Subgroups — C₂×#23758206 #34116558

Order 20096.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23758206	C₂	1	4	yes	maximal, direct product factor
2	C₂×#19097821	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₆×D₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#19097821	C₂²	1	4	yes	complemented
5	C₂×C₈×D₁₅₇	C₄	1	1	yes	characteristic
6	C₂×#14578929	C₂²	1	1	yes	characteristic
7	C₁₆×D₁₅₇	C₄	1	2	yes	—
8	C₁₆×D₁₅₇	C₂²	1	2	yes	—
9	C₂×C₁₆×C₁₅₇	C₄	1	1	yes	characteristic
10	C₂×C₄×D₁₅₇	C₈	1	1	yes	characteristic
11	C₂×#11929434	C₈	1	1	yes	characteristic
12	C₈×D₁₅₇	C₈	1	2	yes	—
13	C₈×D₁₅₇	C₂×C₄	1	2	yes	—
14	#14578929	C₂×C₄	1	1	yes	characteristic
15	#14578929	C₂³	1	1	yes	characteristic
16	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₁₅₇	C₂×C₄	1	2	yes	—
18	C₂²×D₁₅₇	C₁₆	1	1	yes	characteristic
19	C₂×Dic₁₅₇	C₁₆	1	1	yes	characteristic
20	C₄×D₁₅₇	C₁₆	1	2	yes	—
21	C₄×D₁₅₇	C₂×C₈	1	2	yes	—
22	#11929434	C₂×C₈	1	2	yes	—
23	C₂×C₄×C₁₅₇	C₂×C₈	1	1	yes	characteristic
24	C₈×C₁₅₇	C₂×C₈	1	1	yes	characteristic
25	C₈×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₁₅₇	C₂×C₁₆	1	1	yes	characteristic
27	C₂×D₁₅₇	—	2	2	no	—
28	C₂×D₁₅₇	C₂×C₁₆	1	2	yes	—
29	C₄×C₁₅₇	C₂×C₁₆	1	1	yes	characteristic
30	C₄×C₁₅₇	C₂²×C₈	1	1	yes	characteristic
31	Dic₁₅₇	C₂×C₁₆	1	2	yes	—
32	C₂×C₁₅₇	M₆₄	1	2	yes	complemented
33	C₂×C₁₅₇	C₂²×C₁₆	1	1	yes	characteristic
34	D₁₅₇	—	2	2	no	—
35	C₁₅₇	C₂×M₆₄	1	1	yes	characteristic, complemented, Sylow
36	C₂×M₆₄	—	157	1	no	maximal, Sylow
37	C₂²×C₁₆	—	157	1	no	—
38	M₆₄	—	157	4	no	—
39	C₂×C₃₂	—	157	2	no	—
40	C₂²×C₈	—	157	1	no	—
41	C₂×C₁₆	—	157	1	no	—
42	C₂×C₁₆	—	157	2	no	—
43	C₂×C₁₆	—	157	2	no	—
44	C₂×C₁₆	#10597845	1	1	yes	characteristic
45	C₃₂	—	157	4	no	—
46	C₂²×C₄	—	157	1	no	—
47	C₂×C₈	—	157	1	no	—
48	C₂×C₈	—	157	2	no	—
49	C₂×C₈	—	157	2	no	—
50	C₂×C₈	C₂×#10597845	1	1	yes	characteristic
51	C₁₆	—	157	1	no	—
52	C₁₆	—	157	1	no	—
53	C₁₆	C₂×#10597845	1	2	yes	—
54	C₂³	—	157	1	no	—
55	C₂×C₄	—	157	1	no	—
56	C₂×C₄	—	157	2	no	—
57	C₂×C₄	—	157	2	no	—
58	C₂×C₄	D₁₅₇⋊C₈	1	1	yes	characteristic
59	C₈	—	157	2	no	—
60	C₈	D₁₅₇⋊C₈	1	1	yes	characteristic
61	C₈	C₂²×#10597845	1	1	yes	characteristic
62	C₂²	—	314	2	no	—
63	C₂²	—	157	2	no	—
64	C₂²	D₁₅₇⋊C₁₆	1	1	yes	characteristic
65	C₄	—	157	2	no	—
66	C₄	D₁₅₇⋊C₁₆	1	1	yes	characteristic
67	C₄	C₂×D₁₅₇⋊C₈	1	1	yes	characteristic
68	C₂	—	314	2	no	—
69	C₂	#23758206	1	2	yes	direct product factor
70	C₂	C₂×D₁₅₇⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×#10597845
commutator	C₂×C₁₅₇	C₂²×C₁₆
Frattini	C₈	C₂²×#10597845
Fitting	C₂×C₁₆×C₁₅₇	C₄
socle	C₂²×C₁₅₇	C₂×C₁₆
prime radical	C₂²×D₁₅₇	C₁₆
cosocle	#14578929	C₂³
hypercenter	C₂×C₁₆	#10597845
hypocenter	C₁₅₇	C₂×M₆₄
LCS	G ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. #23758206	C₂	direct (N × H)
2. C₂×#19097821	C₂	semidirect
4. #19097821	C₂²	semidirect
32. C₂×C₁₅₇	M₆₄	semidirect
35. C₁₅₇	C₂×M₆₄	semidirect
69. C₂	#23758206	direct (N × H)

Maximal subgroups

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

subgroup	index
#23758206	2
C₂×#19097821	2
C₂×C₁₆×D₁₅₇	2
C₂×M₆₄	157

Subgroups — C2×#23758206 #34116558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23758206 #34116558