Group #28399962 — subgroups

Subgroups — C₂×#20298788 #28399962

Order 14464.

794 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16813603	C₂	1	3	yes	maximal, complemented
2	#20298788	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16813537	C₂	1	3	yes	maximal, complemented
4	C₂×H(5,𝔽₂)×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₁₃	C₂²	1	1	yes	characteristic, complemented
6	#16813603	—	2	6	no	—
7	#16813603	C₂²	1	12	yes	complemented
8	C₂×#14268626	—	2	3	no	—
9	C₂×C₂²⋊Dic₁₁₃	—	2	3	no	—
10	C₂×C₂²⋊Dic₁₁₃	C₂²	1	3	yes	complemented
11	#16813537	C₂²	1	12	yes	complemented
12	H(5,𝔽₂)×C₁₁₃	—	2	6	no	—
13	H(5,𝔽₂)×C₁₁₃	C₂²	1	4	yes	—
14	C₂×#41×C₁₁₃	—	2	3	no	—
15	C₂²×D₄×C₁₁₃	—	2	3	no	—
16	C₂²×D₄×C₁₁₃	C₂²	1	3	yes	—
17	C₂²×Dic₁₁₃	—	2	3	no	—
18	C₂²×Dic₁₁₃	—	2	3	no	—
19	C₂²×Dic₁₁₃	—	2	3	no	—
20	C₂²×Dic₁₁₃	C₂³	1	2	yes	complemented
21	#14268626	—	2	12	no	—
22	C₂²⋊Dic₁₁₃	—	2	12	no	—
23	C₂²⋊Dic₁₁₃	—	2	3	no	—
24	C₂²⋊Dic₁₁₃	C₂³	1	6	yes	complemented
25	C₂⁴×C₁₁₃	—	2	1	no	—
26	C₂⁴×C₁₁₃	D₄	1	3	yes	—
27	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
28	#41×C₁₁₃	—	2	12	no	—
29	#41×C₁₁₃	—	2	12	no	—
30	C₂×Q₈×C₁₁₃	—	2	1	no	—
31	C₂×D₄×C₁₁₃	—	2	3	no	—
32	C₂×D₄×C₁₁₃	—	2	12	no	—
33	C₂×D₄×C₁₁₃	—	2	6	no	—
34	C₂×D₄×C₁₁₃	—	2	12	no	—
35	C₂×D₄×C₁₁₃	—	2	6	no	—
36	C₂×D₄×C₁₁₃	D₄	1	6	yes	—
37	C₂×D₄×C₁₁₃	C₂³	1	6	yes	—
38	C₂²×C₄×C₁₁₃	—	2	3	no	—
39	C₂²×C₄×C₁₁₃	D₄	1	3	yes	—
40	C₂³×C₁₁₃	—	2	4	no	—
41	C₂³×C₁₁₃	—	2	3	no	—
42	C₂³×C₁₁₃	—	2	3	no	—
43	C₂³×C₁₁₃	—	4	2	no	—
44	C₂³×C₁₁₃	—	4	6	no	—
45	C₂³×C₁₁₃	—	8	2	no	—
46	C₂³×C₁₁₃	C₂×D₄	1	3	yes	—
47	C₂³×C₁₁₃	C₂×D₄	1	6	yes	—
48	C₂³×C₁₁₃	C₂×D₄	1	3	yes	—
49	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
50	Q₈×C₁₁₃	—	2	4	no	—
51	D₄×C₁₁₃	—	2	12	no	—
52	D₄×C₁₁₃	—	2	12	no	—
53	D₄×C₁₁₃	—	2	12	no	—
54	C₂×C₄×C₁₁₃	—	2	12	no	—
55	C₂×C₄×C₁₁₃	—	2	3	no	—
56	C₂×C₄×C₁₁₃	—	2	3	no	—
57	C₂×C₄×C₁₁₃	C₂×D₄	1	6	yes	—
58	C₂×Dic₁₁₃	—	2	6	no	—
59	C₂×Dic₁₁₃	—	2	6	no	—
60	C₂×Dic₁₁₃	—	4	6	no	—
61	C₂×Dic₁₁₃	—	4	3	no	—
62	C₂×Dic₁₁₃	—	4	1	no	—
63	C₂²×C₁₁₃	—	2	6	no	—
64	C₂²×C₁₁₃	—	2	6	no	—
65	C₂²×C₁₁₃	—	4	3	no	—
66	C₂²×C₁₁₃	—	4	1	no	—
67	C₂²×C₁₁₃	—	4	6	no	—
68	C₂²×C₁₁₃	—	4	6	no	—
69	C₂²×C₁₁₃	—	8	3	no	—
70	C₂²×C₁₁₃	—	8	4	no	—
71	C₂²×C₁₁₃	#120	1	3	yes	—
72	C₂²×C₁₁₃	#120	1	1	yes	characteristic
73	C₂²×C₁₁₃	C₂²×D₄	1	3	yes	—
74	C₄×C₁₁₃	—	2	6	no	—
75	Dic₁₁₃	—	4	6	no	—
76	Dic₁₁₃	—	4	2	no	—
77	C₂×C₁₁₃	—	2	3	no	—
78	C₂×C₁₁₃	—	2	3	no	—
79	C₂×C₁₁₃	—	4	6	no	—
80	C₂×C₁₁₃	#458	1	2	yes	complemented
81	C₂×C₁₁₃	C₂×#120	1	1	yes	characteristic
82	C₂×#458	—	113	1	no	maximal, Sylow
83	C₁₁₃	C₂×#458	1	1	yes	characteristic, complemented, Sylow
84	C₂×H(5,𝔽₂)	D₁₁₃	1	1	yes	characteristic
85	C₂×#123	—	113	3	no	—
86	#458	—	113	8	no	—
87	C₂×#99	—	113	3	no	—
88	H(5,𝔽₂)	—	2	6	no	—
89	H(5,𝔽₂)	C₂×D₁₁₃	1	4	yes	—
90	C₂×#41	—	2	3	no	—
91	C₂²×D₄	—	2	3	no	—
92	C₂²×D₄	C₂×D₁₁₃	1	3	yes	—
93	C₂³×C₄	—	113	1	no	—
94	#123	—	113	12	no	—
95	#123	—	226	6	no	—
96	C₂×C₄⋊C₄	—	226	3	no	—
97	C₂×C₂²⋊C₄	—	113	3	no	—
98	C₂×C₂²⋊C₄	—	226	3	no	—
99	#99	—	113	12	no	—
100	C₂⁴	—	2	1	no	—
101	C₂⁴	C₂²⋊D₁₁₃	1	3	yes	—
102	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
103	#41	—	2	12	no	—
104	#41	—	2	12	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×D₄	—	2	3	no	—
107	C₂×D₄	—	2	12	no	—
108	C₂×D₄	—	2	6	no	—
109	C₂×D₄	—	2	12	no	—
110	C₂×D₄	—	2	6	no	—
111	C₂×D₄	C₂²⋊D₁₁₃	1	6	yes	—
112	C₂×D₄	C₂²×D₁₁₃	1	6	yes	—
113	C₂²×C₄	—	2	3	no	—
114	C₂²×C₄	—	113	2	no	—
115	C₂²×C₄	—	226	3	no	—
116	C₂²×C₄	—	226	3	no	—
117	C₂²×C₄	—	226	3	no	—
118	C₂²×C₄	C₂²⋊D₁₁₃	1	3	yes	—
119	C₄⋊C₄	—	226	12	no	—
120	C₂²⋊C₄	—	113	6	no	—
121	C₂²⋊C₄	—	226	3	no	—
122	C₂²⋊C₄	—	226	12	no	—
123	C₂³	—	2	4	no	—
124	C₂³	—	2	3	no	—
125	C₂³	—	2	3	no	—
126	C₂³	—	4	2	no	—
127	C₂³	—	4	6	no	—
128	C₂³	—	8	2	no	—
129	C₂³	C₂×C₂²⋊D₁₁₃	1	3	yes	—
130	C₂³	C₂×C₂²⋊D₁₁₃	1	6	yes	—
131	C₂³	C₂×C₂²⋊D₁₁₃	1	3	yes	—
132	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
133	Q₈	—	2	4	no	—
134	D₄	—	2	12	no	—
135	D₄	—	2	12	no	—
136	D₄	—	2	12	no	—
137	C₂×C₄	—	2	12	no	—
138	C₂×C₄	—	2	3	no	—
139	C₂×C₄	—	2	3	no	—
140	C₂×C₄	—	452	6	no	—
141	C₂×C₄	—	452	3	no	—
142	C₂×C₄	—	452	1	no	—
143	C₂×C₄	—	226	6	no	—
144	C₂×C₄	—	226	6	no	—
145	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	6	yes	—
146	C₂²	—	2	6	no	—
147	C₂²	—	2	6	no	—
148	C₂²	—	4	3	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	4	6	no	—
151	C₂²	—	4	6	no	—
152	C₂²	—	8	3	no	—
153	C₂²	—	8	4	no	—
154	C₂²	#16813591	1	3	yes	—
155	C₂²	#16813591	1	1	yes	characteristic
156	C₂²	C₂²×C₂²⋊D₁₁₃	1	3	yes	—
157	C₄	—	2	6	no	—
158	C₄	—	452	6	no	—
159	C₄	—	452	2	no	—
160	C₂	—	2	3	no	—
161	C₂	—	2	3	no	—
162	C₂	—	4	6	no	—
163	C₂	#20298788	1	2	yes	direct product factor
164	C₂	C₂×#16813591	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813603

C₂

yes

maximal, complemented

#20298788

C₂

yes

maximal, direct product factor

C₂×#16813537

C₂

yes

maximal, complemented

C₂×H(5,𝔽₂)×C₁₁₃

C₂

yes

maximal, characteristic

C₂³×Dic₁₁₃

C₂²

yes

characteristic, complemented

#16813603

—

#16813603

C₂²

yes

complemented

C₂×#14268626

—

C₂×C₂²⋊Dic₁₁₃

—

C₂×C₂²⋊Dic₁₁₃

C₂²

yes

complemented

#16813537

C₂²

yes

complemented

H(5,𝔽₂)×C₁₁₃

—

H(5,𝔽₂)×C₁₁₃

C₂²

yes

—

C₂×#41×C₁₁₃

—

C₂²×D₄×C₁₁₃

—

C₂²×D₄×C₁₁₃

C₂²

yes

—

C₂²×Dic₁₁₃

—

C₂²×Dic₁₁₃

—

C₂²×Dic₁₁₃

—

C₂²×Dic₁₁₃

C₂³

yes

complemented

#14268626

—

C₂²⋊Dic₁₁₃

—

C₂²⋊Dic₁₁₃

—

C₂²⋊Dic₁₁₃

C₂³

yes

complemented

C₂⁴×C₁₁₃

—

C₂⁴×C₁₁₃

D₄

yes

—

C₂⁴×C₁₁₃

C₂³

yes

characteristic

#41×C₁₁₃

—

#41×C₁₁₃

—

C₂×Q₈×C₁₁₃

—

C₂×D₄×C₁₁₃

—

C₂×D₄×C₁₁₃

—

C₂×D₄×C₁₁₃

—

C₂×D₄×C₁₁₃

—

C₂×D₄×C₁₁₃

—

C₂×D₄×C₁₁₃

D₄

yes

—

C₂×D₄×C₁₁₃

C₂³

yes

—

C₂²×C₄×C₁₁₃

—

C₂²×C₄×C₁₁₃

D₄

yes

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

C₂×D₄

yes

—

C₂³×C₁₁₃

C₂×D₄

yes

—

C₂³×C₁₁₃

C₂×D₄

yes

—

C₂³×C₁₁₃

C₂⁴

yes

characteristic

Q₈×C₁₁₃

—

D₄×C₁₁₃

—

D₄×C₁₁₃

—

D₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

#120

yes

—

C₂²×C₁₁₃

#120

yes

characteristic

C₂²×C₁₁₃

C₂²×D₄

yes

—

C₄×C₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

—

113

maximal, Sylow

C₁₁₃

C₂×#458

yes

characteristic, complemented, Sylow

C₂×H(5,𝔽₂)

D₁₁₃

yes

characteristic

C₂×#123

—

113

—

#458

—

113

—

C₂×#99

—

113

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₁₁₃

yes

—

C₂×#41

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₁₃

yes

—

C₂³×C₄

—

113

—

#123

—

113

—

#123

—

226

—

C₂×C₄⋊C₄

—

226

—

C₂×C₂²⋊C₄

—

113

—

C₂×C₂²⋊C₄

—

226

—

#99

—

113

—

100

C₂⁴

—

101

C₂⁴

C₂²⋊D₁₁₃

yes

—

102

C₂⁴

C₂²×D₁₁₃

yes

characteristic

103

#41

—

104

#41

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

C₂²⋊D₁₁₃

yes

—

112

C₂×D₄

C₂²×D₁₁₃

yes

—

113

C₂²×C₄

—

114

C₂²×C₄

—

113

—

115

C₂²×C₄

—

226

—

116

C₂²×C₄

—

226

—

117

C₂²×C₄

—

226

—

118

C₂²×C₄

C₂²⋊D₁₁₃

yes

—

119

C₄⋊C₄

—

226

—

120

C₂²⋊C₄

—

113

—

121

C₂²⋊C₄

—

226

—

122

C₂²⋊C₄

—

226

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₂×C₂²⋊D₁₁₃

yes

—

130

C₂³

C₂×C₂²⋊D₁₁₃

yes

—

131

C₂³

C₂×C₂²⋊D₁₁₃

yes

—

132

C₂³

C₂³×D₁₁₃

yes

characteristic

133

Q₈

—

134

D₄

—

135

D₄

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

452

—

141

C₂×C₄

—

452

—

142

C₂×C₄

—

452

—

143

C₂×C₄

—

226

—

144

C₂×C₄

—

226

—

145

C₂×C₄

C₂×C₂²⋊D₁₁₃

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

#16813591

yes

—

155

C₂²

#16813591

yes

characteristic

156

C₂²

C₂²×C₂²⋊D₁₁₃

yes

—

157

C₄

—

158

C₄

—

452

—

159

C₄

—

452

—

160

C₂

—

161

C₂

—

162

C₂

—

163

C₂

#20298788

yes

direct product factor

164

C₂

C₂×#16813591

yes

characteristic

	subgroup	G/H
center	C₂²	#16813591
commutator, cosocle	C₂³×C₁₁₃	C₂⁴
Frattini	C₂³	C₂³×D₁₁₃
Fitting, prime radical	C₂×H(5,𝔽₂)×C₁₁₃	C₂
socle	C₂²×C₁₁₃	#120
hypercenter	C₂×H(5,𝔽₂)	D₁₁₃
hypocenter	C₁₁₃	C₂×#458
LCS	G ▷ C₂³×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×H(5,𝔽₂)

subgroup

G/H

center

C₂²

#16813591

commutator, cosocle

C₂³×C₁₁₃

C₂⁴

Frattini

C₂³

C₂³×D₁₁₃

Fitting, prime radical

C₂×H(5,𝔽₂)×C₁₁₃

C₂

socle

C₂²×C₁₁₃

#120

hypercenter

C₂×H(5,𝔽₂)

D₁₁₃

hypocenter

C₁₁₃

C₂×#458

LCS

G ▷ C₂³×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×H(5,𝔽₂)

N	H ≅ G/N	type
1. C₂×#16813603	C₂	semidirect
2. #20298788	C₂	direct (N × H)
3. C₂×#16813537	C₂	semidirect
5. C₂³×Dic₁₁₃	C₂²	semidirect
7. #16813603	C₂²	semidirect
10. C₂×C₂²⋊Dic₁₁₃	C₂²	semidirect
11. #16813537	C₂²	semidirect
20. C₂²×Dic₁₁₃	C₂³	semidirect
24. C₂²⋊Dic₁₁₃	C₂³	semidirect
80. C₂×C₁₁₃	#458	semidirect
83. C₁₁₃	C₂×#458	semidirect
163. C₂	#20298788	direct (N × H)

H ≅ G/N

type

1. C₂×#16813603

C₂

semidirect

2. #20298788

C₂

direct (N × H)

3. C₂×#16813537

C₂

semidirect

5. C₂³×Dic₁₁₃

C₂²

semidirect

7. #16813603

C₂²

semidirect

10. C₂×C₂²⋊Dic₁₁₃

C₂²

semidirect

11. #16813537

C₂²

semidirect

20. C₂²×Dic₁₁₃

C₂³

semidirect

24. C₂²⋊Dic₁₁₃

C₂³

semidirect

80. C₂×C₁₁₃

#458

semidirect

83. C₁₁₃

C₂×#458

semidirect

163. C₂

#20298788

direct (N × H)

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

subgroup	index
C₂×#16813603	2
#20298788	2
C₂×#16813537	2
C₂×H(5,𝔽₂)×C₁₁₃	2
C₂×#458	113

subgroup

index

C₂×H(5,𝔽₂)×C₁₁₃

113

Subgroups — C₂×#20298788 #28399962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20298788 #28399962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20298788 #28399962