Group #27185280 — subgroups

Subgroups — #27185280

Order 12928.

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	#20265931	C₂	1	10	yes	maximal, complemented
2	#20265925	C₂	1	5	yes	maximal, complemented
3	#20265733	C₂	1	5	yes	maximal, complemented
4	#20265712	C₂	1	10	yes	maximal, complemented
5	#564×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#15692254	C₂²	1	10	yes	—
7	#15692250	C₂²	1	20	yes	—
8	#15692247	C₂²	1	5	yes	—
9	#15692243	C₂²	1	10	yes	—
10	#15692240	C₂²	1	20	yes	—
11	#15692235	C₂²	1	10	yes	—
12	#15692233	C₂²	1	20	yes	—
13	#15692230	C₂²	1	10	yes	—
14	#15692220	C₂²	1	10	yes	—
15	#15692217	C₂²	1	10	yes	—
16	#15692213	C₂²	1	10	yes	—
17	C₄²×D₁₀₁	C₂²	1	5	yes	—
18	#125×C₁₀₁	C₂²	1	5	yes	—
19	#125×C₁₀₁	C₂²	1	10	yes	—
20	C₂×Dic₂₀₂	C₂³	1	10	yes	—
21	C₂×D₄₀₄	C₂³	1	10	yes	—
22	C₂×C₄×D₁₀₁	C₂³	1	5	yes	—
23	C₂×C₄×D₁₀₁	C₂³	1	10	yes	—
24	C₄⋊Dic₁₀₁	C₂³	1	10	yes	—
25	#13108053	C₂³	1	20	yes	—
26	#13108051	C₂³	1	20	yes	—
27	#13108051	C₂³	1	20	yes	—
28	C₄×Dic₁₀₁	C₂³	1	10	yes	—
29	C₄×Dic₁₀₁	C₂³	1	5	yes	—
30	C₄⋊C₄×C₁₀₁	C₂³	1	10	yes	—
31	C₄⋊C₄×C₁₀₁	C₂³	1	10	yes	—
32	C₄⋊C₄×C₁₀₁	C₂³	1	10	yes	—
33	C₄²×C₁₀₁	C₂³	1	5	yes	—
34	C₂²×D₁₀₁	C₂⁴	1	5	yes	—
35	C₂×C₄×C₁₀₁	C₂⁴	1	5	yes	—
36	C₂×C₄×C₁₀₁	C₂⁴	1	10	yes	—
37	C₂×Dic₁₀₁	C₂⁴	1	10	yes	—
38	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
39	D₄₀₄	—	4	10	no	—
40	C₄×D₁₀₁	—	2	10	no	—
41	C₄×D₁₀₁	—	4	10	no	—
42	Dic₂₀₂	—	4	10	no	—
43	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
44	C₂×D₁₀₁	—	2	5	no	—
45	C₂×D₁₀₁	—	2	10	no	—
46	C₄×C₁₀₁	—	2	5	no	—
47	C₄×C₁₀₁	—	2	10	no	—
48	Dic₁₀₁	—	2	10	no	—
49	Dic₁₀₁	2₋¹⁺⁴	1	2	yes	—
50	C₂×C₁₀₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
51	C₂×C₁₀₁	#585	1	2	yes	—
52	D₁₀₁	—	4	5	no	—
53	#3531	—	101	1	no	maximal, Sylow
54	C₁₀₁	#3531	1	1	yes	characteristic, complemented, Sylow
55	#564	D₁₀₁	1	1	yes	characteristic, complemented
56	#563	—	101	10	no	—
57	#562	—	101	5	no	—
58	#533	—	101	5	no	—
59	#529	—	101	10	no	—
60	C₄⋊Q₈	—	101	10	no	—
61	#126	—	101	20	no	—
62	#125	—	101	5	no	—
63	#125	C₂×D₁₀₁	1	10	yes	—
64	#125	C₂×D₁₀₁	1	5	yes	—
65	#124	—	101	10	no	—
66	#123	—	101	20	no	—
67	C₂²⋊Q₈	—	101	10	no	—
68	C₂²⋊Q₈	—	101	20	no	—
69	#121	—	101	10	no	—
70	C₄×Q₈	—	101	10	no	—
71	C₄×D₄	—	101	10	no	—
72	#117	—	101	10	no	—
73	C₂×C₄²	—	101	5	no	—
74	C₂×Q₈	—	101	10	no	—
75	C₂×D₄	—	101	10	no	—
76	C₂²×C₄	—	101	10	no	—
77	C₂²×C₄	—	101	5	no	—
78	C₄⋊C₄	—	101	20	no	—
79	C₄⋊C₄	—	101	10	no	—
80	C₄⋊C₄	C₂²×D₁₀₁	1	10	yes	—
81	C₄⋊C₄	C₂²×D₁₀₁	1	10	yes	—
82	C₄⋊C₄	C₂²×D₁₀₁	1	10	yes	—
83	C₂²⋊C₄	—	101	20	no	—
84	C₂²⋊C₄	—	101	20	no	—
85	C₄²	—	101	10	no	—
86	C₄²	—	101	5	no	—
87	C₄²	C₂²×D₁₀₁	1	5	yes	—
88	C₂³	—	101	5	no	—
89	Q₈	—	404	10	no	—
90	D₄	—	404	10	no	—
91	C₂×C₄	—	101	1	no	—
92	C₂×C₄	—	101	10	no	—
93	C₂×C₄	—	404	10	no	—
94	C₂×C₄	—	202	10	no	—
95	C₂×C₄	C₂³×D₁₀₁	1	5	yes	—
96	C₂×C₄	C₂³×D₁₀₁	1	10	yes	—
97	C₂²	—	202	5	no	—
98	C₂²	—	202	10	no	—
99	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
100	C₄	—	2	5	no	—
101	C₄	—	2	10	no	—
102	C₄	—	101	2	no	—
103	C₄	—	202	10	no	—
104	C₂	—	404	5	no	—
105	C₂	2₋¹⁺⁴×D₁₀₁	1	1	yes	characteristic
106	C₂	#20266014	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265931

C₂

yes

maximal, complemented

#20265925

C₂

yes

maximal, complemented

#20265733

C₂

yes

maximal, complemented

#20265712

C₂

yes

maximal, complemented

#564×C₁₀₁

C₂

yes

maximal, characteristic, complemented

#15692254

C₂²

yes

—

#15692250

C₂²

yes

—

#15692247

C₂²

yes

—

#15692243

C₂²

yes

—

#15692240

C₂²

yes

—

#15692235

C₂²

yes

—

#15692233

C₂²

yes

—

#15692230

C₂²

yes

—

#15692220

C₂²

yes

—

#15692217

C₂²

yes

—

#15692213

C₂²

yes

—

C₄²×D₁₀₁

C₂²

yes

—

#125×C₁₀₁

C₂²

yes

—

#125×C₁₀₁

C₂²

yes

—

C₂×Dic₂₀₂

C₂³

yes

—

C₂×D₄₀₄

C₂³

yes

—

C₂×C₄×D₁₀₁

C₂³

yes

—

C₂×C₄×D₁₀₁

C₂³

yes

—

C₄⋊Dic₁₀₁

C₂³

yes

—

#13108053

C₂³

yes

—

#13108051

C₂³

yes

—

#13108051

C₂³

yes

—

C₄×Dic₁₀₁

C₂³

yes

—

C₄×Dic₁₀₁

C₂³

yes

—

C₄⋊C₄×C₁₀₁

C₂³

yes

—

C₄⋊C₄×C₁₀₁

C₂³

yes

—

C₄⋊C₄×C₁₀₁

C₂³

yes

—

C₄²×C₁₀₁

C₂³

yes

—

C₂²×D₁₀₁

C₂⁴

yes

—

C₂×C₄×C₁₀₁

C₂⁴

yes

—

C₂×C₄×C₁₀₁

C₂⁴

yes

—

C₂×Dic₁₀₁

C₂⁴

yes

—

C₂×Dic₁₀₁

C₂⁴

yes

characteristic

D₄₀₄

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

Dic₂₀₂

—

C₂²×C₁₀₁

C₂⁵

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

2₋¹⁺⁴

yes

—

C₂×C₁₀₁

C₂×2₋¹⁺⁴

yes

characteristic

C₂×C₁₀₁

#585

yes

—

D₁₀₁

—

#3531

—

101

maximal, Sylow

C₁₀₁

#3531

yes

characteristic, complemented, Sylow

#564

D₁₀₁

yes

characteristic, complemented

#563

—

101

—

#562

—

101

—

#533

—

101

—

#529

—

101

—

C₄⋊Q₈

—

101

—

#126

—

101

—

#125

—

101

—

#125

C₂×D₁₀₁

yes

—

#125

C₂×D₁₀₁

yes

—

#124

—

101

—

#123

—

101

—

C₂²⋊Q₈

—

101

—

C₂²⋊Q₈

—

101

—

#121

—

101

—

C₄×Q₈

—

101

—

C₄×D₄

—

101

—

#117

—

101

—

C₂×C₄²

—

101

—

C₂×Q₈

—

101

—

C₂×D₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

101

—

C₄⋊C₄

C₂²×D₁₀₁

yes

—

C₄⋊C₄

C₂²×D₁₀₁

yes

—

C₄⋊C₄

C₂²×D₁₀₁

yes

—

C₂²⋊C₄

—

101

—

C₂²⋊C₄

—

101

—

C₄²

—

101

—

C₄²

—

101

—

C₄²

C₂²×D₁₀₁

yes

—

C₂³

—

101

—

Q₈

—

404

—

D₄

—

404

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

404

—

C₂×C₄

—

202

—

C₂×C₄

C₂³×D₁₀₁

yes

—

C₂×C₄

C₂³×D₁₀₁

yes

—

C₂²

—

202

—

C₂²

—

202

—

C₂²

C₂⁴×D₁₀₁

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

101

—

103

C₄

—

202

—

104

C₂

—

404

—

105

C₂

2₋¹⁺⁴×D₁₀₁

yes

characteristic

106

C₂

#20266014

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₁
commutator, socle, cosocle	C₂²×C₁₀₁	C₂⁵
Fitting	#564×C₁₀₁	C₂
hypercenter	#564	D₁₀₁
hypocenter	C₁₀₁	#3531
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₁₀₁

commutator, socle, cosocle

C₂²×C₁₀₁

C₂⁵

Fitting

#564×C₁₀₁

C₂

D₁₀₁

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
1. #20265931	C₂	semidirect
2. #20265925	C₂	semidirect
3. #20265733	C₂	semidirect
4. #20265712	C₂	semidirect
5. #564×C₁₀₁	C₂	semidirect
54. C₁₀₁	#3531	semidirect
55. #564	D₁₀₁	semidirect

H ≅ G/N

type

1. #20265931

C₂

semidirect

2. #20265925

C₂

semidirect

3. #20265733

C₂

semidirect

4. #20265712

C₂

semidirect

5. #564×C₁₀₁

C₂

semidirect

54. C₁₀₁

#3531

semidirect

55. #564

D₁₀₁

semidirect

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

subgroup	index
#20265931	2
#20265925	2
#20265733	2
#20265712	2
#564×C₁₀₁	2
#3531	101

subgroup

index

#564×C₁₀₁

101

Subgroups — #27185280

Conjugacy classes of subgroups

Complements

Maximal subgroups