Group #23752877 — subgroups

Subgroups — #23752877

Order 9664.

280 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 115 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×Q₈×D₁₅₁	C₂	1	2	yes	maximal, complemented
2	#18012738	C₂	1	2	yes	maximal, complemented
3	#18012717	C₂	1	4	yes	maximal, complemented
4	#18012715	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
7	#18012701	C₂	1	2	yes	maximal
8	C₄⋊Q₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₃₀₂	C₂²	1	4	yes	—
10	C₂×Dic₃₀₂	C₂²	1	1	yes	characteristic
11	Q₈×D₁₅₁	—	2	8	no	—
12	C₂×D₆₀₄	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₅₁	C₂²	1	4	yes	—
14	C₂×C₄×D₁₅₁	C₂²	1	2	yes	—
15	C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
17	Dic₁₅₁⋊C₄	C₂²	1	2	yes	—
18	Dic₁₅₁⋊C₄	C₂²	1	4	yes	—
19	(C₂×D₁₅₁)⋊C₄	C₂²	1	4	yes	—
20	(C₂×D₁₅₁)⋊C₄	C₂²	1	2	yes	—
21	C₄×Dic₁₅₁	C₂²	1	2	yes	—
22	C₂×Q₈×C₁₅₁	C₂²	1	2	yes	—
23	C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
24	C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
25	C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
26	C₂²×D₁₅₁	C₂³	1	2	yes	—
27	Dic₃₀₂	—	2	8	no	—
28	Dic₃₀₂	D₄	1	4	yes	complemented
29	D₆₀₄	Q₈	1	4	yes	complemented
30	C₂×Dic₁₅₁	C₂³	1	2	yes	—
31	C₂×Dic₁₅₁	C₂³	1	2	yes	—
32	C₂×Dic₁₅₁	C₂³	1	2	yes	—
33	C₄×D₁₅₁	—	2	4	no	—
34	C₄×D₁₅₁	—	2	8	no	—
35	Q₈×C₁₅₁	—	2	4	no	—
36	C₂×C₄×C₁₅₁	C₂³	1	4	yes	—
37	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
40	C₂²×C₁₅₁	C₂⁴	1	1	yes	characteristic
41	C₂×D₁₅₁	—	2	2	no	—
42	C₂×D₁₅₁	—	2	2	no	—
43	C₂×D₁₅₁	C₂×Q₈	1	4	yes	complemented
44	C₄×C₁₅₁	—	2	4	no	—
45	C₄×C₁₅₁	—	2	1	no	—
46	C₄×C₁₅₁	C₂×D₄	1	2	yes	—
47	C₄×C₁₅₁	C₂×Q₈	1	2	yes	complemented
48	Dic₁₅₁	—	2	2	no	—
49	Dic₁₅₁	—	2	2	no	—
50	Dic₁₅₁	C₂×D₄	1	4	yes	—
51	C₂×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
52	C₂×C₁₅₁	C₂²×Q₈	1	1	yes	characteristic
53	C₂×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
54	D₁₅₁	—	2	4	no	—
55	C₁₅₁	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
56	D₄×Q₈	—	151	1	no	maximal, Sylow
57	C₂²×Q₈	—	151	2	no	—
58	C₄⋊Q₈	—	151	2	no	—
59	C₄⋊Q₈	D₁₅₁	1	1	yes	characteristic, complemented
60	C₂²⋊Q₈	—	151	2	no	—
61	C₂²⋊Q₈	—	151	4	no	—
62	C₄×Q₈	—	151	1	no	—
63	C₄×D₄	—	151	2	no	—
64	C₄×D₄	—	151	1	no	—
65	C₂×Q₈	—	302	8	no	—
66	C₂×Q₈	—	151	4	no	—
67	C₂×Q₈	—	151	1	no	—
68	C₂×Q₈	C₂×D₁₅₁	1	2	yes	—
69	C₂×D₄	—	151	1	no	—
70	C₂²×C₄	—	151	4	no	—
71	C₂²×C₄	—	151	2	no	—
72	C₄⋊C₄	—	151	2	no	—
73	C₄⋊C₄	—	151	4	no	—
74	C₄⋊C₄	—	151	1	no	—
75	C₄⋊C₄	—	151	1	no	—
76	C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
77	C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
78	C₂²⋊C₄	—	151	4	no	—
79	C₂²⋊C₄	—	151	2	no	—
80	C₄²	—	151	2	no	—
81	C₄²	C₂×D₁₅₁	1	1	yes	characteristic
82	C₂³	—	151	2	no	—
83	Q₈	—	2	4	no	—
84	Q₈	—	302	8	no	—
85	Q₈	—	151	4	no	—
86	D₄	—	151	4	no	—
87	C₂×C₄	—	302	8	no	—
88	C₂×C₄	—	302	4	no	—
89	C₂×C₄	—	151	2	no	—
90	C₂×C₄	—	151	2	no	—
91	C₂×C₄	—	151	2	no	—
92	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
93	C₂×C₄	C₂²×D₁₅₁	1	4	yes	—
94	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
96	C₂²	—	302	2	no	—
97	C₂²	—	302	2	no	—
98	C₂²	—	151	4	no	—
99	C₂²	C₂³×D₁₅₁	1	1	yes	characteristic
100	C₄	—	2	4	no	—
101	C₄	—	2	1	no	—
102	C₄	—	302	2	no	—
103	C₄	—	302	2	no	—
104	C₄	—	151	4	no	—
105	C₄	D₄×D₁₅₁	1	2	yes	—
106	C₄	Q₈×D₁₅₁	1	2	yes	complemented
107	C₂	—	302	4	no	—
108	C₂	C₂×D₄×D₁₅₁	1	1	yes	characteristic
109	C₂	C₂×Q₈×D₁₅₁	1	1	yes	characteristic
110	C₂	#18012787	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₁₅₁

C₂

yes

maximal, complemented

#18012738

C₂

yes

maximal, complemented

#18012717

C₂

yes

maximal, complemented

#18012715

C₂

yes

maximal, complemented

C₄×Dic₃₀₂

C₂

yes

maximal, characteristic, complemented

C₄×D₆₀₄

C₂

yes

maximal, characteristic

#18012701

C₂

yes

maximal

C₄⋊Q₈×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₀₂

C₂²

yes

—

C₂×Dic₃₀₂

C₂²

yes

characteristic

Q₈×D₁₅₁

—

C₂×D₆₀₄

C₂²

yes

characteristic

C₂×C₄×D₁₅₁

C₂²

yes

—

C₂×C₄×D₁₅₁

C₂²

yes

—

C₄⋊Dic₁₅₁

C₂²

yes

characteristic

C₄⋊Dic₁₅₁

C₂²

yes

characteristic

Dic₁₅₁⋊C₄

C₂²

yes

—

Dic₁₅₁⋊C₄

C₂²

yes

—

(C₂×D₁₅₁)⋊C₄

C₂²

yes

—

(C₂×D₁₅₁)⋊C₄

C₂²

yes

—

C₄×Dic₁₅₁

C₂²

yes

—

C₂×Q₈×C₁₅₁

C₂²

yes

—

C₄⋊C₄×C₁₅₁

C₂²

yes

—

C₄⋊C₄×C₁₅₁

C₂²

yes

—

C₄²×C₁₅₁

C₂²

yes

characteristic

C₂²×D₁₅₁

C₂³

yes

—

Dic₃₀₂

—

Dic₃₀₂

D₄

yes

complemented

D₆₀₄

Q₈

yes

complemented

C₂×Dic₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

C₂³

yes

—

C₄×D₁₅₁

—

C₄×D₁₅₁

—

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂³

yes

—

C₂×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₄×C₁₅₁

C₂³

yes

characteristic

C₂²×C₁₅₁

C₂⁴

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

C₂×Q₈

yes

complemented

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×D₄

yes

—

C₄×C₁₅₁

C₂×Q₈

yes

complemented

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

C₂×D₄

yes

—

C₂×C₁₅₁

C₂²×D₄

yes

characteristic

C₂×C₁₅₁

C₂²×Q₈

yes

characteristic

C₂×C₁₅₁

2₋¹⁺⁴

yes

characteristic

D₁₅₁

—

C₁₅₁

D₄×Q₈

yes

characteristic, complemented, Sylow

D₄×Q₈

—

151

maximal, Sylow

C₂²×Q₈

—

151

—

C₄⋊Q₈

—

151

—

C₄⋊Q₈

D₁₅₁

yes

characteristic, complemented

C₂²⋊Q₈

—

151

—

C₂²⋊Q₈

—

151

—

C₄×Q₈

—

151

—

C₄×D₄

—

151

—

C₄×D₄

—

151

—

C₂×Q₈

—

302

—

C₂×Q₈

—

151

—

C₂×Q₈

—

151

—

C₂×Q₈

C₂×D₁₅₁

yes

—

C₂×D₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

C₂×D₁₅₁

yes

—

C₄⋊C₄

C₂×D₁₅₁

yes

—

C₂²⋊C₄

—

151

—

C₂²⋊C₄

—

151

—

C₄²

—

151

—

C₄²

C₂×D₁₅₁

yes

characteristic

C₂³

—

151

—

Q₈

—

Q₈

—

302

—

Q₈

—

151

—

D₄

—

151

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₄

C₂²×D₁₅₁

yes

—

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

C₂²

—

302

—

C₂²

—

302

—

C₂²

—

151

—

C₂²

C₂³×D₁₅₁

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

302

—

103

C₄

—

302

—

104

C₄

—

151

—

105

C₄

D₄×D₁₅₁

yes

—

106

C₄

Q₈×D₁₅₁

yes

complemented

107

C₂

—

302

—

108

C₂

C₂×D₄×D₁₅₁

yes

characteristic

109

C₂

C₂×Q₈×D₁₅₁

yes

characteristic

110

C₂

#18012787

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅₁
commutator, socle, cosocle	C₂²×C₁₅₁	C₂⁴
Fitting	C₄⋊Q₈×C₁₅₁	C₂
prime radical	C₂×D₆₀₄	C₂²
hypercenter	C₄⋊Q₈	D₁₅₁
hypocenter	C₁₅₁	D₄×Q₈
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅₁

commutator, socle, cosocle

C₂²×C₁₅₁

C₂⁴

Fitting

C₄⋊Q₈×C₁₅₁

C₂

prime radical

C₂×D₆₀₄

C₂²

hypercenter

C₄⋊Q₈

D₁₅₁

hypocenter

C₁₅₁

D₄×Q₈

LCS

G ▷ C₂²×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×Q₈×D₁₅₁	C₂	semidirect
2. #18012738	C₂	semidirect
3. #18012717	C₂	semidirect
4. #18012715	C₂	semidirect
5. C₄×Dic₃₀₂	C₂	semidirect
8. C₄⋊Q₈×C₁₅₁	C₂	semidirect
28. Dic₃₀₂	D₄	semidirect
29. D₆₀₄	Q₈	semidirect
43. C₂×D₁₅₁	C₂×Q₈	semidirect
47. C₄×C₁₅₁	C₂×Q₈	semidirect
55. C₁₅₁	D₄×Q₈	semidirect
59. C₄⋊Q₈	D₁₅₁	semidirect
106. C₄	Q₈×D₁₅₁	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₁₅₁

C₂

semidirect

2. #18012738

C₂

semidirect

3. #18012717

C₂

semidirect

4. #18012715

C₂

semidirect

5. C₄×Dic₃₀₂

C₂

semidirect

8. C₄⋊Q₈×C₁₅₁

C₂

semidirect

28. Dic₃₀₂

D₄

semidirect

29. D₆₀₄

Q₈

semidirect

43. C₂×D₁₅₁

C₂×Q₈

semidirect

47. C₄×C₁₅₁

C₂×Q₈

semidirect

55. C₁₅₁

D₄×Q₈

semidirect

59. C₄⋊Q₈

D₁₅₁

semidirect

106. C₄

Q₈×D₁₅₁

semidirect

Subgroups — #23752877

Conjugacy classes of subgroups

Complements