Group #23773002 — subgroups

Subgroups — C₂×#19098821 #23773002

Order 10112.

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

normal subgroups only

This group has 492 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	C₂²×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14579140	C₂	1	2	yes	maximal
3	#19098821	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14579093	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14579086	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₅₈	C₂²	1	2	yes	—
8	C₂×Dic₃₁₆	—	2	4	no	—
9	C₂×Dic₃₁₆	C₂²	1	4	yes	complemented
10	#14579140	—	2	4	no	—
11	#14579140	C₂²	1	8	yes	—
12	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
13	C₂×Dic₇₉⋊C₄	—	2	2	no	—
14	C₂×C₂²⋊Dic₇₉	—	2	2	no	—
15	#14579093	C₂²	1	4	yes	complemented
16	#14579086	C₂²	1	8	yes	complemented
17	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₇₉	—	2	4	no	—
19	C₂×M₁₆×C₇₉	C₂²	1	4	yes	—
20	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₅₈	—	2	4	no	—
23	C₂×Dic₁₅₈	—	2	2	no	—
24	C₂×Dic₁₅₈	—	2	4	no	—
25	C₂×Dic₁₅₈	C₂³	1	4	yes	—
26	C₂²×Dic₇₉	—	2	2	no	—
27	C₂²×Dic₇₉	—	2	2	no	—
28	Dic₃₁₆	—	2	8	no	—
29	C₄⋊Dic₇₉	—	2	2	no	—
30	C₄⋊Dic₇₉	C₂³	1	4	yes	—
31	Dic₇₉⋊C₄	—	2	8	no	—
32	C₂²⋊Dic₇₉	—	2	8	no	—
33	C₂⁴×C₇₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	—	2	2	no	—
35	C₂²×C₄×C₇₉	—	2	2	no	—
36	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₉	D₄	1	2	yes	—
38	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
40	M₁₆×C₇₉	—	2	8	no	—
41	C₂×C₈×C₇₉	—	2	2	no	—
42	C₂×C₈×C₇₉	D₄	1	4	yes	—
43	C₂×C₈×C₇₉	C₂³	1	2	yes	—
44	C₂×C₈×C₇₉	C₂³	1	2	yes	—
45	C₂³×C₇₉	—	2	2	no	—
46	C₂³×C₇₉	—	2	4	no	—
47	C₂³×C₇₉	C₂×D₄	1	2	yes	—
48	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	—	2	2	no	—
50	C₂×C₄×C₇₉	—	2	4	no	—
51	C₂×C₄×C₇₉	—	2	2	no	—
52	C₂×C₄×C₇₉	—	2	2	no	—
53	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇₉	#41	1	2	yes	—
57	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₇₉	—	2	4	no	—
59	C₂×Dic₇₉	—	2	4	no	—
60	C₂×Dic₇₉	—	4	4	no	—
61	C₂×Dic₇₉	—	4	4	no	—
62	Dic₁₅₈	—	2	4	no	—
63	Dic₁₅₈	—	2	4	no	—
64	Dic₁₅₈	—	4	4	no	—
65	C₈×C₇₉	—	2	2	no	—
66	C₈×C₇₉	C₂×D₄	1	4	yes	—
67	C₂²×C₇₉	—	2	2	no	—
68	C₂²×C₇₉	—	2	4	no	—
69	C₂²×C₇₉	—	4	4	no	—
70	C₂²×C₇₉	#121	1	2	yes	—
71	C₂²×C₇₉	#137	1	4	yes	—
72	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₇₉	—	2	2	no	—
74	C₄×C₇₉	#121	1	2	yes	—
75	C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
76	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
77	Dic₇₉	—	4	4	no	—
78	Dic₇₉	—	4	4	no	—
79	C₂×C₇₉	—	4	2	no	—
80	C₂×C₇₉	#470	1	4	yes	complemented
81	C₂×C₇₉	C₂×#121	1	1	yes	characteristic
82	C₂×C₇₉	C₂×#137	1	2	yes	—
83	C₂×#470	—	79	1	no	maximal, Sylow
84	C₇₉	C₂×#470	1	1	yes	characteristic, complemented, Sylow
85	C₂²×Q₁₆	—	79	1	no	—
86	C₂²×M₁₆	D₇₉	1	1	yes	characteristic
87	C₂×C₂²⋊Q₈	—	79	2	no	—
88	#470	—	79	8	no	—
89	C₂×#106	—	79	1	no	—
90	C₂×#103	—	79	2	no	—
91	C₂²×Q₈	—	79	2	no	—
92	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
93	C₂×Q₁₆	—	79	4	no	—
94	C₂×Q₁₆	—	158	4	no	—
95	C₂×M₁₆	—	2	4	no	—
96	C₂×M₁₆	C₂×D₇₉	1	4	yes	—
97	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
98	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
99	C₂²⋊Q₈	—	79	8	no	—
100	C₂²⋊Q₈	—	158	4	no	—
101	C₂×C₄⋊C₄	—	79	2	no	—
102	C₂×C₄⋊C₄	—	158	2	no	—
103	C₂×C₂²⋊C₄	—	158	2	no	—
104	#106	—	79	4	no	—
105	#103	—	79	8	no	—
106	C₂⁴	D₃₁₆	1	1	yes	characteristic
107	C₂×Q₈	—	79	4	no	—
108	C₂×Q₈	—	158	4	no	—
109	C₂×Q₈	—	158	4	no	—
110	C₂×Q₈	—	158	2	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	158	2	no	—
114	C₂²×C₄	—	158	2	no	—
115	C₂²×C₄	D₃₁₆	1	2	yes	—
116	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
118	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
119	Q₁₆	—	158	8	no	—
120	M₁₆	—	2	8	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	C₂²⋊D₇₉	1	4	yes	—
123	C₂×C₈	C₂²×D₇₉	1	2	yes	—
124	C₂×C₈	C₂²×D₇₉	1	2	yes	—
125	C₄⋊C₄	—	79	4	no	—
126	C₄⋊C₄	—	158	8	no	—
127	C₄⋊C₄	—	158	2	no	—
128	C₂²⋊C₄	—	158	8	no	—
129	C₂³	—	2	2	no	—
130	C₂³	—	2	4	no	—
131	C₂³	C₂×D₃₁₆	1	2	yes	—
132	C₂³	C₂×D₃₁₆	1	1	yes	characteristic
133	Q₈	—	316	4	no	—
134	Q₈	—	158	4	no	—
135	Q₈	—	158	4	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	316	4	no	—
141	C₂×C₄	—	316	4	no	—
142	C₂×C₄	—	158	4	no	—
143	C₂×C₄	—	158	4	no	—
144	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
145	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
146	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
147	C₂×C₄	#11929624	1	2	yes	—
148	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	C₂×C₂²⋊D₇₉	1	4	yes	—
151	C₂²	—	2	2	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	4	4	no	—
154	C₂²	#14579135	1	2	yes	—
155	C₂²	#14579194	1	4	yes	—
156	C₂²	C₂²×D₃₁₆	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	316	4	no	—
159	C₄	—	316	4	no	—
160	C₄	#14579135	1	2	yes	—
161	C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
162	C₄	C₂×#11929624	1	1	yes	characteristic
163	C₂	—	4	2	no	—
164	C₂	#19098821	1	4	yes	direct product factor
165	C₂	C₂×#14579135	1	1	yes	characteristic
166	C₂	C₂×#14579194	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#14579140

C₂

yes

maximal

#19098821

C₂

yes

maximal, direct product factor

C₂×#14579093

C₂

yes

maximal, characteristic, complemented

C₂×#14579086

C₂

yes

maximal, complemented

C₂²×M₁₆×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

C₂²

yes

complemented

#14579140

—

#14579140

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×Dic₇₉⋊C₄

—

C₂×C₂²⋊Dic₇₉

—

#14579093

C₂²

yes

complemented

#14579086

C₂²

yes

complemented

C₂³×C₄×C₇₉

C₂²

yes

characteristic

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

C₂²

yes

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂³

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

Dic₃₁₆

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₂⁴×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

—

C₂×C₈×C₇₉

C₂³

yes

—

C₂×C₈×C₇₉

C₂³

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂×D₄

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#121

yes

—

C₂²×C₇₉

#137

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

#121

yes

—

C₄×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

C₂×#41

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×#470

—

maximal, Sylow

C₇₉

C₂×#470

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂²×M₁₆

D₇₉

yes

characteristic

C₂×C₂²⋊Q₈

—

#470

—

C₂×#106

—

C₂×#103

—

C₂²×Q₈

—

C₂³×C₄

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

158

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₉

yes

—

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₂²⋊Q₈

—

100

C₂²⋊Q₈

—

158

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

158

—

103

C₂×C₂²⋊C₄

—

158

—

104

#106

—

105

#103

—

106

C₂⁴

D₃₁₆

yes

characteristic

107

C₂×Q₈

—

108

C₂×Q₈

—

158

—

109

C₂×Q₈

—

158

—

110

C₂×Q₈

—

158

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

158

—

114

C₂²×C₄

—

158

—

115

C₂²×C₄

D₃₁₆

yes

—

116

C₂²×C₄

D₃₁₆

yes

characteristic

117

C₂²×C₄

C₂²×D₇₉

yes

—

118

C₂²×C₄

C₂²×D₇₉

yes

characteristic

119

Q₁₆

—

158

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

C₂²⋊D₇₉

yes

—

123

C₂×C₈

C₂²×D₇₉

yes

—

124

C₂×C₈

C₂²×D₇₉

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

158

—

127

C₄⋊C₄

—

158

—

128

C₂²⋊C₄

—

158

—

129

C₂³

—

130

C₂³

—

131

C₂³

C₂×D₃₁₆

yes

—

132

C₂³

C₂×D₃₁₆

yes

characteristic

133

Q₈

—

316

—

134

Q₈

—

158

—

135

Q₈

—

158

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

316

—

141

C₂×C₄

—

316

—

142

C₂×C₄

—

158

—

143

C₂×C₄

—

158

—

144

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

145

C₂×C₄

C₂×D₃₁₆

yes

—

146

C₂×C₄

C₂×D₃₁₆

yes

characteristic

147

C₂×C₄

#11929624

yes

—

148

C₂×C₄

C₂³×D₇₉

yes

characteristic

149

C₈

—

150

C₈

C₂×C₂²⋊D₇₉

yes

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

#14579135

yes

—

155

C₂²

#14579194

yes

—

156

C₂²

C₂²×D₃₁₆

yes

characteristic

157

C₄

—

158

C₄

—

316

—

159

C₄

—

316

—

160

C₄

#14579135

yes

—

161

C₄

C₂²×C₂²⋊D₇₉

yes

characteristic

162

C₄

C₂×#11929624

yes

characteristic

163

C₂

—

164

C₂

#19098821

yes

direct product factor

165

C₂

C₂×#14579135

yes

characteristic

166

C₂

C₂×#14579194

yes

—

	subgroup	G/H
center	C₂³	C₂×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₂²×M₁₆×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₄
prime radical	C₂⁴×C₇₉	D₄
hypercenter	C₂²×M₁₆	D₇₉
hypocenter	C₇₉	C₂×#470
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

subgroup

G/H

center

C₂³

C₂×D₃₁₆

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₂²×M₁₆×C₇₉

C₂

socle

C₂³×C₇₉

C₂×D₄

prime radical

C₂⁴×C₇₉

D₄

hypercenter

C₂²×M₁₆

D₇₉

hypocenter

C₇₉

C₂×#470

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

N	H ≅ G/N	type
1. C₂²×Dic₃₁₆	C₂	semidirect
3. #19098821	C₂	direct (N × H)
4. C₂×#14579093	C₂	semidirect
5. C₂×#14579086	C₂	semidirect
9. C₂×Dic₃₁₆	C₂²	semidirect
15. #14579093	C₂²	semidirect
16. #14579086	C₂²	semidirect
80. C₂×C₇₉	#470	semidirect
84. C₇₉	C₂×#470	semidirect
164. C₂	#19098821	direct (N × H)

H ≅ G/N

type

1. C₂²×Dic₃₁₆

C₂

semidirect

3. #19098821

C₂

direct (N × H)

4. C₂×#14579093

C₂

semidirect

5. C₂×#14579086

C₂

semidirect

9. C₂×Dic₃₁₆

C₂²

semidirect

15. #14579093

C₂²

semidirect

16. #14579086

C₂²

semidirect

80. C₂×C₇₉

#470

semidirect

84. C₇₉

C₂×#470

semidirect

164. C₂

#19098821

direct (N × H)

Subgroups — C₂×#19098821 #23773002

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19098821 #23773002

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19098821 #23773002