Group #28337398 — subgroups

Subgroups — #28337398

Order 13696.

292 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 145 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×C₄×C₁₀₇⋊C₈	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	C₄×C₁₀₇⋊C₈	—	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	2	no	—
23	C₂×Q₈×C₁₀₇	C₈	1	4	yes	complemented
24	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₀₇	—	2	4	no	—
30	C₄⋊C₄×C₁₀₇	—	2	2	no	—
31	C₄²×C₁₀₇	—	2	2	no	—
32	C₄²×C₁₀₇	—	2	2	no	—
33	C₄²×C₁₀₇	D₄	1	2	yes	—
34	C₄²×C₁₀₇	D₄	1	2	yes	—
35	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
36	Q₈×C₁₀₇	—	2	8	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	C₂×C₈	1	4	yes	—
45	C₂×C₄×C₁₀₇	C₂×C₈	1	2	yes	—
46	C₂×C₄×C₁₀₇	M₁₆	1	2	yes	—
47	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₀₇	#41	1	2	yes	—
51	C₁₀₇⋊C₈	—	2	4	no	—
52	C₁₀₇⋊C₈	—	4	2	no	—
53	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₁₀₇	C₄×Q₈	1	2	yes	—
55	C₂²×C₁₀₇	C₂²×C₈	1	1	yes	characteristic
56	C₂²×C₁₀₇	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₁₀₇	#131	1	2	yes	—
58	C₄×C₁₀₇	—	2	4	no	—
59	C₄×C₁₀₇	—	2	2	no	—
60	C₄×C₁₀₇	—	4	2	no	—
61	C₄×C₁₀₇	C₂²⋊C₈	1	4	yes	—
62	C₄×C₁₀₇	C₂²⋊Q₈	1	2	yes	—
63	C₄×C₁₀₇	#124	1	1	yes	characteristic
64	C₄×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₁₀₇	#391	1	1	yes	characteristic
66	C₂×C₁₀₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₁₀₇	#408	1	1	yes	characteristic
68	C₂×C₁₀₇	C₈×Q₈	1	2	yes	—
69	C₂×C₁₀₇	#446	1	2	yes	—
70	#1984	—	107	1	no	maximal, Sylow
71	C₁₀₇	#1984	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₄×Q₈	D₁₀₇	1	1	yes	characteristic
73	C₂×C₄⋊C₈	—	107	1	no	—
74	C₂×C₄×C₈	—	107	1	no	—
75	#336	—	107	4	no	—
76	C₂²×Q₈	Dic₁₀₇	1	1	yes	characteristic
77	C₂²×C₈	—	107	2	no	—
78	C₂²×C₈	—	107	2	no	—
79	C₄×Q₈	—	2	2	no	—
80	C₄×Q₈	—	2	2	no	—
81	C₂×C₄⋊C₄	Dic₁₀₇	1	1	yes	characteristic
82	C₂×C₄⋊C₄	Dic₁₀₇	1	2	yes	—
83	C₂×C₄²	C₂×D₁₀₇	1	2	yes	—
84	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
85	C₄⋊C₈	—	214	2	no	—
86	C₄×C₈	—	214	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	C₁₀₇⋊C₈	1	4	yes	complemented
90	C₂²×C₄	C₂×Dic₁₀₇	1	2	yes	—
91	C₂²×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₁₀₇	1	2	yes	—
93	C₂²×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
95	C₂×C₈	—	107	4	no	—
96	C₂×C₈	—	214	2	no	—
97	C₂×C₈	—	214	4	no	—
98	C₂×C₈	—	214	2	no	—
99	C₂×C₈	—	214	2	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄²	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
105	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
106	C₂³	C₂²×Dic₁₀₇	1	1	yes	characteristic
107	Q₈	—	2	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	C₂²⋊Dic₁₀₇	1	2	yes	—
114	C₂×C₄	C₂²⋊Dic₁₀₇	1	2	yes	—
115	C₂×C₄	C₂×C₁₀₇⋊C₈	1	4	yes	—
116	C₂×C₄	C₂×C₁₀₇⋊C₈	1	2	yes	—
117	C₂×C₄	#13133802	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₁₀₇	1	2	yes	—
121	C₂×C₄	#13133819	1	2	yes	—
122	C₈	—	428	2	no	—
123	C₈	—	214	4	no	—
124	C₂²	C₂×C₂²⋊Dic₁₀₇	1	1	yes	characteristic
125	C₂²	Q₈×Dic₁₀₇	1	2	yes	—
126	C₂²	C₂²×C₁₀₇⋊C₈	1	1	yes	characteristic
127	C₂²	C₂×#13133802	1	1	yes	characteristic
128	C₂²	#16812116	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	2	2	no	—
131	C₄	—	4	2	no	—
132	C₄	C₂²⋊(C₁₀₇⋊C₈)	1	4	yes	—
133	C₄	#16812085	1	2	yes	—
134	C₄	#16812098	1	1	yes	characteristic
135	C₄	#16812107	1	1	yes	characteristic
136	C₂	#20292534	1	1	yes	characteristic
137	C₂	C₂×C₂²⋊(C₁₀₇⋊C₈)	1	1	yes	characteristic
138	C₂	#20292594	1	1	yes	characteristic
139	C₂	Q₈×C₁₀₇⋊C₈	1	2	yes	—
140	C₂	#20292758	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₀₇⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₄×C₁₀₇⋊C₈

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)

—

C₄×C₁₀₇⋊C₈

—

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

—

C₂×Q₈×C₁₀₇

C₈

yes

complemented

C₂²×C₄×C₁₀₇

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

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

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂×C₈

yes

—

C₂×C₄×C₁₀₇

C₂×C₈

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

C₂²×C₈

yes

characteristic

C₂²×C₁₀₇

C₂×M₁₆

yes

characteristic

C₂²×C₁₀₇

#131

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂²⋊C₈

yes

—

C₄×C₁₀₇

C₂²⋊Q₈

yes

—

C₄×C₁₀₇

#124

yes

characteristic

C₄×C₁₀₇

C₄⋊Q₈

yes

characteristic

C₂×C₁₀₇

#391

yes

characteristic

C₂×C₁₀₇

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₀₇

#408

yes

characteristic

C₂×C₁₀₇

C₈×Q₈

yes

—

C₂×C₁₀₇

#446

yes

—

#1984

—

107

maximal, Sylow

C₁₀₇

#1984

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₁₀₇

yes

characteristic

C₂×C₄⋊C₈

—

107

—

C₂×C₄×C₈

—

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

—

C₄×C₈

—

214

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₁₀₇⋊C₈

yes

complemented

C₂²×C₄

C₂×Dic₁₀₇

yes

—

C₂²×C₄

C₂×Dic₁₀₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₇

yes

—

C₂²×C₄

C₂×Dic₁₀₇

yes

characteristic

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

C₂×C₈

—

107

—

C₂×C₈

—

214

—

C₂×C₈

—

214

—

C₂×C₈

—

214

—

C₂×C₈

—

214

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₂²⋊D₁₀₇

yes

—

105

C₄²

C₂²⋊D₁₀₇

yes

—

106

C₂³

C₂²×Dic₁₀₇

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²⋊Dic₁₀₇

yes

—

114

C₂×C₄

C₂²⋊Dic₁₀₇

yes

—

115

C₂×C₄

C₂×C₁₀₇⋊C₈

yes

—

116

C₂×C₄

C₂×C₁₀₇⋊C₈

yes

—

117

C₂×C₄

#13133802

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

120

C₂×C₄

Q₈×D₁₀₇

yes

—

121

C₂×C₄

#13133819

yes

—

122

C₈

—

428

—

123

C₈

—

214

—

124

C₂²

C₂×C₂²⋊Dic₁₀₇

yes

characteristic

125

C₂²

Q₈×Dic₁₀₇

yes

—

126

C₂²

C₂²×C₁₀₇⋊C₈

yes

characteristic

127

C₂²

C₂×#13133802

yes

characteristic

128

C₂²

#16812116

yes

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₂²⋊(C₁₀₇⋊C₈)

yes

—

133

C₄

#16812085

yes

—

134

C₄

#16812098

yes

characteristic

135

C₄

#16812107

yes

characteristic

136

C₂

#20292534

yes

characteristic

137

C₂

C₂×C₂²⋊(C₁₀₇⋊C₈)

yes

characteristic

138

C₂

#20292594

yes

characteristic

139

C₂

Q₈×C₁₀₇⋊C₈

yes

—

140

C₂

#20292758

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₇
commutator	C₂²×C₁₀₇	C₂²×C₈
Fitting	C₂×C₄×Q₈×C₁₀₇	C₂
socle, prime radical	C₂³×C₁₀₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₂×C₄×Q₈	D₁₀₇
hypocenter	C₁₀₇	#1984
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₇

commutator

C₂²×C₁₀₇

C₂²×C₈

Fitting

C₂×C₄×Q₈×C₁₀₇

C₂

socle, prime radical

C₂³×C₁₀₇

C₂²×C₄

cosocle

C₂²×C₄×C₁₀₇

C₂³

hypercenter

C₂×C₄×Q₈

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
20. C₂×C₁₀₇⋊C₈	Q₈	semidirect
23. C₂×Q₈×C₁₀₇	C₈	semidirect
71. C₁₀₇	#1984	semidirect
89. C₂×Q₈	C₁₀₇⋊C₈	semidirect

H ≅ G/N

type

20. C₂×C₁₀₇⋊C₈

Q₈

semidirect

23. C₂×Q₈×C₁₀₇

C₈

semidirect

71. C₁₀₇

#1984

semidirect

89. C₂×Q₈

C₁₀₇⋊C₈

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×C₄×C₁₀₇⋊C₈	2
#20292382	2
C₂×C₄×Q₈×C₁₀₇	2
#1984	107

subgroup

index

C₂×C₄⋊(C₁₀₇⋊C₈)

C₂×C₄×C₁₀₇⋊C₈

#20292382

C₂×C₄×Q₈×C₁₀₇

#1984

107

Subgroups — #28337398

Conjugacy classes of subgroups

Complements

Maximal subgroups