Subgroups — #34115269

Order 20096.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23758178	C₂	1	2	yes	maximal
2	#23757647	C₂	1	1	yes	maximal, characteristic
3	#19097830	—	2	2	no	—
4	C₂×#14578941	—	2	2	no	—
5	#19097768	C₂²	1	1	yes	characteristic
6	C₄×#11929434	C₄	1	1	yes	characteristic, complemented
7	C₄⋊Q₈×C₁₅₇	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊#10597845	—	4	2	no	—
9	C₂×D₆₂₈	—	4	1	no	—
10	C₂×D₆₂₈	D₄	1	2	yes	—
11	#14578941	—	4	4	no	—
12	C₂×#11929434	—	2	1	no	—
13	C₂×Q₈×C₁₅₇	—	2	1	no	—
14	C₄⋊C₄×C₁₅₇	—	4	1	no	—
15	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
16	C₂×#10597845	—	8	2	no	—
17	C₂²×D₁₅₇	—	2	2	no	—
18	D₆₂₈	—	4	2	no	—
19	D₆₂₈	—	8	2	no	—
20	#11929434	—	2	2	no	—
21	Q₈×C₁₅₇	—	4	2	no	—
22	C₂×C₄×C₁₅₇	—	2	1	no	—
23	C₂×C₄×C₁₅₇	—	4	1	no	—
24	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₁₅₇	#99	1	1	yes	characteristic
26	C₂×D₁₅₇	—	4	2	no	—
27	C₂×D₁₅₇	—	8	2	no	—
28	#10597845	—	16	2	no	—
29	C₄×C₁₅₇	—	2	2	no	—
30	C₄×C₁₅₇	—	2	1	no	—
31	C₄×C₁₅₇	—	8	1	no	—
32	C₂×C₁₅₇	—	2	1	no	—
33	C₂×C₁₅₇	#353	1	1	yes	characteristic
34	D₁₅₇	—	8	2	no	—
35	C₁₅₇	#1409	1	1	yes	characteristic, complemented, Sylow
36	#1409	—	157	1	no	maximal, Sylow
37	#492	—	157	1	no	—
38	#353	—	157	2	no	—
39	C₂×SD₁₆	—	314	2	no	—
40	C₄⋊Q₈	#10597845	1	1	yes	characteristic, complemented
41	#127	—	157	1	no	—
42	#99	—	314	2	no	—
43	C₄×C₈	—	157	1	no	—
44	C₂×Q₈	—	2	1	no	—
45	C₂×D₄	—	628	1	no	—
46	C₂×D₄	—	157	2	no	—
47	SD₁₆	—	628	4	no	—
48	C₂×C₈	—	314	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₂²⋊C₄	—	628	2	no	—
51	C₄²	C₂×#10597845	1	1	yes	characteristic
52	C₂³	—	314	2	no	—
53	Q₈	—	4	2	no	—
54	D₄	—	628	2	no	—
55	D₄	—	1256	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	—	1256	2	no	—
59	C₂×C₄	C₂²⋊#10597845	1	1	yes	characteristic
60	C₈	—	314	2	no	—
61	C₂²	—	628	2	no	—
62	C₂²	—	1256	2	no	—
63	C₂²	#19097829	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	2	1	no	—
66	C₄	—	8	1	no	—
67	C₄	—	2512	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	1256	2	no	—
70	C₂	#23758177	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23758177
commutator	C₄²×C₁₅₇	C₂×C₄
Frattini	C₄²	C₂×#10597845
Fitting	C₄⋊Q₈×C₁₅₇	C₄
socle	C₂×C₁₅₇	#353
prime radical, cosocle	#19097768	C₂²
hypercenter	C₄⋊Q₈	#10597845
hypocenter	C₁₅₇	#1409
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
6. C₄×#11929434	C₄	semidirect
7. C₄⋊Q₈×C₁₅₇	C₄	semidirect
35. C₁₅₇	#1409	semidirect
40. C₄⋊Q₈	#10597845	semidirect

Maximal subgroups

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

subgroup	index
#23758178	2
#23757647	2
#1409	157