Group #23752455 — subgroups

Subgroups — #23752455

Order 9664.

120 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 41 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×#14558987	C₂	1	1	yes	maximal, characteristic, complemented
2	#18012715	C₂	1	1	yes	maximal, characteristic, complemented
3	#18012710	C₂	1	1	yes	maximal, characteristic
4	#18012673	C₂	1	1	yes	maximal, characteristic, complemented
5	#18012663	C₂	1	1	yes	maximal, characteristic, complemented
6	#18012660	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₁₆×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊D₁₅₁	—	2	1	no	—
9	C₂×C₄×D₁₅₁	C₂²	1	1	yes	characteristic
10	#14558987	—	2	2	no	—
11	C₂×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
14	Dic₁₅₁⋊C₄	—	2	1	no	—
15	C₂²⋊Dic₁₅₁	—	2	1	no	—
16	(C₂×D₁₅₁)⋊C₄	—	2	1	no	—
17	C₂×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₁₅₁	C₂²	1	1	yes	characteristic
19	SD₁₆×C₁₅₁	—	2	2	no	—
20	C₂×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
21	C₂²×D₁₅₁	D₄	1	1	yes	characteristic
22	C₂²⋊D₁₅₁	—	4	2	no	—
23	C₂×Dic₁₅₁	—	2	1	no	—
24	C₂×Dic₁₅₁	—	2	1	no	—
25	C₂×Dic₁₅₁	D₄	1	1	yes	characteristic
26	C₄×D₁₅₁	—	2	1	no	—
27	C₄×D₁₅₁	—	2	1	no	—
28	C₁₅₁⋊C₈	—	2	1	no	—
29	C₂³×C₁₅₁	—	2	1	no	—
30	Q₈×C₁₅₁	—	2	1	no	—
31	Q₈×C₁₅₁	—	2	1	no	—
32	D₄×C₁₅₁	—	2	1	no	—
33	D₄×C₁₅₁	—	2	1	no	—
34	C₂×C₄×C₁₅₁	—	2	1	no	—
35	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
36	C₈×C₁₅₁	D₄	1	2	yes	complemented
37	C₂²×C₁₅₁	—	4	1	no	—
38	C₂²×C₁₅₁	—	4	1	no	—
39	C₂²×C₁₅₁	—	4	1	no	—
40	C₂²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
41	C₂×D₁₅₁	—	2	1	no	—
42	C₂×D₁₅₁	—	2	1	no	—
43	C₂×D₁₅₁	—	2	1	no	—
44	C₄×C₁₅₁	—	4	1	no	—
45	C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
46	C₄×C₁₅₁	#41	1	1	yes	characteristic
47	Dic₁₅₁	—	2	1	no	—
48	Dic₁₅₁	—	4	1	no	—
49	Dic₁₅₁	—	4	1	no	—
50	C₂×C₁₅₁	—	8	1	no	—
51	C₂×C₁₅₁	#121	1	1	yes	characteristic
52	C₂×C₁₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
53	C₂×C₁₅₁	#137	1	1	yes	characteristic
54	D₁₅₁	—	4	1	no	—
55	C₁₅₁	#468	1	1	yes	characteristic, complemented, Sylow
56	#468	—	151	1	no	maximal, Sylow
57	C₂×SD₁₆	D₁₅₁	1	1	yes	characteristic, complemented
58	C₂×M₁₆	—	151	1	no	—
59	C₂²⋊Q₈	—	151	1	no	—
60	#121	—	151	1	no	—
61	#107	—	151	1	no	—
62	#103	—	151	1	no	—
63	#102	—	151	1	no	—
64	C₂×Q₈	C₂×D₁₅₁	1	1	yes	characteristic
65	C₂×D₄	—	302	1	no	—
66	C₂×D₄	C₂×D₁₅₁	1	1	yes	characteristic
67	C₂²×C₄	—	151	1	no	—
68	SD₁₆	—	2	2	no	—
69	M₁₆	—	302	2	no	—
70	C₂×C₈	—	151	1	no	—
71	C₂×C₈	C₂×D₁₅₁	1	1	yes	characteristic
72	C₄⋊C₄	—	302	1	no	—
73	C₄⋊C₄	—	151	1	no	—
74	C₄⋊C₄	—	151	1	no	—
75	C₂²⋊C₄	—	302	1	no	—
76	C₂²⋊C₄	—	302	1	no	—
77	C₂³	—	2	1	no	—
78	C₂³	—	151	1	no	—
79	Q₈	—	2	1	no	—
80	Q₈	—	2	1	no	—
81	D₄	—	2	1	no	—
82	D₄	—	2	1	no	—
83	D₄	—	604	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	302	1	no	—
86	C₂×C₄	—	302	1	no	—
87	C₂×C₄	—	302	1	no	—
88	C₂×C₄	—	302	1	no	—
89	C₂×C₄	—	151	1	no	—
90	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
91	C₈	—	302	1	no	—
92	C₈	C₂²⋊D₁₅₁	1	2	yes	complemented
93	C₂²	—	4	1	no	—
94	C₂²	—	4	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	302	1	no	—
97	C₂²	—	302	1	no	—
98	C₂²	—	302	1	no	—
99	C₂²	D₄×D₁₅₁	1	1	yes	characteristic
100	C₄	—	4	1	no	—
101	C₄	—	302	1	no	—
102	C₄	—	604	1	no	—
103	C₄	—	604	1	no	—
104	C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
105	C₄	#14559003	1	1	yes	characteristic
106	C₂	—	8	1	no	—
107	C₂	—	604	1	no	—
108	C₂	#18012710	1	1	yes	characteristic
109	C₂	#18012766	1	1	yes	characteristic
110	C₂	#18012768	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14558987

C₂

yes

maximal, characteristic, complemented

#18012715

C₂

yes

maximal, characteristic, complemented

#18012710

C₂

yes

maximal, characteristic

#18012673

C₂

yes

maximal, characteristic, complemented

#18012663

C₂

yes

maximal, characteristic, complemented

#18012660

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊D₁₅₁

—

C₂×C₄×D₁₅₁

C₂²

yes

characteristic

#14558987

—

C₂×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₅₁

C₂²

yes

characteristic

C₄⋊Dic₁₅₁

C₂²

yes

characteristic

Dic₁₅₁⋊C₄

—

C₂²⋊Dic₁₅₁

—

(C₂×D₁₅₁)⋊C₄

—

C₂×Q₈×C₁₅₁

C₂²

yes

characteristic

C₂×D₄×C₁₅₁

C₂²

yes

characteristic

SD₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

C₂²

yes

characteristic

C₂²×D₁₅₁

D₄

yes

characteristic

C₂²⋊D₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

D₄

yes

characteristic

C₄×D₁₅₁

—

C₄×D₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

—

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂³

yes

characteristic

C₈×C₁₅₁

D₄

yes

complemented

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂×D₄

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₄×C₁₅₁

#41

yes

characteristic

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

D₁₅₁

—

C₁₅₁

#468

yes

characteristic, complemented, Sylow

#468

—

151

maximal, Sylow

C₂×SD₁₆

D₁₅₁

yes

characteristic, complemented

C₂×M₁₆

—

151

—

C₂²⋊Q₈

—

151

—

#121

—

151

—

#107

—

151

—

#103

—

151

—

#102

—

151

—

C₂×Q₈

C₂×D₁₅₁

yes

characteristic

C₂×D₄

—

302

—

C₂×D₄

C₂×D₁₅₁

yes

characteristic

C₂²×C₄

—

151

—

SD₁₆

—

M₁₆

—

302

—

C₂×C₈

—

151

—

C₂×C₈

C₂×D₁₅₁

yes

characteristic

C₄⋊C₄

—

302

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₂²⋊C₄

—

302

—

C₂²⋊C₄

—

302

—

C₂³

—

C₂³

—

151

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

604

—

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

151

—

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

C₈

—

302

—

C₈

C₂²⋊D₁₅₁

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

302

—

C₂²

—

302

—

C₂²

—

302

—

C₂²

D₄×D₁₅₁

yes

characteristic

100

C₄

—

101

C₄

—

302

—

102

C₄

—

604

—

103

C₄

—

604

—

104

C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

105

C₄

#14559003

yes

characteristic

106

C₂

—

107

C₂

—

604

—

108

C₂

#18012710

yes

characteristic

109

C₂

#18012766

yes

characteristic

110

C₂

#18012768

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₅₁
Fitting	C₂×SD₁₆×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂×D₄
prime radical	#18012710	C₂
hypercenter	C₂×SD₁₆	D₁₅₁
hypocenter	C₁₅₁	#468
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×D₁₅₁

commutator, cosocle

C₂×C₄×C₁₅₁

C₂³

Frattini

C₂×C₄

C₂²×D₁₅₁

Fitting

C₂×SD₁₆×C₁₅₁

C₂

socle

C₂²×C₁₅₁

C₂×D₄

prime radical

#18012710

C₂

hypercenter

C₂×SD₁₆

D₁₅₁

hypocenter

C₁₅₁

#468

LCS

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
1. C₂×#14558987	C₂	semidirect
2. #18012715	C₂	semidirect
4. #18012673	C₂	semidirect
5. #18012663	C₂	semidirect
6. #18012660	C₂	semidirect
7. C₂×SD₁₆×C₁₅₁	C₂	semidirect
36. C₈×C₁₅₁	D₄	semidirect
55. C₁₅₁	#468	semidirect
57. C₂×SD₁₆	D₁₅₁	semidirect
92. C₈	C₂²⋊D₁₅₁	semidirect

H ≅ G/N

type

1. C₂×#14558987

C₂

semidirect

2. #18012715

C₂

semidirect

4. #18012673

C₂

semidirect

5. #18012663

C₂

semidirect

6. #18012660

C₂

semidirect

7. C₂×SD₁₆×C₁₅₁

C₂

semidirect

36. C₈×C₁₅₁

D₄

semidirect

55. C₁₅₁

#468

semidirect

57. C₂×SD₁₆

D₁₅₁

semidirect

92. C₈

C₂²⋊D₁₅₁

semidirect

Subgroups — #23752455

Conjugacy classes of subgroups

Complements