Group #23765009 — subgroups

Subgroups — #23765009

Order 10112.

260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 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×C₈×D₇₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579100	C₂	1	2	yes	maximal, complemented
3	C₂²⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₇₉)⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₇₉	C₄	1	1	yes	characteristic
7	C₂²×C₇₉⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×D₇₉	—	2	4	no	—
9	C₂×C₈×D₇₉	C₄	1	1	yes	characteristic
10	C₂×C₈×D₇₉	C₄	1	1	yes	characteristic
11	C₂×C₈×D₇₉	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₇₉	C₂²	1	1	yes	characteristic
13	#14579100	—	2	4	no	—
14	C₂×C₇₉⋊C₁₆	C₂²	1	2	yes	complemented
15	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₇₉	C₂²	1	2	yes	complemented
17	C₂³×D₇₉	C₈	1	1	yes	characteristic
18	C₂²×Dic₇₉	C₈	1	1	yes	characteristic
19	C₂×C₄×D₇₉	—	2	2	no	—
20	C₂×C₄×D₇₉	—	2	2	no	—
21	C₂×C₄×D₇₉	C₈	1	1	yes	characteristic
22	C₂×C₄×D₇₉	C₈	1	1	yes	characteristic
23	C₂×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₇₉⋊C₈	—	2	2	no	—
26	C₂×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₈×D₇₉	—	2	2	no	—
29	C₈×D₇₉	—	2	2	no	—
30	C₈×D₇₉	—	2	2	no	—
31	C₈×D₇₉	D₄	1	4	yes	—
32	C₇₉⋊C₁₆	—	2	2	no	—
33	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₇₉	—	2	2	no	—
35	C₂×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
37	C₁₆×C₇₉	—	2	2	no	—
38	C₂³×C₇₉	C₂×C₈	1	1	yes	characteristic
39	C₂²×D₇₉	—	2	2	no	—
40	C₂²×D₇₉	—	2	2	no	—
41	C₂²×D₇₉	—	2	2	no	—
42	C₂²×D₇₉	C₂×C₈	1	1	yes	characteristic
43	C₂²×D₇₉	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₇₉	—	2	1	no	—
45	C₂×C₄×C₇₉	—	2	1	no	—
46	C₂×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₇₉	—	2	2	no	—
49	C₂×Dic₇₉	C₂×C₈	1	1	yes	characteristic
50	C₂×Dic₇₉	C₂×C₈	1	1	yes	characteristic
51	C₄×D₇₉	—	2	2	no	—
52	C₄×D₇₉	—	2	1	no	—
53	C₄×D₇₉	—	2	1	no	—
54	C₄×D₇₉	—	2	2	no	—
55	C₄×D₇₉	C₂²⋊C₄	1	2	yes	—
56	C₄×D₇₉	M₁₆	1	2	yes	—
57	C₈×C₇₉	—	2	1	no	—
58	C₈×C₇₉	C₂×D₄	1	2	yes	—
59	C₇₉⋊C₈	—	2	1	no	—
60	C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₇₉	—	2	1	no	—
62	C₂²×C₇₉	—	2	1	no	—
63	C₂²×C₇₉	M₃₂	1	2	yes	complemented
64	C₂²×C₇₉	C₂²×C₈	1	1	yes	characteristic
65	C₂×D₇₉	—	2	8	no	—
66	C₂×D₇₉	—	2	1	no	—
67	C₂×D₇₉	—	2	1	no	—
68	C₂×D₇₉	—	2	1	no	—
69	C₂×D₇₉	—	2	1	no	—
70	C₂×D₇₉	C₂²⋊C₈	1	2	yes	—
71	C₂×D₇₉	M₃₂	1	2	yes	complemented
72	C₄×C₇₉	—	2	1	no	—
73	C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
75	Dic₇₉	—	2	1	no	—
76	Dic₇₉	C₂²⋊C₈	1	2	yes	—
77	C₂×C₇₉	—	2	2	no	—
78	C₂×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₇₉	C₂×M₃₂	1	1	yes	characteristic
80	C₂×C₇₉	C₂×M₃₂	1	1	yes	characteristic
81	D₇₉	—	2	2	no	—
82	D₇₉	—	2	2	no	—
83	#2112	—	79	1	no	maximal, Sylow
84	C₇₉	#2112	1	1	yes	characteristic, complemented, Sylow
85	C₂³×C₈	—	79	1	no	—
86	C₂×M₃₂	—	79	2	no	—
87	C₂²⋊C₁₆	—	79	1	no	—
88	C₂²⋊C₁₆	—	79	2	no	—
89	C₂²⋊C₁₆	D₇₉	1	1	yes	characteristic, complemented
90	C₂³×C₄	—	79	1	no	—
91	C₂²×C₈	—	79	1	no	—
92	C₂²×C₈	—	79	1	no	—
93	C₂²×C₈	—	79	1	no	—
94	C₂²×C₈	—	79	1	no	—
95	C₂²×C₈	—	79	1	no	—
96	C₂²×C₈	—	158	4	no	—
97	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
98	M₃₂	—	158	4	no	—
99	C₂×C₁₆	—	79	2	no	—
100	C₂×C₁₆	C₂×D₇₉	1	2	yes	complemented
101	C₂⁴	—	79	1	no	—
102	C₂²×C₄	—	79	1	no	—
103	C₂²×C₄	—	79	1	no	—
104	C₂²×C₄	—	79	1	no	—
105	C₂²×C₄	—	79	1	no	—
106	C₂²×C₄	—	79	1	no	—
107	C₂²×C₄	—	158	2	no	—
108	C₂²×C₄	—	158	2	no	—
109	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	79	1	no	—
112	C₂×C₈	—	79	4	no	—
113	C₂×C₈	—	79	1	no	—
114	C₂×C₈	—	158	2	no	—
115	C₂×C₈	—	158	2	no	—
116	C₂×C₈	—	158	2	no	—
117	C₂×C₈	—	158	2	no	—
118	C₂×C₈	C₄×D₇₉	1	1	yes	characteristic
119	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
120	C₁₆	—	2	2	no	—
121	C₁₆	—	158	2	no	—
122	C₂³	—	79	1	no	—
123	C₂³	—	79	1	no	—
124	C₂³	—	158	2	no	—
125	C₂³	—	158	2	no	—
126	C₂³	—	158	2	no	—
127	C₂³	C₈×D₇₉	1	1	yes	characteristic
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	79	1	no	—
131	C₂×C₄	—	79	2	no	—
132	C₂×C₄	—	79	2	no	—
133	C₂×C₄	—	79	1	no	—
134	C₂×C₄	—	158	2	no	—
135	C₂×C₄	—	158	2	no	—
136	C₂×C₄	—	158	2	no	—
137	C₂×C₄	—	158	1	no	—
138	C₂×C₄	—	158	1	no	—
139	C₂×C₄	C₈×D₇₉	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	79	2	no	—
143	C₈	—	158	1	no	—
144	C₈	D₄×D₇₉	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	79	2	no	—
148	C₂²	—	79	2	no	—
149	C₂²	—	158	8	no	—
150	C₂²	—	158	1	no	—
151	C₂²	—	158	1	no	—
152	C₂²	—	158	1	no	—
153	C₂²	—	158	1	no	—
154	C₂²	#14579100	1	2	yes	complemented
155	C₂²	C₂×C₈×D₇₉	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	79	2	no	—
158	C₄	—	158	1	no	—
159	C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
160	C₄	M₁₆×D₇₉	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	—	158	2	no	—
163	C₂	—	158	2	no	—
164	C₂	C₂²⋊C₈×D₇₉	1	1	yes	characteristic
165	C₂	M₃₂×D₇₉	1	1	yes	characteristic
166	C₂	C₂×#14579100	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₇₉

C₂

yes

maximal, characteristic

C₂×#14579100

C₂

yes

maximal, complemented

C₂²⋊(C₇₉⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

(C₂×D₇₉)⋊C₁₆

C₂

yes

maximal, complemented

C₂²⋊C₁₆×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₇₉

C₄

yes

characteristic

C₂²×C₇₉⋊C₈

C₄

yes

characteristic

C₂×C₈×D₇₉

—

C₂×C₈×D₇₉

C₄

yes

characteristic

C₂×C₈×D₇₉

C₄

yes

characteristic

C₂×C₈×D₇₉

C₂²

yes

characteristic

C₂×C₈×D₇₉

C₂²

yes

characteristic

#14579100

—

C₂×C₇₉⋊C₁₆

C₂²

yes

complemented

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₁₆×C₇₉

C₂²

yes

complemented

C₂³×D₇₉

C₈

yes

characteristic

C₂²×Dic₇₉

C₈

yes

characteristic

C₂×C₄×D₇₉

—

C₂×C₄×D₇₉

—

C₂×C₄×D₇₉

C₈

yes

characteristic

C₂×C₄×D₇₉

C₈

yes

characteristic

C₂×C₄×D₇₉

C₂×C₄

yes

characteristic

C₂×C₄×D₇₉

C₂×C₄

yes

characteristic

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₈×D₇₉

—

C₈×D₇₉

—

C₈×D₇₉

—

C₈×D₇₉

D₄

yes

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₁₆×C₇₉

—

C₂³×C₇₉

C₂×C₈

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

C₂×C₈

yes

characteristic

C₂²×D₇₉

C₂×C₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₈

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×C₈

yes

characteristic

C₂×Dic₇₉

C₂×C₈

yes

characteristic

C₄×D₇₉

—

C₄×D₇₉

—

C₄×D₇₉

—

C₄×D₇₉

—

C₄×D₇₉

C₂²⋊C₄

yes

—

C₄×D₇₉

M₁₆

yes

—

C₈×C₇₉

—

C₈×C₇₉

C₂×D₄

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

M₃₂

yes

complemented

C₂²×C₇₉

C₂²×C₈

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

C₂²⋊C₈

yes

—

C₂×D₇₉

M₃₂

yes

complemented

C₄×C₇₉

—

C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

C₂×M₁₆

yes

characteristic

Dic₇₉

—

Dic₇₉

C₂²⋊C₈

yes

—

C₂×C₇₉

—

C₂×C₇₉

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₇₉

C₂×M₃₂

yes

characteristic

C₂×C₇₉

C₂×M₃₂

yes

characteristic

D₇₉

—

D₇₉

—

#2112

—

maximal, Sylow

C₇₉

#2112

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂×M₃₂

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

D₇₉

yes

characteristic, complemented

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

158

—

C₂²×C₈

C₂×D₇₉

yes

characteristic

M₃₂

—

158

—

C₂×C₁₆

—

100

C₂×C₁₆

C₂×D₇₉

yes

complemented

101

C₂⁴

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

158

—

108

C₂²×C₄

—

158

—

109

C₂²×C₄

C₄×D₇₉

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

158

—

115

C₂×C₈

—

158

—

116

C₂×C₈

—

158

—

117

C₂×C₈

—

158

—

118

C₂×C₈

C₄×D₇₉

yes

characteristic

119

C₂×C₈

C₂²×D₇₉

yes

characteristic

120

C₁₆

—

121

C₁₆

—

158

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

158

—

125

C₂³

—

158

—

126

C₂³

—

158

—

127

C₂³

C₈×D₇₉

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

158

—

135

C₂×C₄

—

158

—

136

C₂×C₄

—

158

—

137

C₂×C₄

—

158

—

138

C₂×C₄

—

158

—

139

C₂×C₄

C₈×D₇₉

yes

characteristic

140

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

158

—

144

C₈

D₄×D₇₉

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

158

—

150

C₂²

—

158

—

151

C₂²

—

158

—

152

C₂²

—

158

—

153

C₂²

—

158

—

154

C₂²

#14579100

yes

complemented

155

C₂²

C₂×C₈×D₇₉

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

158

—

159

C₄

C₂²⋊C₄×D₇₉

yes

characteristic

160

C₄

M₁₆×D₇₉

yes

characteristic

161

C₂

—

162

C₂

—

158

—

163

C₂

—

158

—

164

C₂

C₂²⋊C₈×D₇₉

yes

characteristic

165

C₂

M₃₂×D₇₉

yes

characteristic

166

C₂

C₂×#14579100

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₇₉
commutator, socle	C₂²×C₇₉	C₂²×C₈
Fitting	C₂²⋊C₁₆×C₇₉	C₂
prime radical	C₂³×D₇₉	C₈
cosocle	C₂×C₈×C₇₉	C₂³
hypercenter	C₂²⋊C₁₆	D₇₉
hypocenter	C₇₉	#2112
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₇₉

commutator, socle

C₂²×C₇₉

C₂²×C₈

Fitting

C₂²⋊C₁₆×C₇₉

C₂

prime radical

C₂³×D₇₉

C₈

cosocle

C₂×C₈×C₇₉

C₂³

hypercenter

C₂²⋊C₁₆

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. C₂×#14579100	C₂	semidirect
3. C₂²⋊(C₇₉⋊C₁₆)	C₂	semidirect
4. (C₂×D₇₉)⋊C₁₆	C₂	semidirect
5. C₂²⋊C₁₆×C₇₉	C₂	semidirect
14. C₂×C₇₉⋊C₁₆	C₂²	semidirect
16. C₂×C₁₆×C₇₉	C₂²	semidirect
63. C₂²×C₇₉	M₃₂	semidirect
71. C₂×D₇₉	M₃₂	semidirect
84. C₇₉	#2112	semidirect
89. C₂²⋊C₁₆	D₇₉	semidirect
100. C₂×C₁₆	C₂×D₇₉	semidirect
154. C₂²	#14579100	semidirect

H ≅ G/N

type

2. C₂×#14579100

C₂

semidirect

3. C₂²⋊(C₇₉⋊C₁₆)

C₂

semidirect

4. (C₂×D₇₉)⋊C₁₆

C₂

semidirect

5. C₂²⋊C₁₆×C₇₉

C₂

semidirect

14. C₂×C₇₉⋊C₁₆

C₂²

semidirect

16. C₂×C₁₆×C₇₉

C₂²

semidirect

63. C₂²×C₇₉

M₃₂

semidirect

71. C₂×D₇₉

M₃₂

semidirect

84. C₇₉

#2112

semidirect

89. C₂²⋊C₁₆

D₇₉

semidirect

100. C₂×C₁₆

C₂×D₇₉

semidirect

154. C₂²

#14579100

semidirect

Subgroups — #23765009

Conjugacy classes of subgroups

Complements