Subgroups — #34115169

Order 20096.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23758136	C₂	1	2	yes	maximal
2	C₁₆×Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11929441	—	2	2	no	—
4	C₂×#14578929	C₄	1	1	yes	characteristic
5	C₈×Dic₁₅₇	C₂²	1	1	yes	characteristic
6	C₂×C₁₆×C₁₅₇	C₄	1	1	yes	characteristic
7	C₂×#11929441	—	4	2	no	—
8	C₂×#11929434	Q₈	1	1	yes	characteristic
9	C₄×Dic₁₅₇	D₄	1	1	yes	characteristic
10	#14578929	—	2	1	no	—
11	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₅₇	C₈	1	2	yes	complemented
13	#11929441	—	8	2	no	—
14	C₂×Dic₁₅₇	D₈	1	1	yes	characteristic
15	C₂×Dic₁₅₇	Q₁₆	1	1	yes	characteristic
16	#11929434	—	2	1	no	—
17	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
18	C₈×C₁₅₇	C₂×C₈	1	1	yes	characteristic
19	C₈×C₁₅₇	M₁₆	1	1	yes	characteristic
20	C₂²×C₁₅₇	#107	1	1	yes	characteristic
21	C₄×C₁₅₇	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₁₅₇	#108	1	1	yes	characteristic
23	Dic₁₅₇	—	2	1	no	—
24	Dic₁₅₇	SD₃₂	1	2	yes	—
25	C₂×C₁₅₇	#335	1	1	yes	characteristic
26	C₂×C₁₅₇	#367	1	1	yes	characteristic
27	C₂×C₁₅₇	#368	1	1	yes	characteristic
28	C₁₅₇	#1372	1	1	yes	characteristic, complemented, Sylow
29	#1372	—	157	1	no	maximal, Sylow
30	C₄×C₁₆	—	157	1	no	—
31	#335	—	157	2	no	—
32	C₂×C₁₆	—	157	1	no	—
33	C₂×C₁₆	#10597845	1	1	yes	characteristic
34	C₄⋊C₈	—	314	2	no	—
35	C₄×C₈	—	157	1	no	—
36	C₂×C₈	—	628	2	no	—
37	C₂×C₈	—	157	1	no	—
38	C₂×C₈	C₂×#10597845	1	1	yes	characteristic
39	C₄²	—	157	1	no	—
40	C₁₆	—	314	1	no	—
41	C₁₆	#11929441	1	2	yes	complemented
42	C₂×C₄	—	157	1	no	—
43	C₂×C₄	—	157	1	no	—
44	C₂×C₄	C₄⋊#10597845	1	1	yes	characteristic
45	C₈	—	314	1	no	—
46	C₈	—	1256	2	no	—
47	C₈	C₂×#11929441	1	1	yes	characteristic
48	C₈	#14578964	1	1	yes	characteristic
49	C₂²	#19097840	1	1	yes	characteristic
50	C₄	—	314	1	no	—
51	C₄	—	157	2	no	—
52	C₄	C₄⋊#11929441	1	1	yes	characteristic
53	C₄	#19097842	1	1	yes	characteristic
54	C₂	#23758136	1	1	yes	characteristic
55	C₂	#23758200	1	1	yes	characteristic
56	C₂	#23758202	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19097840
commutator	C₈×C₁₅₇	C₂×C₈
Frattini	C₂×C₈	C₂×#10597845
Fitting	C₂×C₁₆×C₁₅₇	C₄
socle, prime radical	C₂²×C₁₅₇	#107
cosocle	C₈×Dic₁₅₇	C₂²
hypercenter	C₂×C₁₆	#10597845
hypocenter	C₁₅₇	#1372
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₁₅₇	C₈	semidirect
28. C₁₅₇	#1372	semidirect
41. C₁₆	#11929441	semidirect

Maximal subgroups

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

subgroup	index
#23758136	2
C₁₆×Dic₁₅₇	2
#1372	157