Subgroups — #34098279

Order 20096.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757689	C₂	1	1	yes	maximal, characteristic
2	C₂×#19097677	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#19097777	C₄	1	1	yes	characteristic
5	#19097711	C₄	1	1	yes	characteristic
6	#19097710	—	2	1	no	—
7	C₁₆×D₁₅₇	—	2	1	no	—
8	#19097677	—	2	1	no	—
9	C₂×C₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₅₇	—	2	1	no	—
11	#14578954	C₈	1	1	yes	characteristic
12	#14578938	C₈	1	1	yes	characteristic
13	#14578937	—	2	1	no	—
14	C₈×D₁₅₇	—	2	1	no	—
15	#14578929	—	2	1	no	—
16	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₁₅₇	D₄	1	1	yes	characteristic
18	C₁₆×C₁₅₇	D₄	1	1	yes	characteristic
19	Dic₃₁₄	C₁₆	1	1	yes	characteristic
20	D₆₂₈	C₁₆	1	1	yes	characteristic
21	C₂²⋊D₁₅₇	—	2	1	no	—
22	C₄×D₁₅₇	—	2	1	no	—
23	#11929434	—	2	1	no	—
24	C₂×C₄×C₁₅₇	C₂×C₈	1	1	yes	characteristic
25	C₈×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₅₇	M₁₆	1	1	yes	characteristic
27	C₂²×C₁₅₇	M₃₂	1	1	yes	characteristic
28	C₂×D₁₅₇	—	2	1	no	—
29	C₄×C₁₅₇	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₁₅₇	C₂×C₁₆	1	1	yes	characteristic
31	Dic₁₅₇	—	2	1	no	—
32	C₂×C₁₅₇	—	2	1	no	—
33	C₂×C₁₅₇	C₂²⋊C₁₆	1	1	yes	characteristic
34	D₁₅₇	—	4	1	no	—
35	C₁₅₇	#1401	1	1	yes	characteristic, complemented, Sylow
36	#1401	—	157	1	no	maximal, Sylow
37	#504	—	157	1	no	—
38	M₆₄	D₁₅₇	1	1	yes	characteristic, complemented
39	C₂×C₃₂	—	157	1	no	—
40	#131	—	157	1	no	—
41	M₃₂	—	314	1	no	—
42	M₃₂	—	157	1	no	—
43	C₂×C₁₆	—	314	1	no	—
44	C₂×C₁₆	C₂×D₁₅₇	1	1	yes	characteristic
45	C₃₂	—	2	1	no	—
46	C₃₂	—	314	1	no	—
47	#41	—	157	1	no	—
48	M₁₆	—	314	1	no	—
49	M₁₆	—	157	1	no	—
50	C₂×C₈	—	314	1	no	—
51	C₂×C₈	C₄×D₁₅₇	1	1	yes	characteristic
52	C₁₆	—	314	1	no	—
53	C₁₆	C₂²⋊D₁₅₇	1	1	yes	characteristic
54	C₁₆	D₆₂₈	1	1	yes	characteristic
55	Q₈	—	157	1	no	—
56	D₄	—	314	1	no	—
57	D₄	—	157	1	no	—
58	C₂×C₄	—	314	1	no	—
59	C₂×C₄	C₈×D₁₅₇	1	1	yes	characteristic
60	C₈	—	314	1	no	—
61	C₈	#14578931	1	1	yes	characteristic
62	C₈	#14578937	1	1	yes	characteristic
63	C₂²	—	314	1	no	—
64	C₂²	#19097710	1	1	yes	characteristic
65	C₄	—	314	1	no	—
66	C₄	#19097683	1	1	yes	characteristic
67	C₄	C₁₆×D₁₅₇	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	628	1	no	—
70	C₂	#23757052	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₁₅₇
commutator	C₄×C₁₅₇	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₅₇
Fitting	M₆₄×C₁₅₇	C₂
socle	C₂×C₁₅₇	C₂²⋊C₁₆
prime radical	#14578954	C₈
cosocle	C₂×C₁₆×C₁₅₇	C₂²
hypercenter	M₆₄	D₁₅₇
hypocenter	C₁₅₇	#1401
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×#19097677	C₂	semidirect
3. M₆₄×C₁₅₇	C₂	semidirect
35. C₁₅₇	#1401	semidirect
38. M₆₄	D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23757689	2
C₂×#19097677	2
M₆₄×C₁₅₇	2
#1401	157