Group #23762744 — subgroups

Subgroups — #23762744

Order 10112.

188 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 53 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	#19098937	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098840	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098759	C₂	1	1	yes	maximal, characteristic
4	#19098628	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098313	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098306	C₂	1	1	yes	maximal, characteristic, complemented
7	#332×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11929611	—	2	1	no	—
9	C₂×#11929611	—	2	1	no	—
10	C₂×D₆₃₂	—	2	1	no	—
11	C₂×C₈×D₇₉	—	2	1	no	—
12	#14579158	C₂²	1	1	yes	characteristic
13	C₄×D₃₁₆	C₂²	1	1	yes	characteristic
14	#14579094	—	2	1	no	—
15	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
16	#14579083	—	2	1	no	—
17	#14579081	—	2	1	no	—
18	(C₂×D₇₉)⋊C₈	—	2	1	no	—
19	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×D₃₁₆	—	2	1	no	—
24	C₂×D₃₁₆	—	2	1	no	—
25	C₂×D₃₁₆	—	2	1	no	—
26	C₂×D₃₁₆	D₄	1	1	yes	characteristic
27	C₂×C₄×D₇₉	—	2	1	no	—
28	#11929611	—	4	2	no	—
29	#11929611	—	4	2	no	—
30	D₆₃₂	—	4	2	no	—
31	C₂×C₇₉⋊C₈	—	2	1	no	—
32	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
33	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
34	C₈×D₇₉	—	4	2	no	—
35	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
36	(C₂×D₇₉)⋊C₄	—	2	1	no	—
37	C₂×Q₈×C₇₉	—	2	1	no	—
38	C₂×C₈×C₇₉	—	2	1	no	—
39	C₂×C₈×C₇₉	—	2	1	no	—
40	C₄⋊C₄×C₇₉	—	2	1	no	—
41	C₄⋊C₄×C₇₉	—	2	1	no	—
42	C₄²×C₇₉	C₂³	1	1	yes	characteristic
43	C₂²×D₇₉	—	2	1	no	—
44	C₂²×D₇₉	—	4	1	no	—
45	Q₈×C₇₉	—	4	1	no	—
46	Q₈×C₇₉	—	4	1	no	—
47	C₂×C₄×C₇₉	—	4	1	no	—
48	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
51	C₂×Dic₇₉	—	2	1	no	—
52	D₃₁₆	—	2	1	no	—
53	D₃₁₆	—	4	1	no	—
54	D₃₁₆	—	4	1	no	—
55	D₃₁₆	—	4	1	no	—
56	D₃₁₆	—	4	1	no	—
57	D₃₁₆	—	8	1	no	—
58	D₃₁₆	SD₁₆	1	2	yes	complemented
59	C₄×D₇₉	—	4	1	no	—
60	C₄×D₇₉	—	4	1	no	—
61	C₈×C₇₉	—	4	1	no	—
62	C₈×C₇₉	—	4	1	no	—
63	C₇₉⋊C₈	—	2	1	no	—
64	C₇₉⋊C₈	—	4	1	no	—
65	C₇₉⋊C₈	D₈	1	2	yes	complemented
66	C₂²×C₇₉	#121	1	1	yes	characteristic
67	C₂×D₇₉	—	2	2	no	—
68	C₂×D₇₉	—	4	1	no	—
69	C₂×D₇₉	—	4	1	no	—
70	C₂×D₇₉	—	8	1	no	—
71	C₂×D₇₉	—	8	1	no	—
72	C₂×D₇₉	—	8	1	no	—
73	C₄×C₇₉	—	2	1	no	—
74	C₄×C₇₉	—	8	1	no	—
75	C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
76	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
77	C₄×C₇₉	#135	1	1	yes	characteristic
78	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	Dic₇₉	—	4	1	no	—
80	C₂×C₇₉	#459	1	1	yes	characteristic
81	C₂×C₇₉	#465	1	1	yes	characteristic
82	C₂×C₇₉	#471	1	1	yes	characteristic
83	D₇₉	—	4	2	no	—
84	D₇₉	—	16	1	no	—
85	#1665	—	79	1	no	maximal, Sylow
86	C₇₉	#1665	1	1	yes	characteristic, complemented, Sylow
87	#492	—	79	1	no	—
88	#474	—	79	1	no	—
89	#459	—	79	1	no	—
90	C₈×D₄	—	79	1	no	—
91	#334	—	79	1	no	—
92	#332	D₇₉	1	1	yes	characteristic, complemented
93	#331	—	79	1	no	—
94	C₂×SD₁₆	—	158	1	no	—
95	C₂×SD₁₆	—	158	1	no	—
96	C₂×D₈	—	158	1	no	—
97	C₂²×C₈	—	158	1	no	—
98	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
99	#127	—	79	1	no	—
100	C₄×D₄	—	79	1	no	—
101	#107	—	158	1	no	—
102	C₄⋊C₈	—	79	1	no	—
103	C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
104	C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
105	#102	—	158	1	no	—
106	#102	—	158	1	no	—
107	C₂²⋊C₈	—	158	1	no	—
108	C₄×C₈	—	79	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×D₄	—	79	1	no	—
111	C₂×D₄	—	158	1	no	—
112	C₂×D₄	—	158	1	no	—
113	C₂×D₄	—	158	1	no	—
114	C₂²×C₄	—	158	1	no	—
115	SD₁₆	—	316	2	no	—
116	SD₁₆	—	316	2	no	—
117	D₈	—	316	2	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	316	2	no	—
121	C₂×C₈	—	79	1	no	—
122	C₂×C₈	—	79	1	no	—
123	C₂×C₈	—	158	1	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	79	1	no	—
127	C₂²⋊C₄	—	158	1	no	—
128	C₄²	C₂²×D₇₉	1	1	yes	characteristic
129	C₂³	—	316	1	no	—
130	C₂³	—	158	1	no	—
131	Q₈	—	4	1	no	—
132	Q₈	—	4	1	no	—
133	D₄	—	316	1	no	—
134	D₄	—	316	1	no	—
135	D₄	—	316	1	no	—
136	D₄	—	316	1	no	—
137	D₄	—	79	2	no	—
138	D₄	—	632	1	no	—
139	D₄	—	158	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	316	1	no	—
142	C₂×C₄	—	316	1	no	—
143	C₂×C₄	—	158	1	no	—
144	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
145	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
146	C₂×C₄	#11929624	1	1	yes	characteristic
147	C₈	—	4	1	no	—
148	C₈	—	4	1	no	—
149	C₈	—	316	1	no	—
150	C₈	—	79	2	no	—
151	C₈	—	158	1	no	—
152	C₂²	—	316	1	no	—
153	C₂²	—	316	1	no	—
154	C₂²	—	632	1	no	—
155	C₂²	—	632	1	no	—
156	C₂²	—	632	1	no	—
157	C₂²	—	158	2	no	—
158	C₂²	#14579134	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	8	1	no	—
161	C₄	—	316	1	no	—
162	C₄	D₈×D₇₉	1	1	yes	characteristic
163	C₄	SD₁₆×D₇₉	1	1	yes	characteristic
164	C₄	#14579184	1	1	yes	characteristic
165	C₄	#14579188	1	1	yes	characteristic
166	C₂	—	316	2	no	—
167	C₂	—	1264	1	no	—
168	C₂	#19098757	1	1	yes	characteristic
169	C₂	#19098795	1	1	yes	characteristic
170	C₂	#19098829	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098937

C₂

yes

maximal, characteristic, complemented

#19098840

C₂

yes

maximal, characteristic, complemented

#19098759

C₂

yes

maximal, characteristic

#19098628

C₂

yes

maximal, characteristic, complemented

#19098313

C₂

yes

maximal, characteristic, complemented

#19098306

C₂

yes

maximal, characteristic, complemented

#332×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929611

—

C₂×#11929611

—

C₂×D₆₃₂

—

C₂×C₈×D₇₉

—

#14579158

C₂²

yes

characteristic

C₄×D₃₁₆

C₂²

yes

characteristic

#14579094

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

#14579083

—

#14579081

—

(C₂×D₇₉)⋊C₈

—

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

D₄

yes

characteristic

C₂×C₄×D₇₉

—

#11929611

—

#11929611

—

D₆₃₂

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₈×D₇₉

—

C₄⋊Dic₇₉

D₄

yes

characteristic

(C₂×D₇₉)⋊C₄

—

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

characteristic

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

SD₁₆

yes

complemented

C₄×D₇₉

—

C₄×D₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

D₈

yes

complemented

C₂²×C₇₉

#121

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×D₈

yes

characteristic

C₄×C₇₉

C₂×SD₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

D₇₉

—

#1665

—

maximal, Sylow

C₇₉

#1665

yes

characteristic, complemented, Sylow

#492

—

#474

—

#459

—

C₈×D₄

—

#334

—

#332

D₇₉

yes

characteristic, complemented

#331

—

C₂×SD₁₆

—

158

—

C₂×SD₁₆

—

158

—

C₂×D₈

—

158

—

C₂²×C₈

—

158

—

C₄⋊Q₈

C₂×D₇₉

yes

characteristic

#127

—

100

C₄×D₄

—

101

#107

—

158

—

102

C₄⋊C₈

—

103

C₄⋊C₈

C₂×D₇₉

yes

characteristic

104

C₄⋊C₈

C₂×D₇₉

yes

characteristic

105

#102

—

158

—

106

#102

—

158

—

107

C₂²⋊C₈

—

158

—

108

C₄×C₈

—

109

C₂×Q₈

—

110

C₂×D₄

—

111

C₂×D₄

—

158

—

112

C₂×D₄

—

158

—

113

C₂×D₄

—

158

—

114

C₂²×C₄

—

158

—

115

SD₁₆

—

316

—

116

SD₁₆

—

316

—

117

D₈

—

316

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

316

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

158

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

158

—

128

C₄²

C₂²×D₇₉

yes

characteristic

129

C₂³

—

316

—

130

C₂³

—

158

—

131

Q₈

—

132

Q₈

—

133

D₄

—

316

—

134

D₄

—

316

—

135

D₄

—

316

—

136

D₄

—

316

—

137

D₄

—

138

D₄

—

632

—

139

D₄

—

158

—

140

C₂×C₄

—

141

C₂×C₄

—

316

—

142

C₂×C₄

—

316

—

143

C₂×C₄

—

158

—

144

C₂×C₄

D₄×D₇₉

yes

characteristic

145

C₂×C₄

D₄×D₇₉

yes

characteristic

146

C₂×C₄

#11929624

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

316

—

150

C₈

—

151

C₈

—

158

—

152

C₂²

—

316

—

153

C₂²

—

316

—

154

C₂²

—

632

—

155

C₂²

—

632

—

156

C₂²

—

632

—

157

C₂²

—

158

—

158

C₂²

#14579134

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

316

—

162

C₄

D₈×D₇₉

yes

characteristic

163

C₄

SD₁₆×D₇₉

yes

characteristic

164

C₄

#14579184

yes

characteristic

165

C₄

#14579188

yes

characteristic

166

C₂

—

316

—

167

C₂

—

1264

—

168

C₂

#19098757

yes

characteristic

169

C₂

#19098795

yes

characteristic

170

C₂

#19098829

yes

characteristic

	subgroup	G/H
center	C₂²	#14579134
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	#332×C₇₉	C₂
socle	C₂²×C₇₉	#121
prime radical	#19098759	C₂
hypercenter	#332	D₇₉
hypocenter	C₇₉	#1665
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#14579134

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

#332×C₇₉

C₂

C₂²×C₇₉

C₂

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
1. #19098937	C₂	semidirect
2. #19098840	C₂	semidirect
4. #19098628	C₂	semidirect
5. #19098313	C₂	semidirect
6. #19098306	C₂	semidirect
7. #332×C₇₉	C₂	semidirect
58. D₃₁₆	SD₁₆	semidirect
65. C₇₉⋊C₈	D₈	semidirect
86. C₇₉	#1665	semidirect
92. #332	D₇₉	semidirect

H ≅ G/N

type

1. #19098937

C₂

semidirect

2. #19098840

C₂

semidirect

4. #19098628

C₂

semidirect

5. #19098313

C₂

semidirect

6. #19098306

C₂

semidirect

7. #332×C₇₉

C₂

semidirect

58. D₃₁₆

SD₁₆

semidirect

65. C₇₉⋊C₈

D₈

semidirect

86. C₇₉

#1665

semidirect

92. #332

D₇₉

semidirect

Subgroups — #23762744

Conjugacy classes of subgroups

Complements