Group #28337410 — subgroups

Subgroups — #28337410

Order 13696.

292 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₄×C₄⋊Dic₁₀₇	C₂	1	1	yes	maximal, characteristic
3	#20292389	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	C₂×C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₀₇	—	2	1	no	—
8	C₄⋊(C₁₀₇⋊C₈)	—	2	2	no	—
9	C₁₀₇⋊#95	—	2	1	no	—
10	C₂²×Q₈×C₁₀₇	—	2	1	no	—
11	C₄⋊Q₈×C₁₀₇	—	2	2	no	—
12	C₄⋊Q₈×C₁₀₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₀₇	—	2	2	no	—
17	C₂×C₁₀₇⋊C₈	—	2	2	no	—
18	C₂×C₁₀₇⋊C₈	—	2	4	no	—
19	C₄⋊Dic₁₀₇	—	2	2	no	—
20	C₄⋊Dic₁₀₇	Q₈	1	4	yes	complemented
21	C₄×Dic₁₀₇	—	4	2	no	—
22	C₂×Q₈×C₁₀₇	—	2	4	no	—
23	C₂×Q₈×C₁₀₇	—	4	4	no	—
24	C₂²×C₄×C₁₀₇	—	2	2	no	—
25	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₀₇	—	2	4	no	—
29	C₄⋊C₄×C₁₀₇	—	2	2	no	—
30	C₄⋊C₄×C₁₀₇	C₂×C₄	1	4	yes	—
31	C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
32	C₄²×C₁₀₇	D₄	1	2	yes	—
33	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₁₀₇	—	4	8	no	—
35	C₂×C₄×C₁₀₇	—	2	2	no	—
36	C₂×C₄×C₁₀₇	—	2	4	no	—
37	C₂×C₄×C₁₀₇	—	4	4	no	—
38	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₀₇	SD₁₆	1	2	yes	—
41	C₂×C₄×C₁₀₇	Q₁₆	1	2	yes	—
42	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₁₀₇	#41	1	2	yes	—
46	C₂×Dic₁₀₇	—	2	4	no	—
47	C₂×Dic₁₀₇	—	4	2	no	—
48	C₂×Dic₁₀₇	—	4	2	no	—
49	C₁₀₇⋊C₈	—	4	2	no	—
50	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₀₇	C₂²⋊Q₈	1	2	yes	—
52	C₂²×C₁₀₇	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₁₀₇	C₂×Q₁₆	1	1	yes	characteristic
54	C₂²×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₀₇	—	2	2	no	—
56	C₄×C₁₀₇	—	4	4	no	—
57	C₄×C₁₀₇	#103	1	4	yes	—
58	C₄×C₁₀₇	C₄×Q₈	1	2	yes	—
59	C₄×C₁₀₇	#124	1	1	yes	characteristic
60	C₄×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
61	Dic₁₀₇	—	4	4	no	—
62	C₂×C₁₀₇	#391	1	1	yes	characteristic
63	C₂×C₁₀₇	C₂×#103	1	1	yes	characteristic
64	C₂×C₁₀₇	#418	1	1	yes	characteristic
65	C₂×C₁₀₇	#476	1	2	yes	—
66	C₂×C₁₀₇	#477	1	2	yes	—
67	#1987	—	107	1	no	maximal, Sylow
68	C₁₀₇	#1987	1	1	yes	characteristic, complemented, Sylow
69	C₂×C₄⋊Q₈	D₁₀₇	1	1	yes	characteristic
70	C₂×C₄⋊C₈	—	107	1	no	—
71	C₄×C₄⋊C₄	—	107	1	no	—
72	#340	—	107	4	no	—
73	C₂²×Q₈	—	2	1	no	—
74	C₂²×C₈	—	107	2	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	Dic₁₀₇	1	4	yes	complemented
77	C₂×C₄⋊C₄	—	2	1	no	—
78	C₂×C₄⋊C₄	—	107	2	no	—
79	C₂×C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
80	C₂×C₄²	—	214	1	no	—
81	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
82	C₄⋊C₈	—	214	2	no	—
83	#95	—	214	1	no	—
84	C₂×Q₈	—	2	4	no	—
85	C₂×Q₈	—	4	4	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	214	2	no	—
88	C₂²×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
89	C₂²×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
91	C₂×C₈	—	214	2	no	—
92	C₂×C₈	—	214	4	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	107	4	no	—
96	C₄⋊C₄	—	214	2	no	—
97	C₄⋊C₄	C₂×Dic₁₀₇	1	4	yes	—
98	C₄²	—	428	2	no	—
99	C₄²	C₂×Dic₁₀₇	1	2	yes	—
100	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
101	C₂³	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
102	Q₈	—	4	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	428	2	no	—
107	C₂×C₄	—	428	2	no	—
108	C₂×C₄	—	214	4	no	—
109	C₂×C₄	C₂²⋊Dic₁₀₇	1	2	yes	—
110	C₂×C₄	C₂²⋊Dic₁₀₇	1	2	yes	—
111	C₂×C₄	#13133805	1	2	yes	—
112	C₂×C₄	#13133808	1	2	yes	—
113	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₁₀₇	1	2	yes	—
116	C₂×C₄	#13133819	1	2	yes	—
117	C₈	—	428	2	no	—
118	C₂²	C₂×C₂²⋊Dic₁₀₇	1	1	yes	characteristic
119	C₂²	#16812085	1	2	yes	—
120	C₂²	C₂×#13133805	1	1	yes	characteristic
121	C₂²	C₂×#13133808	1	1	yes	characteristic
122	C₂²	#16812135	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	428	4	no	—
126	C₄	#16812033	1	4	yes	—
127	C₄	Q₈×Dic₁₀₇	1	2	yes	—
128	C₄	#16812098	1	1	yes	characteristic
129	C₄	#16812107	1	1	yes	characteristic
130	C₂	#20292534	1	1	yes	characteristic
131	C₂	C₂×#16812033	1	1	yes	characteristic
132	C₂	#20292632	1	1	yes	characteristic
133	C₂	#20292914	1	2	yes	—
134	C₂	#20292919	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₀₇⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₀₇

C₂

yes

maximal, characteristic

#20292389

C₂

yes

maximal

C₂×C₄⋊Q₈×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×C₁₀₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₀₇

C₂²

yes

—

C₂×C₄×Dic₁₀₇

—

C₄⋊(C₁₀₇⋊C₈)

—

C₁₀₇⋊#95

—

C₂²×Q₈×C₁₀₇

—

C₄⋊Q₈×C₁₀₇

—

C₄⋊Q₈×C₁₀₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₂²×Dic₁₀₇

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

Q₈

yes

complemented

C₄×Dic₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

D₄

yes

characteristic

C₂²×C₄×C₁₀₇

D₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

C₂×C₄

yes

—

C₄²×C₁₀₇

C₂×C₄

yes

—

C₄²×C₁₀₇

D₄

yes

—

C₂³×C₁₀₇

C₂×D₄

yes

characteristic

Q₈×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

SD₁₆

yes

—

C₂×C₄×C₁₀₇

Q₁₆

yes

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×Q₈

yes

—

C₂×C₄×C₁₀₇

#41

yes

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₇

C₂²⋊Q₈

yes

—

C₂²×C₁₀₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₇

C₂×Q₁₆

yes

characteristic

C₂²×C₁₀₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#103

yes

—

C₄×C₁₀₇

C₄×Q₈

yes

—

C₄×C₁₀₇

#124

yes

characteristic

C₄×C₁₀₇

C₄⋊Q₈

yes

characteristic

Dic₁₀₇

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

C₂×C₁₀₇

#477

yes

—

#1987

—

107

maximal, Sylow

C₁₀₇

#1987

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₁₀₇

yes

characteristic

C₂×C₄⋊C₈

—

107

—

C₄×C₄⋊C₄

—

107

—

#340

—

107

—

C₂²×Q₈

—

C₂²×C₈

—

107

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₁₀₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

107

—

C₂×C₄⋊C₄

C₂×D₁₀₇

yes

—

C₂×C₄²

—

214

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

214

—

#95

—

214

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

214

—

C₂²×C₄

C₂²⋊D₁₀₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₇

yes

characteristic

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

C₂×C₈

—

214

—

C₂×C₈

—

214

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

214

—

C₄⋊C₄

C₂×Dic₁₀₇

yes

—

C₄²

—

428

—

C₄²

C₂×Dic₁₀₇

yes

—

100

C₄²

C₂²⋊D₁₀₇

yes

—

101

C₂³

C₂×C₂²⋊D₁₀₇

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

428

—

107

C₂×C₄

—

428

—

108

C₂×C₄

—

214

—

109

C₂×C₄

C₂²⋊Dic₁₀₇

yes

—

110

C₂×C₄

C₂²⋊Dic₁₀₇

yes

—

111

C₂×C₄

#13133805

yes

—

112

C₂×C₄

#13133808

yes

—

113

C₂×C₄

C₂²×Dic₁₀₇

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

115

C₂×C₄

Q₈×D₁₀₇

yes

—

116

C₂×C₄

#13133819

yes

—

117

C₈

—

428

—

118

C₂²

C₂×C₂²⋊Dic₁₀₇

yes

characteristic

119

C₂²

#16812085

yes

—

120

C₂²

C₂×#13133805

yes

characteristic

121

C₂²

C₂×#13133808

yes

characteristic

122

C₂²

#16812135

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

428

—

126

C₄

#16812033

yes

—

127

C₄

Q₈×Dic₁₀₇

yes

—

128

C₄

#16812098

yes

characteristic

129

C₄

#16812107

yes

characteristic

130

C₂

#20292534

yes

characteristic

131

C₂

C₂×#16812033

yes

characteristic

132

C₂

#20292632

yes

characteristic

133

C₂

#20292914

yes

—

134

C₂

#20292919

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₇
commutator	C₂×C₄×C₁₀₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	C₂×C₄⋊Q₈×C₁₀₇	C₂
socle, prime radical	C₂³×C₁₀₇	C₂×D₄
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₀₇
hypocenter	C₁₀₇	#1987
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₇

commutator

C₂×C₄×C₁₀₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

C₂×C₄⋊Q₈×C₁₀₇

C₂

socle, prime radical

C₂³×C₁₀₇

C₂×D₄

cosocle

C₂²×C₄×C₁₀₇

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₀₇

hypocenter

C₁₀₇

#1987

LCS

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₁₀₇	C₄	semidirect
20. C₄⋊Dic₁₀₇	Q₈	semidirect
68. C₁₀₇	#1987	semidirect
76. C₄⋊Q₈	Dic₁₀₇	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₁₀₇

C₄

semidirect

20. C₄⋊Dic₁₀₇

Q₈

semidirect

68. C₁₀₇

#1987

semidirect

76. C₄⋊Q₈

Dic₁₀₇

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
C₄×C₄⋊Dic₁₀₇	2
#20292389	2
C₂×C₄⋊Q₈×C₁₀₇	2
#1987	107

subgroup

index

C₂×C₄⋊(C₁₀₇⋊C₈)

C₄×C₄⋊Dic₁₀₇

#20292389

C₂×C₄⋊Q₈×C₁₀₇

#1987

107

Subgroups — #28337410

Conjugacy classes of subgroups

Complements

Maximal subgroups