Subgroups — #34098130

Order 20096.

100 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	#23757031	C₂	1	1	yes	maximal, characteristic
3	#470×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14578938	—	2	1	no	—
5	#19097711	—	2	1	no	—
6	C₂×Q₁₆×C₁₅₇	C₄	1	1	yes	characteristic
7	C₂×M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
8	C₂²⋊Q₈×C₁₅₇	—	2	1	no	—
9	#106×C₁₅₇	C₄	1	1	yes	characteristic
10	#103×C₁₅₇	—	2	1	no	—
11	#14578938	—	4	1	no	—
12	#14578938	—	4	1	no	—
13	C₂×#11929434	—	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	#11929434	—	4	2	no	—
25	C₂³×C₁₅₇	SD₁₆	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	D₈	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	C₂×C₁₅₇	—	2	1	no	—
45	C₂×C₁₅₇	—	4	1	no	—
46	C₂×C₁₅₇	#327	1	1	yes	characteristic
47	C₁₅₇	#1342	1	1	yes	characteristic, complemented, Sylow
48	#1342	—	157	1	no	maximal, Sylow
49	#470	D₁₅₇	1	1	yes	characteristic
50	#349	—	157	1	no	—
51	#338	—	157	1	no	—
52	C₂×Q₁₆	Dic₁₅₇	1	1	yes	characteristic
53	C₂×M₁₆	—	314	1	no	—
54	C₂×M₁₆	C₂×D₁₅₇	1	1	yes	characteristic
55	C₂²⋊Q₈	—	2	1	no	—
56	M₃₂	—	314	1	no	—
57	#106	Dic₁₅₇	1	1	yes	characteristic
58	#103	—	2	1	no	—
59	C₂×Q₈	—	2	1	no	—
60	C₂²×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
61	Q₁₆	—	4	1	no	—
62	M₁₆	—	4	1	no	—
63	M₁₆	—	628	1	no	—
64	M₁₆	—	628	1	no	—
65	C₂×C₈	—	314	2	no	—
66	C₂×C₈	C₂×Dic₁₅₇	1	1	yes	characteristic
67	C₂×C₈	C₂²⋊D₁₅₇	1	1	yes	characteristic
68	C₄⋊C₄	—	2	1	no	—
69	C₄⋊C₄	—	4	1	no	—
70	C₂²⋊C₄	—	4	1	no	—
71	C₁₆	—	628	1	no	—
72	C₂³	#14578942	1	1	yes	characteristic
73	Q₈	—	4	1	no	—
74	Q₈	—	4	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	4	1	no	—
79	C₂×C₄	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
80	C₂×C₄	#14578939	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	—	2	1	no	—
83	C₈	—	628	2	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	#19097692	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	8	1	no	—
89	C₄	—	8	1	no	—
90	C₄	#19097686	1	1	yes	characteristic
91	C₄	#19097698	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	4	1	no	—
94	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	SD₁₆
cosocle	C₂×M₁₆×C₁₅₇	C₂²
hypercenter	#470	D₁₅₇
hypocenter	C₁₅₇	#1342
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
47. C₁₅₇	#1342	semidirect

Maximal subgroups

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

subgroup	index
#23757055	2
#23757031	2
#470×C₁₅₇	2
#1342	157