Group #28337402 — subgroups

Subgroups — #28337402

Order 13696.

280 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 121 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×C₄⋊(C₁₀₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812020	C₂	1	1	yes	maximal, characteristic
3	#20292382	C₂	1	4	yes	maximal
4	C₂×C₄×Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₀₇⋊C₈	C₂²	1	2	yes	—
6	C₂²×C₁₀₇⋊C₈	C₂²	1	2	yes	—
7	C₄⋊(C₁₀₇⋊C₈)	—	2	2	no	—
8	#16812020	—	2	2	no	—
9	C₂²×Q₈×C₁₀₇	C₄	1	1	yes	characteristic
10	C₄×Q₈×C₁₀₇	—	2	2	no	—
11	C₄×Q₈×C₁₀₇	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₀₇	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₀₇	C₄	1	2	yes	—
14	C₂×C₄²×C₁₀₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
16	C₂×C₁₀₇⋊C₈	—	2	2	no	—
17	C₂×C₁₀₇⋊C₈	—	2	2	no	—
18	C₂×C₁₀₇⋊C₈	—	2	2	no	—
19	C₂×C₁₀₇⋊C₈	—	2	4	no	—
20	C₂×C₁₀₇⋊C₈	Q₈	1	4	yes	complemented
21	C₂×Q₈×C₁₀₇	—	2	2	no	—
22	C₂×Q₈×C₁₀₇	—	2	4	no	—
23	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₀₇	—	2	4	no	—
29	C₄⋊C₄×C₁₀₇	—	2	2	no	—
30	C₄²×C₁₀₇	—	2	2	no	—
31	C₄²×C₁₀₇	—	2	2	no	—
32	C₄²×C₁₀₇	D₄	1	2	yes	—
33	C₄²×C₁₀₇	D₄	1	2	yes	—
34	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₁₀₇	—	2	8	no	—
36	C₂×C₄×C₁₀₇	—	2	2	no	—
37	C₂×C₄×C₁₀₇	—	2	2	no	—
38	C₂×C₄×C₁₀₇	—	2	2	no	—
39	C₂×C₄×C₁₀₇	—	2	2	no	—
40	C₂×C₄×C₁₀₇	—	2	2	no	—
41	C₂×C₄×C₁₀₇	—	2	4	no	—
42	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₀₇	M₁₆	1	4	yes	—
45	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₁₀₇	#41	1	2	yes	—
49	C₁₀₇⋊C₈	—	2	4	no	—
50	C₁₀₇⋊C₈	—	4	2	no	—
51	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₀₇	C₄×Q₈	1	2	yes	—
53	C₂²×C₁₀₇	C₂×M₁₆	1	2	yes	—
54	C₂²×C₁₀₇	#131	1	2	yes	—
55	C₄×C₁₀₇	—	2	2	no	—
56	C₄×C₁₀₇	—	2	4	no	—
57	C₄×C₁₀₇	—	2	2	no	—
58	C₄×C₁₀₇	—	4	2	no	—
59	C₄×C₁₀₇	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₁₀₇	#124	1	1	yes	characteristic
61	C₄×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₁₀₇	#391	1	1	yes	characteristic
63	C₂×C₁₀₇	#407	1	1	yes	characteristic
64	C₂×C₁₀₇	#408	1	1	yes	characteristic
65	C₂×C₁₀₇	#446	1	4	yes	—
66	#1985	—	107	1	no	maximal, Sylow
67	C₁₀₇	#1985	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄×Q₈	D₁₀₇	1	1	yes	characteristic
69	C₂×C₄⋊C₈	—	107	1	no	—
70	C₂×#97	—	107	1	no	—
71	#336	—	107	4	no	—
72	C₂²×Q₈	Dic₁₀₇	1	1	yes	characteristic
73	C₂²×C₈	—	107	2	no	—
74	C₂²×C₈	—	107	2	no	—
75	C₄×Q₈	—	2	2	no	—
76	C₄×Q₈	—	2	2	no	—
77	C₂×C₄⋊C₄	Dic₁₀₇	1	1	yes	characteristic
78	C₂×C₄⋊C₄	Dic₁₀₇	1	2	yes	—
79	C₂×C₄²	C₂×D₁₀₇	1	2	yes	—
80	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
81	C₄⋊C₈	—	214	2	no	—
82	#97	—	214	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	4	no	—
85	C₂²×C₄	C₂×Dic₁₀₇	1	2	yes	—
86	C₂²×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₁₀₇	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
90	C₂×C₈	—	107	4	no	—
91	C₂×C₈	—	214	2	no	—
92	C₂×C₈	—	214	4	no	—
93	C₂×C₈	—	214	2	no	—
94	C₂×C₈	—	214	2	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	2	2	no	—
99	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
100	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
101	C₂³	C₂²×Dic₁₀₇	1	1	yes	characteristic
102	Q₈	—	2	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	C₂²⋊Dic₁₀₇	1	2	yes	—
110	C₂×C₄	C₂²⋊Dic₁₀₇	1	2	yes	—
111	C₂×C₄	#13133802	1	4	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
113	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
114	C₂×C₄	Q₈×D₁₀₇	1	2	yes	—
115	C₂×C₄	#13133819	1	2	yes	—
116	C₈	—	428	2	no	—
117	C₈	—	214	4	no	—
118	C₂²	C₂×C₂²⋊Dic₁₀₇	1	1	yes	characteristic
119	C₂²	Q₈×Dic₁₀₇	1	2	yes	—
120	C₂²	C₂×#13133802	1	2	yes	—
121	C₂²	#16812116	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	2	4	no	—
124	C₄	—	2	2	no	—
125	C₄	—	4	2	no	—
126	C₄	#16812085	1	2	yes	—
127	C₄	#16812098	1	1	yes	characteristic
128	C₄	#16812107	1	1	yes	characteristic
129	C₂	#20292534	1	1	yes	characteristic
130	C₂	#20292591	1	1	yes	characteristic
131	C₂	#20292594	1	1	yes	characteristic
132	C₂	#20292758	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₀₇⋊C₈)

C₂

yes

maximal, characteristic

C₂×#16812020

C₂

yes

maximal, characteristic

#20292382

C₂

yes

maximal

C₂×C₄×Q₈×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×C₁₀₇⋊C₈

C₂²

yes

—

C₂²×C₁₀₇⋊C₈

C₂²

yes

—

C₄⋊(C₁₀₇⋊C₈)

—

#16812020

—

C₂²×Q₈×C₁₀₇

C₄

yes

characteristic

C₄×Q₈×C₁₀₇

—

C₄×Q₈×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

C₄

yes

characteristic

C₂×C₄⋊C₄×C₁₀₇

C₄

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

Q₈

yes

complemented

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

D₄

yes

—

C₄²×C₁₀₇

D₄

yes

—

C₂³×C₁₀₇

C₂²×C₄

yes

characteristic

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

M₁₆

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×Q₈

yes

—

C₂×C₄×C₁₀₇

#41

yes

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₇

C₄×Q₈

yes

—

C₂²×C₁₀₇

C₂×M₁₆

yes

—

C₂²×C₁₀₇

#131

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂²⋊Q₈

yes

—

C₄×C₁₀₇

#124

yes

characteristic

C₄×C₁₀₇

C₄⋊Q₈

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

#1985

—

107

maximal, Sylow

C₁₀₇

#1985

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₁₀₇

yes

characteristic

C₂×C₄⋊C₈

—

107

—

C₂×#97

—

107

—

#336

—

107

—

C₂²×Q₈

Dic₁₀₇

yes

characteristic

C₂²×C₈

—

107

—

C₂²×C₈

—

107

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₁₀₇

yes

characteristic

C₂×C₄⋊C₄

Dic₁₀₇

yes

—

C₂×C₄²

C₂×D₁₀₇

yes

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

214

—

#97

—

214

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₁₀₇

yes

—

C₂²×C₄

C₂×Dic₁₀₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₇

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

C₂×C₈

—

107

—

C₂×C₈

—

214

—

C₂×C₈

—

214

—

C₂×C₈

—

214

—

C₂×C₈

—

214

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₁₀₇

yes

—

100

C₄²

C₂²⋊D₁₀₇

yes

—

101

C₂³

C₂²×Dic₁₀₇

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²⋊Dic₁₀₇

yes

—

110

C₂×C₄

C₂²⋊Dic₁₀₇

yes

—

111

C₂×C₄

#13133802

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

113

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

114

C₂×C₄

Q₈×D₁₀₇

yes

—

115

C₂×C₄

#13133819

yes

—

116

C₈

—

428

—

117

C₈

—

214

—

118

C₂²

C₂×C₂²⋊Dic₁₀₇

yes

characteristic

119

C₂²

Q₈×Dic₁₀₇

yes

—

120

C₂²

C₂×#13133802

yes

—

121

C₂²

#16812116

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#16812085

yes

—

127

C₄

#16812098

yes

characteristic

128

C₄

#16812107

yes

characteristic

129

C₂

#20292534

yes

characteristic

130

C₂

#20292591

yes

characteristic

131

C₂

#20292594

yes

characteristic

132

C₂

#20292758

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₇
commutator, socle, prime radical	C₂³×C₁₀₇	C₂²×C₄
Fitting	C₂×C₄×Q₈×C₁₀₇	C₂
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₂×C₄×Q₈	D₁₀₇
hypocenter	C₁₀₇	#1985
LCS	G ▷ C₂³×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₇

commutator, socle, prime radical

C₂³×C₁₀₇

C₂²×C₄

Fitting

C₂×C₄×Q₈×C₁₀₇

C₂

cosocle

C₂²×C₄×C₁₀₇

C₂³

hypercenter

C₂×C₄×Q₈

D₁₀₇

C₁₀₇

G ▷ C₂³×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
20. C₂×C₁₀₇⋊C₈	Q₈	semidirect
67. C₁₀₇	#1985	semidirect

H ≅ G/N

type

20. C₂×C₁₀₇⋊C₈

Q₈

semidirect

67. C₁₀₇

#1985

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₀₇⋊C₈)	2
C₂×#16812020	2
#20292382	2
C₂×C₄×Q₈×C₁₀₇	2
#1985	107

subgroup

index

C₂×C₄⋊(C₁₀₇⋊C₈)

C₂×#16812020

#20292382

C₂×C₄×Q₈×C₁₀₇

#1985

107

Subgroups — #28337402

Conjugacy classes of subgroups

Complements

Maximal subgroups