Group #28337286 — subgroups

Subgroups — #28337286

Order 13696.

412 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×#16812030	C₂	1	4	yes	maximal, complemented
2	C₂×#16812020	C₂	1	1	yes	maximal, characteristic, complemented
3	#20292493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#127×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₀₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
7	#16812030	—	2	8	no	—
8	#16812020	—	2	2	no	—
9	C₁₀₇⋊#95	—	2	2	no	—
10	C₂²×D₄×C₁₀₇	—	2	2	no	—
11	C₂²×D₄×C₁₀₇	C₂²	1	2	yes	—
12	#127×C₁₀₇	—	2	2	no	—
13	#127×C₁₀₇	C₄	1	4	yes	complemented
14	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₇	—	2	2	no	—
16	C₂×C₁₀₇⋊C₈	—	2	2	no	—
17	C₂×C₁₀₇⋊C₈	—	2	2	no	—
18	C₂×C₁₀₇⋊C₈	D₄	1	4	yes	complemented
19	C₄⋊Dic₁₀₇	—	2	4	no	—
20	C₂⁴×C₁₀₇	—	2	2	no	—
21	C₂×D₄×C₁₀₇	—	2	8	no	—
22	C₂×D₄×C₁₀₇	—	2	4	no	—
23	C₂×D₄×C₁₀₇	—	2	2	no	—
24	C₂×D₄×C₁₀₇	—	2	4	no	—
25	C₂×D₄×C₁₀₇	—	4	8	no	—
26	C₂×D₄×C₁₀₇	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
28	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
29	C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
30	C₄²×C₁₀₇	D₄	1	2	yes	—
31	C₂³×C₁₀₇	—	2	4	no	—
32	C₂³×C₁₀₇	—	4	2	no	—
33	C₂³×C₁₀₇	—	4	2	no	—
34	C₂³×C₁₀₇	—	4	8	no	—
35	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
36	D₄×C₁₀₇	—	2	8	no	—
37	D₄×C₁₀₇	—	4	16	no	—
38	D₄×C₁₀₇	—	4	4	no	—
39	C₂×C₄×C₁₀₇	—	2	4	no	—
40	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₇	#41	1	2	yes	—
45	C₂×Dic₁₀₇	—	4	2	no	—
46	C₂×Dic₁₀₇	—	4	4	no	—
47	C₁₀₇⋊C₈	—	2	4	no	—
48	C₂²×C₁₀₇	—	4	4	no	—
49	C₂²×C₁₀₇	—	4	8	no	—
50	C₂²×C₁₀₇	—	8	4	no	—
51	C₂²×C₁₀₇	—	8	4	no	—
52	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₁₀₇	#124	1	1	yes	characteristic
54	C₂²×C₁₀₇	#127	1	1	yes	characteristic
55	C₂²×C₁₀₇	AGL(1,ℤ/8)	1	4	yes	—
56	C₄×C₁₀₇	—	2	4	no	—
57	C₄×C₁₀₇	C₄×D₄	1	2	yes	—
58	C₄×C₁₀₇	#121	1	2	yes	—
59	Dic₁₀₇	—	8	2	no	—
60	C₂×C₁₀₇	—	8	4	no	—
61	C₂×C₁₀₇	#390	1	1	yes	characteristic
62	C₂×C₁₀₇	#418	1	2	yes	—
63	C₂×C₁₀₇	#490	1	2	yes	—
64	C₂×C₁₀₇	#496	1	2	yes	—
65	#1961	—	107	1	no	maximal, Sylow
66	C₁₀₇	#1961	1	1	yes	characteristic, complemented, Sylow
67	C₂×#127	D₁₀₇	1	1	yes	characteristic
68	C₂×#102	—	107	4	no	—
69	C₂×#97	—	107	1	no	—
70	#382	—	107	1	no	—
71	C₂²×D₄	—	2	2	no	—
72	C₂²×D₄	C₂×D₁₀₇	1	2	yes	—
73	C₂²×C₈	—	107	2	no	—
74	#127	—	2	2	no	—
75	#127	Dic₁₀₇	1	4	yes	complemented
76	C₂×C₄⋊C₄	—	107	2	no	—
77	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
78	#102	—	214	8	no	—
79	#97	—	214	2	no	—
80	#95	—	214	2	no	—
81	C₂⁴	—	2	2	no	—
82	C₂×D₄	—	2	8	no	—
83	C₂×D₄	—	2	4	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	4	no	—
86	C₂×D₄	—	4	8	no	—
87	C₂×D₄	C₂×Dic₁₀₇	1	4	yes	—
88	C₂²×C₄	—	214	2	no	—
89	C₂²×C₄	C₂²⋊D₁₀₇	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
91	C₂×C₈	—	107	4	no	—
92	C₂×C₈	—	214	2	no	—
93	C₂×C₈	—	214	2	no	—
94	C₄⋊C₄	—	214	4	no	—
95	C₄²	C₂×Dic₁₀₇	1	2	yes	—
96	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
97	C₂³	—	2	4	no	—
98	C₂³	—	4	2	no	—
99	C₂³	—	4	2	no	—
100	C₂³	—	4	8	no	—
101	C₂³	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
102	D₄	—	2	8	no	—
103	D₄	—	4	16	no	—
104	D₄	—	4	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	428	2	no	—
107	C₂×C₄	—	428	4	no	—
108	C₂×C₄	C₂²⋊Dic₁₀₇	1	4	yes	—
109	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
110	C₂×C₄	D₄×D₁₀₇	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
112	C₂×C₄	#13133817	1	2	yes	—
113	C₈	—	214	4	no	—
114	C₂²	—	4	8	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	8	4	no	—
117	C₂²	—	8	4	no	—
118	C₂²	C₂×C₂²⋊Dic₁₀₇	1	1	yes	characteristic
119	C₂²	#16812096	1	1	yes	characteristic
120	C₂²	#16812105	1	1	yes	characteristic
121	C₂²	#16812135	1	4	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	856	2	no	—
124	C₄	D₄×Dic₁₀₇	1	2	yes	—
125	C₄	#16812080	1	2	yes	—
126	C₂	—	8	4	no	—
127	C₂	#20292530	1	1	yes	characteristic
128	C₂	#20292632	1	2	yes	—
129	C₂	#20292991	1	2	yes	—
130	C₂	#20293014	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812030

C₂

yes

maximal, complemented

C₂×#16812020

C₂

yes

maximal, characteristic, complemented

#20292493

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×C₁₀₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₀₇

C₂²

yes

—

#16812030

—

#16812020

—

C₁₀₇⋊#95

—

C₂²×D₄×C₁₀₇

—

C₂²×D₄×C₁₀₇

C₂²

yes

—

#127×C₁₀₇

—

#127×C₁₀₇

C₄

yes

complemented

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₂²×Dic₁₀₇

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

D₄

yes

complemented

C₄⋊Dic₁₀₇

—

C₂⁴×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₄²×C₁₀₇

C₂×C₄

yes

—

C₄²×C₁₀₇

D₄

yes

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

C₂×D₄

yes

characteristic

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

#41

yes

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×C₂²⋊C₄

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₄×D₄

yes

—

C₄×C₁₀₇

#121

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#390

yes

characteristic

C₂×C₁₀₇

#418

yes

—

C₂×C₁₀₇

#490

yes

—

C₂×C₁₀₇

#496

yes

—

#1961

—

107

maximal, Sylow

C₁₀₇

#1961

yes

characteristic, complemented, Sylow

C₂×#127

D₁₀₇

yes

characteristic

C₂×#102

—

107

—

C₂×#97

—

107

—

#382

—

107

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₀₇

yes

—

C₂²×C₈

—

107

—

#127

—

#127

Dic₁₀₇

yes

complemented

C₂×C₄⋊C₄

—

107

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

#102

—

214

—

#97

—

214

—

#95

—

214

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₁₀₇

yes

—

C₂²×C₄

—

214

—

C₂²×C₄

C₂²⋊D₁₀₇

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

C₂×C₈

—

107

—

C₂×C₈

—

214

—

C₂×C₈

—

214

—

C₄⋊C₄

—

214

—

C₄²

C₂×Dic₁₀₇

yes

—

C₄²

C₂²⋊D₁₀₇

yes

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×C₂²⋊D₁₀₇

yes

characteristic

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

428

—

107

C₂×C₄

—

428

—

108

C₂×C₄

C₂²⋊Dic₁₀₇

yes

—

109

C₂×C₄

C₂²×Dic₁₀₇

yes

characteristic

110

C₂×C₄

D₄×D₁₀₇

yes

—

111

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

112

C₂×C₄

#13133817

yes

—

113

C₈

—

214

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂×C₂²⋊Dic₁₀₇

yes

characteristic

119

C₂²

#16812096

yes

characteristic

120

C₂²

#16812105

yes

characteristic

121

C₂²

#16812135

yes

—

122

C₄

—

123

C₄

—

856

—

124

C₄

D₄×Dic₁₀₇

yes

—

125

C₄

#16812080

yes

—

126

C₂

—

127

C₂

#20292530

yes

characteristic

128

C₂

#20292632

yes

—

129

C₂

#20292991

yes

—

130

C₂

#20293014

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₇
commutator	C₂×C₄×C₁₀₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting, prime radical	C₂×#127×C₁₀₇	C₂
socle	C₂³×C₁₀₇	C₂×D₄
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₂×#127	D₁₀₇
hypocenter	C₁₀₇	#1961
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₇

commutator

C₂×C₄×C₁₀₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting, prime radical

C₂×#127×C₁₀₇

C₂

socle

C₂³×C₁₀₇

C₂×D₄

cosocle

C₂²×C₄×C₁₀₇

C₂³

D₁₀₇

C₁₀₇

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#16812030	C₂	semidirect
2. C₂×#16812020	C₂	semidirect
3. #20292493	C₂	semidirect
13. #127×C₁₀₇	C₄	semidirect
18. C₂×C₁₀₇⋊C₈	D₄	semidirect
66. C₁₀₇	#1961	semidirect
75. #127	Dic₁₀₇	semidirect

H ≅ G/N

type

1. C₂×#16812030

C₂

semidirect

2. C₂×#16812020

C₂

semidirect

3. #20292493

C₂

semidirect

13. #127×C₁₀₇

C₄

semidirect

18. C₂×C₁₀₇⋊C₈

D₄

semidirect

66. C₁₀₇

#1961

semidirect

75. #127

Dic₁₀₇

semidirect

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

subgroup	index
C₂×#16812030	2
C₂×#16812020	2
#20292493	2
C₂×#127×C₁₀₇	2
#1961	107

subgroup

index

C₂×#127×C₁₀₇

107

Subgroups — #28337286

Conjugacy classes of subgroups

Complements

Maximal subgroups