Group #28336453 — subgroups

Subgroups — #28336453

Order 13696.

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

normal subgroups only

This group has 336 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, complemented
2	C₂×#16812026	C₂	1	2	yes	maximal
3	#20292396	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×D₄×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	#16812026	—	2	4	no	—
9	C₂²×D₄×C₁₀₇	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₀₇	C₄	1	2	yes	—
11	C₄×D₄×C₁₀₇	—	2	2	no	—
12	C₄×D₄×C₁₀₇	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×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	C₂⁴×C₁₀₇	C₂×C₄	1	2	yes	—
20	C₂×D₄×C₁₀₇	—	2	2	no	—
21	C₂×D₄×C₁₀₇	—	2	2	no	—
22	C₂×D₄×C₁₀₇	D₄	1	2	yes	—
23	C₂×D₄×C₁₀₇	Q₈	1	2	yes	—
24	C₂²×C₄×C₁₀₇	—	2	2	no	—
25	C₂²×C₄×C₁₀₇	—	2	2	no	—
26	C₂²×C₄×C₁₀₇	—	4	4	no	—
27	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₁₀₇	—	2	2	no	—
32	C₂²⋊C₄×C₁₀₇	—	2	4	no	—
33	C₄²×C₁₀₇	D₄	1	2	yes	—
34	C₄²×C₁₀₇	D₄	1	2	yes	—
35	C₂³×C₁₀₇	—	2	2	no	—
36	C₂³×C₁₀₇	—	2	2	no	—
37	C₂³×C₁₀₇	—	4	2	no	—
38	C₂³×C₁₀₇	—	4	2	no	—
39	C₂³×C₁₀₇	C₄⋊C₄	1	4	yes	—
40	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
41	D₄×C₁₀₇	—	2	4	no	—
42	D₄×C₁₀₇	—	2	4	no	—
43	C₂×C₄×C₁₀₇	—	2	2	no	—
44	C₂×C₄×C₁₀₇	—	2	2	no	—
45	C₂×C₄×C₁₀₇	—	4	1	no	—
46	C₂×C₄×C₁₀₇	—	4	4	no	—
47	C₂×C₄×C₁₀₇	—	4	4	no	—
48	C₂×C₄×C₁₀₇	—	4	2	no	—
49	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₁₀₇	#41	1	2	yes	—
56	C₁₀₇⋊C₈	—	4	4	no	—
57	C₂²×C₁₀₇	—	2	1	no	—
58	C₂²×C₁₀₇	—	2	1	no	—
59	C₂²×C₁₀₇	—	2	4	no	—
60	C₂²×C₁₀₇	—	4	4	no	—
61	C₂²×C₁₀₇	—	4	4	no	—
62	C₂²×C₁₀₇	—	4	2	no	—
63	C₂²×C₁₀₇	—	4	2	no	—
64	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₁₀₇	#117	1	1	yes	characteristic
67	C₄×C₁₀₇	—	2	1	no	—
68	C₄×C₁₀₇	—	2	2	no	—
69	C₄×C₁₀₇	—	4	2	no	—
70	C₄×C₁₀₇	#100	1	2	yes	complemented
71	C₄×C₁₀₇	#121	1	2	yes	—
72	C₄×C₁₀₇	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₁₀₇	—	2	1	no	—
74	C₂×C₁₀₇	—	2	1	no	—
75	C₂×C₁₀₇	—	4	4	no	—
76	C₂×C₁₀₇	#380	1	1	yes	characteristic
77	C₂×C₁₀₇	C₂×#100	1	1	yes	characteristic
78	C₂×C₁₀₇	#413	1	1	yes	characteristic
79	#1800	—	107	1	no	maximal, Sylow
80	C₁₀₇	#1800	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₄×D₄	D₁₀₇	1	1	yes	characteristic
82	#423	—	107	2	no	—
83	C₂×#100	—	107	2	no	—
84	#343	—	107	2	no	—
85	C₂²×D₄	C₂×D₁₀₇	1	1	yes	characteristic
86	C₂³×C₄	Dic₁₀₇	1	2	yes	—
87	C₂×M₁₆	—	107	4	no	—
88	C₄×D₄	—	2	2	no	—
89	C₄×D₄	—	2	2	no	—
90	C₂×C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
91	C₂×C₂²⋊C₄	Dic₁₀₇	1	2	yes	—
92	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
93	C₄⋊C₈	—	214	2	no	—
94	C₄⋊C₈	—	214	2	no	—
95	#100	—	214	4	no	—
96	C₂⁴	C₂×Dic₁₀₇	1	2	yes	—
97	C₂×D₄	—	2	2	no	—
98	C₂×D₄	—	2	2	no	—
99	C₂×D₄	Dic₂₁₄	1	2	yes	—
100	C₂×D₄	D₄₂₈	1	2	yes	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	4	4	no	—
104	C₂²×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
105	C₂²×C₄	C₂×Dic₁₀₇	1	2	yes	—
106	C₂²×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
107	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
108	M₁₆	—	214	4	no	—
109	M₁₆	—	214	4	no	—
110	C₂×C₈	—	214	4	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₂²⋊C₄	—	2	4	no	—
113	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
114	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
115	C₂³	—	2	2	no	—
116	C₂³	—	2	2	no	—
117	C₂³	—	4	2	no	—
118	C₂³	—	4	2	no	—
119	C₂³	C₄⋊Dic₁₀₇	1	4	yes	—
120	C₂³	C₂²×Dic₁₀₇	1	1	yes	characteristic
121	D₄	—	2	4	no	—
122	D₄	—	2	4	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	4	no	—
127	C₂×C₄	—	4	4	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	C₂²⋊Dic₁₀₇	1	2	yes	—
130	C₂×C₄	C₂²⋊Dic₁₀₇	1	1	yes	characteristic
131	C₂×C₄	C₂²⋊Dic₁₀₇	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
133	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
134	C₂×C₄	C₂×Dic₂₁₄	1	1	yes	characteristic
135	C₂×C₄	#13133818	1	2	yes	—
136	C₈	—	428	4	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	4	no	—
140	C₂²	—	4	4	no	—
141	C₂²	—	4	4	no	—
142	C₂²	—	4	2	no	—
143	C₂²	—	4	2	no	—
144	C₂²	C₂×C₂²⋊Dic₁₀₇	1	1	yes	characteristic
145	C₂²	C₂×C₄⋊Dic₁₀₇	1	1	yes	characteristic
146	C₂²	#16812068	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	1	no	—
149	C₄	—	4	2	no	—
150	C₄	#16812026	1	2	yes	complemented
151	C₄	#16812083	1	2	yes	—
152	C₄	#16812088	1	2	yes	—
153	C₂	—	2	1	no	—
154	C₂	—	2	1	no	—
155	C₂	—	4	4	no	—
156	C₂	#20292487	1	1	yes	characteristic
157	C₂	C₂×#16812026	1	1	yes	characteristic
158	C₂	#20292611	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20292651

C₂

yes

maximal, complemented

C₂×#16812026

C₂

yes

maximal

#20292396

C₂

yes

maximal, complemented

C₂×C₄×D₄×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133802

C₂²

yes

—

C₄⋊(C₁₀₇⋊C₈)

—

C₄⋊(C₁₀₇⋊C₈)

—

#16812026

—

C₂²×D₄×C₁₀₇

C₂²

yes

characteristic

C₂³×C₄×C₁₀₇

C₄

yes

—

C₄×D₄×C₁₀₇

—

C₄×D₄×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₁₀₇

C₄

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

#13133802

—

#13133802

—

C₂×C₁₀₇⋊C₈

—

C₂⁴×C₁₀₇

C₂×C₄

yes

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

D₄

yes

—

C₂×D₄×C₁₀₇

Q₈

yes

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

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

—

C₂²⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

D₄

yes

—

C₄²×C₁₀₇

D₄

yes

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

C₄⋊C₄

yes

—

C₂³×C₁₀₇

C₂²×C₄

yes

characteristic

D₄×C₁₀₇

—

D₄×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

characteristic

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

characteristic

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

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

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

#100

yes

complemented

C₄×C₁₀₇

#121

yes

—

C₄×C₁₀₇

C₂²⋊Q₈

yes

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

107

maximal, Sylow

C₁₀₇

#1800

yes

characteristic, complemented, Sylow

C₂×C₄×D₄

D₁₀₇

yes

characteristic

#423

—

107

—

C₂×#100

—

107

—

#343

—

107

—

C₂²×D₄

C₂×D₁₀₇

yes

characteristic

C₂³×C₄

Dic₁₀₇

yes

—

C₂×M₁₆

—

107

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

C₂×D₁₀₇

yes

characteristic

C₂×C₂²⋊C₄

Dic₁₀₇

yes

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

214

—

C₄⋊C₈

—

214

—

#100

—

214

—

C₂⁴

C₂×Dic₁₀₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₂₁₄

yes

—

100

C₂×D₄

D₄₂₈

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂×Dic₁₀₇

yes

characteristic

105

C₂²×C₄

C₂×Dic₁₀₇

yes

—

106

C₂²×C₄

C₂×Dic₁₀₇

yes

characteristic

107

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

108

M₁₆

—

214

—

109

M₁₆

—

214

—

110

C₂×C₈

—

214

—

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

C₂²⋊D₁₀₇

yes

—

114

C₄²

C₂²⋊D₁₀₇

yes

—

115

C₂³

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

119

C₂³

C₄⋊Dic₁₀₇

yes

—

120

C₂³

C₂²×Dic₁₀₇

yes

characteristic

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂²⋊Dic₁₀₇

yes

—

130

C₂×C₄

C₂²⋊Dic₁₀₇

yes

characteristic

131

C₂×C₄

C₂²⋊Dic₁₀₇

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

133

C₂×C₄

C₂×D₄₂₈

yes

characteristic

134

C₂×C₄

C₂×Dic₂₁₄

yes

characteristic

135

C₂×C₄

#13133818

yes

—

136

C₈

—

428

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

C₂×C₂²⋊Dic₁₀₇

yes

characteristic

145

C₂²

C₂×C₄⋊Dic₁₀₇

yes

characteristic

146

C₂²

#16812068

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#16812026

yes

complemented

151

C₄

#16812083

yes

—

152

C₄

#16812088

yes

—

153

C₂

—

154

C₂

—

155

C₂

—

156

C₂

#20292487

yes

characteristic

157

C₂

C₂×#16812026

yes

characteristic

158

C₂

#20292611

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₀₇
commutator	C₂³×C₁₀₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	C₂×C₄×D₄×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₀₇	C₂²
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₂×C₄×D₄	D₁₀₇
hypocenter	C₁₀₇	#1800
LCS	G ▷ C₂³×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×D₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₀₇

commutator

C₂³×C₁₀₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

C₂×C₄×D₄×C₁₀₇

C₂

socle

C₂²×C₁₀₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₄×C₁₀₇

C₂²

cosocle

C₂²×C₄×C₁₀₇

C₂³

hypercenter

C₂×C₄×D₄

D₁₀₇

hypocenter

C₁₀₇

#1800

LCS

G ▷ C₂³×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. #20292651	C₂	semidirect
3. #20292396	C₂	semidirect
70. C₄×C₁₀₇	#100	semidirect
80. C₁₀₇	#1800	semidirect
150. C₄	#16812026	semidirect

H ≅ G/N

type

1. #20292651

C₂

semidirect

3. #20292396

C₂

semidirect

70. C₄×C₁₀₇

#100

semidirect

80. C₁₀₇

#1800

semidirect

150. C₄

#16812026

semidirect

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

subgroup	index
#20292651	2
C₂×#16812026	2
#20292396	2
C₂×C₄×D₄×C₁₀₇	2
#1800	107

subgroup

index

C₂×C₄×D₄×C₁₀₇

107

Subgroups — #28336453

Conjugacy classes of subgroups

Complements

Maximal subgroups