Subgroups — #34115253

Order 20096.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23758205	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19097821	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757690	C₂	1	1	yes	maximal, characteristic
4	#19097821	—	2	1	no	—
5	#19097821	—	2	1	no	—
6	#19097711	—	2	1	no	—
7	#19097711	C₄	1	1	yes	characteristic
8	C₂×#14578929	—	2	1	no	—
9	C₂×#14578929	C₂²	1	1	yes	characteristic
10	#131×C₁₅₇	C₄	1	1	yes	characteristic
11	#14578929	—	2	1	no	—
12	#14578929	D₄	1	1	yes	characteristic
13	#14578929	D₄	1	1	yes	characteristic
14	#41×C₁₅₇	C₈	1	1	yes	characteristic
15	M₁₆×C₁₅₇	—	2	1	no	—
16	M₁₆×C₁₅₇	C₈	1	1	yes	characteristic
17	C₂×C₈×C₁₅₇	—	2	1	no	—
18	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
19	Q₈×C₁₅₇	C₁₆	1	1	yes	characteristic
20	D₄×C₁₅₇	—	2	1	no	—
21	D₄×C₁₅₇	C₁₆	1	1	yes	characteristic
22	C₂×C₄×C₁₅₇	—	2	1	no	—
23	C₂×C₄×C₁₅₇	C₂×C₈	1	1	yes	characteristic
24	C₈×C₁₅₇	—	2	1	no	—
25	C₈×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₅₇	M₁₆	1	1	yes	characteristic
27	C₂²×C₁₅₇	—	2	1	no	—
28	C₂²×C₁₅₇	M₃₂	1	1	yes	characteristic
29	C₄×C₁₅₇	—	2	1	no	—
30	C₄×C₁₅₇	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₁₅₇	C₂×C₁₆	1	1	yes	characteristic
32	C₂×C₁₅₇	—	2	1	no	—
33	C₂×C₁₅₇	—	4	1	no	—
34	C₂×C₁₅₇	C₂²⋊C₁₆	1	1	yes	characteristic
35	C₁₅₇	#1401	1	1	yes	characteristic, complemented, Sylow
36	#1401	—	157	1	no	maximal, Sylow
37	#504	—	157	1	no	—
38	M₆₄	—	157	1	no	—
39	C₂×C₃₂	—	157	1	no	—
40	#131	#10597845	1	1	yes	characteristic
41	M₃₂	—	314	1	no	—
42	M₃₂	—	157	1	no	—
43	C₂×C₁₆	—	314	1	no	—
44	C₂×C₁₆	—	157	1	no	—
45	C₃₂	—	314	1	no	—
46	C₃₂	—	314	1	no	—
47	#41	#11929441	1	1	yes	characteristic
48	M₁₆	—	2	1	no	—
49	M₁₆	#11929441	1	1	yes	characteristic
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	C₂×#10597845	1	1	yes	characteristic
52	C₁₆	—	314	1	no	—
53	C₁₆	—	157	1	no	—
54	C₁₆	—	157	1	no	—
55	Q₈	#14578957	1	1	yes	characteristic
56	D₄	—	2	1	no	—
57	D₄	#14578957	1	1	yes	characteristic
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₂×#11929441	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	C₂²⋊#10597845	1	1	yes	characteristic
62	C₈	#14578963	1	1	yes	characteristic
63	C₂²	—	2	1	no	—
64	C₂²	#19097845	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	C₂²⋊#11929441	1	1	yes	characteristic
67	C₄	C₂×#14578957	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	C₂²⋊#14578957	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊#10597845
commutator	C₄×C₁₅₇	C₂×C₁₆
Frattini	C₂×C₈	C₂×#10597845
Fitting	#131×C₁₅₇	C₄
socle	C₂×C₁₅₇	C₂²⋊C₁₆
prime radical	#41×C₁₅₇	C₈
cosocle	C₂×#14578929	C₂²
hypercenter	#131	#10597845
hypocenter	C₁₅₇	#1401
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
1. #23758205	C₂	semidirect
2. C₂×#19097821	C₂	semidirect
35. C₁₅₇	#1401	semidirect

Maximal subgroups

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

subgroup	index
#23758205	2
C₂×#19097821	2
#23757690	2
#1401	157