Group #28335540 — subgroups

Subgroups — #28335540

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	#20293001	C₂	1	1	yes	maximal, characteristic
3	#20292828	C₂	1	2	yes	maximal
4	#20292357	C₂	1	2	yes	maximal, complemented
5	#333×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13133807	—	2	2	no	—
7	C₂×#13133805	—	2	2	no	—
8	#16812108	—	2	2	no	—
9	#16812107	—	2	2	no	—
10	#16812106	C₂²	1	1	yes	characteristic
11	C₄×Dic₂₁₄	C₂²	1	2	yes	—
12	#16812074	—	2	2	no	—
13	#16812031	—	2	2	no	—
14	C₄×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₀₇	C₂²	1	2	yes	—
17	C₂×Dic₂₁₄	—	2	2	no	—
18	C₂×Dic₂₁₄	D₄	1	2	yes	—
19	C₂×D₄₂₈	—	2	1	no	—
20	C₂×D₄₂₈	—	2	2	no	—
21	#13133807	—	4	4	no	—
22	#13133805	—	4	4	no	—
23	C₂×C₁₀₇⋊C₈	—	2	1	no	—
24	C₄⋊Dic₁₀₇	D₄	1	2	yes	—
25	Dic₁₀₇⋊C₄	—	2	4	no	—
26	Dic₁₀₇⋊C₄	—	2	2	no	—
27	C₄×Dic₁₀₇	—	2	2	no	—
28	C₄×Dic₁₀₇	—	2	2	no	—
29	C₂×Q₈×C₁₀₇	D₄	1	2	yes	—
30	C₂×C₈×C₁₀₇	—	2	2	no	—
31	C₄⋊C₄×C₁₀₇	—	2	2	no	—
32	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
33	C₂²×D₁₀₇	—	4	1	no	—
34	Q₈×C₁₀₇	—	4	2	no	—
35	C₂×C₄×C₁₀₇	—	2	2	no	—
36	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₁₀₇	—	2	2	no	—
39	C₂×Dic₁₀₇	—	2	2	no	—
40	D₄₂₈	—	4	2	no	—
41	D₄₂₈	—	4	2	no	—
42	D₄₂₈	—	4	2	no	—
43	Dic₂₁₄	—	2	2	no	—
44	Dic₂₁₄	—	8	2	no	—
45	Dic₂₁₄	SD₁₆	1	4	yes	complemented
46	C₈×C₁₀₇	—	4	2	no	—
47	C₁₀₇⋊C₈	—	2	2	no	—
48	C₂²×C₁₀₇	#120	1	1	yes	characteristic
49	C₂×D₁₀₇	—	8	2	no	—
50	C₂×D₁₀₇	—	8	1	no	—
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	#448	1	2	yes	—
58	C₂×C₁₀₇	#453	1	1	yes	characteristic
59	D₁₀₇	—	16	1	no	—
60	#1623	—	107	1	no	maximal, Sylow
61	C₁₀₇	#1623	1	1	yes	characteristic, complemented, Sylow
62	Q₈²	—	107	1	no	—
63	#492	—	107	1	no	—
64	#460	—	107	2	no	—
65	#333	D₁₀₇	1	1	yes	characteristic, complemented
66	Q₈⋊C₈	—	107	2	no	—
67	C₂×SD₁₆	—	214	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	—	107	1	no	—
73	C₄×Q₈	—	107	2	no	—
74	C₄×Q₈	—	214	2	no	—
75	C₄⋊C₈	C₂×D₁₀₇	1	2	yes	—
76	#102	—	214	2	no	—
77	C₄×C₈	—	107	1	no	—
78	C₂×Q₈	—	107	2	no	—
79	C₂×Q₈	—	214	2	no	—
80	C₂×Q₈	D₄₂₈	1	2	yes	—
81	C₂×D₄	—	214	2	no	—
82	C₂×D₄	—	214	1	no	—
83	SD₁₆	—	428	4	no	—
84	SD₁₆	—	428	4	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	214	1	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	—	428	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	856	2	no	—
97	Q₈	—	107	4	no	—
98	Q₈	—	214	2	no	—
99	D₄	—	428	2	no	—
100	D₄	—	428	2	no	—
101	D₄	—	428	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×D₄₂₈	1	1	yes	characteristic
107	C₈	—	4	2	no	—
108	C₈	—	214	2	no	—
109	C₂²	—	856	2	no	—
110	C₂²	—	856	1	no	—
111	C₂²	#16812079	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	#16812141	1	2	yes	—
118	C₂	—	1712	1	no	—
119	C₂	#20292767	1	2	yes	—
120	C₂	#20292797	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20293357

C₂

yes

maximal, characteristic, complemented

#20293001

C₂

yes

maximal, characteristic

#20292828

C₂

yes

maximal

#20292357

C₂

yes

maximal, complemented

#333×C₁₀₇

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₂₁₄

C₂²

yes

—

#16812074

—

#16812031

—

C₄×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₇

C₂²

yes

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

D₄

yes

—

C₂×D₄₂₈

—

C₂×D₄₂₈

—

#13133807

—

#13133805

—

C₂×C₁₀₇⋊C₈

—

C₄⋊Dic₁₀₇

D₄

yes

—

Dic₁₀₇⋊C₄

—

Dic₁₀₇⋊C₄

—

C₄×Dic₁₀₇

—

C₄×Dic₁₀₇

—

C₂×Q₈×C₁₀₇

D₄

yes

—

C₂×C₈×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

C₂³

yes

characteristic

C₂²×D₁₀₇

—

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

D₄₂₈

—

D₄₂₈

—

D₄₂₈

—

Dic₂₁₄

—

Dic₂₁₄

—

Dic₂₁₄

SD₁₆

yes

complemented

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

#120

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×SD₁₆

yes

—

C₄×C₁₀₇

#137

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

#448

yes

—

C₂×C₁₀₇

#453

yes

characteristic

D₁₀₇

—

#1623

—

107

maximal, Sylow

C₁₀₇

#1623

yes

characteristic, complemented, Sylow

Q₈²

—

107

—

#492

—

107

—

#460

—

107

—

#333

D₁₀₇

yes

characteristic, complemented

Q₈⋊C₈

—

107

—

C₂×SD₁₆

—

214

—

C₂×SD₁₆

—

214

—

C₄⋊Q₈

—

214

—

C₄⋊Q₈

—

214

—

C₄⋊Q₈

C₂×D₁₀₇

yes

characteristic

#127

—

107

—

C₄×Q₈

—

107

—

C₄×Q₈

—

214

—

C₄⋊C₈

C₂×D₁₀₇

yes

—

#102

—

214

—

C₄×C₈

—

107

—

C₂×Q₈

—

107

—

C₂×Q₈

—

214

—

C₂×Q₈

D₄₂₈

yes

—

C₂×D₄

—

214

—

C₂×D₄

—

214

—

SD₁₆

—

428

—

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

—

428

—

Q₈

—

Q₈

—

856

—

Q₈

—

107

—

Q₈

—

214

—

D₄

—

428

—

100

D₄

—

428

—

101

D₄

—

428

—

102

C₂×C₄

—

103

C₂×C₄

—

214

—

104

C₂×C₄

—

214

—

105

C₂×C₄

D₄×D₁₀₇

yes

—

106

C₂×C₄

C₂×D₄₂₈

yes

characteristic

107

C₈

—

108

C₈

—

214

—

109

C₂²

—

856

—

110

C₂²

—

856

—

111

C₂²

#16812079

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

428

—

115

C₄

—

214

—

116

C₄

SD₁₆×D₁₀₇

yes

—

117

C₄

#16812141

yes

—

118

C₂

—

1712

—

119

C₂

#20292767

yes

—

120

C₂

#20292797

yes

characteristic

	subgroup	G/H
center	C₂²	#16812079
commutator, cosocle	C₄²×C₁₀₇	C₂³
Frattini	C₄²	C₂²×D₁₀₇
Fitting	#333×C₁₀₇	C₂
socle	C₂²×C₁₀₇	#120
prime radical	#16812106	C₂²
hypercenter	#333	D₁₀₇
hypocenter	C₁₀₇	#1623
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#16812079

commutator, cosocle

C₄²×C₁₀₇

C₂³

Frattini

C₄²

C₂²×D₁₀₇

Fitting

#333×C₁₀₇

C₂

C₂²×C₁₀₇

C₂²

D₁₀₇

C₁₀₇

G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
1. #20293357	C₂	semidirect
4. #20292357	C₂	semidirect
5. #333×C₁₀₇	C₂	semidirect
45. Dic₂₁₄	SD₁₆	semidirect
61. C₁₀₇	#1623	semidirect
65. #333	D₁₀₇	semidirect

H ≅ G/N

type

1. #20293357

C₂

semidirect

4. #20292357

C₂

semidirect

5. #333×C₁₀₇

C₂

semidirect

45. Dic₂₁₄

SD₁₆

semidirect

61. C₁₀₇

#1623

semidirect

65. #333

D₁₀₇

semidirect

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

subgroup	index
#20293357	2
#20293001	2
#20292828	2
#20292357	2
#333×C₁₀₇	2
#1623	107

subgroup

index

#333×C₁₀₇

107

Subgroups — #28335540

Conjugacy classes of subgroups

Complements

Maximal subgroups