Group #28336457 — subgroups

Subgroups — #28336457

Order 13696.

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

normal subgroups only

This group has 272 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	#20292651	C₂	1	2	yes	maximal
2	C₂×#16812028	C₂	1	2	yes	maximal
3	#20292396	C₂	1	2	yes	maximal
4	C₂×C₄×Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133802	C₂²	1	4	yes	—
6	C₄⋊(C₁₀₇⋊C₈)	—	2	2	no	—
7	C₄⋊(C₁₀₇⋊C₈)	—	2	2	no	—
8	#16812028	—	2	4	no	—
9	C₂²×Q₈×C₁₀₇	C₂²	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	2	yes	—
13	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₀₇	C₄	1	2	yes	—
15	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
16	#13133802	—	2	4	no	—
17	#13133802	—	2	4	no	—
18	C₂×C₁₀₇⋊C₈	—	2	4	no	—
19	C₂×Q₈×C₁₀₇	—	2	2	no	—
20	C₂×Q₈×C₁₀₇	—	2	2	no	—
21	C₂×Q₈×C₁₀₇	D₄	1	2	yes	—
22	C₂×Q₈×C₁₀₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₀₇	C₂×C₄	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	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	4	no	—
36	Q₈×C₁₀₇	—	2	4	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	—	4	1	no	—
42	C₂×C₄×C₁₀₇	—	4	2	no	—
43	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₇	C₄⋊C₄	1	4	yes	—
47	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₀₇	#41	1	2	yes	—
51	C₁₀₇⋊C₈	—	4	4	no	—
52	C₂²×C₁₀₇	—	2	1	no	—
53	C₂²×C₁₀₇	—	2	1	no	—
54	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₀₇	#117	1	1	yes	characteristic
57	C₄×C₁₀₇	—	2	4	no	—
58	C₄×C₁₀₇	—	2	1	no	—
59	C₄×C₁₀₇	—	2	2	no	—
60	C₄×C₁₀₇	—	4	2	no	—
61	C₄×C₁₀₇	#101	1	2	yes	complemented
62	C₄×C₁₀₇	#121	1	2	yes	—
63	C₄×C₁₀₇	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₁₀₇	—	2	1	no	—
65	C₂×C₁₀₇	—	2	1	no	—
66	C₂×C₁₀₇	#380	1	1	yes	characteristic
67	C₂×C₁₀₇	C₂×#101	1	1	yes	characteristic
68	C₂×C₁₀₇	#413	1	1	yes	characteristic
69	#1801	—	107	1	no	maximal, Sylow
70	C₁₀₇	#1801	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄×Q₈	D₁₀₇	1	1	yes	characteristic
72	#423	—	107	2	no	—
73	C₂×#101	—	107	2	no	—
74	#343	—	107	2	no	—
75	C₂²×Q₈	C₂×D₁₀₇	1	1	yes	characteristic
76	C₂×M₁₆	—	107	4	no	—
77	C₄×Q₈	—	2	2	no	—
78	C₄×Q₈	—	2	2	no	—
79	C₂×C₄⋊C₄	Dic₁₀₇	1	2	yes	—
80	C₂×C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
81	C₂×C₄²	Dic₁₀₇	1	2	yes	—
82	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
83	C₄⋊C₈	—	214	2	no	—
84	C₄⋊C₈	—	214	2	no	—
85	#101	—	214	4	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	Dic₂₁₄	1	2	yes	—
89	C₂×Q₈	D₄₂₈	1	2	yes	—
90	C₂²×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
91	C₂²×C₄	C₂×Dic₁₀₇	1	2	yes	—
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	M₁₆	—	214	4	no	—
96	M₁₆	—	214	4	no	—
97	C₂×C₈	—	214	4	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
103	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
104	C₂³	C₂²×Dic₁₀₇	1	1	yes	characteristic
105	Q₈	—	2	4	no	—
106	Q₈	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₂²⋊Dic₁₀₇	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊Dic₁₀₇	1	2	yes	—
115	C₂×C₄	C₂²⋊Dic₁₀₇	1	1	yes	characteristic
116	C₂×C₄	C₄⋊Dic₁₀₇	1	4	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
118	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
119	C₂×C₄	C₂×Dic₂₁₄	1	1	yes	characteristic
120	C₂×C₄	#13133818	1	2	yes	—
121	C₈	—	428	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×C₂²⋊Dic₁₀₇	1	1	yes	characteristic
125	C₂²	C₂×C₄⋊Dic₁₀₇	1	1	yes	characteristic
126	C₂²	#16812068	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	4	no	—
129	C₄	—	2	1	no	—
130	C₄	—	4	2	no	—
131	C₄	#16812028	1	2	yes	complemented
132	C₄	#16812083	1	2	yes	—
133	C₄	#16812088	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#20292487	1	1	yes	characteristic
137	C₂	C₂×#16812028	1	1	yes	characteristic
138	C₂	#20292611	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20292651

C₂

yes

maximal

C₂×#16812028

C₂

yes

maximal

#20292396

C₂

yes

maximal

C₂×C₄×Q₈×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133802

C₂²

yes

—

C₄⋊(C₁₀₇⋊C₈)

—

C₄⋊(C₁₀₇⋊C₈)

—

#16812028

—

C₂²×Q₈×C₁₀₇

C₂²

yes

characteristic

C₄×Q₈×C₁₀₇

—

C₄×Q₈×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

C₄

yes

—

C₂×C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₂×C₄²×C₁₀₇

C₄

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

#13133802

—

#13133802

—

C₂×C₁₀₇⋊C₈

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

D₄

yes

—

C₂×Q₈×C₁₀₇

Q₈

yes

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂×C₄

yes

—

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

—

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

characteristic

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₇

#41

yes

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₇

#117

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#101

yes

complemented

C₄×C₁₀₇

#121

yes

—

C₄×C₁₀₇

C₂²⋊Q₈

yes

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

107

maximal, Sylow

C₁₀₇

#1801

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₁₀₇

yes

characteristic

#423

—

107

—

C₂×#101

—

107

—

#343

—

107

—

C₂²×Q₈

C₂×D₁₀₇

yes

characteristic

C₂×M₁₆

—

107

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₁₀₇

yes

—

C₂×C₄⋊C₄

C₂×D₁₀₇

yes

characteristic

C₂×C₄²

Dic₁₀₇

yes

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

214

—

C₄⋊C₈

—

214

—

#101

—

214

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₂₁₄

yes

—

C₂×Q₈

D₄₂₈

yes

—

C₂²×C₄

C₂×Dic₁₀₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₇

yes

—

C₂²×C₄

C₂×Dic₁₀₇

yes

—

C₂²×C₄

C₂×Dic₁₀₇

yes

characteristic

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

M₁₆

—

214

—

M₁₆

—

214

—

C₂×C₈

—

214

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²⋊D₁₀₇

yes

—

103

C₄²

C₂²⋊D₁₀₇

yes

—

104

C₂³

C₂²×Dic₁₀₇

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²⋊Dic₁₀₇

yes

characteristic

114

C₂×C₄

C₂²⋊Dic₁₀₇

yes

—

115

C₂×C₄

C₂²⋊Dic₁₀₇

yes

characteristic

116

C₂×C₄

C₄⋊Dic₁₀₇

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

118

C₂×C₄

C₂×D₄₂₈

yes

characteristic

119

C₂×C₄

C₂×Dic₂₁₄

yes

characteristic

120

C₂×C₄

#13133818

yes

—

121

C₈

—

428

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×C₂²⋊Dic₁₀₇

yes

characteristic

125

C₂²

C₂×C₄⋊Dic₁₀₇

yes

characteristic

126

C₂²

#16812068

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#16812028

yes

complemented

132

C₄

#16812083

yes

—

133

C₄

#16812088

yes

—

134

C₂

—

135

C₂

—

136

C₂

#20292487

yes

characteristic

137

C₂

C₂×#16812028

yes

characteristic

138

C₂

#20292611

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₀₇
commutator, prime radical	C₂³×C₁₀₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	C₂×C₄×Q₈×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₂×C₄×Q₈	D₁₀₇
hypocenter	C₁₀₇	#1801
LCS	G ▷ C₂³×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₀₇

commutator, prime radical

C₂³×C₁₀₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

C₂×C₄×Q₈×C₁₀₇

C₂

socle

C₂²×C₁₀₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₀₇

C₂³

hypercenter

C₂×C₄×Q₈

D₁₀₇

hypocenter

C₁₀₇

#1801

LCS

G ▷ C₂³×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
61. C₄×C₁₀₇	#101	semidirect
70. C₁₀₇	#1801	semidirect
131. C₄	#16812028	semidirect

H ≅ G/N

type

61. C₄×C₁₀₇

#101

semidirect

70. C₁₀₇

#1801

semidirect

131. C₄

#16812028

semidirect

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

subgroup	index
#20292651	2
C₂×#16812028	2
#20292396	2
C₂×C₄×Q₈×C₁₀₇	2
#1801	107

subgroup

index

C₂×C₄×Q₈×C₁₀₇

107

Subgroups — #28336457

Conjugacy classes of subgroups

Complements

Maximal subgroups