Subgroups — #34098240

Order 20096.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19097707	C₂	1	2	yes	maximal
2	C₂×M₃₂×C₁₅₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#14578938	—	2	2	no	—
4	#19097707	—	2	2	no	—
5	#19097707	C₄	1	4	yes	—
6	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
7	M₃₂×C₁₅₇	—	2	2	no	—
8	C₂×C₁₆×C₁₅₇	C₄	1	2	yes	—
9	#14578938	—	2	4	no	—
10	#14578938	—	4	2	no	—
11	C₂×#11929434	—	4	2	no	—
12	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₅₇	C₂×C₄	1	2	yes	—
14	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
16	C₂×C₈×C₁₅₇	Q₈	1	1	yes	characteristic
17	C₁₆×C₁₅₇	—	2	2	no	—
18	#11929434	—	4	4	no	—
19	C₂³×C₁₅₇	Q₁₆	1	1	yes	characteristic
20	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
21	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₅₇	D₈	1	1	yes	characteristic
23	C₂×C₄×C₁₅₇	SD₁₆	1	2	yes	—
24	C₈×C₁₅₇	C₄²	1	1	yes	characteristic
25	C₈×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₅₇	C₄⋊C₄	1	2	yes	—
27	C₂²×C₁₅₇	—	4	1	no	—
28	C₂²×C₁₅₇	#103	1	2	yes	—
29	C₂²×C₁₅₇	#107	1	1	yes	characteristic
30	C₄×C₁₅₇	#95	1	1	yes	characteristic
31	C₄×C₁₅₇	#102	1	2	yes	—
32	C₄×C₁₅₇	#106	1	1	yes	characteristic
33	C₂×C₁₅₇	—	2	2	no	—
34	C₂×C₁₅₇	—	2	1	no	—
35	C₂×C₁₅₇	#340	1	1	yes	characteristic
36	C₁₅₇	#1385	1	1	yes	characteristic, complemented, Sylow
37	#1385	—	157	1	no	maximal, Sylow
38	C₂×M₃₂	D₁₅₇	1	1	yes	characteristic
39	C₂×#108	—	157	2	no	—
40	C₂×M₁₆	—	314	2	no	—
41	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
42	M₃₂	—	2	2	no	—
43	C₂×C₁₆	Dic₁₅₇	1	2	yes	—
44	#108	—	314	2	no	—
45	#108	—	157	4	no	—
46	C₂²×C₄	D₆₂₈	1	1	yes	characteristic
47	M₁₆	—	314	4	no	—
48	M₁₆	—	628	2	no	—
49	C₂×C₈	—	628	2	no	—
50	C₂×C₈	C₄×D₁₅₇	1	2	yes	—
51	C₂×C₈	C₂×Dic₁₅₇	1	1	yes	characteristic
52	C₂×C₈	C₂²⋊D₁₅₇	1	2	yes	—
53	C₂×C₈	Dic₃₁₄	1	1	yes	characteristic
54	C₁₆	—	2	2	no	—
55	C₂³	Dic₆₂₈	1	1	yes	characteristic
56	C₂×C₄	#14578931	1	2	yes	—
57	C₂×C₄	C₄⋊Dic₁₅₇	1	1	yes	characteristic
58	C₂×C₄	D₁₂₅₆	1	1	yes	characteristic
59	C₂×C₄	#14578943	1	2	yes	—
60	C₈	—	628	4	no	—
61	C₈	C₄×Dic₁₅₇	1	1	yes	characteristic
62	C₈	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
63	C₈	#14578933	1	2	yes	—
64	C₂²	—	4	1	no	—
65	C₂²	#19097696	1	2	yes	—
66	C₂²	#19097705	1	1	yes	characteristic
67	C₄	#19097678	1	1	yes	characteristic
68	C₄	#19097693	1	2	yes	—
69	C₄	#19097703	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	—	2	1	no	—
72	C₂	#23757036	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19097703
commutator	C₈×C₁₅₇	C₄²
Frattini	C₂²×C₈	C₂×D₁₅₇
Fitting	C₂×M₃₂×C₁₅₇	C₂
socle	C₂×C₁₅₇	#340
prime radical	C₂³×C₁₅₇	Q₁₆
cosocle	C₂²×C₈×C₁₅₇	C₂²
hypercenter	C₂×M₃₂	D₁₅₇
hypocenter	C₁₅₇	#1385
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
36. C₁₅₇	#1385	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19097707	2
C₂×M₃₂×C₁₅₇	2
#1385	157