Subgroups — #34098310

Order 20096.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757627	C₂	1	1	yes	maximal, characteristic
2	#23757070	C₂	1	1	yes	maximal, characteristic
3	#354×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#19097763	C₄	1	1	yes	characteristic, complemented
5	#19097695	—	2	1	no	—
6	#19097692	—	2	1	no	—
7	#19097690	—	2	1	no	—
8	C₄×#11929434	C₄	1	1	yes	characteristic, complemented
9	#124×C₁₅₇	C₂²	1	1	yes	characteristic
10	#99×C₁₅₇	—	2	1	no	—
11	#14578938	—	4	1	no	—
12	C₂×#11929434	—	2	1	no	—
13	C₄⋊Dic₁₅₇	—	2	1	no	—
14	#14578933	—	4	1	no	—
15	C₂×Q₈×C₁₅₇	D₄	1	1	yes	characteristic
16	C₂×D₄×C₁₅₇	D₄	1	1	yes	characteristic
17	C₂²⋊C₄×C₁₅₇	—	4	1	no	—
18	C₂²⋊C₄×C₁₅₇	—	4	1	no	—
19	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₅₇	—	4	1	no	—
21	#11929434	—	4	1	no	—
22	#11929434	—	8	1	no	—
23	C₂³×C₁₅₇	—	2	1	no	—
24	Q₈×C₁₅₇	—	4	1	no	—
25	D₄×C₁₅₇	—	4	1	no	—
26	C₂×C₄×C₁₅₇	—	2	1	no	—
27	C₂×C₄×C₁₅₇	—	2	1	no	—
28	C₂×C₄×C₁₅₇	—	8	1	no	—
29	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₁₅₇	—	4	1	no	—
31	C₂²×C₁₅₇	—	8	1	no	—
32	C₂²×C₁₅₇	#99	1	1	yes	characteristic
33	C₄×C₁₅₇	—	2	1	no	—
34	C₄×C₁₅₇	—	4	1	no	—
35	C₄×C₁₅₇	—	4	1	no	—
36	C₄×C₁₅₇	—	16	1	no	—
37	Dic₁₅₇	—	8	1	no	—
38	C₂×C₁₅₇	—	2	1	no	—
39	C₂×C₁₅₇	—	8	1	no	—
40	C₂×C₁₅₇	#354	1	1	yes	characteristic
41	C₁₅₇	#1414	1	1	yes	characteristic, complemented, Sylow
42	#1414	—	157	1	no	maximal, Sylow
43	#488	—	157	1	no	—
44	#355	—	157	1	no	—
45	#354	D₁₅₇	1	1	yes	characteristic
46	#125	—	157	1	no	—
47	#124	C₂×D₁₅₇	1	1	yes	characteristic
48	#103	—	314	1	no	—
49	#102	—	314	1	no	—
50	#101	—	314	1	no	—
51	#99	—	2	1	no	—
52	C₄×C₈	—	157	1	no	—
53	C₂×Q₈	D₆₂₈	1	1	yes	characteristic
54	C₂×D₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
55	M₁₆	—	628	1	no	—
56	C₂×C₈	—	314	1	no	—
57	C₄⋊C₄	—	314	1	no	—
58	C₄⋊C₄	—	628	1	no	—
59	C₂²⋊C₄	—	4	1	no	—
60	C₂²⋊C₄	—	4	1	no	—
61	C₄²	C₄×D₁₅₇	1	1	yes	characteristic
62	C₂³	—	2	1	no	—
63	Q₈	—	4	1	no	—
64	D₄	—	4	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	8	1	no	—
68	C₂×C₄	—	628	1	no	—
69	C₂×C₄	#14578931	1	1	yes	characteristic
70	C₈	—	628	1	no	—
71	C₈	—	1256	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	8	1	no	—
74	C₂²	#19097685	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	16	1	no	—
79	C₄	—	1256	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	#23757067	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23757067
commutator	C₄²×C₁₅₇	C₂×C₄
Frattini	#124	C₂×D₁₅₇
Fitting	#354×C₁₅₇	C₂
socle	C₂×C₁₅₇	#354
prime radical	C₂×D₄×C₁₅₇	D₄
cosocle	#124×C₁₅₇	C₂²
hypercenter	#354	D₁₅₇
hypocenter	C₁₅₇	#1414
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #354

Complements

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

N	H ≅ G/N	type
4. #19097763	C₄	semidirect
8. C₄×#11929434	C₄	semidirect
41. C₁₅₇	#1414	semidirect

Maximal subgroups

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

subgroup	index
#23757627	2
#23757070	2
#354×C₁₅₇	2
#1414	157