Group #34083258 — subgroups

Subgroups — #34083258

Order 19328.

930 conjugacy classes of subgroups; only the 451 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 451 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	#23752966	C₂	1	10	yes	maximal, complemented
2	#23752960	C₂	1	5	yes	maximal, complemented
3	#23752768	C₂	1	5	yes	maximal, complemented
4	#23752747	C₂	1	10	yes	maximal, complemented
5	#564×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#18012738	C₂²	1	10	yes	—
7	#18012734	C₂²	1	20	yes	—
8	#18012731	C₂²	1	5	yes	—
9	#18012727	C₂²	1	10	yes	—
10	#18012724	C₂²	1	20	yes	—
11	#18012719	C₂²	1	10	yes	—
12	#18012717	C₂²	1	20	yes	—
13	#18012714	C₂²	1	10	yes	—
14	#18012704	C₂²	1	10	yes	—
15	#18012701	C₂²	1	10	yes	—
16	#18012697	C₂²	1	10	yes	—
17	C₄²×D₁₅₁	C₂²	1	5	yes	—
18	#125×C₁₅₁	C₂²	1	5	yes	—
19	#125×C₁₅₁	C₂²	1	10	yes	—
20	C₂×Dic₃₀₂	C₂³	1	10	yes	—
21	C₂×D₆₀₄	C₂³	1	10	yes	—
22	C₂×C₄×D₁₅₁	C₂³	1	5	yes	—
23	C₂×C₄×D₁₅₁	C₂³	1	10	yes	—
24	C₄⋊Dic₁₅₁	C₂³	1	10	yes	—
25	Dic₁₅₁⋊C₄	C₂³	1	20	yes	—
26	(C₂×D₁₅₁)⋊C₄	C₂³	1	20	yes	—
27	(C₂×D₁₅₁)⋊C₄	C₂³	1	20	yes	—
28	C₄×Dic₁₅₁	C₂³	1	10	yes	—
29	C₄×Dic₁₅₁	C₂³	1	5	yes	—
30	C₄⋊C₄×C₁₅₁	C₂³	1	10	yes	—
31	C₄⋊C₄×C₁₅₁	C₂³	1	10	yes	—
32	C₄⋊C₄×C₁₅₁	C₂³	1	10	yes	—
33	C₄²×C₁₅₁	C₂³	1	5	yes	—
34	C₂²×D₁₅₁	C₂⁴	1	5	yes	—
35	Dic₃₀₂	—	4	10	no	—
36	D₆₀₄	—	4	10	no	—
37	C₂×Dic₁₅₁	C₂⁴	1	10	yes	—
38	C₂×Dic₁₅₁	C₂⁴	1	1	yes	characteristic
39	C₄×D₁₅₁	—	2	10	no	—
40	C₄×D₁₅₁	—	4	10	no	—
41	C₂×C₄×C₁₅₁	C₂⁴	1	5	yes	—
42	C₂×C₄×C₁₅₁	C₂⁴	1	10	yes	—
43	C₂²×C₁₅₁	C₂⁵	1	1	yes	characteristic
44	C₂×D₁₅₁	—	2	5	no	—
45	C₂×D₁₅₁	—	2	10	no	—
46	C₄×C₁₅₁	—	2	5	no	—
47	C₄×C₁₅₁	—	2	10	no	—
48	Dic₁₅₁	—	2	10	no	—
49	Dic₁₅₁	2₋¹⁺⁴	1	2	yes	—
50	C₂×C₁₅₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
51	C₂×C₁₅₁	#585	1	2	yes	—
52	D₁₅₁	—	4	5	no	—
53	C₁₅₁	#3531	1	1	yes	characteristic, complemented, Sylow
54	#3531	—	151	1	no	maximal, Sylow
55	#564	D₁₅₁	1	1	yes	characteristic, complemented
56	#563	—	151	10	no	—
57	#562	—	151	5	no	—
58	#533	—	151	5	no	—
59	#529	—	151	10	no	—
60	C₄⋊Q₈	—	151	10	no	—
61	#126	—	151	20	no	—
62	#125	—	151	5	no	—
63	#125	C₂×D₁₅₁	1	10	yes	—
64	#125	C₂×D₁₅₁	1	5	yes	—
65	#124	—	151	10	no	—
66	#123	—	151	20	no	—
67	C₂²⋊Q₈	—	151	10	no	—
68	C₂²⋊Q₈	—	151	20	no	—
69	#121	—	151	10	no	—
70	C₄×Q₈	—	151	10	no	—
71	C₄×D₄	—	151	10	no	—
72	#117	—	151	10	no	—
73	C₂×C₄²	—	151	5	no	—
74	C₂×Q₈	—	151	10	no	—
75	C₂×D₄	—	151	10	no	—
76	C₂²×C₄	—	151	5	no	—
77	C₂²×C₄	—	151	10	no	—
78	C₄⋊C₄	—	151	20	no	—
79	C₄⋊C₄	—	151	10	no	—
80	C₄⋊C₄	C₂²×D₁₅₁	1	10	yes	—
81	C₄⋊C₄	C₂²×D₁₅₁	1	10	yes	—
82	C₄⋊C₄	C₂²×D₁₅₁	1	10	yes	—
83	C₂²⋊C₄	—	151	20	no	—
84	C₂²⋊C₄	—	151	20	no	—
85	C₄²	—	151	10	no	—
86	C₄²	—	151	5	no	—
87	C₄²	C₂²×D₁₅₁	1	5	yes	—
88	C₂³	—	151	5	no	—
89	Q₈	—	604	10	no	—
90	D₄	—	604	10	no	—
91	C₂×C₄	—	302	10	no	—
92	C₂×C₄	—	604	10	no	—
93	C₂×C₄	—	151	10	no	—
94	C₂×C₄	—	151	1	no	—
95	C₂×C₄	C₂³×D₁₅₁	1	5	yes	—
96	C₂×C₄	C₂³×D₁₅₁	1	10	yes	—
97	C₂²	—	302	5	no	—
98	C₂²	—	302	10	no	—
99	C₂²	C₂⁴×D₁₅₁	1	1	yes	characteristic
100	C₄	—	2	5	no	—
101	C₄	—	2	10	no	—
102	C₄	—	302	10	no	—
103	C₄	—	151	2	no	—
104	C₂	—	604	5	no	—
105	C₂	2₋¹⁺⁴×D₁₅₁	1	1	yes	characteristic
106	C₂	#23753049	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23752966

C₂

yes

maximal, complemented

#23752960

C₂

yes

maximal, complemented

#23752768

C₂

yes

maximal, complemented

#23752747

C₂

yes

maximal, complemented

#564×C₁₅₁

C₂

yes

maximal, characteristic, complemented

#18012738

C₂²

yes

—

#18012734

C₂²

yes

—

#18012731

C₂²

yes

—

#18012727

C₂²

yes

—

#18012724

C₂²

yes

—

#18012719

C₂²

yes

—

#18012717

C₂²

yes

—

#18012714

C₂²

yes

—

#18012704

C₂²

yes

—

#18012701

C₂²

yes

—

#18012697

C₂²

yes

—

C₄²×D₁₅₁

C₂²

yes

—

#125×C₁₅₁

C₂²

yes

—

#125×C₁₅₁

C₂²

yes

—

C₂×Dic₃₀₂

C₂³

yes

—

C₂×D₆₀₄

C₂³

yes

—

C₂×C₄×D₁₅₁

C₂³

yes

—

C₂×C₄×D₁₅₁

C₂³

yes

—

C₄⋊Dic₁₅₁

C₂³

yes

—

Dic₁₅₁⋊C₄

C₂³

yes

—

(C₂×D₁₅₁)⋊C₄

C₂³

yes

—

(C₂×D₁₅₁)⋊C₄

C₂³

yes

—

C₄×Dic₁₅₁

C₂³

yes

—

C₄×Dic₁₅₁

C₂³

yes

—

C₄⋊C₄×C₁₅₁

C₂³

yes

—

C₄⋊C₄×C₁₅₁

C₂³

yes

—

C₄⋊C₄×C₁₅₁

C₂³

yes

—

C₄²×C₁₅₁

C₂³

yes

—

C₂²×D₁₅₁

C₂⁴

yes

—

Dic₃₀₂

—

D₆₀₄

—

C₂×Dic₁₅₁

C₂⁴

yes

—

C₂×Dic₁₅₁

C₂⁴

yes

characteristic

C₄×D₁₅₁

—

C₄×D₁₅₁

—

C₂×C₄×C₁₅₁

C₂⁴

yes

—

C₂×C₄×C₁₅₁

C₂⁴

yes

—

C₂²×C₁₅₁

C₂⁵

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

2₋¹⁺⁴

yes

—

C₂×C₁₅₁

C₂×2₋¹⁺⁴

yes

characteristic

C₂×C₁₅₁

#585

yes

—

D₁₅₁

—

C₁₅₁

#3531

yes

characteristic, complemented, Sylow

#3531

—

151

maximal, Sylow

#564

D₁₅₁

yes

characteristic, complemented

#563

—

151

—

#562

—

151

—

#533

—

151

—

#529

—

151

—

C₄⋊Q₈

—

151

—

#126

—

151

—

#125

—

151

—

#125

C₂×D₁₅₁

yes

—

#125

C₂×D₁₅₁

yes

—

#124

—

151

—

#123

—

151

—

C₂²⋊Q₈

—

151

—

C₂²⋊Q₈

—

151

—

#121

—

151

—

C₄×Q₈

—

151

—

C₄×D₄

—

151

—

#117

—

151

—

C₂×C₄²

—

151

—

C₂×Q₈

—

151

—

C₂×D₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

C₂²×D₁₅₁

yes

—

C₄⋊C₄

C₂²×D₁₅₁

yes

—

C₄⋊C₄

C₂²×D₁₅₁

yes

—

C₂²⋊C₄

—

151

—

C₂²⋊C₄

—

151

—

C₄²

—

151

—

C₄²

—

151

—

C₄²

C₂²×D₁₅₁

yes

—

C₂³

—

151

—

Q₈

—

604

—

D₄

—

604

—

C₂×C₄

—

302

—

C₂×C₄

—

604

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

C₂³×D₁₅₁

yes

—

C₂×C₄

C₂³×D₁₅₁

yes

—

C₂²

—

302

—

C₂²

—

302

—

C₂²

C₂⁴×D₁₅₁

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

302

—

103

C₄

—

151

—

104

C₂

—

604

—

105

C₂

2₋¹⁺⁴×D₁₅₁

yes

characteristic

106

C₂

#23753049

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₅₁
commutator, socle, cosocle	C₂²×C₁₅₁	C₂⁵
Fitting	#564×C₁₅₁	C₂
hypercenter	#564	D₁₅₁
hypocenter	C₁₅₁	#3531
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₁₅₁

commutator, socle, cosocle

C₂²×C₁₅₁

C₂⁵

Fitting

#564×C₁₅₁

C₂

D₁₅₁

C₁₅₁

G ▷ C₂²×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
1. #23752966	C₂	semidirect
2. #23752960	C₂	semidirect
3. #23752768	C₂	semidirect
4. #23752747	C₂	semidirect
5. #564×C₁₅₁	C₂	semidirect
53. C₁₅₁	#3531	semidirect
55. #564	D₁₅₁	semidirect

H ≅ G/N

type

1. #23752966

C₂

semidirect

2. #23752960

C₂

semidirect

3. #23752768

C₂

semidirect

4. #23752747

C₂

semidirect

5. #564×C₁₅₁

C₂

semidirect

53. C₁₅₁

#3531

semidirect

55. #564

D₁₅₁

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23752966	2
#23752960	2
#23752768	2
#23752747	2
#564×C₁₅₁	2
#3531	151

subgroup

index

#564×C₁₅₁

151

Subgroups — #34083258

Conjugacy classes of subgroups

Complements

Maximal subgroups