Group #28335537 — subgroups

Subgroups — #28335537

Order 13696.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#20293357	C₂	1	1	yes	maximal, characteristic, complemented
2	#20292824	C₂	1	2	yes	maximal
3	#20292376	C₂	1	1	yes	maximal, characteristic, complemented
4	#20292359	C₂	1	2	yes	maximal, complemented
5	#492×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133806	—	2	2	no	—
7	#16812108	—	2	2	no	—
8	#16812107	—	2	2	no	—
9	C₄×Dic₂₁₄	C₂²	1	2	yes	—
10	#16812074	—	2	2	no	—
11	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	2	yes	—
12	#16812030	—	2	2	no	—
13	C₂×SD₁₆×C₁₀₇	—	2	2	no	—
14	C₄⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
15	#127×C₁₀₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₁₄	—	2	2	no	—
18	C₂×Dic₂₁₄	D₄	1	2	yes	—
19	#13133806	—	4	4	no	—
20	C₂×C₁₀₇⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₀₇	D₄	1	2	yes	—
22	Dic₁₀₇⋊C₄	—	2	4	no	—
23	Dic₁₀₇⋊C₄	—	2	2	no	—
24	C₄×Dic₁₀₇	—	2	2	no	—
25	C₄×Dic₁₀₇	—	2	2	no	—
26	C₂×Q₈×C₁₀₇	D₄	1	2	yes	—
27	C₂×D₄×C₁₀₇	—	2	2	no	—
28	C₂×D₄×C₁₀₇	—	2	1	no	—
29	SD₁₆×C₁₀₇	—	4	4	no	—
30	C₂×C₈×C₁₀₇	—	2	1	no	—
31	C₄⋊C₄×C₁₀₇	—	2	2	no	—
32	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
33	C₂³×C₁₀₇	—	4	1	no	—
34	Q₈×C₁₀₇	—	4	2	no	—
35	D₄×C₁₀₇	—	4	2	no	—
36	D₄×C₁₀₇	—	4	2	no	—
37	D₄×C₁₀₇	—	4	2	no	—
38	C₂×C₄×C₁₀₇	—	2	2	no	—
39	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₁₀₇	—	2	2	no	—
42	C₂×Dic₁₀₇	—	2	2	no	—
43	Dic₂₁₄	—	2	2	no	—
44	Dic₂₁₄	—	8	2	no	—
45	Dic₂₁₄	SD₁₆	1	4	yes	complemented
46	C₈×C₁₀₇	—	2	2	no	—
47	C₁₀₇⋊C₈	—	4	2	no	—
48	C₂²×C₁₀₇	—	8	2	no	—
49	C₂²×C₁₀₇	—	8	1	no	—
50	C₂²×C₁₀₇	#120	1	1	yes	characteristic
51	C₄×C₁₀₇	—	2	1	no	—
52	C₄×C₁₀₇	—	4	2	no	—
53	C₄×C₁₀₇	C₂×SD₁₆	1	2	yes	—
54	C₄×C₁₀₇	#137	1	2	yes	—
55	Dic₁₀₇	—	2	4	no	—
56	Dic₁₀₇	—	4	2	no	—
57	C₂×C₁₀₇	—	16	1	no	—
58	C₂×C₁₀₇	#448	1	2	yes	—
59	C₂×C₁₀₇	#453	1	1	yes	characteristic
60	#1623	—	107	1	no	maximal, Sylow
61	C₁₀₇	#1623	1	1	yes	characteristic, complemented, Sylow
62	Q₈²	—	107	1	no	—
63	#492	D₁₀₇	1	1	yes	characteristic
64	#460	—	107	2	no	—
65	#333	—	107	1	no	—
66	Q₈⋊C₈	—	107	2	no	—
67	C₂×SD₁₆	—	2	2	no	—
68	C₂×SD₁₆	—	214	2	no	—
69	C₄⋊Q₈	—	214	2	no	—
70	C₄⋊Q₈	—	214	2	no	—
71	C₄⋊Q₈	C₂×D₁₀₇	1	1	yes	characteristic
72	#127	C₂×D₁₀₇	1	1	yes	characteristic
73	C₄×Q₈	—	107	2	no	—
74	C₄×Q₈	—	214	2	no	—
75	C₄⋊C₈	—	107	2	no	—
76	#102	—	214	2	no	—
77	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
78	C₂×Q₈	—	107	2	no	—
79	C₂×Q₈	—	214	2	no	—
80	C₂×Q₈	C₂²⋊D₁₀₇	1	2	yes	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	1	no	—
83	SD₁₆	—	4	4	no	—
84	SD₁₆	—	428	4	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	214	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	107	2	no	—
89	C₄⋊C₄	—	214	4	no	—
90	C₄⋊C₄	—	214	2	no	—
91	C₄²	—	214	2	no	—
92	C₄²	—	214	2	no	—
93	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
94	C₂³	—	4	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	856	2	no	—
97	Q₈	—	107	4	no	—
98	Q₈	—	214	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	214	2	no	—
104	C₂×C₄	—	214	2	no	—
105	C₂×C₄	D₄×D₁₀₇	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	428	2	no	—
109	C₂²	—	8	2	no	—
110	C₂²	—	8	1	no	—
111	C₂²	#16812077	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	428	2	no	—
115	C₄	—	214	4	no	—
116	C₄	SD₁₆×D₁₀₇	1	2	yes	—
117	C₄	#16812138	1	2	yes	—
118	C₂	—	16	1	no	—
119	C₂	#20292764	1	2	yes	—
120	C₂	#20292794	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20293357

C₂

yes

maximal, characteristic, complemented

#20292824

C₂

yes

maximal

#20292376

C₂

yes

maximal, characteristic, complemented

#20292359

C₂

yes

maximal, complemented

#492×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133806

—

#16812108

—

#16812107

—

C₄×Dic₂₁₄

C₂²

yes

—

#16812074

—

C₄⋊(C₁₀₇⋊C₈)

C₂²

yes

—

#16812030

—

C₂×SD₁₆×C₁₀₇

—

C₄⋊Q₈×C₁₀₇

C₂²

yes

characteristic

#127×C₁₀₇

C₂²

yes

characteristic

C₄×C₈×C₁₀₇

C₂²

yes

characteristic

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

D₄

yes

—

#13133806

—

C₂×C₁₀₇⋊C₈

—

C₄⋊Dic₁₀₇

D₄

yes

—

Dic₁₀₇⋊C₄

—

Dic₁₀₇⋊C₄

—

C₄×Dic₁₀₇

—

C₄×Dic₁₀₇

—

C₂×Q₈×C₁₀₇

D₄

yes

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

SD₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

C₂³

yes

characteristic

C₂³×C₁₀₇

—

Q₈×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

Dic₂₁₄

—

Dic₂₁₄

—

Dic₂₁₄

SD₁₆

yes

complemented

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#120

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×SD₁₆

yes

—

C₄×C₁₀₇

#137

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#448

yes

—

C₂×C₁₀₇

#453

yes

characteristic

#1623

—

107

maximal, Sylow

C₁₀₇

#1623

yes

characteristic, complemented, Sylow

Q₈²

—

107

—

#492

D₁₀₇

yes

characteristic

#460

—

107

—

#333

—

107

—

Q₈⋊C₈

—

107

—

C₂×SD₁₆

—

C₂×SD₁₆

—

214

—

C₄⋊Q₈

—

214

—

C₄⋊Q₈

—

214

—

C₄⋊Q₈

C₂×D₁₀₇

yes

characteristic

#127

C₂×D₁₀₇

yes

characteristic

C₄×Q₈

—

107

—

C₄×Q₈

—

214

—

C₄⋊C₈

—

107

—

#102

—

214

—

C₄×C₈

C₂×D₁₀₇

yes

characteristic

C₂×Q₈

—

107

—

C₂×Q₈

—

214

—

C₂×Q₈

C₂²⋊D₁₀₇

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

428

—

C₂×C₈

—

C₂×C₈

—

214

—

C₄⋊C₄

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

C₄²

—

214

—

C₄²

—

214

—

C₄²

C₂²×D₁₀₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

856

—

Q₈

—

107

—

Q₈

—

214

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

214

—

104

C₂×C₄

—

214

—

105

C₂×C₄

D₄×D₁₀₇

yes

—

106

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

107

C₈

—

108

C₈

—

428

—

109

C₂²

—

110

C₂²

—

111

C₂²

#16812077

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

428

—

115

C₄

—

214

—

116

C₄

SD₁₆×D₁₀₇

yes

—

117

C₄

#16812138

yes

—

118

C₂

—

119

C₂

#20292764

yes

—

120

C₂

#20292794

yes

characteristic

	subgroup	G/H
center	C₂²	#16812077
commutator, cosocle	C₄²×C₁₀₇	C₂³
Frattini	C₄²	C₂²×D₁₀₇
Fitting	#492×C₁₀₇	C₂
socle	C₂²×C₁₀₇	#120
prime radical	#127×C₁₀₇	C₂²
hypercenter	#492	D₁₀₇
hypocenter	C₁₀₇	#1623
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#16812077

commutator, cosocle

C₄²×C₁₀₇

C₂³

Frattini

C₄²

C₂²×D₁₀₇

Fitting

#492×C₁₀₇

C₂

socle

C₂²×C₁₀₇

#120

prime radical

#127×C₁₀₇

C₂²

D₁₀₇

C₁₀₇

G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #20293357	C₂	semidirect
3. #20292376	C₂	semidirect
4. #20292359	C₂	semidirect
45. Dic₂₁₄	SD₁₆	semidirect
61. C₁₀₇	#1623	semidirect

H ≅ G/N

type

1. #20293357

C₂

semidirect

3. #20292376

C₂

semidirect

4. #20292359

C₂

semidirect

45. Dic₂₁₄

SD₁₆

semidirect

61. C₁₀₇

#1623

semidirect

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

subgroup	index
#20293357	2
#20292824	2
#20292376	2
#20292359	2
#492×C₁₀₇	2
#1623	107

subgroup

index

#492×C₁₀₇

107

Subgroups — #28335537

Conjugacy classes of subgroups

Complements

Maximal subgroups