#23757393	2
#23757391	2
#23757387	2
#23757320	2
#104×D₁₅₇	2
C₄×#14578937	2
#443×C₁₅₇	2
#2956	157