Group #23757762 — subgroups

Subgroups — C₂×#19097742 #23757762

Order 10048.

426 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 135 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×C₂²⋊D₁₅₇	C₂	1	2	yes	maximal, complemented
2	C₂²×C₄×D₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#19097742	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊Dic₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊Dic₁₅₇	C₂	1	2	yes	maximal, complemented
6	C₂²×D₄×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₅₇	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊D₁₅₇	—	2	8	no	—
9	C₂×C₂²⋊D₁₅₇	C₂²	1	8	yes	complemented
10	C₂²×Dic₁₅₇	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₅₇	C₂²	1	2	yes	—
12	C₂×C₄×D₁₅₇	—	2	4	no	—
13	C₂×C₄×D₁₅₇	C₂²	1	4	yes	complemented
14	C₄⋊Dic₁₅₇	C₂²	1	4	yes	complemented
15	C₂²⋊Dic₁₅₇	C₂²	1	8	yes	complemented
16	C₂⁴×C₁₅₇	C₂²	1	2	yes	—
17	C₂×D₄×C₁₅₇	—	2	4	no	—
18	C₂×D₄×C₁₅₇	C₂²	1	4	yes	complemented
19	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
20	C₂²×D₁₅₇	—	2	2	no	—
21	C₂²×D₁₅₇	—	2	2	no	—
22	C₂²×D₁₅₇	D₄	1	4	yes	complemented
23	C₂²×D₁₅₇	C₂³	1	2	yes	—
24	C₂²⋊D₁₅₇	—	2	16	no	—
25	C₂×Dic₁₅₇	—	2	2	no	—
26	C₂×Dic₁₅₇	—	2	4	no	—
27	C₂×Dic₁₅₇	C₂³	1	2	yes	—
28	C₂×Dic₁₅₇	C₂³	1	4	yes	—
29	C₄×D₁₅₇	—	2	8	no	—
30	C₂³×C₁₅₇	—	2	4	no	—
31	C₂³×C₁₅₇	—	2	4	no	—
32	C₂³×C₁₅₇	—	2	4	no	—
33	C₂³×C₁₅₇	C₂³	1	4	yes	—
34	C₂³×C₁₅₇	C₂³	1	1	yes	characteristic
35	D₄×C₁₅₇	—	2	8	no	—
36	C₂×C₄×C₁₅₇	D₄	1	4	yes	complemented
37	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
38	C₂²×C₁₅₇	—	2	4	no	—
39	C₂²×C₁₅₇	—	2	4	no	—
40	C₂²×C₁₅₇	—	2	4	no	—
41	C₂²×C₁₅₇	—	4	8	no	—
42	C₂²×C₁₅₇	C₂×D₄	1	2	yes	—
43	C₂²×C₁₅₇	C₂×D₄	1	2	yes	—
44	C₂²×C₁₅₇	#41	1	2	yes	—
45	C₂²×C₁₅₇	C₂⁴	1	1	yes	characteristic
46	C₂×D₁₅₇	—	2	8	no	—
47	C₂×D₁₅₇	—	2	2	no	—
48	C₂×D₁₅₇	—	2	2	no	—
49	C₂×D₁₅₇	C₂×D₄	1	4	yes	complemented
50	C₄×C₁₅₇	C₂×D₄	1	4	yes	complemented
51	Dic₁₅₇	—	2	2	no	—
52	Dic₁₅₇	—	2	4	no	—
53	C₂×C₁₅₇	—	4	4	no	—
54	C₂×C₁₅₇	#121	1	4	yes	complemented
55	C₂×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
56	C₂×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
57	C₂×C₁₅₇	C₂×#41	1	1	yes	characteristic
58	D₁₅₇	—	2	4	no	—
59	C₁₅₇	C₂×#121	1	1	yes	characteristic, complemented, Sylow
60	C₂×#121	—	157	1	no	maximal, Sylow
61	C₂²×D₄	—	157	2	no	—
62	C₂²×D₄	D₁₅₇	1	1	yes	characteristic, complemented
63	C₂³×C₄	—	157	1	no	—
64	#121	—	157	8	no	—
65	C₂×C₄⋊C₄	—	157	1	no	—
66	C₂×C₂²⋊C₄	—	157	2	no	—
67	C₂⁴	—	157	1	no	—
68	C₂⁴	C₂×D₁₅₇	1	2	yes	—
69	C₂×D₄	—	2	4	no	—
70	C₂×D₄	—	314	8	no	—
71	C₂×D₄	—	157	8	no	—
72	C₂×D₄	C₂×D₁₅₇	1	4	yes	complemented
73	C₂²×C₄	—	314	4	no	—
74	C₂²×C₄	—	157	1	no	—
75	C₂²×C₄	—	157	2	no	—
76	C₂²×C₄	—	157	4	no	—
77	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
78	C₄⋊C₄	—	157	4	no	—
79	C₂²⋊C₄	—	157	8	no	—
80	C₂³	—	2	4	no	—
81	C₂³	—	2	4	no	—
82	C₂³	—	2	4	no	—
83	C₂³	—	314	2	no	—
84	C₂³	—	314	2	no	—
85	C₂³	—	157	4	no	—
86	C₂³	—	157	2	no	—
87	C₂³	C₂²×D₁₅₇	1	4	yes	—
88	C₂³	C₂²×D₁₅₇	1	1	yes	characteristic
89	D₄	—	2	8	no	—
90	D₄	—	314	16	no	—
91	C₂×C₄	—	314	4	no	—
92	C₂×C₄	—	314	2	no	—
93	C₂×C₄	—	314	8	no	—
94	C₂×C₄	—	157	2	no	—
95	C₂×C₄	—	157	4	no	—
96	C₂×C₄	C₂²⋊D₁₅₇	1	4	yes	complemented
97	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
98	C₂²	—	2	4	no	—
99	C₂²	—	2	4	no	—
100	C₂²	—	2	4	no	—
101	C₂²	—	4	8	no	—
102	C₂²	—	314	8	no	—
103	C₂²	—	314	2	no	—
104	C₂²	—	314	2	no	—
105	C₂²	—	157	4	no	—
106	C₂²	D₄×D₁₅₇	1	2	yes	—
107	C₂²	C₂×C₂²⋊D₁₅₇	1	2	yes	—
108	C₂²	#14578953	1	2	yes	—
109	C₂²	C₂³×D₁₅₇	1	1	yes	characteristic
110	C₄	—	314	4	no	—
111	C₄	—	314	2	no	—
112	C₄	C₂×C₂²⋊D₁₅₇	1	4	yes	complemented
113	C₂	—	4	4	no	—
114	C₂	—	314	4	no	—
115	C₂	#19097742	1	4	yes	direct product factor
116	C₂	C₂×D₄×D₁₅₇	1	1	yes	characteristic
117	C₂	C₂²×C₂²⋊D₁₅₇	1	1	yes	characteristic
118	C₂	C₂×#14578953	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₂²⋊D₁₅₇

C₂

yes

maximal, complemented

C₂²×C₄×D₁₅₇

C₂

yes

maximal, characteristic, complemented

#19097742

C₂

yes

maximal, direct product factor

C₂×C₄⋊Dic₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Dic₁₅₇

C₂

yes

maximal, complemented

C₂²×D₄×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₅₇

C₂²

yes

characteristic

C₂×C₂²⋊D₁₅₇

—

C₂×C₂²⋊D₁₅₇

C₂²

yes

complemented

C₂²×Dic₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

C₂²

yes

—

C₂×C₄×D₁₅₇

—

C₂×C₄×D₁₅₇

C₂²

yes

complemented

C₄⋊Dic₁₅₇

C₂²

yes

complemented

C₂²⋊Dic₁₅₇

C₂²

yes

complemented

C₂⁴×C₁₅₇

C₂²

yes

—

C₂×D₄×C₁₅₇

—

C₂×D₄×C₁₅₇

C₂²

yes

complemented

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

C₂²×D₁₅₇

—

C₂²×D₁₅₇

—

C₂²×D₁₅₇

D₄

yes

complemented

C₂²×D₁₅₇

C₂³

yes

—

C₂²⋊D₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

—

C₄×D₁₅₇

—

C₂³×C₁₅₇

—

C₂³×C₁₅₇

—

C₂³×C₁₅₇

—

C₂³×C₁₅₇

C₂³

yes

—

C₂³×C₁₅₇

C₂³

yes

characteristic

D₄×C₁₅₇

—

C₂×C₄×C₁₅₇

D₄

yes

complemented

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

#41

yes

—

C₂²×C₁₅₇

C₂⁴

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

C₂×D₄

yes

complemented

C₄×C₁₅₇

C₂×D₄

yes

complemented

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#121

yes

complemented

C₂×C₁₅₇

C₂²×D₄

yes

characteristic

C₂×C₁₅₇

C₂²×D₄

yes

characteristic

C₂×C₁₅₇

C₂×#41

yes

characteristic

D₁₅₇

—

C₁₅₇

C₂×#121

yes

characteristic, complemented, Sylow

C₂×#121

—

157

maximal, Sylow

C₂²×D₄

—

157

—

C₂²×D₄

D₁₅₇

yes

characteristic, complemented

C₂³×C₄

—

157

—

#121

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₂²⋊C₄

—

157

—

C₂⁴

—

157

—

C₂⁴

C₂×D₁₅₇

yes

—

C₂×D₄

—

C₂×D₄

—

314

—

C₂×D₄

—

157

—

C₂×D₄

C₂×D₁₅₇

yes

complemented

C₂²×C₄

—

314

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

C₄⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

314

—

C₂³

—

314

—

C₂³

—

157

—

C₂³

—

157

—

C₂³

C₂²×D₁₅₇

yes

—

C₂³

C₂²×D₁₅₇

yes

characteristic

D₄

—

D₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

C₂²⋊D₁₅₇

yes

complemented

C₂×C₄

C₂²×D₁₅₇

yes

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

314

—

103

C₂²

—

314

—

104

C₂²

—

314

—

105

C₂²

—

157

—

106

C₂²

D₄×D₁₅₇

yes

—

107

C₂²

C₂×C₂²⋊D₁₅₇

yes

—

108

C₂²

#14578953

yes

—

109

C₂²

C₂³×D₁₅₇

yes

characteristic

110

C₄

—

314

—

111

C₄

—

314

—

112

C₄

C₂×C₂²⋊D₁₅₇

yes

complemented

113

C₂

—

114

C₂

—

314

—

115

C₂

#19097742

yes

direct product factor

116

C₂

C₂×D₄×D₁₅₇

yes

characteristic

117

C₂

C₂²×C₂²⋊D₁₅₇

yes

characteristic

118

C₂

C₂×#14578953

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₅₇
commutator, cosocle	C₂²×C₁₅₇	C₂⁴
Frattini	C₂²	C₂³×D₁₅₇
Fitting	C₂²×D₄×C₁₅₇	C₂
socle	C₂³×C₁₅₇	C₂³
hypercenter	C₂²×D₄	D₁₅₇
hypocenter	C₁₅₇	C₂×#121
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

subgroup

G/H

center

C₂³

C₂²×D₁₅₇

commutator, cosocle

C₂²×C₁₅₇

C₂⁴

Frattini

C₂²

C₂³×D₁₅₇

Fitting

C₂²×D₄×C₁₅₇

C₂

socle

C₂³×C₁₅₇

C₂³

hypercenter

C₂²×D₄

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂³ ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂²×C₂²⋊D₁₅₇	C₂	semidirect
2. C₂²×C₄×D₁₅₇	C₂	semidirect
3. #19097742	C₂	direct (N × H)
4. C₂×C₄⋊Dic₁₅₇	C₂	semidirect
5. C₂×C₂²⋊Dic₁₅₇	C₂	semidirect
6. C₂²×D₄×C₁₅₇	C₂	semidirect
9. C₂×C₂²⋊D₁₅₇	C₂²	semidirect
13. C₂×C₄×D₁₅₇	C₂²	semidirect
14. C₄⋊Dic₁₅₇	C₂²	semidirect
15. C₂²⋊Dic₁₅₇	C₂²	semidirect
18. C₂×D₄×C₁₅₇	C₂²	semidirect
22. C₂²×D₁₅₇	D₄	semidirect
36. C₂×C₄×C₁₅₇	D₄	semidirect
49. C₂×D₁₅₇	C₂×D₄	semidirect
50. C₄×C₁₅₇	C₂×D₄	semidirect
54. C₂×C₁₅₇	#121	semidirect
59. C₁₅₇	C₂×#121	semidirect
62. C₂²×D₄	D₁₅₇	semidirect
72. C₂×D₄	C₂×D₁₅₇	semidirect
96. C₂×C₄	C₂²⋊D₁₅₇	semidirect
112. C₄	C₂×C₂²⋊D₁₅₇	semidirect
115. C₂	#19097742	direct (N × H)

H ≅ G/N

type

1. C₂²×C₂²⋊D₁₅₇

C₂

semidirect

2. C₂²×C₄×D₁₅₇

C₂

semidirect

3. #19097742

C₂

direct (N × H)

4. C₂×C₄⋊Dic₁₅₇

C₂

semidirect

5. C₂×C₂²⋊Dic₁₅₇

C₂

semidirect

6. C₂²×D₄×C₁₅₇

C₂

semidirect

9. C₂×C₂²⋊D₁₅₇

C₂²

semidirect

13. C₂×C₄×D₁₅₇

C₂²

semidirect

14. C₄⋊Dic₁₅₇

C₂²

semidirect

15. C₂²⋊Dic₁₅₇

C₂²

semidirect

18. C₂×D₄×C₁₅₇

C₂²

semidirect

22. C₂²×D₁₅₇

D₄

semidirect

36. C₂×C₄×C₁₅₇

D₄

semidirect

49. C₂×D₁₅₇

C₂×D₄

semidirect

50. C₄×C₁₅₇

C₂×D₄

semidirect

54. C₂×C₁₅₇

#121

semidirect

59. C₁₅₇

C₂×#121

semidirect

62. C₂²×D₄

D₁₅₇

semidirect

72. C₂×D₄

C₂×D₁₅₇

semidirect

96. C₂×C₄

C₂²⋊D₁₅₇

semidirect

112. C₄

C₂×C₂²⋊D₁₅₇

semidirect

115. C₂

#19097742

direct (N × H)

Subgroups — C₂×#19097742 #23757762

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19097742 #23757762

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19097742 #23757762