Group #23752882 — subgroups

Subgroups — #23752882

Order 9664.

312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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	C₂×#14559003	C₂	1	2	yes	maximal, complemented
2	#18012735	C₂	1	2	yes	maximal, complemented
3	#18012711	C₂	1	4	yes	maximal, complemented
4	#18012710	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₃₀₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₆₀₄	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×Dic₁₅₁	C₂	1	2	yes	maximal, complemented
8	#127×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#14559003	—	2	8	no	—
10	C₂×Dic₃₀₂	C₂²	1	1	yes	characteristic
11	C₂×D₆₀₄	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₅₁	C₂²	1	4	yes	—
13	C₂×C₂²⋊D₁₅₁	C₂²	1	4	yes	—
14	C₂²×Dic₁₅₁	C₂²	1	4	yes	—
15	C₂×C₄×D₁₅₁	C₂²	1	2	yes	—
16	C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
18	Dic₁₅₁⋊C₄	C₂²	1	2	yes	—
19	C₂²⋊Dic₁₅₁	C₂²	1	4	yes	—
20	(C₂×D₁₅₁)⋊C₄	C₂²	1	2	yes	—
21	C₄×Dic₁₅₁	C₂²	1	2	yes	—
22	C₂×D₄×C₁₅₁	C₂²	1	2	yes	—
23	C₂×D₄×C₁₅₁	C₂²	1	2	yes	—
24	C₂×D₄×C₁₅₁	C₂²	1	2	yes	—
25	C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
26	C₂²×D₁₅₁	C₂³	1	2	yes	—
27	Dic₃₀₂	D₄	1	4	yes	complemented
28	D₆₀₄	—	2	2	no	—
29	C₂²⋊D₁₅₁	—	2	8	no	—
30	C₂²⋊D₁₅₁	—	4	4	no	—
31	C₂×Dic₁₅₁	—	2	8	no	—
32	C₂×Dic₁₅₁	C₂³	1	2	yes	—
33	C₂×Dic₁₅₁	C₂³	1	2	yes	—
34	C₂×Dic₁₅₁	C₂³	1	2	yes	—
35	C₄×D₁₅₁	—	2	4	no	—
36	C₂³×C₁₅₁	C₂³	1	4	yes	—
37	D₄×C₁₅₁	—	2	4	no	—
38	D₄×C₁₅₁	—	2	4	no	—
39	D₄×C₁₅₁	—	4	2	no	—
40	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
43	C₂²×C₁₅₁	—	2	4	no	—
44	C₂²×C₁₅₁	—	2	4	no	—
45	C₂²×C₁₅₁	—	2	4	no	—
46	C₂²×C₁₅₁	C₂⁴	1	1	yes	characteristic
47	C₂×D₁₅₁	—	2	2	no	—
48	C₂×D₁₅₁	—	2	2	no	—
49	C₂×D₁₅₁	—	2	2	no	—
50	C₄×C₁₅₁	—	2	1	no	—
51	C₄×C₁₅₁	C₂×D₄	1	2	yes	—
52	C₄×C₁₅₁	#41	1	2	yes	complemented
53	Dic₁₅₁	—	2	2	no	—
54	Dic₁₅₁	—	2	2	no	—
55	Dic₁₅₁	C₂×D₄	1	4	yes	—
56	C₂×C₁₅₁	—	4	4	no	—
57	C₂×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
58	C₂×C₁₅₁	C₂×#41	1	1	yes	characteristic
59	C₂×C₁₅₁	H(5,𝔽₂)	1	1	yes	characteristic
60	D₁₅₁	—	4	2	no	—
61	C₁₅₁	#550	1	1	yes	characteristic, complemented, Sylow
62	#550	—	151	1	no	maximal, Sylow
63	C₂×#41	—	151	2	no	—
64	#127	—	151	2	no	—
65	#127	D₁₅₁	1	1	yes	characteristic, complemented
66	#121	—	151	2	no	—
67	#121	—	151	4	no	—
68	C₄×Q₈	—	151	1	no	—
69	C₄×D₄	—	151	2	no	—
70	C₄×D₄	—	151	1	no	—
71	#41	—	302	8	no	—
72	C₂×Q₈	—	151	1	no	—
73	C₂×D₄	—	151	4	no	—
74	C₂×D₄	—	151	4	no	—
75	C₂×D₄	—	151	1	no	—
76	C₂×D₄	C₂×D₁₅₁	1	2	yes	—
77	C₂×D₄	C₂×D₁₅₁	1	2	yes	—
78	C₂×D₄	C₂×D₁₅₁	1	2	yes	—
79	C₂²×C₄	—	151	4	no	—
80	C₂²×C₄	—	151	2	no	—
81	C₄⋊C₄	—	151	2	no	—
82	C₄⋊C₄	—	151	1	no	—
83	C₄⋊C₄	—	151	1	no	—
84	C₂²⋊C₄	—	151	4	no	—
85	C₂²⋊C₄	—	151	2	no	—
86	C₄²	—	151	2	no	—
87	C₄²	C₂×D₁₅₁	1	1	yes	characteristic
88	C₂³	—	151	2	no	—
89	C₂³	C₂²×D₁₅₁	1	4	yes	—
90	Q₈	—	151	4	no	—
91	D₄	—	2	4	no	—
92	D₄	—	2	4	no	—
93	D₄	—	4	2	no	—
94	D₄	—	302	8	no	—
95	D₄	—	302	2	no	—
96	D₄	—	604	4	no	—
97	C₂×C₄	—	302	8	no	—
98	C₂×C₄	—	302	4	no	—
99	C₂×C₄	—	151	2	no	—
100	C₂×C₄	—	151	2	no	—
101	C₂×C₄	—	151	2	no	—
102	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
103	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
105	C₂²	—	2	4	no	—
106	C₂²	—	2	4	no	—
107	C₂²	—	2	4	no	—
108	C₂²	—	302	2	no	—
109	C₂²	—	302	2	no	—
110	C₂²	—	302	2	no	—
111	C₂²	C₂³×D₁₅₁	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	302	2	no	—
114	C₄	—	302	2	no	—
115	C₄	—	151	4	no	—
116	C₄	D₄×D₁₅₁	1	2	yes	—
117	C₄	#14559003	1	2	yes	complemented
118	C₂	—	4	4	no	—
119	C₂	—	604	2	no	—
120	C₂	C₂×D₄×D₁₅₁	1	1	yes	characteristic
121	C₂	C₂×#14559003	1	1	yes	characteristic
122	C₂	#18012784	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14559003

C₂

yes

maximal, complemented

#18012735

C₂

yes

maximal, complemented

#18012711

C₂

yes

maximal, complemented

#18012710

C₂

yes

maximal, complemented

C₄×Dic₃₀₂

C₂

yes

maximal, characteristic, complemented

C₄×D₆₀₄

C₂

yes

maximal, characteristic, complemented

D₄×Dic₁₅₁

C₂

yes

maximal, complemented

#127×C₁₅₁

C₂

yes

maximal, characteristic, complemented

#14559003

—

C₂×Dic₃₀₂

C₂²

yes

characteristic

C₂×D₆₀₄

C₂²

yes

characteristic

C₂×C₂²⋊D₁₅₁

C₂²

yes

—

C₂×C₂²⋊D₁₅₁

C₂²

yes

—

C₂²×Dic₁₅₁

C₂²

yes

—

C₂×C₄×D₁₅₁

C₂²

yes

—

C₄⋊Dic₁₅₁

C₂²

yes

characteristic

C₄⋊Dic₁₅₁

C₂²

yes

characteristic

Dic₁₅₁⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₅₁

C₂²

yes

—

(C₂×D₁₅₁)⋊C₄

C₂²

yes

—

C₄×Dic₁₅₁

C₂²

yes

—

C₂×D₄×C₁₅₁

C₂²

yes

—

C₂×D₄×C₁₅₁

C₂²

yes

—

C₂×D₄×C₁₅₁

C₂²

yes

—

C₄²×C₁₅₁

C₂²

yes

characteristic

C₂²×D₁₅₁

C₂³

yes

—

Dic₃₀₂

D₄

yes

complemented

D₆₀₄

—

C₂²⋊D₁₅₁

—

C₂²⋊D₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

C₂³

yes

—

C₄×D₁₅₁

—

C₂³×C₁₅₁

C₂³

yes

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₄×C₁₅₁

C₂³

yes

characteristic

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂⁴

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×D₄

yes

—

C₄×C₁₅₁

#41

yes

complemented

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

C₂×D₄

yes

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₂²×D₄

yes

characteristic

C₂×C₁₅₁

C₂×#41

yes

characteristic

C₂×C₁₅₁

H(5,𝔽₂)

yes

characteristic

D₁₅₁

—

C₁₅₁

#550

yes

characteristic, complemented, Sylow

#550

—

151

maximal, Sylow

C₂×#41

—

151

—

#127

—

151

—

#127

D₁₅₁

yes

characteristic, complemented

#121

—

151

—

#121

—

151

—

C₄×Q₈

—

151

—

C₄×D₄

—

151

—

C₄×D₄

—

151

—

#41

—

302

—

C₂×Q₈

—

151

—

C₂×D₄

—

151

—

C₂×D₄

—

151

—

C₂×D₄

—

151

—

C₂×D₄

C₂×D₁₅₁

yes

—

C₂×D₄

C₂×D₁₅₁

yes

—

C₂×D₄

C₂×D₁₅₁

yes

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₂²⋊C₄

—

151

—

C₂²⋊C₄

—

151

—

C₄²

—

151

—

C₄²

C₂×D₁₅₁

yes

characteristic

C₂³

—

151

—

C₂³

C₂²×D₁₅₁

yes

—

Q₈

—

151

—

D₄

—

D₄

—

D₄

—

D₄

—

302

—

D₄

—

302

—

D₄

—

604

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

151

—

100

C₂×C₄

—

151

—

101

C₂×C₄

—

151

—

102

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

103

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

104

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

302

—

109

C₂²

—

302

—

110

C₂²

—

302

—

111

C₂²

C₂³×D₁₅₁

yes

characteristic

112

C₄

—

113

C₄

—

302

—

114

C₄

—

302

—

115

C₄

—

151

—

116

C₄

D₄×D₁₅₁

yes

—

117

C₄

#14559003

yes

complemented

118

C₂

—

119

C₂

—

604

—

120

C₂

C₂×D₄×D₁₅₁

yes

characteristic

121

C₂

C₂×#14559003

yes

characteristic

122

C₂

#18012784

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅₁
commutator, socle, cosocle	C₂²×C₁₅₁	C₂⁴
Fitting	#127×C₁₅₁	C₂
hypercenter	#127	D₁₅₁
hypocenter	C₁₅₁	#550
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ #127

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅₁

commutator, socle, cosocle

C₂²×C₁₅₁

C₂⁴

Fitting

#127×C₁₅₁

C₂

D₁₅₁

C₁₅₁

G ▷ C₂²×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂² ◁ #127

N	H ≅ G/N	type
1. C₂×#14559003	C₂	semidirect
2. #18012735	C₂	semidirect
3. #18012711	C₂	semidirect
4. #18012710	C₂	semidirect
5. C₄×Dic₃₀₂	C₂	semidirect
6. C₄×D₆₀₄	C₂	semidirect
7. D₄×Dic₁₅₁	C₂	semidirect
8. #127×C₁₅₁	C₂	semidirect
27. Dic₃₀₂	D₄	semidirect
52. C₄×C₁₅₁	#41	semidirect
61. C₁₅₁	#550	semidirect
65. #127	D₁₅₁	semidirect
117. C₄	#14559003	semidirect

H ≅ G/N

type

1. C₂×#14559003

C₂

semidirect

2. #18012735

C₂

semidirect

3. #18012711

C₂

semidirect

4. #18012710

C₂

semidirect

5. C₄×Dic₃₀₂

C₂

semidirect

6. C₄×D₆₀₄

C₂

semidirect

7. D₄×Dic₁₅₁

C₂

semidirect

8. #127×C₁₅₁

C₂

semidirect

27. Dic₃₀₂

D₄

semidirect

52. C₄×C₁₅₁

#41

semidirect

61. C₁₅₁

#550

semidirect

65. #127

D₁₅₁

semidirect

117. C₄

#14559003

semidirect

Subgroups — #23752882

Conjugacy classes of subgroups

Complements