Group #23752867 — subgroups

Subgroups — #23752867

Order 9664.

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

normal subgroups only

This group has 292 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×#14559004	C₂	1	2	yes	maximal, complemented
2	#18012739	C₂	1	1	yes	maximal, characteristic, complemented
3	#18012721	C₂	1	4	yes	maximal, complemented
4	#18012718	C₂	1	2	yes	maximal, complemented
5	#18012710	C₂	1	2	yes	maximal, complemented
6	C₄×D₆₀₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₄×D₁₅₁	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#14559004	—	2	8	no	—
10	C₂×Dic₃₀₂	C₂²	1	2	yes	—
11	C₂×D₆₀₄	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₅₁	C₂²	1	4	yes	—
13	C₂×C₄×D₁₅₁	C₂²	1	2	yes	—
14	C₂×C₄×D₁₅₁	C₂²	1	4	yes	—
15	C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
16	C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
18	Dic₁₅₁⋊C₄	C₂²	1	4	yes	—
19	C₂²⋊Dic₁₅₁	C₂²	1	4	yes	—
20	(C₂×D₁₅₁)⋊C₄	C₂²	1	2	yes	—
21	C₂×D₄×C₁₅₁	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₁₅₁	C₂²	1	2	yes	—
23	C₄⋊C₄×C₁₅₁	C₂²	1	1	yes	characteristic
24	C₂²⋊C₄×C₁₅₁	C₂²	1	2	yes	—
25	C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
26	C₂²×D₁₅₁	C₂³	1	2	yes	—
27	Dic₃₀₂	—	2	4	no	—
28	D₆₀₄	D₄	1	4	yes	complemented
29	C₂²⋊D₁₅₁	—	2	8	no	—
30	C₂×Dic₁₅₁	C₂³	1	2	yes	—
31	C₂×Dic₁₅₁	C₂³	1	4	yes	—
32	C₄×D₁₅₁	—	2	8	no	—
33	C₄×D₁₅₁	—	2	4	no	—
34	C₂³×C₁₅₁	C₂³	1	2	yes	—
35	D₄×C₁₅₁	—	2	2	no	—
36	C₂×C₄×C₁₅₁	—	2	4	no	—
37	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₅₁	C₂³	1	2	yes	—
39	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
41	C₂²×C₁₅₁	—	2	2	no	—
42	C₂²×C₁₅₁	—	2	2	no	—
43	C₂²×C₁₅₁	—	2	2	no	—
44	C₂²×C₁₅₁	C₂⁴	1	1	yes	characteristic
45	C₂×D₁₅₁	—	2	2	no	—
46	C₂×D₁₅₁	—	2	2	no	—
47	C₂×D₁₅₁	C₂×D₄	1	4	yes	—
48	C₄×C₁₅₁	—	2	2	no	—
49	C₄×C₁₅₁	—	2	1	no	—
50	C₄×C₁₅₁	C₂×D₄	1	2	yes	—
51	C₄×C₁₅₁	#41	1	2	yes	complemented
52	Dic₁₅₁	—	2	4	no	—
53	Dic₁₅₁	—	2	2	no	—
54	C₂×C₁₅₁	—	4	2	no	—
55	C₂×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
56	C₂×C₁₅₁	C₂×#41	1	1	yes	characteristic
57	C₂×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
58	D₁₅₁	—	2	4	no	—
59	C₁₅₁	#547	1	1	yes	characteristic, complemented, Sylow
60	#547	—	151	1	no	maximal, Sylow
61	C₂×#41	—	151	2	no	—
62	C₄⋊Q₈	—	151	1	no	—
63	#123	—	151	4	no	—
64	C₂²⋊Q₈	—	151	2	no	—
65	#121	—	151	2	no	—
66	C₄×D₄	—	151	1	no	—
67	C₄×D₄	D₁₅₁	1	1	yes	characteristic, complemented
68	C₂×C₄⋊C₄	—	151	2	no	—
69	#41	—	302	8	no	—
70	C₂×Q₈	—	151	2	no	—
71	C₂×D₄	—	151	4	no	—
72	C₂×D₄	—	151	1	no	—
73	C₂×D₄	C₂×D₁₅₁	1	1	yes	characteristic
74	C₂²×C₄	—	151	2	no	—
75	C₂²×C₄	—	151	4	no	—
76	C₂²×C₄	C₂×D₁₅₁	1	2	yes	—
77	C₄⋊C₄	—	151	4	no	—
78	C₄⋊C₄	—	151	2	no	—
79	C₄⋊C₄	—	151	1	no	—
80	C₄⋊C₄	—	151	2	no	—
81	C₄⋊C₄	C₂×D₁₅₁	1	1	yes	characteristic
82	C₂²⋊C₄	—	151	4	no	—
83	C₂²⋊C₄	—	151	2	no	—
84	C₂²⋊C₄	C₂×D₁₅₁	1	2	yes	—
85	C₄²	C₂×D₁₅₁	1	1	yes	characteristic
86	C₂³	—	151	2	no	—
87	C₂³	C₂²×D₁₅₁	1	2	yes	—
88	Q₈	—	302	4	no	—
89	D₄	—	2	2	no	—
90	D₄	—	302	8	no	—
91	D₄	—	151	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	302	8	no	—
94	C₂×C₄	—	302	4	no	—
95	C₂×C₄	—	151	2	no	—
96	C₂×C₄	—	151	4	no	—
97	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₁₅₁	1	2	yes	—
99	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
101	C₂²	—	2	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	2	2	no	—
104	C₂²	—	302	2	no	—
105	C₂²	—	302	2	no	—
106	C₂²	—	151	4	no	—
107	C₂²	C₂³×D₁₅₁	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	302	4	no	—
111	C₄	—	302	2	no	—
112	C₄	D₄×D₁₅₁	1	2	yes	—
113	C₄	#14559004	1	2	yes	complemented
114	C₂	—	4	2	no	—
115	C₂	—	302	4	no	—
116	C₂	C₂×D₄×D₁₅₁	1	1	yes	characteristic
117	C₂	C₂×#14559004	1	1	yes	characteristic
118	C₂	#18012786	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14559004

C₂

yes

maximal, complemented

#18012739

C₂

yes

maximal, characteristic, complemented

#18012721

C₂

yes

maximal, complemented

#18012718

C₂

yes

maximal, complemented

#18012710

C₂

yes

maximal, complemented

C₄×D₆₀₄

C₂

yes

maximal, characteristic, complemented

C₄⋊C₄×D₁₅₁

C₂

yes

maximal, complemented

C₄×D₄×C₁₅₁

C₂

yes

maximal, characteristic, complemented

#14559004

—

C₂×Dic₃₀₂

C₂²

yes

—

C₂×D₆₀₄

C₂²

yes

characteristic

C₂×C₂²⋊D₁₅₁

C₂²

yes

—

C₂×C₄×D₁₅₁

C₂²

yes

—

C₂×C₄×D₁₅₁

C₂²

yes

—

C₄⋊Dic₁₅₁

C₂²

yes

—

C₄⋊Dic₁₅₁

C₂²

yes

characteristic

C₄⋊Dic₁₅₁

C₂²

yes

—

Dic₁₅₁⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₅₁

C₂²

yes

—

(C₂×D₁₅₁)⋊C₄

C₂²

yes

—

C₂×D₄×C₁₅₁

C₂²

yes

characteristic

C₂²×C₄×C₁₅₁

C₂²

yes

—

C₄⋊C₄×C₁₅₁

C₂²

yes

characteristic

C₂²⋊C₄×C₁₅₁

C₂²

yes

—

C₄²×C₁₅₁

C₂²

yes

characteristic

C₂²×D₁₅₁

C₂³

yes

—

Dic₃₀₂

—

D₆₀₄

D₄

yes

complemented

C₂²⋊D₁₅₁

—

C₂×Dic₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

C₂³

yes

—

C₄×D₁₅₁

—

C₄×D₁₅₁

—

C₂³×C₁₅₁

C₂³

yes

—

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₄×C₁₅₁

C₂³

yes

—

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

C₂×D₄

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×D₄

yes

—

C₄×C₁₅₁

#41

yes

complemented

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₂²×D₄

yes

characteristic

C₂×C₁₅₁

C₂×#41

yes

characteristic

C₂×C₁₅₁

2₋¹⁺⁴

yes

characteristic

D₁₅₁

—

C₁₅₁

#547

yes

characteristic, complemented, Sylow

#547

—

151

maximal, Sylow

C₂×#41

—

151

—

C₄⋊Q₈

—

151

—

#123

—

151

—

C₂²⋊Q₈

—

151

—

#121

—

151

—

C₄×D₄

—

151

—

C₄×D₄

D₁₅₁

yes

characteristic, complemented

C₂×C₄⋊C₄

—

151

—

#41

—

302

—

C₂×Q₈

—

151

—

C₂×D₄

—

151

—

C₂×D₄

—

151

—

C₂×D₄

C₂×D₁₅₁

yes

characteristic

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₂×D₁₅₁

yes

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

C₂×D₁₅₁

yes

characteristic

C₂²⋊C₄

—

151

—

C₂²⋊C₄

—

151

—

C₂²⋊C₄

C₂×D₁₅₁

yes

—

C₄²

C₂×D₁₅₁

yes

characteristic

C₂³

—

151

—

C₂³

C₂²×D₁₅₁

yes

—

Q₈

—

302

—

D₄

—

D₄

—

302

—

D₄

—

151

—

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₄

C₂²×D₁₅₁

yes

—

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

100

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

302

—

105

C₂²

—

302

—

106

C₂²

—

151

—

107

C₂²

C₂³×D₁₅₁

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

302

—

111

C₄

—

302

—

112

C₄

D₄×D₁₅₁

yes

—

113

C₄

#14559004

yes

complemented

114

C₂

—

115

C₂

—

302

—

116

C₂

C₂×D₄×D₁₅₁

yes

characteristic

117

C₂

C₂×#14559004

yes

characteristic

118

C₂

#18012786

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅₁
commutator, socle, cosocle	C₂²×C₁₅₁	C₂⁴
Fitting	C₄×D₄×C₁₅₁	C₂
hypercenter	C₄×D₄	D₁₅₁
hypocenter	C₁₅₁	#547
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅₁

commutator, socle, cosocle

C₂²×C₁₅₁

C₂⁴

Fitting

C₄×D₄×C₁₅₁

C₂

hypercenter

C₄×D₄

D₁₅₁

C₁₅₁

G ▷ C₂²×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. C₂×#14559004	C₂	semidirect
2. #18012739	C₂	semidirect
3. #18012721	C₂	semidirect
4. #18012718	C₂	semidirect
5. #18012710	C₂	semidirect
6. C₄×D₆₀₄	C₂	semidirect
7. C₄⋊C₄×D₁₅₁	C₂	semidirect
8. C₄×D₄×C₁₅₁	C₂	semidirect
28. D₆₀₄	D₄	semidirect
51. C₄×C₁₅₁	#41	semidirect
59. C₁₅₁	#547	semidirect
67. C₄×D₄	D₁₅₁	semidirect
113. C₄	#14559004	semidirect

H ≅ G/N

type

1. C₂×#14559004

C₂

semidirect

2. #18012739

C₂

semidirect

3. #18012721

C₂

semidirect

4. #18012718

C₂

semidirect

5. #18012710

C₂

semidirect

6. C₄×D₆₀₄

C₂

semidirect

7. C₄⋊C₄×D₁₅₁

C₂

semidirect

8. C₄×D₄×C₁₅₁

C₂

semidirect

28. D₆₀₄

D₄

semidirect

51. C₄×C₁₅₁

#41

semidirect

59. C₁₅₁

#547

semidirect

67. C₄×D₄

D₁₅₁

semidirect

113. C₄

#14559004

semidirect

Subgroups — #23752867

Conjugacy classes of subgroups

Complements