Subgroups — #34098006

Order 20096.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757330	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11929434	—	2	2	no	—
4	C₂²⋊#11929434	—	2	2	no	—
5	#19097682	C₄	1	4	yes	—
6	C₂²×C₈×C₁₅₇	C₄	1	2	yes	—
7	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
8	C₄⋊C₈×C₁₅₇	—	2	2	no	—
9	C₂×#11929434	—	2	4	no	—
10	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
12	C₂²×C₄×C₁₅₇	Q₈	1	1	yes	characteristic
13	C₂×C₈×C₁₅₇	—	2	2	no	—
14	C₂×C₈×C₁₅₇	—	4	2	no	—
15	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
16	C₄²×C₁₅₇	D₄	1	2	yes	—
17	#11929434	—	4	4	no	—
18	C₂³×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₅₇	—	2	2	no	—
20	C₂×C₄×C₁₅₇	—	2	1	no	—
21	C₂×C₄×C₁₅₇	—	2	1	no	—
22	C₂×C₄×C₁₅₇	—	4	1	no	—
23	C₂×C₄×C₁₅₇	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
25	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
27	C₈×C₁₅₇	—	4	2	no	—
28	C₂²×C₁₅₇	—	2	1	no	—
29	C₂²×C₁₅₇	—	2	1	no	—
30	C₂²×C₁₅₇	#95	1	1	yes	characteristic
31	C₂²×C₁₅₇	#108	1	2	yes	—
32	C₄×C₁₅₇	—	2	2	no	—
33	C₄×C₁₅₇	—	2	2	no	—
34	C₄×C₁₅₇	—	2	1	no	—
35	C₄×C₁₅₇	#100	1	1	yes	characteristic
36	C₄×C₁₅₇	#101	1	1	yes	characteristic
37	C₂×C₁₅₇	—	2	2	no	—
38	C₂×C₁₅₇	H(3,ℤ/4)	1	1	yes	characteristic
39	C₂×C₁₅₇	#341	1	1	yes	characteristic
40	C₂×C₁₅₇	#343	1	1	yes	characteristic
41	C₁₅₇	#1290	1	1	yes	characteristic, complemented, Sylow
42	#1290	—	157	1	no	maximal, Sylow
43	#432	—	157	2	no	—
44	C₂×C₄⋊C₈	D₁₅₇	1	1	yes	characteristic
45	C₂²×C₈	Dic₁₅₇	1	2	yes	—
46	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
47	C₄⋊C₈	—	2	2	no	—
48	C₄⋊C₈	—	314	2	no	—
49	C₂²⋊C₈	—	314	2	no	—
50	#97	—	157	4	no	—
51	C₂²×C₄	C₂×Dic₁₅₇	1	1	yes	characteristic
52	C₂²×C₄	D₆₂₈	1	1	yes	characteristic
53	C₂²×C₄	Dic₃₁₄	1	1	yes	characteristic
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	4	2	no	—
56	C₂×C₈	—	314	4	no	—
57	C₄²	C₄×D₁₅₇	1	2	yes	—
58	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
59	C₂³	C₄⋊Dic₁₅₇	1	1	yes	characteristic
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	C₄×Dic₁₅₇	1	1	yes	characteristic
65	C₂×C₄	#14578931	1	2	yes	—
66	C₂×C₄	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
67	C₂×C₄	#14578933	1	2	yes	—
68	C₈	—	4	2	no	—
69	C₈	—	628	4	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	2	1	no	—
72	C₂²	#19097678	1	1	yes	characteristic
73	C₂²	#19097707	1	2	yes	—
74	C₄	—	2	2	no	—
75	C₄	—	2	2	no	—
76	C₄	—	2	1	no	—
77	C₄	#19097688	1	1	yes	characteristic
78	C₄	#19097690	1	1	yes	characteristic
79	C₂	—	2	2	no	—
80	C₂	#23757030	1	1	yes	characteristic
81	C₂	#23757038	1	1	yes	characteristic
82	C₂	#23757042	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19097678
commutator	C₂×C₄×C₁₅₇	C₄²
Frattini	C₂×C₄²	C₂×D₁₅₇
Fitting	C₂×C₄⋊C₈×C₁₅₇	C₂
socle	C₂²×C₁₅₇	#95
prime radical	C₂³×C₁₅₇	C₄⋊C₄
cosocle	C₂×C₄²×C₁₅₇	C₂²
hypercenter	C₂×C₄⋊C₈	D₁₅₇
hypocenter	C₁₅₇	#1290
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
41. C₁₅₇	#1290	semidirect

Maximal subgroups

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

subgroup	index
#23757330	2
C₂×C₄⋊C₈×C₁₅₇	2
#1290	157