Subgroups — #34115166

Order 20096.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14578957	C₂	1	2	yes	maximal
2	C₈×#11929434	C₂	1	1	yes	maximal, characteristic
3	C₂×#14578957	—	2	2	no	—
4	C₄×#11929434	C₄	1	1	yes	characteristic
5	C₄×#11929434	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₁₅₇	C₄	1	1	yes	characteristic
7	#14578957	—	4	2	no	—
8	C₂×#11929434	C₈	1	1	yes	characteristic
9	C₂×#11929434	C₈	1	1	yes	characteristic
10	C₂×#11929434	D₄	1	1	yes	characteristic
11	C₂×#11929434	Q₈	1	1	yes	characteristic
12	C₂×C₈×C₁₅₇	—	2	1	no	—
13	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
14	#11929434	—	2	1	no	—
15	#11929434	—	2	1	no	—
16	#11929434	C₁₆	1	2	yes	complemented
17	#11929434	D₈	1	1	yes	characteristic
18	#11929434	Q₁₆	1	1	yes	characteristic
19	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₅₇	C₂×C₈	1	1	yes	characteristic
21	C₂×C₄×C₁₅₇	M₁₆	1	1	yes	characteristic
22	C₈×C₁₅₇	—	2	2	no	—
23	C₂²×C₁₅₇	C₄⋊C₈	1	1	yes	characteristic
24	C₄×C₁₅₇	—	2	1	no	—
25	C₄×C₁₅₇	#107	1	1	yes	characteristic
26	C₄×C₁₅₇	#108	1	1	yes	characteristic
27	C₄×C₁₅₇	C₂×C₁₆	1	1	yes	characteristic
28	C₄×C₁₅₇	M₃₂	1	1	yes	characteristic
29	C₂×C₁₅₇	#335	1	1	yes	characteristic
30	C₂×C₁₅₇	C₄⋊C₁₆	1	1	yes	characteristic
31	C₂×C₁₅₇	#364	1	1	yes	characteristic
32	C₁₅₇	#1370	1	1	yes	characteristic, complemented, Sylow
33	#1370	—	157	1	no	maximal, Sylow
34	C₄⋊C₁₆	—	157	2	no	—
35	C₈²	—	157	1	no	—
36	C₂×C₁₆	—	314	2	no	—
37	C₄×C₈	—	157	1	no	—
38	C₄×C₈	—	157	1	no	—
39	C₄×C₈	#10597845	1	1	yes	characteristic
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	—	157	1	no	—
42	C₂×C₈	—	157	1	no	—
43	C₂×C₈	—	157	1	no	—
44	C₂×C₈	—	157	1	no	—
45	C₄²	C₂×#10597845	1	1	yes	characteristic
46	C₁₆	—	628	2	no	—
47	C₂×C₄	C₄⋊#10597845	1	1	yes	characteristic
48	C₂×C₄	D₁₅₇⋊C₈	1	1	yes	characteristic
49	C₂×C₄	#14578963	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₈	—	314	1	no	—
52	C₈	—	314	1	no	—
53	C₈	—	157	1	no	—
54	C₈	—	157	1	no	—
55	C₈	—	157	2	no	—
56	C₂²	#19097838	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	#19097840	1	1	yes	characteristic
59	C₄	#19097841	1	1	yes	characteristic
60	C₄	D₁₅₇⋊C₁₆	1	1	yes	characteristic
61	C₄	#19097846	1	1	yes	characteristic
62	C₂	#23758137	1	1	yes	characteristic
63	C₂	#23758194	1	1	yes	characteristic
64	C₂	#23758196	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#10597845
commutator	C₄×C₁₅₇	C₂×C₁₆
Frattini	C₄²	C₂×#10597845
Fitting	C₄×C₈×C₁₅₇	C₄
socle, prime radical	C₂²×C₁₅₇	C₄⋊C₈
cosocle	C₄×#11929434	C₂²
hypercenter	C₄×C₈	#10597845
hypocenter	C₁₅₇	#1370
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
16. #11929434	C₁₆	semidirect
32. C₁₅₇	#1370	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14578957	2
C₈×#11929434	2
#1370	157