Group #23763513 — subgroups

Subgroups — #23763513

Order 10112.

236 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 73 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	#19099040	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579079	C₂	1	1	yes	maximal, characteristic
3	C₂×#14579079	C₂	1	1	yes	maximal, characteristic
4	#19098335	C₂	1	1	yes	maximal, characteristic
5	#19098334	C₂	1	1	yes	maximal, characteristic
6	#19098333	C₂	1	1	yes	maximal, characteristic
7	#344×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
9	C₂×#11929608	C₂²	1	1	yes	characteristic
10	C₂×#11929608	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₅₈	—	2	2	no	—
12	#14579126	—	2	2	no	—
13	#14579120	C₂²	1	1	yes	characteristic
14	#14579117	C₄	1	4	yes	—
15	#14579079	—	2	2	no	—
16	#14579079	—	2	2	no	—
17	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
19	#117×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₅₈	—	2	1	no	—
21	C₂×Dic₁₅₈	—	2	1	no	—
22	C₂×Dic₁₅₈	—	2	1	no	—
23	C₂×Dic₁₅₈	—	2	1	no	—
24	C₂×Dic₁₅₈	D₄	1	2	yes	—
25	C₂×Dic₁₅₈	D₄	1	2	yes	—
26	C₂²×Dic₇₉	C₂×C₄	1	2	yes	—
27	#11929608	—	2	1	no	—
28	#11929608	—	2	1	no	—
29	#11929608	—	2	1	no	—
30	#11929608	—	2	1	no	—
31	C₂×C₇₉⋊C₈	—	2	1	no	—
32	C₂×C₇₉⋊C₈	—	2	1	no	—
33	C₄⋊Dic₇₉	—	2	1	no	—
34	Dic₇₉⋊C₄	—	2	4	no	—
35	C₂²⋊Dic₇₉	C₂×C₄	1	2	yes	—
36	C₄×Dic₇₉	—	2	4	no	—
37	C₄×Dic₇₉	—	2	1	no	—
38	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
39	M₁₆×C₇₉	—	2	1	no	—
40	M₁₆×C₇₉	—	2	1	no	—
41	M₁₆×C₇₉	—	2	1	no	—
42	M₁₆×C₇₉	—	2	1	no	—
43	C₂×C₈×C₇₉	—	2	1	no	—
44	C₂×C₈×C₇₉	—	2	1	no	—
45	C₄⋊C₄×C₇₉	—	2	1	no	—
46	C₂²⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
47	C₄²×C₇₉	—	2	1	no	—
48	C₂³×C₇₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	—	2	2	no	—
50	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
53	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
56	C₂×Dic₇₉	—	2	2	no	—
57	C₂×Dic₇₉	—	2	2	no	—
58	C₂×Dic₇₉	—	2	2	no	—
59	C₂×Dic₇₉	C₂²⋊C₄	1	4	yes	—
60	Dic₁₅₈	—	2	2	no	—
61	Dic₁₅₈	—	2	2	no	—
62	Dic₁₅₈	—	2	2	no	—
63	Dic₁₅₈	—	2	2	no	—
64	C₈×C₇₉	—	4	1	no	—
65	C₈×C₇₉	—	4	1	no	—
66	C₇₉⋊C₈	—	4	1	no	—
67	C₇₉⋊C₈	—	4	1	no	—
68	C₂²×C₇₉	—	4	1	no	—
69	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₇₉	#117	1	1	yes	characteristic
71	C₂²×C₇₉	#117	1	1	yes	characteristic
72	C₄×C₇₉	—	4	2	no	—
73	C₄×C₇₉	#123	1	1	yes	characteristic
74	C₄×C₇₉	#123	1	1	yes	characteristic
75	C₄×C₇₉	#123	1	1	yes	characteristic
76	C₄×C₇₉	#123	1	1	yes	characteristic
77	Dic₇₉	—	2	4	no	—
78	Dic₇₉	—	4	2	no	—
79	C₂×C₇₉	—	2	1	no	—
80	C₂×C₇₉	—	2	1	no	—
81	C₂×C₇₉	—	2	1	no	—
82	C₂×C₇₉	#381	1	1	yes	characteristic
83	#1830	—	79	1	no	maximal, Sylow
84	C₇₉	#1830	1	1	yes	characteristic, complemented, Sylow
85	#519	—	79	1	no	—
86	C₂×#101	—	79	1	no	—
87	C₂×#101	—	79	1	no	—
88	#344	—	79	1	no	—
89	#344	—	79	1	no	—
90	#344	—	79	1	no	—
91	#344	D₇₉	1	1	yes	characteristic
92	C₂²×Q₈	—	79	1	no	—
93	C₂×M₁₆	—	79	1	no	—
94	C₂×M₁₆	—	79	1	no	—
95	C₂×M₁₆	C₂×D₇₉	1	1	yes	characteristic
96	C₂×M₁₆	C₂×D₇₉	1	1	yes	characteristic
97	C₄×Q₈	—	158	2	no	—
98	C₄×Q₈	—	158	2	no	—
99	#117	—	79	1	no	—
100	#117	—	79	4	no	—
101	#117	C₂×D₇₉	1	1	yes	characteristic
102	#101	—	158	2	no	—
103	#101	—	158	2	no	—
104	C₂×Q₈	—	79	2	no	—
105	C₂×Q₈	—	79	2	no	—
106	C₂×Q₈	—	158	1	no	—
107	C₂×Q₈	—	158	1	no	—
108	C₂×Q₈	—	158	1	no	—
109	C₂×Q₈	—	158	1	no	—
110	C₂²×C₄	—	79	2	no	—
111	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
112	M₁₆	—	2	1	no	—
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	2	1	no	—
116	M₁₆	—	158	1	no	—
117	M₁₆	—	158	1	no	—
118	M₁₆	—	158	1	no	—
119	M₁₆	—	158	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	158	1	no	—
123	C₂×C₈	—	158	1	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	158	4	no	—
126	C₄⋊C₄	—	158	1	no	—
127	C₂²⋊C₄	—	79	2	no	—
128	C₂²⋊C₄	C₄×D₇₉	1	2	yes	—
129	C₄²	—	2	1	no	—
130	C₄²	—	158	4	no	—
131	C₄²	—	158	1	no	—
132	C₂³	C₂×C₄×D₇₉	1	1	yes	characteristic
133	Q₈	—	158	2	no	—
134	Q₈	—	158	2	no	—
135	Q₈	—	158	2	no	—
136	Q₈	—	158	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	79	4	no	—
139	C₂×C₄	—	158	2	no	—
140	C₂×C₄	—	158	2	no	—
141	C₂×C₄	—	158	2	no	—
142	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
143	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
144	C₂×C₄	#11929623	1	1	yes	characteristic
145	C₂×C₄	#11929624	1	1	yes	characteristic
146	C₂×C₄	#11929624	1	1	yes	characteristic
147	C₂×C₄	#11929625	1	1	yes	characteristic
148	C₈	—	4	1	no	—
149	C₈	—	4	1	no	—
150	C₈	—	316	1	no	—
151	C₈	—	316	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
154	C₂²	#14579118	1	1	yes	characteristic
155	C₂²	#14579119	1	1	yes	characteristic
156	C₄	—	4	2	no	—
157	C₄	—	316	2	no	—
158	C₄	—	158	4	no	—
159	C₄	#14579143	1	1	yes	characteristic
160	C₄	#14579143	1	1	yes	characteristic
161	C₄	#14579146	1	1	yes	characteristic
162	C₄	#14579146	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	#19098425	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099040

C₂

yes

maximal, characteristic

C₂×#14579079

C₂

yes

maximal, characteristic

C₂×#14579079

C₂

yes

maximal, characteristic

#19098335

C₂

yes

maximal, characteristic

#19098334

C₂

yes

maximal, characteristic

#19098333

C₂

yes

maximal, characteristic

#344×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

characteristic

C₂×#11929608

C₂²

yes

characteristic

C₂×#11929608

C₂²

yes

characteristic

C₄×Dic₁₅₈

—

#14579126

—

#14579120

C₂²

yes

characteristic

#14579117

C₄

yes

—

#14579079

—

#14579079

—

C₂×M₁₆×C₇₉

C₂²

yes

characteristic

C₂×M₁₆×C₇₉

C₂²

yes

characteristic

#117×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

D₄

yes

—

C₂×Dic₁₅₈

D₄

yes

—

C₂²×Dic₇₉

C₂×C₄

yes

—

—

—

—

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

C₂×C₄

yes

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

C₂×C₄

yes

—

C₄²×C₇₉

—

C₂³×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂²⋊C₄

yes

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₉

yes

characteristic

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₉

yes

characteristic

C₂×M₁₆

C₂×D₇₉

yes

characteristic

C₄×Q₈

—

158

—

C₄×Q₈

—

158

—

#117

—

100

#117

—

101

#117

C₂×D₇₉

yes

characteristic

102

#101

—

158

—

103

#101

—

158

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

158

—

107

C₂×Q₈

—

158

—

108

C₂×Q₈

—

158

—

109

C₂×Q₈

—

158

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²×D₇₉

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

158

—

117

M₁₆

—

158

—

118

M₁₆

—

158

—

119

M₁₆

—

158

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

158

—

123

C₂×C₈

—

158

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

158

—

126

C₄⋊C₄

—

158

—

127

C₂²⋊C₄

—

128

C₂²⋊C₄

C₄×D₇₉

yes

—

129

C₄²

—

130

C₄²

—

158

—

131

C₄²

—

158

—

132

C₂³

C₂×C₄×D₇₉

yes

characteristic

133

Q₈

—

158

—

134

Q₈

—

158

—

135

Q₈

—

158

—

136

Q₈

—

158

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

158

—

140

C₂×C₄

—

158

—

141

C₂×C₄

—

158

—

142

C₂×C₄

D₄×D₇₉

yes

characteristic

143

C₂×C₄

D₄×D₇₉

yes

characteristic

144

C₂×C₄

#11929623

yes

characteristic

145

C₂×C₄

#11929624

yes

characteristic

146

C₂×C₄

#11929624

yes

characteristic

147

C₂×C₄

#11929625

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

316

—

151

C₈

—

316

—

152

C₂²

—

153

C₂²

C₂²⋊C₄×D₇₉

yes

characteristic

154

C₂²

#14579118

yes

characteristic

155

C₂²

#14579119

yes

characteristic

156

C₄

—

157

C₄

—

316

—

158

C₄

—

158

—

159

C₄

#14579143

yes

characteristic

160

C₄

#14579143

yes

characteristic

161

C₄

#14579146

yes

characteristic

162

C₄

#14579146

yes

characteristic

163

C₂

—

164

C₂

—

165

C₂

—

166

C₂

#19098425

yes

characteristic

	subgroup	G/H
center	C₂	#19098425
commutator, prime radical	C₂³×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#344×C₇₉	C₂
socle	C₂×C₇₉	#381
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#344	D₇₉
hypocenter	C₇₉	#1830
LCS	G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₂

#19098425

commutator, prime radical

C₂³×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#344×C₇₉

C₂

socle

C₂×C₇₉

#381

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
84. C₇₉	#1830	semidirect

H ≅ G/N

type

84. C₇₉

#1830

semidirect

Subgroups — #23763513

Conjugacy classes of subgroups

Complements