Group #23762865 — subgroups

Subgroups — #23762865

Order 10112.

182 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19098936	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098843	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098758	C₂	1	1	yes	maximal, characteristic
4	#19098316	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098310	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098306	C₂	1	1	yes	maximal, characteristic, complemented
7	#436×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11929613	—	2	1	no	—
9	C₂×#11929613	—	2	1	no	—
10	C₂×#11929609	—	2	1	no	—
11	#14579161	C₂²	1	1	yes	characteristic
12	#14579158	C₂²	1	1	yes	characteristic
13	#14579094	—	2	1	no	—
14	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
16	#14579082	—	2	1	no	—
17	#14579081	—	2	1	no	—
18	C₂×M₁₆×C₇₉	—	2	1	no	—
19	C₄×D₄×C₇₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂²⋊C₈×C₇₉	—	2	1	no	—
22	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₅₈	—	2	1	no	—
24	C₂×D₃₁₆	—	2	1	no	—
25	C₂×D₃₁₆	—	2	1	no	—
26	C₂×D₃₁₆	—	2	1	no	—
27	#11929613	—	4	2	no	—
28	#11929613	—	4	2	no	—
29	#11929609	—	4	2	no	—
30	C₂×C₇₉⋊C₈	—	2	1	no	—
31	C₂×C₇₉⋊C₈	—	2	1	no	—
32	C₄⋊Dic₇₉	—	2	1	no	—
33	C₄⋊Dic₇₉	—	2	1	no	—
34	C₂×D₄×C₇₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	—	2	1	no	—
36	M₁₆×C₇₉	—	4	2	no	—
37	C₂×C₈×C₇₉	—	2	1	no	—
38	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
41	C₂²⋊C₄×C₇₉	—	2	1	no	—
42	C₄²×C₇₉	C₂³	1	1	yes	characteristic
43	C₂³×C₇₉	—	2	1	no	—
44	C₂²×D₇₉	—	4	1	no	—
45	D₄×C₇₉	—	2	1	no	—
46	D₄×C₇₉	—	2	1	no	—
47	C₂×C₄×C₇₉	—	2	1	no	—
48	C₂×C₄×C₇₉	—	4	1	no	—
49	C₂×C₄×C₇₉	—	4	1	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×Dic₇₉	—	4	1	no	—
54	D₃₁₆	—	4	1	no	—
55	D₃₁₆	—	4	1	no	—
56	D₃₁₆	—	4	1	no	—
57	D₃₁₆	—	4	1	no	—
58	D₃₁₆	—	8	1	no	—
59	Dic₁₅₈	—	4	1	no	—
60	Dic₁₅₈	—	4	1	no	—
61	C₈×C₇₉	—	2	1	no	—
62	C₈×C₇₉	—	4	1	no	—
63	C₈×C₇₉	D₈	1	2	yes	complemented
64	C₇₉⋊C₈	—	4	1	no	—
65	C₇₉⋊C₈	—	4	1	no	—
66	C₂²×C₇₉	—	4	1	no	—
67	C₂²×C₇₉	—	4	1	no	—
68	C₂²×C₇₉	—	4	1	no	—
69	C₂²×C₇₉	#121	1	1	yes	characteristic
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	—	4	1	no	—
75	C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
76	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₄×C₇₉	#137	1	1	yes	characteristic
79	Dic₇₉	—	8	1	no	—
80	C₂×C₇₉	—	8	1	no	—
81	C₂×C₇₉	#459	1	1	yes	characteristic
82	C₂×C₇₉	#468	1	1	yes	characteristic
83	C₂×C₇₉	#471	1	1	yes	characteristic
84	D₇₉	—	16	1	no	—
85	#1685	—	79	1	no	maximal, Sylow
86	C₇₉	#1685	1	1	yes	characteristic, complemented, Sylow
87	#492	—	79	1	no	—
88	#474	—	79	1	no	—
89	#459	—	79	1	no	—
90	#436	D₇₉	1	1	yes	characteristic, complemented
91	#335	—	79	1	no	—
92	#332	—	79	1	no	—
93	#331	—	79	1	no	—
94	C₂×SD₁₆	—	158	1	no	—
95	C₂×SD₁₆	—	158	1	no	—
96	C₂×D₈	—	158	1	no	—
97	C₂×M₁₆	—	2	1	no	—
98	C₄⋊Q₈	—	79	1	no	—
99	#127	—	79	1	no	—
100	C₄×D₄	C₂×D₇₉	1	1	yes	characteristic
101	#107	—	158	1	no	—
102	C₄⋊C₈	—	79	1	no	—
103	C₄⋊C₈	—	79	1	no	—
104	C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
105	#102	—	158	1	no	—
106	#102	—	158	1	no	—
107	C₂²⋊C₈	—	2	1	no	—
108	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
109	C₂×Q₈	—	158	1	no	—
110	C₂×D₄	—	158	1	no	—
111	C₂×D₄	—	158	1	no	—
112	C₂×D₄	—	158	1	no	—
113	C₂×D₄	D₃₁₆	1	1	yes	characteristic
114	C₂²×C₄	—	2	1	no	—
115	SD₁₆	—	316	2	no	—
116	SD₁₆	—	316	2	no	—
117	D₈	—	316	2	no	—
118	M₁₆	—	4	2	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	158	1	no	—
121	C₂×C₈	—	158	1	no	—
122	C₂×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
123	C₂×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
124	C₄⋊C₄	—	158	1	no	—
125	C₄⋊C₄	—	158	1	no	—
126	C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
127	C₂²⋊C₄	—	2	1	no	—
128	C₄²	C₂²×D₇₉	1	1	yes	characteristic
129	C₂³	—	2	1	no	—
130	C₂³	—	316	1	no	—
131	Q₈	—	316	1	no	—
132	Q₈	—	316	1	no	—
133	D₄	—	2	1	no	—
134	D₄	—	2	1	no	—
135	D₄	—	316	1	no	—
136	D₄	—	316	1	no	—
137	D₄	—	316	1	no	—
138	D₄	—	316	1	no	—
139	D₄	—	632	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	316	1	no	—
144	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
145	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
146	C₂×C₄	#11929624	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	4	1	no	—
149	C₈	—	316	1	no	—
150	C₈	—	316	1	no	—
151	C₈	#11929609	1	2	yes	complemented
152	C₂²	—	4	1	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	632	1	no	—
156	C₂²	—	632	1	no	—
157	C₂²	—	632	1	no	—
158	C₂²	#14579135	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	632	1	no	—
162	C₄	C₂×#11929609	1	1	yes	characteristic
163	C₄	#14579186	1	1	yes	characteristic
164	C₄	#14579189	1	1	yes	characteristic
165	C₄	#14579194	1	1	yes	characteristic
166	C₂	—	8	1	no	—
167	C₂	—	1264	1	no	—
168	C₂	#19098758	1	1	yes	characteristic
169	C₂	#19098809	1	1	yes	characteristic
170	C₂	#19098826	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098936

C₂

yes

maximal, characteristic, complemented

#19098843

C₂

yes

maximal, characteristic, complemented

#19098758

C₂

yes

maximal, characteristic

#19098316

C₂

yes

maximal, characteristic, complemented

#19098310

C₂

yes

maximal, characteristic, complemented

#19098306

C₂

yes

maximal, characteristic, complemented

#436×C₇₉

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

#14579158

C₂²

yes

characteristic

#14579094

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

#14579082

—

#14579081

—

C₂×M₁₆×C₇₉

—

C₄×D₄×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₂²⋊C₈×C₇₉

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

#11929613

—

#11929613

—

#11929609

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂×D₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

D₄

yes

characteristic

C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

C₂²×D₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×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

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

D₈

yes

complemented

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#121

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×D₈

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

#1685

—

maximal, Sylow

C₇₉

#1685

yes

characteristic, complemented, Sylow

—

—

—

D₇₉

yes

characteristic, complemented

#335

—

#332

—

#331

—

C₂×SD₁₆

—

158

—

C₂×SD₁₆

—

158

—

C₂×D₈

—

158

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

100

C₄×D₄

C₂×D₇₉

yes

characteristic

101

#107

—

158

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

C₂×D₇₉

yes

characteristic

105

#102

—

158

—

106

#102

—

158

—

107

C₂²⋊C₈

—

108

C₄×C₈

C₂×D₇₉

yes

characteristic

109

C₂×Q₈

—

158

—

110

C₂×D₄

—

158

—

111

C₂×D₄

—

158

—

112

C₂×D₄

—

158

—

113

C₂×D₄

D₃₁₆

yes

characteristic

114

C₂²×C₄

—

115

SD₁₆

—

316

—

116

SD₁₆

—

316

—

117

D₈

—

316

—

118

M₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

158

—

121

C₂×C₈

—

158

—

122

C₂×C₈

C₂²⋊D₇₉

yes

characteristic

123

C₂×C₈

C₂²⋊D₇₉

yes

characteristic

124

C₄⋊C₄

—

158

—

125

C₄⋊C₄

—

158

—

126

C₄⋊C₄

D₃₁₆

yes

characteristic

127

C₂²⋊C₄

—

128

C₄²

C₂²×D₇₉

yes

characteristic

129

C₂³

—

130

C₂³

—

316

—

131

Q₈

—

316

—

132

Q₈

—

316

—

133

D₄

—

134

D₄

—

135

D₄

—

316

—

136

D₄

—

316

—

137

D₄

—

316

—

138

D₄

—

316

—

139

D₄

—

632

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

316

—

144

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

145

C₂×C₄

C₂×D₃₁₆

yes

characteristic

146

C₂×C₄

#11929624

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

316

—

150

C₈

—

316

—

151

C₈

#11929609

yes

complemented

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

632

—

156

C₂²

—

632

—

157

C₂²

—

632

—

158

C₂²

#14579135

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

632

—

162

C₄

C₂×#11929609

yes

characteristic

163

C₄

#14579186

yes

characteristic

164

C₄

#14579189

yes

characteristic

165

C₄

#14579194

yes

characteristic

166

C₂

—

167

C₂

—

1264

—

168

C₂

#19098758

yes

characteristic

169

C₂

#19098809

yes

characteristic

170

C₂

#19098826

yes

characteristic

	subgroup	G/H
center	C₂²	#14579135
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	#436×C₇₉	C₂
socle	C₂²×C₇₉	#121
prime radical	#19098758	C₂
hypercenter	#436	D₇₉
hypocenter	C₇₉	#1685
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #436

subgroup

G/H

center

C₂²

#14579135

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

#436×C₇₉

C₂

C₂²×C₇₉

C₂

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #436

N	H ≅ G/N	type
1. #19098936	C₂	semidirect
2. #19098843	C₂	semidirect
4. #19098316	C₂	semidirect
5. #19098310	C₂	semidirect
6. #19098306	C₂	semidirect
7. #436×C₇₉	C₂	semidirect
63. C₈×C₇₉	D₈	semidirect
86. C₇₉	#1685	semidirect
90. #436	D₇₉	semidirect
151. C₈	#11929609	semidirect

H ≅ G/N

type

1. #19098936

C₂

semidirect

2. #19098843

C₂

semidirect

4. #19098316

C₂

semidirect

5. #19098310

C₂

semidirect

6. #19098306

C₂

semidirect

7. #436×C₇₉

C₂

semidirect

63. C₈×C₇₉

D₈

semidirect

86. C₇₉

#1685

semidirect

90. #436

D₇₉

semidirect

151. C₈

#11929609

semidirect

Subgroups — #23762865

Conjugacy classes of subgroups

Complements