Group #23752782 — subgroups

Subgroups — #23752782

Order 9664.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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	1	yes	maximal, characteristic
2	C₂×Q₈×D₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#18012739	C₂	1	1	yes	maximal, characteristic, complemented
4	#18012737	C₂	1	1	yes	maximal, characteristic, complemented
5	#18012726	C₂	1	1	yes	maximal, characteristic, complemented
6	#18012721	C₂	1	4	yes	maximal, complemented
7	#18012716	C₂	1	4	yes	maximal, complemented
8	#18012696	C₂	1	1	yes	maximal, characteristic, complemented
9	#124×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#14559003	—	2	4	no	—
11	C₂×Dic₃₀₂	C₂²	1	2	yes	—
12	C₂×Dic₃₀₂	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₀₂	C₂²	1	1	yes	characteristic
14	Q₈×D₁₅₁	—	2	4	no	—
15	C₂×C₂²⋊D₁₅₁	C₂²	1	2	yes	—
16	C₂²×Dic₁₅₁	C₂²	1	2	yes	—
17	C₂×C₄×D₁₅₁	C₂²	1	2	yes	—
18	C₂×C₄×D₁₅₁	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₁₅₁	C₂²	1	4	yes	—
20	Dic₁₅₁⋊C₄	C₂²	1	2	yes	—
21	Dic₁₅₁⋊C₄	C₂²	1	4	yes	—
22	C₂²⋊Dic₁₅₁	C₂²	1	4	yes	—
23	(C₂×D₁₅₁)⋊C₄	C₂²	1	2	yes	—
24	C₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
26	C₂×D₄×C₁₅₁	C₂²	1	1	yes	characteristic
27	C₂²⋊C₄×C₁₅₁	C₂²	1	4	yes	—
28	C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
29	C₂²×D₁₅₁	C₂³	1	1	yes	characteristic
30	Dic₃₀₂	—	2	2	no	—
31	Dic₃₀₂	—	2	2	no	—
32	Dic₃₀₂	—	2	4	no	—
33	C₂²⋊D₁₅₁	—	2	4	no	—
34	C₂×Dic₁₅₁	—	2	4	no	—
35	C₂×Dic₁₅₁	C₂³	1	2	yes	—
36	C₂×Dic₁₅₁	C₂³	1	2	yes	—
37	C₂×Dic₁₅₁	C₂³	1	2	yes	—
38	C₂×Dic₁₅₁	C₂³	1	1	yes	characteristic
39	C₄×D₁₅₁	—	2	4	no	—
40	C₄×D₁₅₁	D₄	1	4	yes	—
41	C₂³×C₁₅₁	C₂³	1	2	yes	—
42	Q₈×C₁₅₁	—	2	2	no	—
43	D₄×C₁₅₁	—	2	2	no	—
44	C₂×C₄×C₁₅₁	C₂³	1	2	yes	—
45	C₂×C₄×C₁₅₁	C₂³	1	2	yes	—
46	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
47	C₂²×C₁₅₁	—	2	2	no	—
48	C₂²×C₁₅₁	—	2	4	no	—
49	C₂²×C₁₅₁	C₂⁴	1	1	yes	characteristic
50	C₂×D₁₅₁	—	2	2	no	—
51	C₂×D₁₅₁	C₂×D₄	1	2	yes	—
52	C₄×C₁₅₁	—	2	2	no	—
53	C₄×C₁₅₁	—	2	2	no	—
54	C₄×C₁₅₁	C₂×D₄	1	2	yes	—
55	Dic₁₅₁	—	2	2	no	—
56	Dic₁₅₁	—	2	2	no	—
57	Dic₁₅₁	—	2	2	no	—
58	Dic₁₅₁	C₂×D₄	1	2	yes	—
59	C₂×C₁₅₁	—	4	2	no	—
60	C₂×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
61	C₂×C₁₅₁	2₋¹⁺⁴	1	2	yes	—
62	D₁₅₁	—	2	2	no	—
63	C₁₅₁	#536	1	1	yes	characteristic, complemented, Sylow
64	#536	—	151	1	no	maximal, Sylow
65	C₂×#41	—	151	1	no	—
66	C₂²×Q₈	—	151	1	no	—
67	C₄⋊Q₈	—	151	1	no	—
68	C₄⋊Q₈	—	151	1	no	—
69	#124	—	151	1	no	—
70	#124	D₁₅₁	1	1	yes	characteristic, complemented
71	#123	—	151	4	no	—
72	C₂²⋊Q₈	—	151	4	no	—
73	#117	—	151	1	no	—
74	#41	—	302	4	no	—
75	C₂×Q₈	—	302	4	no	—
76	C₂×Q₈	—	151	2	no	—
77	C₂×Q₈	—	151	1	no	—
78	C₂×Q₈	—	151	1	no	—
79	C₂×Q₈	C₂×D₁₅₁	1	1	yes	characteristic
80	C₂×D₄	—	151	2	no	—
81	C₂×D₄	C₂×D₁₅₁	1	1	yes	characteristic
82	C₂²×C₄	—	151	2	no	—
83	C₂²×C₄	—	151	2	no	—
84	C₂²×C₄	—	151	1	no	—
85	C₄⋊C₄	—	151	2	no	—
86	C₄⋊C₄	—	151	4	no	—
87	C₄⋊C₄	—	151	4	no	—
88	C₂²⋊C₄	—	151	4	no	—
89	C₂²⋊C₄	—	151	2	no	—
90	C₂²⋊C₄	C₂×D₁₅₁	1	4	yes	—
91	C₄²	—	151	1	no	—
92	C₄²	C₂×D₁₅₁	1	1	yes	characteristic
93	C₂³	—	151	1	no	—
94	C₂³	C₂²×D₁₅₁	1	2	yes	—
95	Q₈	—	2	2	no	—
96	Q₈	—	302	4	no	—
97	Q₈	—	302	2	no	—
98	Q₈	—	302	2	no	—
99	D₄	—	2	2	no	—
100	D₄	—	302	4	no	—
101	C₂×C₄	—	302	4	no	—
102	C₂×C₄	—	302	4	no	—
103	C₂×C₄	—	151	2	no	—
104	C₂×C₄	—	151	2	no	—
105	C₂×C₄	—	151	2	no	—
106	C₂×C₄	—	151	1	no	—
107	C₂×C₄	—	151	4	no	—
108	C₂×C₄	C₂²×D₁₅₁	1	2	yes	—
109	C₂×C₄	C₂²×D₁₅₁	1	2	yes	—
110	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	302	2	no	—
114	C₂²	—	151	2	no	—
115	C₂²	C₂³×D₁₅₁	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	302	2	no	—
119	C₄	—	302	2	no	—
120	C₄	—	302	2	no	—
121	C₄	—	151	2	no	—
122	C₄	D₄×D₁₅₁	1	2	yes	—
123	C₂	—	4	2	no	—
124	C₂	—	302	2	no	—
125	C₂	C₂×D₄×D₁₅₁	1	1	yes	characteristic
126	C₂	#18012786	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14559003

C₂

yes

maximal, characteristic

C₂×Q₈×D₁₅₁

C₂

yes

maximal, characteristic, complemented

#18012739

C₂

yes

maximal, characteristic, complemented

#18012737

C₂

yes

maximal, characteristic, complemented

#18012726

C₂

yes

maximal, characteristic, complemented

#18012721

C₂

yes

maximal, complemented

#18012716

C₂

yes

maximal, complemented

#18012696

C₂

yes

maximal, characteristic, complemented

#124×C₁₅₁

C₂

yes

maximal, characteristic, complemented

#14559003

—

C₂×Dic₃₀₂

C₂²

yes

—

C₂×Dic₃₀₂

C₂²

yes

characteristic

C₂×Dic₃₀₂

C₂²

yes

characteristic

Q₈×D₁₅₁

—

C₂×C₂²⋊D₁₅₁

C₂²

yes

—

C₂²×Dic₁₅₁

C₂²

yes

—

C₂×C₄×D₁₅₁

C₂²

yes

—

C₂×C₄×D₁₅₁

C₂²

yes

characteristic

C₄⋊Dic₁₅₁

C₂²

yes

—

Dic₁₅₁⋊C₄

C₂²

yes

—

Dic₁₅₁⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₅₁

C₂²

yes

—

(C₂×D₁₅₁)⋊C₄

C₂²

yes

—

C₄×Dic₁₅₁

C₂²

yes

characteristic

C₂×Q₈×C₁₅₁

C₂²

yes

characteristic

C₂×D₄×C₁₅₁

C₂²

yes

characteristic

C₂²⋊C₄×C₁₅₁

C₂²

yes

—

C₄²×C₁₅₁

C₂²

yes

characteristic

C₂²×D₁₅₁

C₂³

yes

characteristic

Dic₃₀₂

—

Dic₃₀₂

—

Dic₃₀₂

—

C₂²⋊D₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

C₂³

yes

characteristic

C₄×D₁₅₁

—

C₄×D₁₅₁

D₄

yes

—

C₂³×C₁₅₁

C₂³

yes

—

Q₈×C₁₅₁

—

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂³

yes

—

C₂×C₄×C₁₅₁

C₂³

yes

—

C₂×C₄×C₁₅₁

C₂³

yes

characteristic

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂⁴

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

C₂×D₄

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×D₄

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

C₂×D₄

yes

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₂²×D₄

yes

characteristic

C₂×C₁₅₁

2₋¹⁺⁴

yes

—

D₁₅₁

—

C₁₅₁

#536

yes

characteristic, complemented, Sylow

#536

—

151

maximal, Sylow

C₂×#41

—

151

—

C₂²×Q₈

—

151

—

C₄⋊Q₈

—

151

—

C₄⋊Q₈

—

151

—

#124

—

151

—

#124

D₁₅₁

yes

characteristic, complemented

#123

—

151

—

C₂²⋊Q₈

—

151

—

#117

—

151

—

#41

—

302

—

C₂×Q₈

—

302

—

C₂×Q₈

—

151

—

C₂×Q₈

—

151

—

C₂×Q₈

—

151

—

C₂×Q₈

C₂×D₁₅₁

yes

characteristic

C₂×D₄

—

151

—

C₂×D₄

C₂×D₁₅₁

yes

characteristic

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₂²⋊C₄

—

151

—

C₂²⋊C₄

—

151

—

C₂²⋊C₄

C₂×D₁₅₁

yes

—

C₄²

—

151

—

C₄²

C₂×D₁₅₁

yes

characteristic

C₂³

—

151

—

C₂³

C₂²×D₁₅₁

yes

—

Q₈

—

Q₈

—

302

—

Q₈

—

302

—

Q₈

—

302

—

D₄

—

100

D₄

—

302

—

101

C₂×C₄

—

302

—

102

C₂×C₄

—

302

—

103

C₂×C₄

—

151

—

104

C₂×C₄

—

151

—

105

C₂×C₄

—

151

—

106

C₂×C₄

—

151

—

107

C₂×C₄

—

151

—

108

C₂×C₄

C₂²×D₁₅₁

yes

—

109

C₂×C₄

C₂²×D₁₅₁

yes

—

110

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

302

—

114

C₂²

—

151

—

115

C₂²

C₂³×D₁₅₁

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

302

—

119

C₄

—

302

—

120

C₄

—

302

—

121

C₄

—

151

—

122

C₄

D₄×D₁₅₁

yes

—

123

C₂

—

124

C₂

—

302

—

125

C₂

C₂×D₄×D₁₅₁

yes

characteristic

126

C₂

#18012786

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅₁
commutator, socle, cosocle	C₂²×C₁₅₁	C₂⁴
Fitting	#124×C₁₅₁	C₂
prime radical	C₂×#14559003	C₂
hypercenter	#124	D₁₅₁
hypocenter	C₁₅₁	#536
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅₁

commutator, socle, cosocle

C₂²×C₁₅₁

C₂⁴

Fitting

#124×C₁₅₁

C₂

C₂

D₁₅₁

C₁₅₁

G ▷ C₂²×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
2. C₂×Q₈×D₁₅₁	C₂	semidirect
3. #18012739	C₂	semidirect
4. #18012737	C₂	semidirect
5. #18012726	C₂	semidirect
6. #18012721	C₂	semidirect
7. #18012716	C₂	semidirect
8. #18012696	C₂	semidirect
9. #124×C₁₅₁	C₂	semidirect
63. C₁₅₁	#536	semidirect
70. #124	D₁₅₁	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₁₅₁

C₂

semidirect

3. #18012739

C₂

semidirect

4. #18012737

C₂

semidirect

5. #18012726

C₂

semidirect

6. #18012721

C₂

semidirect

7. #18012716

C₂

semidirect

8. #18012696

C₂

semidirect

9. #124×C₁₅₁

C₂

semidirect

63. C₁₅₁

#536

semidirect

70. #124

D₁₅₁

semidirect

Subgroups — #23752782

Conjugacy classes of subgroups

Complements