Group #23757837 — subgroups

Subgroups — #23757837

Order 10048.

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

normal subgroups only

This group has 334 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	C₂²×D₆₂₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#19097767	C₂	1	2	yes	maximal, complemented
3	#19097760	C₂	1	2	yes	maximal, complemented
4	#19097745	C₂	1	4	yes	maximal, complemented
5	#19097739	C₂	1	4	yes	maximal, complemented
6	#19097730	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₅₇	C₂²	1	2	yes	—
9	C₂×D₆₂₈	—	2	4	no	—
10	C₂×D₆₂₈	C₂²	1	2	yes	—
11	C₂×D₆₂₈	C₂²	1	2	yes	—
12	C₂×C₂²⋊D₁₅₇	C₂²	1	8	yes	complemented
13	C₄⋊Dic₁₅₇	C₂²	1	2	yes	complemented
14	C₂²⋊Dic₁₅₇	C₂²	1	2	yes	—
15	#14578931	C₂²	1	8	yes	complemented
16	C₄×Dic₁₅₇	C₂²	1	2	yes	complemented
17	#41×C₁₅₇	—	2	4	no	—
18	C₂×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic, complemented
19	C₂×D₄×C₁₅₇	C₂²	1	2	yes	—
20	C₂×D₄×C₁₅₇	C₂²	1	1	yes	characteristic, complemented
21	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₁₅₇	C₂²	1	2	yes	complemented
23	C₂²×D₁₅₇	—	2	4	no	—
24	C₂²×D₁₅₇	—	4	4	no	—
25	C₂²×D₁₅₇	C₂³	1	4	yes	—
26	D₆₂₈	—	2	4	no	—
27	D₆₂₈	—	2	4	no	—
28	C₂²⋊D₁₅₇	—	4	8	no	—
29	C₂×Dic₁₅₇	C₂³	1	4	yes	—
30	C₂³×C₁₅₇	C₂³	1	2	yes	—
31	C₂³×C₁₅₇	C₂³	1	1	yes	characteristic
32	Q₈×C₁₅₇	—	2	2	no	—
33	D₄×C₁₅₇	—	2	4	no	—
34	D₄×C₁₅₇	—	2	2	no	—
35	C₂×C₄×C₁₅₇	—	2	4	no	—
36	C₂×C₄×C₁₅₇	D₄	1	4	yes	complemented
37	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
38	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
40	C₂²×C₁₅₇	—	2	2	no	—
41	C₂²×C₁₅₇	—	2	2	no	—
42	C₂²×C₁₅₇	—	2	4	no	—
43	C₂²×C₁₅₇	C₂×D₄	1	2	yes	—
44	C₂²×C₁₅₇	C₂⁴	1	1	yes	characteristic
45	C₂×D₁₅₇	—	2	4	no	—
46	C₂×D₁₅₇	—	2	8	no	—
47	C₂×D₁₅₇	—	4	8	no	—
48	C₄×C₁₅₇	—	2	2	no	—
49	C₄×C₁₅₇	C₂×D₄	1	2	yes	—
50	C₄×C₁₅₇	C₂×D₄	1	2	yes	—
51	Dic₁₅₇	—	2	4	no	—
52	C₂×C₁₅₇	—	2	2	no	—
53	C₂×C₁₅₇	—	4	2	no	—
54	C₂×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₁₅₇	H(5,𝔽₂)	1	2	yes	—
56	D₁₅₇	—	4	4	no	—
57	C₁₅₇	#535	1	1	yes	characteristic, complemented, Sylow
58	#535	—	157	1	no	maximal, Sylow
59	C₂×#41	D₁₅₇	1	1	yes	characteristic, complemented
60	C₂²×D₄	—	157	1	no	—
61	#127	—	157	2	no	—
62	#124	—	157	2	no	—
63	#121	—	157	4	no	—
64	#120	—	157	4	no	—
65	#117	—	157	1	no	—
66	C₂⁴	—	157	2	no	—
67	#41	—	2	4	no	—
68	C₂×Q₈	C₂×D₁₅₇	1	1	yes	characteristic, complemented
69	C₂×D₄	—	314	4	no	—
70	C₂×D₄	—	157	8	no	—
71	C₂×D₄	—	157	2	no	—
72	C₂×D₄	—	157	2	no	—
73	C₂×D₄	C₂×D₁₅₇	1	2	yes	—
74	C₂×D₄	C₂×D₁₅₇	1	1	yes	characteristic, complemented
75	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
76	C₂²×C₄	C₂×D₁₅₇	1	2	yes	complemented
77	C₄⋊C₄	—	157	2	no	—
78	C₂²⋊C₄	—	157	2	no	—
79	C₂²⋊C₄	—	157	8	no	—
80	C₄²	—	157	2	no	—
81	C₂³	—	314	4	no	—
82	C₂³	—	628	4	no	—
83	C₂³	—	157	4	no	—
84	C₂³	C₂²×D₁₅₇	1	2	yes	—
85	C₂³	C₂²×D₁₅₇	1	1	yes	characteristic
86	Q₈	—	2	2	no	—
87	D₄	—	2	4	no	—
88	D₄	—	2	2	no	—
89	D₄	—	314	4	no	—
90	D₄	—	314	4	no	—
91	D₄	—	628	8	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	157	4	no	—
94	C₂×C₄	C₂²⋊D₁₅₇	1	4	yes	complemented
95	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
96	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	2	4	no	—
101	C₂²	—	314	4	no	—
102	C₂²	—	314	8	no	—
103	C₂²	—	628	8	no	—
104	C₂²	C₂×C₂²⋊D₁₅₇	1	2	yes	—
105	C₂²	C₂³×D₁₅₇	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	314	4	no	—
108	C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
109	C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	4	2	no	—
112	C₂	—	628	4	no	—
113	C₂	C₂²×C₂²⋊D₁₅₇	1	1	yes	characteristic
114	C₂	#19097817	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₆₂₈

C₂

yes

maximal, characteristic, complemented

#19097767

C₂

yes

maximal, complemented

#19097760

C₂

yes

maximal, complemented

#19097745

C₂

yes

maximal, complemented

#19097739

C₂

yes

maximal, complemented

#19097730

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₅₇

C₂²

yes

—

C₂×D₆₂₈

—

C₂×D₆₂₈

C₂²

yes

—

C₂×D₆₂₈

C₂²

yes

—

C₂×C₂²⋊D₁₅₇

C₂²

yes

complemented

C₄⋊Dic₁₅₇

C₂²

yes

complemented

C₂²⋊Dic₁₅₇

C₂²

yes

—

#14578931

C₂²

yes

complemented

C₄×Dic₁₅₇

C₂²

yes

complemented

#41×C₁₅₇

—

C₂×Q₈×C₁₅₇

C₂²

yes

characteristic, complemented

C₂×D₄×C₁₅₇

C₂²

yes

—

C₂×D₄×C₁₅₇

C₂²

yes

characteristic, complemented

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

C₂²×C₄×C₁₅₇

C₂²

yes

complemented

C₂²×D₁₅₇

—

C₂²×D₁₅₇

—

C₂²×D₁₅₇

C₂³

yes

—

D₆₂₈

—

D₆₂₈

—

C₂²⋊D₁₅₇

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂³×C₁₅₇

C₂³

yes

—

C₂³×C₁₅₇

C₂³

yes

characteristic

Q₈×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

D₄

yes

complemented

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

C₂⁴

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×D₄

yes

—

C₄×C₁₅₇

C₂×D₄

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂²×D₄

yes

characteristic

C₂×C₁₅₇

H(5,𝔽₂)

yes

—

D₁₅₇

—

C₁₅₇

#535

yes

characteristic, complemented, Sylow

#535

—

157

maximal, Sylow

C₂×#41

D₁₅₇

yes

characteristic, complemented

C₂²×D₄

—

157

—

#127

—

157

—

#124

—

157

—

#121

—

157

—

#120

—

157

—

#117

—

157

—

C₂⁴

—

157

—

#41

—

C₂×Q₈

C₂×D₁₅₇

yes

characteristic, complemented

C₂×D₄

—

314

—

C₂×D₄

—

157

—

C₂×D₄

—

157

—

C₂×D₄

—

157

—

C₂×D₄

C₂×D₁₅₇

yes

—

C₂×D₄

C₂×D₁₅₇

yes

characteristic, complemented

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

C₂²×C₄

C₂×D₁₅₇

yes

complemented

C₄⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₄²

—

157

—

C₂³

—

314

—

C₂³

—

628

—

C₂³

—

157

—

C₂³

C₂²×D₁₅₇

yes

—

C₂³

C₂²×D₁₅₇

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

314

—

D₄

—

314

—

D₄

—

628

—

C₂×C₄

—

C₂×C₄

—

157

—

C₂×C₄

C₂²⋊D₁₅₇

yes

complemented

C₂×C₄

C₂²×D₁₅₇

yes

—

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

314

—

102

C₂²

—

314

—

103

C₂²

—

628

—

104

C₂²

C₂×C₂²⋊D₁₅₇

yes

—

105

C₂²

C₂³×D₁₅₇

yes

characteristic

106

C₄

—

107

C₄

—

314

—

108

C₄

C₂×C₂²⋊D₁₅₇

yes

—

109

C₄

C₂×C₂²⋊D₁₅₇

yes

—

110

C₂

—

111

C₂

—

112

C₂

—

628

—

113

C₂

C₂²×C₂²⋊D₁₅₇

yes

characteristic

114

C₂

#19097817

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅₇
commutator, socle, cosocle	C₂²×C₁₅₇	C₂⁴
Fitting	C₂×#41×C₁₅₇	C₂
hypercenter	C₂×#41	D₁₅₇
hypocenter	C₁₅₇	#535
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂×#41

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅₇

commutator, socle, cosocle

C₂²×C₁₅₇

C₂⁴

Fitting

C₂×#41×C₁₅₇

C₂

hypercenter

C₂×#41

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂² ◁ C₂×#41

N	H ≅ G/N	type
1. C₂²×D₆₂₈	C₂	semidirect
2. #19097767	C₂	semidirect
3. #19097760	C₂	semidirect
4. #19097745	C₂	semidirect
5. #19097739	C₂	semidirect
6. #19097730	C₂	semidirect
7. C₂×#41×C₁₅₇	C₂	semidirect
12. C₂×C₂²⋊D₁₅₇	C₂²	semidirect
13. C₄⋊Dic₁₅₇	C₂²	semidirect
15. #14578931	C₂²	semidirect
16. C₄×Dic₁₅₇	C₂²	semidirect
18. C₂×Q₈×C₁₅₇	C₂²	semidirect
20. C₂×D₄×C₁₅₇	C₂²	semidirect
22. C₂²×C₄×C₁₅₇	C₂²	semidirect
36. C₂×C₄×C₁₅₇	D₄	semidirect
57. C₁₅₇	#535	semidirect
59. C₂×#41	D₁₅₇	semidirect
68. C₂×Q₈	C₂×D₁₅₇	semidirect
74. C₂×D₄	C₂×D₁₅₇	semidirect
76. C₂²×C₄	C₂×D₁₅₇	semidirect
94. C₂×C₄	C₂²⋊D₁₅₇	semidirect

H ≅ G/N

type

1. C₂²×D₆₂₈

C₂

semidirect

2. #19097767

C₂

semidirect

3. #19097760

C₂

semidirect

4. #19097745

C₂

semidirect

5. #19097739

C₂

semidirect

6. #19097730

C₂

semidirect

7. C₂×#41×C₁₅₇

C₂

semidirect

12. C₂×C₂²⋊D₁₅₇

C₂²

semidirect

13. C₄⋊Dic₁₅₇

C₂²

semidirect

15. #14578931

C₂²

semidirect

16. C₄×Dic₁₅₇

C₂²

semidirect

18. C₂×Q₈×C₁₅₇

C₂²

semidirect

20. C₂×D₄×C₁₅₇

C₂²

semidirect

22. C₂²×C₄×C₁₅₇

C₂²

semidirect

36. C₂×C₄×C₁₅₇

D₄

semidirect

57. C₁₅₇

#535

semidirect

59. C₂×#41

D₁₅₇

semidirect

68. C₂×Q₈

C₂×D₁₅₇

semidirect

74. C₂×D₄

C₂×D₁₅₇

semidirect

76. C₂²×C₄

C₂×D₁₅₇

semidirect

94. C₂×C₄

C₂²⋊D₁₅₇

semidirect

Subgroups — #23757837

Conjugacy classes of subgroups

Complements