Group #28350609 — subgroups

Subgroups — #28350609

Order 13696.

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	#20293381	C₂	1	10	yes	maximal, complemented
2	#20293375	C₂	1	5	yes	maximal, complemented
3	#20293183	C₂	1	5	yes	maximal, complemented
4	#20293162	C₂	1	10	yes	maximal, complemented
5	#564×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812108	C₂²	1	10	yes	—
7	#16812104	C₂²	1	20	yes	—
8	#16812101	C₂²	1	5	yes	—
9	#16812097	C₂²	1	10	yes	—
10	#16812094	C₂²	1	20	yes	—
11	#16812089	C₂²	1	10	yes	—
12	#16812087	C₂²	1	20	yes	—
13	#16812084	C₂²	1	10	yes	—
14	#16812074	C₂²	1	10	yes	—
15	#16812071	C₂²	1	10	yes	—
16	#16812067	C₂²	1	10	yes	—
17	C₄²×D₁₀₇	C₂²	1	5	yes	—
18	#125×C₁₀₇	C₂²	1	10	yes	—
19	#125×C₁₀₇	C₂²	1	5	yes	—
20	C₂×Dic₂₁₄	C₂³	1	10	yes	—
21	C₂×D₄₂₈	C₂³	1	10	yes	—
22	C₂×C₄×D₁₀₇	C₂³	1	5	yes	—
23	C₂×C₄×D₁₀₇	C₂³	1	10	yes	—
24	C₄⋊Dic₁₀₇	C₂³	1	10	yes	—
25	Dic₁₀₇⋊C₄	C₂³	1	20	yes	—
26	(C₂×D₁₀₇)⋊C₄	C₂³	1	20	yes	—
27	(C₂×D₁₀₇)⋊C₄	C₂³	1	20	yes	—
28	C₄×Dic₁₀₇	C₂³	1	10	yes	—
29	C₄×Dic₁₀₇	C₂³	1	5	yes	—
30	C₄⋊C₄×C₁₀₇	C₂³	1	10	yes	—
31	C₄⋊C₄×C₁₀₇	C₂³	1	10	yes	—
32	C₄⋊C₄×C₁₀₇	C₂³	1	10	yes	—
33	C₄²×C₁₀₇	C₂³	1	5	yes	—
34	C₂²×D₁₀₇	C₂⁴	1	5	yes	—
35	C₂×C₄×C₁₀₇	C₂⁴	1	5	yes	—
36	C₂×C₄×C₁₀₇	C₂⁴	1	10	yes	—
37	C₂×Dic₁₀₇	C₂⁴	1	10	yes	—
38	C₂×Dic₁₀₇	C₂⁴	1	1	yes	characteristic
39	D₄₂₈	—	4	10	no	—
40	C₄×D₁₀₇	—	2	10	no	—
41	C₄×D₁₀₇	—	4	10	no	—
42	Dic₂₁₄	—	4	10	no	—
43	C₂²×C₁₀₇	C₂⁵	1	1	yes	characteristic
44	C₂×D₁₀₇	—	2	5	no	—
45	C₂×D₁₀₇	—	2	10	no	—
46	C₄×C₁₀₇	—	2	5	no	—
47	C₄×C₁₀₇	—	2	10	no	—
48	Dic₁₀₇	—	2	10	no	—
49	Dic₁₀₇	2₋¹⁺⁴	1	2	yes	—
50	C₂×C₁₀₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
51	C₂×C₁₀₇	#585	1	2	yes	—
52	D₁₀₇	—	4	5	no	—
53	#3531	—	107	1	no	maximal, Sylow
54	C₁₀₇	#3531	1	1	yes	characteristic, complemented, Sylow
55	#564	D₁₀₇	1	1	yes	characteristic, complemented
56	#563	—	107	10	no	—
57	#562	—	107	5	no	—
58	#533	—	107	5	no	—
59	#529	—	107	10	no	—
60	C₄⋊Q₈	—	107	10	no	—
61	#126	—	107	20	no	—
62	#125	—	107	5	no	—
63	#125	C₂×D₁₀₇	1	10	yes	—
64	#125	C₂×D₁₀₇	1	5	yes	—
65	#124	—	107	10	no	—
66	#123	—	107	20	no	—
67	C₂²⋊Q₈	—	107	10	no	—
68	C₂²⋊Q₈	—	107	20	no	—
69	#121	—	107	10	no	—
70	C₄×Q₈	—	107	10	no	—
71	C₄×D₄	—	107	10	no	—
72	#117	—	107	10	no	—
73	C₂×C₄²	—	107	5	no	—
74	C₂×Q₈	—	107	10	no	—
75	C₂×D₄	—	107	10	no	—
76	C₂²×C₄	—	107	10	no	—
77	C₂²×C₄	—	107	5	no	—
78	C₄⋊C₄	—	107	10	no	—
79	C₄⋊C₄	—	107	20	no	—
80	C₄⋊C₄	C₂²×D₁₀₇	1	10	yes	—
81	C₄⋊C₄	C₂²×D₁₀₇	1	10	yes	—
82	C₄⋊C₄	C₂²×D₁₀₇	1	10	yes	—
83	C₂²⋊C₄	—	107	20	no	—
84	C₂²⋊C₄	—	107	20	no	—
85	C₄²	—	107	10	no	—
86	C₄²	—	107	5	no	—
87	C₄²	C₂²×D₁₀₇	1	5	yes	—
88	C₂³	—	107	5	no	—
89	Q₈	—	428	10	no	—
90	D₄	—	428	10	no	—
91	C₂×C₄	—	107	1	no	—
92	C₂×C₄	—	107	10	no	—
93	C₂×C₄	—	428	10	no	—
94	C₂×C₄	—	214	10	no	—
95	C₂×C₄	C₂³×D₁₀₇	1	5	yes	—
96	C₂×C₄	C₂³×D₁₀₇	1	10	yes	—
97	C₂²	—	214	5	no	—
98	C₂²	—	214	10	no	—
99	C₂²	C₂⁴×D₁₀₇	1	1	yes	characteristic
100	C₄	—	2	5	no	—
101	C₄	—	2	10	no	—
102	C₄	—	107	2	no	—
103	C₄	—	214	10	no	—
104	C₂	—	428	5	no	—
105	C₂	2₋¹⁺⁴×D₁₀₇	1	1	yes	characteristic
106	C₂	#20293464	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20293381

C₂

yes

maximal, complemented

#20293375

C₂

yes

maximal, complemented

#20293183

C₂

yes

maximal, complemented

#20293162

C₂

yes

maximal, complemented

#564×C₁₀₇

C₂

yes

maximal, characteristic, complemented

#16812108

C₂²

yes

—

#16812104

C₂²

yes

—

#16812101

C₂²

yes

—

#16812097

C₂²

yes

—

#16812094

C₂²

yes

—

#16812089

C₂²

yes

—

#16812087

C₂²

yes

—

#16812084

C₂²

yes

—

#16812074

C₂²

yes

—

#16812071

C₂²

yes

—

#16812067

C₂²

yes

—

C₄²×D₁₀₇

C₂²

yes

—

#125×C₁₀₇

C₂²

yes

—

#125×C₁₀₇

C₂²

yes

—

C₂×Dic₂₁₄

C₂³

yes

—

C₂×D₄₂₈

C₂³

yes

—

C₂×C₄×D₁₀₇

C₂³

yes

—

C₂×C₄×D₁₀₇

C₂³

yes

—

C₄⋊Dic₁₀₇

C₂³

yes

—

Dic₁₀₇⋊C₄

C₂³

yes

—

(C₂×D₁₀₇)⋊C₄

C₂³

yes

—

(C₂×D₁₀₇)⋊C₄

C₂³

yes

—

C₄×Dic₁₀₇

C₂³

yes

—

C₄×Dic₁₀₇

C₂³

yes

—

C₄⋊C₄×C₁₀₇

C₂³

yes

—

C₄⋊C₄×C₁₀₇

C₂³

yes

—

C₄⋊C₄×C₁₀₇

C₂³

yes

—

C₄²×C₁₀₇

C₂³

yes

—

C₂²×D₁₀₇

C₂⁴

yes

—

C₂×C₄×C₁₀₇

C₂⁴

yes

—

C₂×C₄×C₁₀₇

C₂⁴

yes

—

C₂×Dic₁₀₇

C₂⁴

yes

—

C₂×Dic₁₀₇

C₂⁴

yes

characteristic

D₄₂₈

—

C₄×D₁₀₇

—

C₄×D₁₀₇

—

Dic₂₁₄

—

C₂²×C₁₀₇

C₂⁵

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

2₋¹⁺⁴

yes

—

C₂×C₁₀₇

C₂×2₋¹⁺⁴

yes

characteristic

C₂×C₁₀₇

#585

yes

—

D₁₀₇

—

#3531

—

107

maximal, Sylow

C₁₀₇

#3531

yes

characteristic, complemented, Sylow

#564

D₁₀₇

yes

characteristic, complemented

#563

—

107

—

#562

—

107

—

#533

—

107

—

#529

—

107

—

C₄⋊Q₈

—

107

—

#126

—

107

—

#125

—

107

—

#125

C₂×D₁₀₇

yes

—

#125

C₂×D₁₀₇

yes

—

#124

—

107

—

#123

—

107

—

C₂²⋊Q₈

—

107

—

C₂²⋊Q₈

—

107

—

#121

—

107

—

C₄×Q₈

—

107

—

C₄×D₄

—

107

—

#117

—

107

—

C₂×C₄²

—

107

—

C₂×Q₈

—

107

—

C₂×D₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

C₂²×D₁₀₇

yes

—

C₄⋊C₄

C₂²×D₁₀₇

yes

—

C₄⋊C₄

C₂²×D₁₀₇

yes

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₄²

—

107

—

C₄²

—

107

—

C₄²

C₂²×D₁₀₇

yes

—

C₂³

—

107

—

Q₈

—

428

—

D₄

—

428

—

C₂×C₄

—

107

—

C₂×C₄

—

107

—

C₂×C₄

—

428

—

C₂×C₄

—

214

—

C₂×C₄

C₂³×D₁₀₇

yes

—

C₂×C₄

C₂³×D₁₀₇

yes

—

C₂²

—

214

—

C₂²

—

214

—

C₂²

C₂⁴×D₁₀₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

107

—

103

C₄

—

214

—

104

C₂

—

428

—

105

C₂

2₋¹⁺⁴×D₁₀₇

yes

characteristic

106

C₂

#20293464

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₇
commutator, socle, cosocle	C₂²×C₁₀₇	C₂⁵
Fitting	#564×C₁₀₇	C₂
hypercenter	#564	D₁₀₇
hypocenter	C₁₀₇	#3531
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₁₀₇

commutator, socle, cosocle

C₂²×C₁₀₇

C₂⁵

Fitting

#564×C₁₀₇

C₂

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
1. #20293381	C₂	semidirect
2. #20293375	C₂	semidirect
3. #20293183	C₂	semidirect
4. #20293162	C₂	semidirect
5. #564×C₁₀₇	C₂	semidirect
54. C₁₀₇	#3531	semidirect
55. #564	D₁₀₇	semidirect

H ≅ G/N

type

1. #20293381

C₂

semidirect

2. #20293375

C₂

semidirect

3. #20293183

C₂

semidirect

4. #20293162

C₂

semidirect

5. #564×C₁₀₇

C₂

semidirect

54. C₁₀₇

#3531

semidirect

55. #564

D₁₀₇

semidirect

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

subgroup	index
#20293381	2
#20293375	2
#20293183	2
#20293162	2
#564×C₁₀₇	2
#3531	107

subgroup

index

#564×C₁₀₇

107

Subgroups — #28350609

Conjugacy classes of subgroups

Complements

Maximal subgroups