Group #23757828 — subgroups

Subgroups — #23757828

Order 10048.

222 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19097771	C₂	1	1	yes	maximal, characteristic, complemented
2	#19097762	C₂	1	2	yes	maximal, complemented
3	#19097751	C₂	1	2	yes	maximal
4	#19097747	C₂	1	2	yes	maximal
5	Q₈×Dic₁₅₇	C₂	1	2	yes	maximal, complemented
6	#19097736	C₂	1	2	yes	maximal, complemented
7	#19097729	C₂	1	2	yes	maximal
8	C₄²×D₁₅₇	C₂	1	1	yes	maximal, characteristic
9	C₄⋊Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₃₁₄	C₂²	1	2	yes	—
11	C₂×C₄×D₁₅₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₅₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₅₇	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
15	C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
16	C₄⋊Dic₁₅₇	C₂²	1	4	yes	—
17	#14578933	C₂²	1	4	yes	—
18	#14578931	C₂²	1	4	yes	—
19	C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
20	C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
21	C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
22	C₄×Dic₁₅₇	C₂²	1	4	yes	—
23	C₂×Q₈×C₁₅₇	C₂²	1	2	yes	—
24	C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
25	C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
26	C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
27	C₂²×D₁₅₇	C₂³	1	1	yes	characteristic
28	Dic₃₁₄	—	2	4	no	—
29	C₂×Dic₁₅₇	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₅₇	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₅₇	C₂³	1	4	yes	—
32	C₂×Dic₁₅₇	C₂³	1	1	yes	characteristic
33	C₄×D₁₅₇	—	2	2	no	—
34	C₄×D₁₅₇	—	2	2	no	—
35	C₄×D₁₅₇	Q₈	1	4	yes	complemented
36	Q₈×C₁₅₇	—	2	4	no	—
37	C₂×C₄×C₁₅₇	C₂³	1	4	yes	—
38	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
41	C₂²×C₁₅₇	C₂⁴	1	1	yes	characteristic
42	C₂×D₁₅₇	—	2	1	no	—
43	C₂×D₁₅₇	—	2	1	no	—
44	C₂×D₁₅₇	C₂×Q₈	1	2	yes	—
45	C₄×C₁₅₇	—	2	4	no	—
46	C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
47	C₄×C₁₅₇	#41	1	2	yes	—
48	C₄×C₁₅₇	#41	1	2	yes	—
49	Dic₁₅₇	—	2	1	no	—
50	Dic₁₅₇	—	2	1	no	—
51	Dic₁₅₇	—	2	4	no	—
52	Dic₁₅₇	C₂×Q₈	1	2	yes	—
53	C₂×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
54	C₂×C₁₅₇	C₂×#41	1	1	yes	characteristic
55	C₂×C₁₅₇	C₂×#41	1	1	yes	characteristic
56	D₁₅₇	—	2	2	no	—
57	C₁₅₇	#533	1	1	yes	characteristic, complemented, Sylow
58	#533	—	157	1	no	maximal, Sylow
59	C₄⋊Q₈	—	157	1	no	—
60	C₄⋊Q₈	D₁₅₇	1	1	yes	characteristic, complemented
61	#125	—	157	2	no	—
62	C₂²⋊Q₈	—	157	2	no	—
63	C₂²⋊Q₈	—	157	2	no	—
64	C₄×Q₈	—	157	2	no	—
65	C₄×Q₈	—	157	2	no	—
66	#117	—	157	2	no	—
67	C₂×C₄²	—	157	1	no	—
68	C₂×Q₈	—	157	2	no	—
69	C₂×Q₈	C₂×D₁₅₇	1	2	yes	—
70	C₂²×C₄	—	157	1	no	—
71	C₂²×C₄	—	157	1	no	—
72	C₂²×C₄	—	157	1	no	—
73	C₄⋊C₄	—	157	2	no	—
74	C₄⋊C₄	—	157	4	no	—
75	C₄⋊C₄	—	157	4	no	—
76	C₄⋊C₄	—	157	2	no	—
77	C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
78	C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
79	C₂²⋊C₄	—	157	4	no	—
80	C₄²	—	157	1	no	—
81	C₄²	—	157	1	no	—
82	C₄²	—	157	1	no	—
83	C₄²	—	157	4	no	—
84	C₄²	C₂×D₁₅₇	1	1	yes	characteristic
85	C₂³	—	157	1	no	—
86	Q₈	—	2	4	no	—
87	Q₈	—	314	4	no	—
88	C₂×C₄	—	314	2	no	—
89	C₂×C₄	—	314	2	no	—
90	C₂×C₄	—	157	1	no	—
91	C₂×C₄	—	157	1	no	—
92	C₂×C₄	—	157	4	no	—
93	C₂×C₄	—	157	1	no	—
94	C₂×C₄	—	157	4	no	—
95	C₂×C₄	C₂²×D₁₅₇	1	4	yes	—
96	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
99	C₂²	—	314	1	no	—
100	C₂²	—	314	1	no	—
101	C₂²	—	157	2	no	—
102	C₂²	C₂³×D₁₅₇	1	1	yes	characteristic
103	C₄	—	2	4	no	—
104	C₄	—	314	1	no	—
105	C₄	—	314	1	no	—
106	C₄	—	314	4	no	—
107	C₄	—	157	2	no	—
108	C₄	Q₈×D₁₅₇	1	2	yes	—
109	C₄	#14578953	1	2	yes	—
110	C₄	#14578955	1	2	yes	—
111	C₂	—	314	2	no	—
112	C₂	C₂×Q₈×D₁₅₇	1	1	yes	characteristic
113	C₂	C₂×#14578953	1	1	yes	characteristic
114	C₂	C₂×#14578955	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19097771

C₂

yes

maximal, characteristic, complemented

#19097762

C₂

yes

maximal, complemented

#19097751

C₂

yes

maximal

#19097747

C₂

yes

maximal

Q₈×Dic₁₅₇

C₂

yes

maximal, complemented

#19097736

C₂

yes

maximal, complemented

#19097729

C₂

yes

maximal

C₄²×D₁₅₇

C₂

yes

maximal, characteristic

C₄⋊Q₈×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₁₄

C₂²

yes

—

C₂×C₄×D₁₅₇

C₂²

yes

characteristic

C₂×C₄×D₁₅₇

C₂²

yes

characteristic

C₂×C₄×D₁₅₇

C₂²

yes

characteristic

C₄⋊Dic₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

C₂²

yes

—

#14578933

C₂²

yes

—

#14578931

C₂²

yes

—

C₄×Dic₁₅₇

C₂²

yes

characteristic

C₄×Dic₁₅₇

C₂²

yes

characteristic

C₄×Dic₁₅₇

C₂²

yes

characteristic

C₄×Dic₁₅₇

C₂²

yes

—

C₂×Q₈×C₁₅₇

C₂²

yes

—

C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₄²×C₁₅₇

C₂²

yes

characteristic

C₂²×D₁₅₇

C₂³

yes

characteristic

Dic₃₁₄

—

C₂×Dic₁₅₇

C₂³

yes

characteristic

C₂×Dic₁₅₇

C₂³

yes

characteristic

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

characteristic

C₄×D₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

Q₈

yes

complemented

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂²×C₁₅₇

C₂⁴

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

C₂×Q₈

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×Q₈

yes

—

C₄×C₁₅₇

#41

yes

—

C₄×C₁₅₇

#41

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

C₂×Q₈

yes

—

C₂×C₁₅₇

C₂²×Q₈

yes

characteristic

C₂×C₁₅₇

C₂×#41

yes

characteristic

C₂×C₁₅₇

C₂×#41

yes

characteristic

D₁₅₇

—

C₁₅₇

#533

yes

characteristic, complemented, Sylow

#533

—

157

maximal, Sylow

C₄⋊Q₈

—

157

—

C₄⋊Q₈

D₁₅₇

yes

characteristic, complemented

#125

—

157

—

C₂²⋊Q₈

—

157

—

C₂²⋊Q₈

—

157

—

C₄×Q₈

—

157

—

C₄×Q₈

—

157

—

#117

—

157

—

C₂×C₄²

—

157

—

C₂×Q₈

—

157

—

C₂×Q₈

C₂×D₁₅₇

yes

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

C₂×D₁₅₇

yes

—

C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂²⋊C₄

—

157

—

C₄²

—

157

—

C₄²

—

157

—

C₄²

—

157

—

C₄²

—

157

—

C₄²

C₂×D₁₅₇

yes

characteristic

C₂³

—

157

—

Q₈

—

Q₈

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

C₂²×D₁₅₇

yes

—

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₂²

—

314

—

100

C₂²

—

314

—

101

C₂²

—

157

—

102

C₂²

C₂³×D₁₅₇

yes

characteristic

103

C₄

—

104

C₄

—

314

—

105

C₄

—

314

—

106

C₄

—

314

—

107

C₄

—

157

—

108

C₄

Q₈×D₁₅₇

yes

—

109

C₄

#14578953

yes

—

110

C₄

#14578955

yes

—

111

C₂

—

314

—

112

C₂

C₂×Q₈×D₁₅₇

yes

characteristic

113

C₂

C₂×#14578953

yes

characteristic

114

C₂

C₂×#14578955

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅₇
commutator, socle, cosocle	C₂²×C₁₅₇	C₂⁴
Fitting	C₄⋊Q₈×C₁₅₇	C₂
prime radical	C₂²×D₁₅₇	C₂³
hypercenter	C₄⋊Q₈	D₁₅₇
hypocenter	C₁₅₇	#533
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅₇

commutator, socle, cosocle

C₂²×C₁₅₇

C₂⁴

Fitting

C₄⋊Q₈×C₁₅₇

C₂

prime radical

C₂²×D₁₅₇

C₂³

hypercenter

C₄⋊Q₈

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #19097771	C₂	semidirect
2. #19097762	C₂	semidirect
5. Q₈×Dic₁₅₇	C₂	semidirect
6. #19097736	C₂	semidirect
9. C₄⋊Q₈×C₁₅₇	C₂	semidirect
35. C₄×D₁₅₇	Q₈	semidirect
57. C₁₅₇	#533	semidirect
60. C₄⋊Q₈	D₁₅₇	semidirect

H ≅ G/N

type

1. #19097771

C₂

semidirect

2. #19097762

C₂

semidirect

5. Q₈×Dic₁₅₇

C₂

semidirect

6. #19097736

C₂

semidirect

9. C₄⋊Q₈×C₁₅₇

C₂

semidirect

35. C₄×D₁₅₇

Q₈

semidirect

57. C₁₅₇

#533

semidirect

60. C₄⋊Q₈

D₁₅₇

semidirect

Subgroups — #23757828

Conjugacy classes of subgroups

Complements