Subgroups — #34098170

Order 20096.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757642	C₂	1	1	yes	maximal, characteristic
2	#23757051	C₂	1	1	yes	maximal, characteristic
3	#347×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#19097771	C₄	1	1	yes	characteristic
5	#19097763	C₄	1	1	yes	characteristic
6	#19097711	—	2	1	no	—
7	#19097696	—	2	1	no	—
8	#19097696	—	2	1	no	—
9	M₃₂×C₁₅₇	—	2	1	no	—
10	#97×C₁₅₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₁₄	—	2	1	no	—
12	C₄⋊Dic₁₅₇	—	2	1	no	—
13	C₄⋊Dic₁₅₇	—	4	1	no	—
14	#14578933	—	4	1	no	—
15	#14578929	—	2	2	no	—
16	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
17	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
18	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₁₅₇	—	2	2	no	—
20	Dic₃₁₄	—	4	1	no	—
21	Dic₃₁₄	—	4	1	no	—
22	C₂×Dic₁₅₇	—	4	1	no	—
23	C₂×Dic₁₅₇	—	4	1	no	—
24	C₂×C₄×C₁₅₇	—	2	1	no	—
25	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₅₇	—	2	1	no	—
27	C₈×C₁₅₇	—	2	1	no	—
28	C₂²×C₁₅₇	#100	1	1	yes	characteristic
29	C₄×C₁₅₇	—	4	1	no	—
30	C₄×C₁₅₇	#104	1	1	yes	characteristic
31	C₄×C₁₅₇	#104	1	1	yes	characteristic
32	Dic₁₅₇	—	8	1	no	—
33	Dic₁₅₇	—	8	1	no	—
34	C₂×C₁₅₇	—	2	1	no	—
35	C₂×C₁₅₇	#329	1	1	yes	characteristic
36	C₁₅₇	#1356	1	1	yes	characteristic, complemented, Sylow
37	#1356	—	157	1	no	maximal, Sylow
38	#491	—	157	1	no	—
39	#347	—	157	1	no	—
40	#347	D₁₅₇	1	1	yes	characteristic
41	C₄⋊Q₈	—	157	1	no	—
42	#125	—	157	1	no	—
43	M₃₂	—	2	1	no	—
44	M₃₂	—	314	1	no	—
45	#103	—	314	1	no	—
46	#103	—	314	1	no	—
47	#97	C₂×D₁₅₇	1	1	yes	characteristic
48	C₂×Q₈	—	314	1	no	—
49	C₂×C₈	C₂²⋊D₁₅₇	1	1	yes	characteristic
50	C₂×C₈	D₆₂₈	1	1	yes	characteristic
51	C₄⋊C₄	—	314	1	no	—
52	C₄⋊C₄	—	628	1	no	—
53	C₄⋊C₄	—	628	1	no	—
54	C₄²	C₄×D₁₅₇	1	1	yes	characteristic
55	C₁₆	—	2	2	no	—
56	C₁₆	—	314	2	no	—
57	Q₈	—	628	1	no	—
58	Q₈	—	628	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	628	1	no	—
61	C₂×C₄	—	628	1	no	—
62	C₂×C₄	#14578931	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	2	1	no	—
65	C₂²	#19097687	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	—	1256	1	no	—
68	C₄	—	1256	1	no	—
69	C₄	#19097697	1	1	yes	characteristic
70	C₄	#19097699	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#23757012	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23757012
commutator	C₄²×C₁₅₇	C₂×C₄
Frattini	#97	C₂×D₁₅₇
Fitting	#347×C₁₅₇	C₂
socle	C₂×C₁₅₇	#329
prime radical	C₂²×C₁₅₇	#100
cosocle	#97×C₁₅₇	C₂²
hypercenter	#347	D₁₅₇
hypocenter	C₁₅₇	#1356
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
36. C₁₅₇	#1356	semidirect

Maximal subgroups

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

subgroup	index
#23757642	2
#23757051	2
#347×C₁₅₇	2
#1356	157