Group #28345532 — subgroups

Subgroups — #28345532

Order 13696.

394 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 197 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	#20292699	C₂	1	8	yes	maximal, complemented
2	#20292684	C₂	1	2	yes	maximal, complemented
3	#20292594	C₂	1	4	yes	maximal, complemented
4	#532×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133802	C₂²	1	4	yes	—
6	C₂²×C₁₀₇⋊C₈	C₂²	1	4	yes	—
7	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	4	yes	—
8	C₂²⋊(C₁₀₇⋊C₈)	C₂²	1	4	yes	—
9	C₂²⋊(C₁₀₇⋊C₈)	C₂²	1	8	yes	—
10	#16812020	C₂²	1	4	yes	—
11	C₂×#41×C₁₀₇	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₀₇	C₄	1	1	yes	characteristic
13	#127×C₁₀₇	C₄	1	1	yes	characteristic
14	#124×C₁₀₇	C₄	1	2	yes	—
15	#121×C₁₀₇	C₄	1	4	yes	—
16	C₄×D₄×C₁₀₇	C₂²	1	4	yes	—
17	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
18	#13133802	—	4	4	no	—
19	C₂×C₁₀₇⋊C₈	—	4	4	no	—
20	C₂×C₁₀₇⋊C₈	C₂³	1	8	yes	—
21	#41×C₁₀₇	—	4	2	no	—
22	#41×C₁₀₇	—	4	2	no	—
23	C₂×Q₈×C₁₀₇	C₂×C₄	1	2	yes	—
24	C₂×D₄×C₁₀₇	C₂×C₄	1	4	yes	—
25	C₂×D₄×C₁₀₇	C₂×C₄	1	4	yes	—
26	C₂×D₄×C₁₀₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₁₀₇	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₀₇	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₁₀₇	C₂×C₄	1	8	yes	—
32	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₀₇	C₂³	1	2	yes	—
35	C₂³×C₁₀₇	C₂²×C₄	1	4	yes	—
36	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₁₀₇	—	4	2	no	—
38	D₄×C₁₀₇	—	4	4	no	—
39	D₄×C₁₀₇	—	4	2	no	—
40	D₄×C₁₀₇	—	4	4	no	—
41	C₂×C₄×C₁₀₇	—	2	2	no	—
42	C₂×C₄×C₁₀₇	—	2	4	no	—
43	C₂×C₄×C₁₀₇	—	2	4	no	—
44	C₂×C₄×C₁₀₇	—	4	2	no	—
45	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₁₀₇	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
50	C₁₀₇⋊C₈	—	2	8	no	—
51	C₂²×C₁₀₇	—	2	1	no	—
52	C₂²×C₁₀₇	—	2	4	no	—
53	C₂²×C₁₀₇	—	2	2	no	—
54	C₂²×C₁₀₇	—	2	8	no	—
55	C₂²×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
56	C₄×C₁₀₇	—	2	1	no	—
57	C₄×C₁₀₇	—	2	4	no	—
58	C₄×C₁₀₇	—	2	2	no	—
59	C₄×C₁₀₇	—	2	2	no	—
60	C₄×C₁₀₇	H(5,𝔽₂)	1	2	yes	—
61	C₂×C₁₀₇	—	4	1	no	—
62	C₂×C₁₀₇	—	4	4	no	—
63	C₂×C₁₀₇	#518	1	1	yes	characteristic
64	C₂×C₁₀₇	#568	1	2	yes	—
65	#2980	—	107	1	no	maximal, Sylow
66	C₁₀₇	#2980	1	1	yes	characteristic, complemented, Sylow
67	#532	D₁₀₇	1	1	yes	characteristic
68	#435	—	107	8	no	—
69	#432	—	107	2	no	—
70	#408	—	107	4	no	—
71	C₂×#41	C₂×D₁₀₇	1	2	yes	—
72	C₂×M₁₆	—	107	4	no	—
73	C₂²×C₈	—	107	4	no	—
74	C₄⋊Q₈	Dic₁₀₇	1	1	yes	characteristic
75	#127	Dic₁₀₇	1	1	yes	characteristic
76	#124	Dic₁₀₇	1	2	yes	—
77	#121	Dic₁₀₇	1	4	yes	—
78	C₄×D₄	C₂×D₁₀₇	1	4	yes	—
79	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
80	C₄⋊C₈	—	107	4	no	—
81	C₂²⋊C₈	—	107	4	no	—
82	C₂²⋊C₈	—	107	8	no	—
83	#97	—	107	4	no	—
84	#41	—	4	2	no	—
85	#41	—	4	2	no	—
86	C₂×Q₈	C₂×Dic₁₀₇	1	2	yes	—
87	C₂×D₄	C₂×Dic₁₀₇	1	4	yes	—
88	C₂×D₄	C₂×Dic₁₀₇	1	4	yes	—
89	C₂×D₄	C₂×Dic₁₀₇	1	2	yes	—
90	C₂²×C₄	C₂×Dic₁₀₇	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
92	C₂²×C₄	C₂²×D₁₀₇	1	4	yes	—
93	M₁₆	—	428	4	no	—
94	C₂×C₈	—	107	8	no	—
95	C₂×C₈	—	428	4	no	—
96	C₄⋊C₄	C₂×Dic₁₀₇	1	4	yes	—
97	C₂²⋊C₄	C₂×Dic₁₀₇	1	8	yes	—
98	C₄²	C₂×Dic₁₀₇	1	1	yes	characteristic
99	C₄²	C₂×Dic₁₀₇	1	1	yes	characteristic
100	C₄²	C₂²×D₁₀₇	1	2	yes	—
101	C₂³	C₂²×Dic₁₀₇	1	4	yes	—
102	C₂³	C₂²×Dic₁₀₇	1	1	yes	characteristic
103	Q₈	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	4	no	—
106	D₄	—	4	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
112	C₂×C₄	C₂²×Dic₁₀₇	1	2	yes	—
113	C₂×C₄	C₂²×Dic₁₀₇	1	2	yes	—
114	C₂×C₄	C₂²×Dic₁₀₇	1	4	yes	—
115	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
116	C₈	—	214	8	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	8	no	—
119	C₂²	—	2	1	no	—
120	C₂²	—	2	4	no	—
121	C₂²	C₂³×Dic₁₀₇	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	4	no	—
126	C₄	#16812154	1	2	yes	—
127	C₂	—	4	1	no	—
128	C₂	—	4	4	no	—
129	C₂	#20293102	1	1	yes	characteristic
130	C₂	#20293394	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20292699

C₂

yes

maximal, complemented

#20292684

C₂

yes

maximal, complemented

#20292594

C₂

yes

maximal, complemented

#532×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133802

C₂²

yes

—

C₂²×C₁₀₇⋊C₈

C₂²

yes

—

C₄⋊(C₁₀₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₁₀₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₁₀₇⋊C₈)

C₂²

yes

—

#16812020

C₂²

yes

—

C₂×#41×C₁₀₇

C₂²

yes

—

C₄⋊Q₈×C₁₀₇

C₄

yes

characteristic

#127×C₁₀₇

C₄

yes

characteristic

#124×C₁₀₇

C₄

yes

—

#121×C₁₀₇

C₄

yes

—

C₄×D₄×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

#13133802

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

C₂³

yes

—

#41×C₁₀₇

—

#41×C₁₀₇

—

C₂×Q₈×C₁₀₇

C₂×C₄

yes

—

C₂×D₄×C₁₀₇

C₂×C₄

yes

—

C₂×D₄×C₁₀₇

C₂×C₄

yes

—

C₂×D₄×C₁₀₇

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₀₇

C₂×C₄

yes

—

C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₄²×C₁₀₇

C₂³

yes

—

C₂³×C₁₀₇

C₂²×C₄

yes

—

C₂³×C₁₀₇

C₂²×C₄

yes

characteristic

Q₈×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂²×C₄

yes

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

—

C₂×C₄×C₁₀₇

C₂⁴

yes

characteristic

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂³×C₄

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

H(5,𝔽₂)

yes

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#518

yes

characteristic

C₂×C₁₀₇

#568

yes

—

#2980

—

107

maximal, Sylow

C₁₀₇

#2980

yes

characteristic, complemented, Sylow

#532

D₁₀₇

yes

characteristic

#435

—

107

—

#432

—

107

—

#408

—

107

—

C₂×#41

C₂×D₁₀₇

yes

—

C₂×M₁₆

—

107

—

C₂²×C₈

—

107

—

C₄⋊Q₈

Dic₁₀₇

yes

characteristic

#127

Dic₁₀₇

yes

characteristic

#124

Dic₁₀₇

yes

—

#121

Dic₁₀₇

yes

—

C₄×D₄

C₂×D₁₀₇

yes

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

107

—

C₂²⋊C₈

—

107

—

C₂²⋊C₈

—

107

—

#97

—

107

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₁₀₇

yes

—

C₂×D₄

C₂×Dic₁₀₇

yes

—

C₂×D₄

C₂×Dic₁₀₇

yes

—

C₂×D₄

C₂×Dic₁₀₇

yes

—

C₂²×C₄

C₂×Dic₁₀₇

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

C₂²×C₄

C₂²×D₁₀₇

yes

—

M₁₆

—

428

—

C₂×C₈

—

107

—

C₂×C₈

—

428

—

C₄⋊C₄

C₂×Dic₁₀₇

yes

—

C₂²⋊C₄

C₂×Dic₁₀₇

yes

—

C₄²

C₂×Dic₁₀₇

yes

characteristic

C₄²

C₂×Dic₁₀₇

yes

characteristic

100

C₄²

C₂²×D₁₀₇

yes

—

101

C₂³

C₂²×Dic₁₀₇

yes

—

102

C₂³

C₂²×Dic₁₀₇

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²×Dic₁₀₇

yes

characteristic

112

C₂×C₄

C₂²×Dic₁₀₇

yes

—

113

C₂×C₄

C₂²×Dic₁₀₇

yes

—

114

C₂×C₄

C₂²×Dic₁₀₇

yes

—

115

C₂×C₄

C₂³×D₁₀₇

yes

characteristic

116

C₈

—

214

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂³×Dic₁₀₇

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#16812154

yes

—

127

C₂

—

128

C₂

—

129

C₂

#20293102

yes

characteristic

130

C₂

#20293394

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂³×C₄
Fitting, prime radical	#532×C₁₀₇	C₂
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	#532	D₁₀₇
hypocenter	C₁₀₇	#2980
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂³×C₄

Fitting, prime radical

#532×C₁₀₇

C₂

cosocle

C₂×C₄×C₁₀₇

C₂⁴

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. #20292699	C₂	semidirect
2. #20292684	C₂	semidirect
3. #20292594	C₂	semidirect
66. C₁₀₇	#2980	semidirect

H ≅ G/N

type

1. #20292699

C₂

semidirect

2. #20292684

C₂

semidirect

3. #20292594

C₂

semidirect

66. C₁₀₇

#2980

semidirect

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

subgroup	index
#20292699	2
#20292684	2
#20292594	2
#532×C₁₀₇	2
#2980	107

subgroup

index

#532×C₁₀₇

107

Subgroups — #28345532

Conjugacy classes of subgroups

Complements

Maximal subgroups