Subgroups — #34100308

Order 20096.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19097707	C₂	1	4	yes	maximal
2	#23757297	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14578938	C₂²	1	4	yes	—
5	#19097707	—	2	8	no	—
6	C₄⋊#11929434	—	2	4	no	—
7	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₁₅₇	C₄	1	2	yes	—
10	C₄×C₈×C₁₅₇	C₄	1	2	yes	—
11	#14578938	—	2	8	no	—
12	C₂×#11929434	—	2	4	no	—
13	C₂²×C₄×C₁₅₇	Q₈	1	2	yes	—
14	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
15	C₂×C₈×C₁₅₇	C₂×C₄	1	4	yes	—
16	C₂×C₈×C₁₅₇	D₄	1	4	yes	—
17	C₂×C₈×C₁₅₇	Q₈	1	4	yes	—
18	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₁₅₇	D₄	1	2	yes	—
21	#11929434	—	4	4	no	—
22	C₂³×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
23	C₂×C₄×C₁₅₇	—	2	4	no	—
24	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
28	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
30	C₈×C₁₅₇	C₄⋊C₄	1	8	yes	—
31	C₂²×C₁₅₇	—	2	2	no	—
32	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₁₅₇	C₄⋊Q₈	1	2	yes	—
34	C₄×C₁₅₇	—	2	2	no	—
35	C₄×C₁₅₇	#108	1	4	yes	complemented
36	C₄×C₁₅₇	C₂×C₄⋊C₄	1	2	yes	—
37	C₄×C₁₅₇	#127	1	1	yes	characteristic
38	C₄×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₁₅₇	—	2	2	no	—
40	C₂×C₁₅₇	#384	1	1	yes	characteristic
41	C₂×C₁₅₇	C₂×#108	1	2	yes	—
42	C₁₅₇	#1848	1	1	yes	characteristic, complemented, Sylow
43	#1848	—	157	1	no	maximal, Sylow
44	C₂×#108	—	157	4	no	—
45	#423	—	157	2	no	—
46	C₂×C₄×C₈	D₁₅₇	1	1	yes	characteristic
47	C₂×M₁₆	—	157	4	no	—
48	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
49	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
50	#108	—	314	8	no	—
51	C₄⋊C₈	—	314	4	no	—
52	C₄×C₈	Dic₁₅₇	1	2	yes	—
53	C₄×C₈	Dic₁₅₇	1	2	yes	—
54	C₂²×C₄	Dic₃₁₄	1	2	yes	—
55	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
56	M₁₆	—	314	8	no	—
57	C₂×C₈	—	314	4	no	—
58	C₂×C₈	C₂×Dic₁₅₇	1	4	yes	—
59	C₂×C₈	D₆₂₈	1	4	yes	—
60	C₂×C₈	Dic₃₁₄	1	4	yes	—
61	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
62	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
63	C₄²	D₆₂₈	1	2	yes	—
64	C₂³	C₂×Dic₃₁₄	1	1	yes	characteristic
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₄⋊Dic₁₅₇	1	2	yes	—
67	C₂×C₄	C₄⋊Dic₁₅₇	1	2	yes	—
68	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
69	C₂×C₄	C₂×D₆₂₈	1	2	yes	—
70	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
71	C₂×C₄	C₂×Dic₃₁₄	1	2	yes	—
72	C₈	—	628	4	no	—
73	C₈	C₄⋊Dic₁₅₇	1	8	yes	—
74	C₂²	—	2	2	no	—
75	C₂²	C₂×C₄⋊Dic₁₅₇	1	1	yes	characteristic
76	C₂²	#19097771	1	2	yes	—
77	C₄	—	2	2	no	—
78	C₄	#19097707	1	4	yes	complemented
79	C₄	C₂×C₄⋊Dic₁₅₇	1	2	yes	—
80	C₄	#19097768	1	1	yes	characteristic
81	C₄	#19097771	1	1	yes	characteristic
82	C₂	—	2	2	no	—
83	C₂	#23757143	1	1	yes	characteristic
84	C₂	C₂×#19097707	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₂₈
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	C₂×C₄×C₈×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₅₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₅₇
hypocenter	C₁₅₇	#1848
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
35. C₄×C₁₅₇	#108	semidirect
42. C₁₅₇	#1848	semidirect
78. C₄	#19097707	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
#23757297	2
C₂×C₄×C₈×C₁₅₇	2
#1848	157