Group #23752124 — subgroups

Subgroups — #23752124

Order 9664.

202 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 63 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×Dic₃₀₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Dic₁₅₁	C₂	1	2	yes	maximal
3	C₁₅₁⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₅₁⋊#95	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₄×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₀₂	—	2	4	no	—
7	C₂×Dic₃₀₂	—	2	2	no	—
8	C₂²×Dic₁₅₁	C₂²	1	2	yes	—
9	C₂²×Dic₁₅₁	C₂²	1	2	yes	—
10	C₂²⋊Dic₁₅₁	—	2	4	no	—
11	C₂⁴×C₁₅₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₁₅₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₁₅₁	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
15	Dic₃₀₂	—	2	8	no	—
16	C₂×Dic₁₅₁	—	2	2	no	—
17	C₂×Dic₁₅₁	—	2	2	no	—
18	C₂×Dic₁₅₁	—	2	2	no	—
19	C₂×Dic₁₅₁	—	2	2	no	—
20	C₂×Dic₁₅₁	—	2	2	no	—
21	C₂×Dic₁₅₁	D₄	1	4	yes	—
22	C₂³×C₁₅₁	—	2	2	no	—
23	C₂³×C₁₅₁	—	2	1	no	—
24	C₂³×C₁₅₁	—	2	2	no	—
25	C₂³×C₁₅₁	—	2	1	no	—
26	C₂³×C₁₅₁	Q₈	1	2	yes	complemented
27	C₂³×C₁₅₁	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₁₅₁	—	2	2	no	—
29	C₂×C₄×C₁₅₁	—	2	1	no	—
30	C₂×C₄×C₁₅₁	—	2	2	no	—
31	C₂×C₄×C₁₅₁	D₄	1	2	yes	—
32	C₂²×C₁₅₁	—	2	4	no	—
33	C₂²×C₁₅₁	—	2	2	no	—
34	C₂²×C₁₅₁	—	4	1	no	—
35	C₂²×C₁₅₁	—	4	2	no	—
36	C₂²×C₁₅₁	—	4	1	no	—
37	C₂²×C₁₅₁	C₂×D₄	1	2	yes	—
38	C₂²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₁₅₁	#41	1	2	yes	—
41	C₂²×C₁₅₁	#41	1	1	yes	characteristic
42	C₄×C₁₅₁	—	2	2	no	—
43	C₄×C₁₅₁	—	4	1	no	—
44	Dic₁₅₁	—	2	4	no	—
45	Dic₁₅₁	—	4	2	no	—
46	C₂×C₁₅₁	—	4	2	no	—
47	C₂×C₁₅₁	#120	1	1	yes	characteristic
48	C₂×C₁₅₁	C₂²⋊Q₈	1	2	yes	—
49	C₂×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂×C₁₅₁	#124	1	2	yes	—
51	C₂×C₁₅₁	#124	1	1	yes	characteristic
52	C₁₅₁	#393	1	1	yes	characteristic, complemented, Sylow
53	#393	—	151	1	no	maximal, Sylow
54	C₂²×Q₈	—	151	1	no	—
55	C₂×C₂²⋊C₄	—	151	2	no	—
56	C₂×C₂²⋊C₄	D₁₅₁	1	1	yes	characteristic
57	#95	—	151	1	no	—
58	#95	—	151	2	no	—
59	C₂⁴	C₂×D₁₅₁	1	1	yes	characteristic
60	C₂×Q₈	—	302	4	no	—
61	C₂×Q₈	—	302	2	no	—
62	C₂²×C₄	—	151	2	no	—
63	C₂²×C₄	—	151	2	no	—
64	C₂²×C₄	C₂×D₁₅₁	1	1	yes	characteristic
65	C₂²×C₄	C₂×D₁₅₁	1	1	yes	characteristic
66	C₂²⋊C₄	—	2	2	no	—
67	C₂²⋊C₄	—	302	4	no	—
68	C₂³	—	2	2	no	—
69	C₂³	—	2	1	no	—
70	C₂³	—	2	2	no	—
71	C₂³	—	2	1	no	—
72	C₂³	Dic₃₀₂	1	2	yes	complemented
73	C₂³	C₂²×D₁₅₁	1	1	yes	characteristic
74	Q₈	—	302	8	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	302	2	no	—
79	C₂×C₄	—	302	2	no	—
80	C₂×C₄	—	302	2	no	—
81	C₂×C₄	—	302	2	no	—
82	C₂×C₄	—	302	2	no	—
83	C₂×C₄	—	151	4	no	—
84	C₂×C₄	C₂²⋊D₁₅₁	1	2	yes	—
85	C₂²	—	2	4	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	4	1	no	—
88	C₂²	—	4	2	no	—
89	C₂²	—	4	1	no	—
90	C₂²	D₄×D₁₅₁	1	2	yes	—
91	C₂²	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
92	C₂²	C₂×Dic₃₀₂	1	1	yes	characteristic
93	C₂²	#14559003	1	2	yes	—
94	C₂²	#14559004	1	1	yes	characteristic
95	C₄	—	2	2	no	—
96	C₄	—	4	1	no	—
97	C₄	—	302	4	no	—
98	C₄	—	604	2	no	—
99	C₂	—	4	2	no	—
100	C₂	#18012707	1	1	yes	characteristic
101	C₂	#18012716	1	2	yes	—
102	C₂	#18012718	1	1	yes	characteristic
103	C₂	#18012725	1	2	yes	—
104	C₂	#18012726	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₀₂

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Dic₁₅₁

C₂

yes

maximal

C₁₅₁⋊#95

C₂

yes

maximal, characteristic, complemented

C₁₅₁⋊#95

C₂

yes

maximal, complemented

C₂×C₂²⋊C₄×C₁₅₁

C₂

yes

maximal, characteristic

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂²×Dic₁₅₁

C₂²

yes

—

C₂²×Dic₁₅₁

C₂²

yes

—

C₂²⋊Dic₁₅₁

—

C₂⁴×C₁₅₁

C₂²

yes

characteristic

C₂²×C₄×C₁₅₁

C₂²

yes

characteristic

C₂²×C₄×C₁₅₁

C₂²

yes

characteristic

C₂²⋊C₄×C₁₅₁

—

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

D₄

yes

—

C₂³×C₁₅₁

—

C₂³×C₁₅₁

—

C₂³×C₁₅₁

—

C₂³×C₁₅₁

—

C₂³×C₁₅₁

Q₈

yes

complemented

C₂³×C₁₅₁

C₂³

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

D₄

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂×D₄

yes

—

C₂²×C₁₅₁

C₂×D₄

yes

characteristic

C₂²×C₁₅₁

C₂×Q₈

yes

characteristic

C₂²×C₁₅₁

#41

yes

—

C₂²×C₁₅₁

#41

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#120

yes

characteristic

C₂×C₁₅₁

C₂²⋊Q₈

yes

—

C₂×C₁₅₁

C₂²⋊Q₈

yes

characteristic

C₂×C₁₅₁

#124

yes

—

C₂×C₁₅₁

#124

yes

characteristic

C₁₅₁

#393

yes

characteristic, complemented, Sylow

#393

—

151

maximal, Sylow

C₂²×Q₈

—

151

—

C₂×C₂²⋊C₄

—

151

—

C₂×C₂²⋊C₄

D₁₅₁

yes

characteristic

#95

—

151

—

#95

—

151

—

C₂⁴

C₂×D₁₅₁

yes

characteristic

C₂×Q₈

—

302

—

C₂×Q₈

—

302

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₂×D₁₅₁

yes

characteristic

C₂²×C₄

C₂×D₁₅₁

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

302

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

Dic₃₀₂

yes

complemented

C₂³

C₂²×D₁₅₁

yes

characteristic

Q₈

—

302

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

151

—

C₂×C₄

C₂²⋊D₁₅₁

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

D₄×D₁₅₁

yes

—

C₂²

C₂×C₂²⋊D₁₅₁

yes

characteristic

C₂²

C₂×Dic₃₀₂

yes

characteristic

C₂²

#14559003

yes

—

C₂²

#14559004

yes

characteristic

C₄

—

C₄

—

C₄

—

302

—

C₄

—

604

—

C₂

—

100

C₂

#18012707

yes

characteristic

101

C₂

#18012716

yes

—

102

C₂

#18012718

yes

characteristic

103

C₂

#18012725

yes

—

104

C₂

#18012726

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₅₁
commutator, socle, cosocle	C₂³×C₁₅₁	C₂³
Fitting	C₂×C₂²⋊C₄×C₁₅₁	C₂
prime radical	C₂⁴×C₁₅₁	C₂²
hypercenter	C₂×C₂²⋊C₄	D₁₅₁
hypocenter	C₁₅₁	#393
LCS	G ▷ C₂³×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₅₁

commutator, socle, cosocle

C₂³×C₁₅₁

C₂³

Fitting

C₂×C₂²⋊C₄×C₁₅₁

C₂

prime radical

C₂⁴×C₁₅₁

C₂²

hypercenter

C₂×C₂²⋊C₄

D₁₅₁

C₁₅₁

G ▷ C₂³×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

N	H ≅ G/N	type
1. C₂²×Dic₃₀₂	C₂	semidirect
3. C₁₅₁⋊#95	C₂	semidirect
4. C₁₅₁⋊#95	C₂	semidirect
26. C₂³×C₁₅₁	Q₈	semidirect
52. C₁₅₁	#393	semidirect
72. C₂³	Dic₃₀₂	semidirect

H ≅ G/N

type

1. C₂²×Dic₃₀₂

C₂

semidirect

3. C₁₅₁⋊#95

C₂

semidirect

4. C₁₅₁⋊#95

C₂

semidirect

26. C₂³×C₁₅₁

Q₈

semidirect

52. C₁₅₁

#393

semidirect

72. C₂³

Dic₃₀₂

semidirect

Subgroups — #23752124

Conjugacy classes of subgroups

Complements