Subgroups — #34098124

Order 20096.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757055	C₂	1	1	yes	maximal, characteristic
2	#23757030	C₂	1	1	yes	maximal, characteristic
3	#470×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#19097730	—	2	1	no	—
5	#19097711	—	2	1	no	—
6	C₂×Q₁₆×C₁₅₇	C₄	1	1	yes	characteristic, complemented
7	C₂×M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
8	C₂²⋊Q₈×C₁₅₇	—	2	1	no	—
9	#106×C₁₅₇	C₄	1	1	yes	characteristic, complemented
10	#103×C₁₅₇	—	2	1	no	—
11	C₄⋊Dic₁₅₇	—	4	1	no	—
12	C₂²⋊Dic₁₅₇	—	4	1	no	—
13	C₄×Dic₁₅₇	—	2	2	no	—
14	#14578929	—	4	1	no	—
15	C₂×Q₈×C₁₅₇	—	2	1	no	—
16	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
17	Q₁₆×C₁₅₇	—	4	1	no	—
18	M₁₆×C₁₅₇	—	4	1	no	—
19	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
21	C₄⋊C₄×C₁₅₇	—	2	1	no	—
22	C₄⋊C₄×C₁₅₇	—	4	1	no	—
23	C₂²⋊C₄×C₁₅₇	—	4	1	no	—
24	C₂×Dic₁₅₇	—	4	2	no	—
25	C₂³×C₁₅₇	D₈	1	1	yes	characteristic
26	Q₈×C₁₅₇	—	4	1	no	—
27	Q₈×C₁₅₇	—	4	1	no	—
28	C₂×C₄×C₁₅₇	—	2	1	no	—
29	C₂×C₄×C₁₅₇	—	2	1	no	—
30	C₂×C₄×C₁₅₇	—	4	1	no	—
31	C₂×C₄×C₁₅₇	—	4	1	no	—
32	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₅₇	SD₁₆	1	1	yes	characteristic
34	C₈×C₁₅₇	—	2	1	no	—
35	C₈×C₁₅₇	—	2	1	no	—
36	C₂²×C₁₅₇	—	2	1	no	—
37	C₂²×C₁₅₇	—	4	1	no	—
38	C₂²×C₁₅₇	#102	1	1	yes	characteristic
39	C₄×C₁₅₇	—	2	1	no	—
40	C₄×C₁₅₇	—	8	1	no	—
41	C₄×C₁₅₇	—	8	1	no	—
42	C₄×C₁₅₇	#99	1	1	yes	characteristic
43	C₄×C₁₅₇	#104	1	1	yes	characteristic
44	Dic₁₅₇	—	8	2	no	—
45	C₂×C₁₅₇	—	2	1	no	—
46	C₂×C₁₅₇	—	4	1	no	—
47	C₂×C₁₅₇	#327	1	1	yes	characteristic
48	C₁₅₇	#1340	1	1	yes	characteristic, complemented, Sylow
49	#1340	—	157	1	no	maximal, Sylow
50	#470	D₁₅₇	1	1	yes	characteristic
51	#349	—	157	1	no	—
52	H(3,ℤ/4)	—	157	1	no	—
53	C₂×Q₁₆	Dic₁₅₇	1	1	yes	characteristic, complemented
54	C₂×M₁₆	C₂×D₁₅₇	1	1	yes	characteristic
55	C₂²⋊Q₈	—	2	1	no	—
56	#117	—	314	1	no	—
57	M₃₂	—	314	1	no	—
58	#106	Dic₁₅₇	1	1	yes	characteristic, complemented
59	#103	—	2	1	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂²×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
62	Q₁₆	—	4	1	no	—
63	M₁₆	—	4	1	no	—
64	C₂×C₈	C₂×Dic₁₅₇	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₁₅₇	1	1	yes	characteristic
66	C₄⋊C₄	—	2	1	no	—
67	C₄⋊C₄	—	4	1	no	—
68	C₄⋊C₄	—	628	1	no	—
69	C₂²⋊C₄	—	4	1	no	—
70	C₂²⋊C₄	—	628	1	no	—
71	C₄²	—	314	2	no	—
72	C₁₆	—	628	1	no	—
73	C₂³	#14578939	1	1	yes	characteristic
74	Q₈	—	4	1	no	—
75	Q₈	—	4	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	4	1	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	—	628	2	no	—
81	C₂×C₄	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
82	C₂×C₄	#14578942	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	2	1	no	—
85	C₂²	—	2	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	#19097692	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	8	1	no	—
90	C₄	—	8	1	no	—
91	C₄	—	1256	2	no	—
92	C₄	#19097686	1	1	yes	characteristic
93	C₄	#19097698	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	4	1	no	—
96	C₂	#23757007	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23757007
commutator	C₂×C₈×C₁₅₇	C₂×C₄
Frattini	C₂×M₁₆	C₂×D₁₅₇
Fitting	#470×C₁₅₇	C₂
socle	C₂×C₁₅₇	#327
prime radical	C₂³×C₁₅₇	D₈
cosocle	C₂×M₁₆×C₁₅₇	C₂²
hypercenter	#470	D₁₅₇
hypocenter	C₁₅₇	#1340
LCS	G ▷ C₂×C₈×C₁₅₇ ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #470

Complements

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

N	H ≅ G/N	type
6. C₂×Q₁₆×C₁₅₇	C₄	semidirect
9. #106×C₁₅₇	C₄	semidirect
48. C₁₅₇	#1340	semidirect
53. C₂×Q₁₆	Dic₁₅₇	semidirect
58. #106	Dic₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23757055	2
#23757030	2
#470×C₁₅₇	2
#1340	157