Group #23752174 — subgroups

Subgroups — #23752174

Order 9664.

190 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 61 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×C₄×D₁₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#14558987	C₂	1	2	yes	maximal, complemented
3	C₂²⋊(C₁₅₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₁₅₁)⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₅₁	C₄	1	1	yes	characteristic
7	C₂²×Dic₁₅₁	C₄	1	1	yes	characteristic
8	C₂×C₄×D₁₅₁	—	2	4	no	—
9	C₂×C₄×D₁₅₁	C₄	1	1	yes	characteristic
10	C₂×C₄×D₁₅₁	C₄	1	1	yes	characteristic
11	C₂×C₄×D₁₅₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₅₁	C₂²	1	1	yes	characteristic
13	#14558987	—	2	4	no	—
14	C₂×C₁₅₁⋊C₈	C₂²	1	2	yes	complemented
15	C₂²×C₄×C₁₅₁	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₁₅₁	C₂²	1	2	yes	complemented
17	C₂²×D₁₅₁	—	2	2	no	—
18	C₂²×D₁₅₁	—	2	2	no	—
19	C₂²×D₁₅₁	—	2	2	no	—
20	C₂²×D₁₅₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×D₁₅₁	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₁₅₁	—	2	2	no	—
23	C₂×Dic₁₅₁	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₁₅₁	C₂×C₄	1	1	yes	characteristic
25	C₄×D₁₅₁	—	2	2	no	—
26	C₄×D₁₅₁	—	2	2	no	—
27	C₄×D₁₅₁	—	2	2	no	—
28	C₄×D₁₅₁	D₄	1	4	yes	—
29	C₁₅₁⋊C₈	—	2	2	no	—
30	C₂³×C₁₅₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₅₁	—	2	2	no	—
32	C₂×C₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
34	C₈×C₁₅₁	—	2	2	no	—
35	C₂²×C₁₅₁	—	2	1	no	—
36	C₂²×C₁₅₁	—	2	1	no	—
37	C₂²×C₁₅₁	M₁₆	1	2	yes	complemented
38	C₂²×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×D₁₅₁	—	2	8	no	—
40	C₂×D₁₅₁	—	2	1	no	—
41	C₂×D₁₅₁	—	2	1	no	—
42	C₂×D₁₅₁	—	2	1	no	—
43	C₂×D₁₅₁	—	2	1	no	—
44	C₂×D₁₅₁	C₂²⋊C₄	1	2	yes	—
45	C₂×D₁₅₁	M₁₆	1	2	yes	complemented
46	C₄×C₁₅₁	—	2	1	no	—
47	C₄×C₁₅₁	C₂×D₄	1	2	yes	—
48	Dic₁₅₁	—	2	1	no	—
49	Dic₁₅₁	C₂²⋊C₄	1	2	yes	—
50	C₂×C₁₅₁	—	2	2	no	—
51	C₂×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₁₅₁	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₁₅₁	C₂×M₁₆	1	1	yes	characteristic
54	D₁₅₁	—	2	2	no	—
55	D₁₅₁	—	2	2	no	—
56	C₁₅₁	#407	1	1	yes	characteristic, complemented, Sylow
57	#407	—	151	1	no	maximal, Sylow
58	C₂³×C₄	—	151	1	no	—
59	C₂×M₁₆	—	151	2	no	—
60	C₂²⋊C₈	—	151	1	no	—
61	C₂²⋊C₈	—	151	2	no	—
62	C₂²⋊C₈	D₁₅₁	1	1	yes	characteristic, complemented
63	C₂⁴	—	151	1	no	—
64	C₂²×C₄	—	302	4	no	—
65	C₂²×C₄	—	151	1	no	—
66	C₂²×C₄	—	151	1	no	—
67	C₂²×C₄	—	151	1	no	—
68	C₂²×C₄	—	151	1	no	—
69	C₂²×C₄	—	151	1	no	—
70	C₂²×C₄	C₂×D₁₅₁	1	1	yes	characteristic
71	M₁₆	—	302	4	no	—
72	C₂×C₈	—	151	2	no	—
73	C₂×C₈	C₂×D₁₅₁	1	2	yes	complemented
74	C₂³	—	302	2	no	—
75	C₂³	—	302	2	no	—
76	C₂³	—	302	2	no	—
77	C₂³	—	151	1	no	—
78	C₂³	—	151	1	no	—
79	C₂³	C₄×D₁₅₁	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	302	2	no	—
82	C₂×C₄	—	302	2	no	—
83	C₂×C₄	—	302	2	no	—
84	C₂×C₄	—	302	2	no	—
85	C₂×C₄	—	151	1	no	—
86	C₂×C₄	—	151	1	no	—
87	C₂×C₄	—	151	4	no	—
88	C₂×C₄	C₄×D₁₅₁	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	302	2	no	—
92	C₂²	—	2	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	302	8	no	—
95	C₂²	—	302	1	no	—
96	C₂²	—	302	1	no	—
97	C₂²	—	302	1	no	—
98	C₂²	—	302	1	no	—
99	C₂²	—	151	2	no	—
100	C₂²	—	151	2	no	—
101	C₂²	#14558987	1	2	yes	complemented
102	C₂²	C₂×C₄×D₁₅₁	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	302	1	no	—
105	C₄	—	151	2	no	—
106	C₄	D₄×D₁₅₁	1	2	yes	—
107	C₂	—	2	2	no	—
108	C₂	—	302	2	no	—
109	C₂	—	302	2	no	—
110	C₂	C₂²⋊C₄×D₁₅₁	1	1	yes	characteristic
111	C₂	M₁₆×D₁₅₁	1	1	yes	characteristic
112	C₂	C₂×#14558987	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₁₅₁

C₂

yes

maximal, characteristic

C₂×#14558987

C₂

yes

maximal, complemented

C₂²⋊(C₁₅₁⋊C₈)

C₂

yes

maximal, characteristic, complemented

(C₂×D₁₅₁)⋊C₈

C₂

yes

maximal, complemented

C₂²⋊C₈×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₅₁

C₄

yes

characteristic

C₂²×Dic₁₅₁

C₄

yes

characteristic

C₂×C₄×D₁₅₁

—

C₂×C₄×D₁₅₁

C₄

yes

characteristic

C₂×C₄×D₁₅₁

C₄

yes

characteristic

C₂×C₄×D₁₅₁

C₂²

yes

characteristic

C₂×C₄×D₁₅₁

C₂²

yes

characteristic

#14558987

—

C₂×C₁₅₁⋊C₈

C₂²

yes

complemented

C₂²×C₄×C₁₅₁

C₂²

yes

characteristic

C₂×C₈×C₁₅₁

C₂²

yes

complemented

C₂²×D₁₅₁

—

C₂²×D₁₅₁

—

C₂²×D₁₅₁

—

C₂²×D₁₅₁

C₂×C₄

yes

characteristic

C₂²×D₁₅₁

C₂×C₄

yes

characteristic

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₂×C₄

yes

characteristic

C₂×Dic₁₅₁

C₂×C₄

yes

characteristic

C₄×D₁₅₁

—

C₄×D₁₅₁

—

C₄×D₁₅₁

—

C₄×D₁₅₁

D₄

yes

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂×C₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂³

yes

characteristic

C₈×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

M₁₆

yes

complemented

C₂²×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

C₂²⋊C₄

yes

—

C₂×D₁₅₁

M₁₆

yes

complemented

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×D₄

yes

—

Dic₁₅₁

—

Dic₁₅₁

C₂²⋊C₄

yes

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₅₁

C₂×M₁₆

yes

characteristic

C₂×C₁₅₁

C₂×M₁₆

yes

characteristic

D₁₅₁

—

D₁₅₁

—

C₁₅₁

#407

yes

characteristic, complemented, Sylow

#407

—

151

maximal, Sylow

C₂³×C₄

—

151

—

C₂×M₁₆

—

151

—

C₂²⋊C₈

—

151

—

C₂²⋊C₈

—

151

—

C₂²⋊C₈

D₁₅₁

yes

characteristic, complemented

C₂⁴

—

151

—

C₂²×C₄

—

302

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₂×D₁₅₁

yes

characteristic

M₁₆

—

302

—

C₂×C₈

—

151

—

C₂×C₈

C₂×D₁₅₁

yes

complemented

C₂³

—

302

—

C₂³

—

302

—

C₂³

—

302

—

C₂³

—

151

—

C₂³

—

151

—

C₂³

C₄×D₁₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

C₄×D₁₅₁

yes

characteristic

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

C₈

—

C₈

—

302

—

C₂²

—

C₂²

—

C₂²

—

302

—

C₂²

—

302

—

C₂²

—

302

—

C₂²

—

302

—

C₂²

—

302

—

C₂²

—

151

—

100

C₂²

—

151

—

101

C₂²

#14558987

yes

complemented

102

C₂²

C₂×C₄×D₁₅₁

yes

characteristic

103

C₄

—

104

C₄

—

302

—

105

C₄

—

151

—

106

C₄

D₄×D₁₅₁

yes

—

107

C₂

—

108

C₂

—

302

—

109

C₂

—

302

—

110

C₂

C₂²⋊C₄×D₁₅₁

yes

characteristic

111

C₂

M₁₆×D₁₅₁

yes

characteristic

112

C₂

C₂×#14558987

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₅₁
commutator, socle	C₂²×C₁₅₁	C₂²×C₄
Fitting	C₂²⋊C₈×C₁₅₁	C₂
prime radical	C₂³×D₁₅₁	C₄
cosocle	C₂×C₄×C₁₅₁	C₂³
hypercenter	C₂²⋊C₈	D₁₅₁
hypocenter	C₁₅₁	#407
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×D₁₅₁

commutator, socle

C₂²×C₁₅₁

C₂²×C₄

Fitting

C₂²⋊C₈×C₁₅₁

C₂

prime radical

C₂³×D₁₅₁

C₄

cosocle

C₂×C₄×C₁₅₁

C₂³

hypercenter

C₂²⋊C₈

D₁₅₁

C₁₅₁

G ▷ C₂²×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×#14558987	C₂	semidirect
3. C₂²⋊(C₁₅₁⋊C₈)	C₂	semidirect
4. (C₂×D₁₅₁)⋊C₈	C₂	semidirect
5. C₂²⋊C₈×C₁₅₁	C₂	semidirect
14. C₂×C₁₅₁⋊C₈	C₂²	semidirect
16. C₂×C₈×C₁₅₁	C₂²	semidirect
37. C₂²×C₁₅₁	M₁₆	semidirect
45. C₂×D₁₅₁	M₁₆	semidirect
56. C₁₅₁	#407	semidirect
62. C₂²⋊C₈	D₁₅₁	semidirect
73. C₂×C₈	C₂×D₁₅₁	semidirect
101. C₂²	#14558987	semidirect

H ≅ G/N

type

2. C₂×#14558987

C₂

semidirect

3. C₂²⋊(C₁₅₁⋊C₈)

C₂

semidirect

4. (C₂×D₁₅₁)⋊C₈

C₂

semidirect

5. C₂²⋊C₈×C₁₅₁

C₂

semidirect

14. C₂×C₁₅₁⋊C₈

C₂²

semidirect

16. C₂×C₈×C₁₅₁

C₂²

semidirect

37. C₂²×C₁₅₁

M₁₆

semidirect

45. C₂×D₁₅₁

M₁₆

semidirect

56. C₁₅₁

#407

semidirect

62. C₂²⋊C₈

D₁₅₁

semidirect

73. C₂×C₈

C₂×D₁₅₁

semidirect

101. C₂²

#14558987

semidirect

Subgroups — #23752174

Conjugacy classes of subgroups

Complements