Group #23772111 — subgroups

Subgroups — #23772111

Order 10112.

500 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 189 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×#14579169	C₂	1	2	yes	maximal, complemented
2	#19099113	C₂	1	1	yes	maximal, characteristic
3	#19098640	C₂	1	4	yes	maximal, complemented
4	#19098628	C₂	1	4	yes	maximal, complemented
5	#19098531	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#423×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11929624	C₂²	1	2	yes	—
9	#14579169	—	2	8	no	—
10	C₂×#11929607	C₂²	1	4	yes	—
11	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₇₉	C₂²	1	4	yes	—
14	#14579161	C₄	1	1	yes	characteristic
15	#14579158	C₄	1	1	yes	characteristic
16	#14579149	C₄	1	2	yes	—
17	#14579135	C₄	1	4	yes	—
18	C₄×D₃₁₆	C₂²	1	4	yes	—
19	(C₂×D₇₉)⋊C₈	C₂²	1	8	yes	—
20	C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
21	C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
22	C₂×M₁₆×C₇₉	C₂²	1	2	yes	—
23	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₇₉	C₂²	1	4	yes	—
25	#11929624	—	2	4	no	—
26	#11929624	—	2	4	no	—
27	C₂×Dic₁₅₈	C₂×C₄	1	2	yes	—
28	C₂×D₃₁₆	C₂×C₄	1	2	yes	—
29	C₂×D₃₁₆	C₂×C₄	1	4	yes	—
30	C₂×C₂²⋊D₇₉	C₂×C₄	1	4	yes	—
31	C₂×C₄×D₇₉	C₂³	1	4	yes	—
32	#11929607	—	2	8	no	—
33	C₂×C₇₉⋊C₈	—	2	2	no	—
34	C₂×C₇₉⋊C₈	D₄	1	4	yes	complemented
35	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
36	C₂×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
37	C₂×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
38	C₈×D₇₉	—	2	8	no	—
39	C₄⋊Dic₇₉	C₂×C₄	1	4	yes	—
40	(C₂×D₇₉)⋊C₄	C₂×C₄	1	8	yes	—
41	C₂²×C₄×C₇₉	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
43	M₁₆×C₇₉	—	2	4	no	—
44	C₂×C₈×C₇₉	C₂³	1	4	yes	—
45	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₇₉	C₂³	1	2	yes	—
48	C₂³×C₇₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×D₇₉	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₇₉	—	2	4	no	—
51	C₂×C₄×C₇₉	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₇₉	#41	1	2	yes	—
56	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
57	C₂²⋊D₇₉	—	2	8	no	—
58	C₂×Dic₇₉	C₂²×C₄	1	4	yes	—
59	D₃₁₆	—	2	4	no	—
60	D₃₁₆	—	2	8	no	—
61	C₄×D₇₉	—	2	4	no	—
62	C₄×D₇₉	—	2	4	no	—
63	Dic₁₅₈	—	2	4	no	—
64	C₈×C₇₉	—	2	4	no	—
65	C₇₉⋊C₈	—	2	2	no	—
66	C₇₉⋊C₈	C₂×D₄	1	2	yes	—
67	C₇₉⋊C₈	C₂×D₄	1	2	yes	—
68	C₂²×C₇₉	—	2	2	no	—
69	C₂²×C₇₉	C₄×D₄	1	2	yes	—
70	C₂²×C₇₉	C₂³×C₄	1	1	yes	characteristic
71	C₂×D₇₉	—	2	4	no	—
72	C₂×D₇₉	—	2	8	no	—
73	C₄×C₇₉	—	2	2	no	—
74	C₄×C₇₉	C₄×D₄	1	2	yes	—
75	C₄×C₇₉	#131	1	4	yes	complemented
76	C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
77	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
78	Dic₇₉	—	2	4	no	—
79	C₂×C₇₉	—	2	2	no	—
80	C₂×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
81	C₂×C₇₉	C₂×#131	1	2	yes	—
82	D₇₉	—	4	4	no	—
83	#2931	—	79	1	no	maximal, Sylow
84	C₇₉	#2931	1	1	yes	characteristic, complemented, Sylow
85	C₂×#131	—	79	2	no	—
86	#532	—	79	1	no	—
87	#436	—	79	4	no	—
88	C₈×D₄	—	79	4	no	—
89	#423	D₇₉	1	1	yes	characteristic, complemented
90	#408	—	79	2	no	—
91	C₂×C₄×C₈	—	79	1	no	—
92	C₂×#41	—	79	2	no	—
93	#131	—	158	8	no	—
94	C₂×M₁₆	—	79	4	no	—
95	C₂×M₁₆	C₂×D₇₉	1	2	yes	—
96	C₂²×C₈	—	79	1	no	—
97	C₂²×C₈	—	79	1	no	—
98	C₂²×C₈	—	79	4	no	—
99	C₄⋊Q₈	—	79	1	no	—
100	#127	—	79	1	no	—
101	#124	—	79	2	no	—
102	#121	—	79	4	no	—
103	C₄×D₄	—	79	4	no	—
104	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
105	C₄⋊C₈	C₂×D₇₉	1	4	yes	—
106	C₂²⋊C₈	—	79	8	no	—
107	C₄×C₈	—	79	2	no	—
108	C₄×C₈	—	79	2	no	—
109	#41	—	158	4	no	—
110	#41	—	158	4	no	—
111	C₂×Q₈	—	79	2	no	—
112	C₂×D₄	—	79	4	no	—
113	C₂×D₄	—	79	4	no	—
114	C₂×D₄	—	79	2	no	—
115	C₂²×C₄	—	79	4	no	—
116	C₂²×C₄	C₄×D₇₉	1	2	yes	—
117	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
118	M₁₆	—	2	4	no	—
119	M₁₆	—	158	8	no	—
120	C₂×C₈	—	79	4	no	—
121	C₂×C₈	—	79	1	no	—
122	C₂×C₈	—	79	1	no	—
123	C₂×C₈	—	79	2	no	—
124	C₂×C₈	—	158	8	no	—
125	C₂×C₈	—	158	2	no	—
126	C₂×C₈	C₂²×D₇₉	1	4	yes	—
127	C₄⋊C₄	—	79	4	no	—
128	C₂²⋊C₄	—	79	8	no	—
129	C₄²	C₄×D₇₉	1	1	yes	characteristic
130	C₄²	C₄×D₇₉	1	1	yes	characteristic
131	C₄²	C₂²×D₇₉	1	2	yes	—
132	C₂³	—	79	4	no	—
133	C₂³	C₂×C₄×D₇₉	1	1	yes	characteristic
134	Q₈	—	158	4	no	—
135	D₄	—	158	8	no	—
136	D₄	—	158	4	no	—
137	D₄	—	158	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	79	4	no	—
140	C₂×C₄	—	158	4	no	—
141	C₂×C₄	—	158	4	no	—
142	C₂×C₄	C₂×C₄×D₇₉	1	2	yes	—
143	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
144	C₂×C₄	C₂×C₄×D₇₉	1	2	yes	—
145	C₂×C₄	D₄×D₇₉	1	2	yes	—
146	C₂×C₄	#11929625	1	2	yes	—
147	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
148	C₈	—	2	4	no	—
149	C₈	—	79	2	no	—
150	C₈	—	79	2	no	—
151	C₈	—	158	2	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	158	4	no	—
154	C₂²	—	158	8	no	—
155	C₂²	#14579123	1	2	yes	—
156	C₂²	C₂²×C₄×D₇₉	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	158	4	no	—
159	C₄	#14579123	1	2	yes	—
160	C₄	#14579169	1	4	yes	complemented
161	C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
162	C₄	C₂×#11929625	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	—	316	4	no	—
165	C₂	C₂×#14579123	1	1	yes	characteristic
166	C₂	C₂×#14579169	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579169

C₂

yes

maximal, complemented

#19099113

C₂

yes

maximal, characteristic

#19098640

C₂

yes

maximal, complemented

#19098628

C₂

yes

maximal, complemented

#19098531

C₂

yes

maximal, complemented

C₂×C₄×C₇₉⋊C₈

C₂

yes

maximal, characteristic, complemented

#423×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929624

C₂²

yes

—

#14579169

—

C₂×#11929607

C₂²

yes

—

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₇₉

C₂²

yes

—

#14579161

C₄

yes

characteristic

#14579158

C₄

yes

characteristic

#14579149

C₄

yes

—

#14579135

C₄

yes

—

C₄×D₃₁₆

C₂²

yes

—

(C₂×D₇₉)⋊C₈

C₂²

yes

—

C₄×C₇₉⋊C₈

C₂²

yes

—

C₄×C₇₉⋊C₈

C₂²

yes

—

C₂×M₁₆×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

—

#11929624

—

#11929624

—

C₂×Dic₁₅₈

C₂×C₄

yes

—

C₂×D₃₁₆

C₂×C₄

yes

—

C₂×D₃₁₆

C₂×C₄

yes

—

C₂×C₂²⋊D₇₉

C₂×C₄

yes

—

C₂×C₄×D₇₉

C₂³

yes

—

#11929607

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₄

yes

complemented

C₂×C₇₉⋊C₈

C₂³

yes

—

C₂×C₇₉⋊C₈

C₂³

yes

characteristic

C₂×C₇₉⋊C₈

C₂³

yes

characteristic

C₈×D₇₉

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

(C₂×D₇₉)⋊C₄

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

—

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

C₂³

yes

—

C₂³×C₇₉

C₂²×C₄

yes

characteristic

C₂²×D₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

C₂²×C₄

yes

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

C₂×D₄

yes

—

C₇₉⋊C₈

C₂×D₄

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

C₄×D₄

yes

—

C₂²×C₇₉

C₂³×C₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₄×D₄

yes

—

C₄×C₇₉

#131

yes

complemented

C₄×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

C₂×#41

yes

characteristic

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₂×C₄×D₄

yes

characteristic

C₂×C₇₉

C₂×#131

yes

—

D₇₉

—

#2931

—

maximal, Sylow

C₇₉

#2931

yes

characteristic, complemented, Sylow

C₂×#131

—

#532

—

#436

—

C₈×D₄

—

#423

D₇₉

yes

characteristic, complemented

#408

—

C₂×C₄×C₈

—

C₂×#41

—

#131

—

158

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₉

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

100

#127

—

101

#124

—

102

#121

—

103

C₄×D₄

—

104

C₂×C₄²

C₂×D₇₉

yes

characteristic

105

C₄⋊C₈

C₂×D₇₉

yes

—

106

C₂²⋊C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

#41

—

158

—

110

#41

—

158

—

111

C₂×Q₈

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₄×D₇₉

yes

—

117

C₂²×C₄

C₂²×D₇₉

yes

characteristic

118

M₁₆

—

119

M₁₆

—

158

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

158

—

125

C₂×C₈

—

158

—

126

C₂×C₈

C₂²×D₇₉

yes

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

C₄×D₇₉

yes

characteristic

130

C₄²

C₄×D₇₉

yes

characteristic

131

C₄²

C₂²×D₇₉

yes

—

132

C₂³

—

133

C₂³

C₂×C₄×D₇₉

yes

characteristic

134

Q₈

—

158

—

135

D₄

—

158

—

136

D₄

—

158

—

137

D₄

—

158

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

158

—

141

C₂×C₄

—

158

—

142

C₂×C₄

C₂×C₄×D₇₉

yes

—

143

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

144

C₂×C₄

C₂×C₄×D₇₉

yes

—

145

C₂×C₄

D₄×D₇₉

yes

—

146

C₂×C₄

#11929625

yes

—

147

C₂×C₄

C₂³×D₇₉

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

158

—

152

C₂²

—

153

C₂²

—

158

—

154

C₂²

—

158

—

155

C₂²

#14579123

yes

—

156

C₂²

C₂²×C₄×D₇₉

yes

characteristic

157

C₄

—

158

C₄

—

158

—

159

C₄

#14579123

yes

—

160

C₄

#14579169

yes

complemented

161

C₄

C₂×D₄×D₇₉

yes

characteristic

162

C₄

C₂×#11929625

yes

characteristic

163

C₂

—

164

C₂

—

316

—

165

C₂

C₂×#14579123

yes

characteristic

166

C₂

C₂×#14579169

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₉
commutator, socle	C₂²×C₇₉	C₂³×C₄
Fitting	#423×C₇₉	C₂
prime radical	#19099113	C₂
cosocle	C₂×C₄×C₇₉	C₂⁴
hypercenter	#423	D₇₉
hypocenter	C₇₉	#2931
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₇₉

commutator, socle

C₂²×C₇₉

C₂³×C₄

Fitting

#423×C₇₉

C₂

prime radical

#19099113

C₂

cosocle

C₂×C₄×C₇₉

C₂⁴

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
1. C₂×#14579169	C₂	semidirect
3. #19098640	C₂	semidirect
4. #19098628	C₂	semidirect
5. #19098531	C₂	semidirect
6. C₂×C₄×C₇₉⋊C₈	C₂	semidirect
7. #423×C₇₉	C₂	semidirect
34. C₂×C₇₉⋊C₈	D₄	semidirect
75. C₄×C₇₉	#131	semidirect
84. C₇₉	#2931	semidirect
89. #423	D₇₉	semidirect
160. C₄	#14579169	semidirect

H ≅ G/N

type

1. C₂×#14579169

C₂

semidirect

3. #19098640

C₂

semidirect

4. #19098628

C₂

semidirect

5. #19098531

C₂

semidirect

6. C₂×C₄×C₇₉⋊C₈

C₂

semidirect

7. #423×C₇₉

C₂

semidirect

34. C₂×C₇₉⋊C₈

D₄

semidirect

75. C₄×C₇₉

#131

semidirect

84. C₇₉

#2931

semidirect

89. #423

D₇₉

semidirect

160. C₄

#14579169

semidirect

Subgroups — #23772111

Conjugacy classes of subgroups

Complements