Group #23762534 — subgroups

Subgroups — #23762534

Order 10112.

214 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 61 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	#19098942	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098858	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098853	C₂	1	1	yes	maximal, characteristic
4	#19098312	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	#554×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#11929614	—	2	2	no	—
9	#14579161	C₂²	1	1	yes	characteristic
10	#14579094	—	2	1	no	—
11	#14579092	—	2	1	no	—
12	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
14	#14579085	—	2	1	no	—
15	#14579082	—	2	1	no	—
16	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
18	#125×C₇₉	—	2	1	no	—
19	C₂²⋊Q₈×C₇₉	—	2	1	no	—
20	C₂²⋊Q₈×C₇₉	—	2	2	no	—
21	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₄×D₄×C₇₉	—	2	1	no	—
23	C₄×D₄×C₇₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
25	C₂×Dic₁₅₈	—	2	1	no	—
26	#11929614	—	4	4	no	—
27	C₂×C₇₉⋊C₈	—	2	1	no	—
28	C₂×C₇₉⋊C₈	—	2	1	no	—
29	C₂×C₇₉⋊C₈	—	2	1	no	—
30	C₄⋊Dic₇₉	—	2	1	no	—
31	C₄⋊Dic₇₉	—	2	1	no	—
32	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
33	C₂×Q₈×C₇₉	D₄	1	2	yes	—
34	C₂×D₄×C₇₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	—	2	2	no	—
36	C₂²×C₄×C₇₉	—	2	1	no	—
37	C₄⋊C₄×C₇₉	—	2	2	no	—
38	C₄⋊C₄×C₇₉	—	2	2	no	—
39	C₄⋊C₄×C₇₉	—	2	1	no	—
40	C₄⋊C₄×C₇₉	—	2	1	no	—
41	C₄⋊C₄×C₇₉	—	2	1	no	—
42	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
43	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
44	C₂²⋊C₄×C₇₉	—	2	2	no	—
45	C₂²⋊C₄×C₇₉	—	2	1	no	—
46	C₄²×C₇₉	—	2	1	no	—
47	C₄²×C₇₉	C₂³	1	1	yes	characteristic
48	C₂³×C₇₉	—	2	1	no	—
49	Q₈×C₇₉	—	2	1	no	—
50	Q₈×C₇₉	—	2	1	no	—
51	Q₈×C₇₉	—	4	2	no	—
52	D₄×C₇₉	—	2	1	no	—
53	D₄×C₇₉	Q₁₆	1	2	yes	complemented
54	C₂×C₄×C₇₉	—	2	1	no	—
55	C₂×C₄×C₇₉	—	2	1	no	—
56	C₂×C₄×C₇₉	—	2	1	no	—
57	C₂×C₄×C₇₉	—	2	2	no	—
58	C₂×C₄×C₇₉	—	4	1	no	—
59	C₂×C₄×C₇₉	—	4	1	no	—
60	C₂×C₄×C₇₉	—	8	2	no	—
61	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
64	C₂×Dic₇₉	—	4	1	no	—
65	Dic₁₅₈	—	4	2	no	—
66	C₇₉⋊C₈	—	2	2	no	—
67	C₇₉⋊C₈	—	4	1	no	—
68	C₇₉⋊C₈	—	4	1	no	—
69	C₂²×C₇₉	—	2	2	no	—
70	C₂²×C₇₉	—	4	1	no	—
71	C₂²×C₇₉	—	4	1	no	—
72	C₂²×C₇₉	#120	1	1	yes	characteristic
73	C₄×C₇₉	—	2	1	no	—
74	C₄×C₇₉	—	4	1	no	—
75	C₄×C₇₉	—	4	1	no	—
76	C₄×C₇₉	—	4	1	no	—
77	C₄×C₇₉	—	4	2	no	—
78	C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
79	C₄×C₇₉	#135	1	1	yes	characteristic
80	C₄×C₇₉	#137	1	1	yes	characteristic
81	C₄×C₇₉	#137	1	1	yes	characteristic
82	Dic₇₉	—	8	1	no	—
83	C₂×C₇₉	—	4	2	no	—
84	C₂×C₇₉	#451	1	1	yes	characteristic
85	C₂×C₇₉	#452	1	1	yes	characteristic
86	C₂×C₇₉	#453	1	1	yes	characteristic
87	#1632	—	79	1	no	maximal, Sylow
88	C₇₉	#1632	1	1	yes	characteristic, complemented, Sylow
89	#554	D₇₉	1	1	yes	characteristic
90	#494	—	79	1	no	—
91	#477	—	79	1	no	—
92	#476	—	79	1	no	—
93	#333	—	79	1	no	—
94	Q₈⋊C₈	—	79	1	no	—
95	D₄⋊C₈	—	79	1	no	—
96	C₂×Q₁₆	—	158	2	no	—
97	C₄⋊Q₈	—	79	1	no	—
98	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
99	#125	—	2	1	no	—
100	C₂²⋊Q₈	—	2	1	no	—
101	C₂²⋊Q₈	—	2	2	no	—
102	C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
103	C₄×D₄	—	2	1	no	—
104	C₄×D₄	C₂×D₇₉	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	2	1	no	—
106	#107	—	158	1	no	—
107	#106	—	158	1	no	—
108	C₄⋊C₈	—	79	1	no	—
109	C₄⋊C₈	—	79	1	no	—
110	#103	—	158	1	no	—
111	#102	—	158	1	no	—
112	C₄×C₈	—	79	1	no	—
113	C₂×Q₈	—	158	1	no	—
114	C₂×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
115	C₂×Q₈	C₂²⋊D₇₉	1	2	yes	—
116	C₂×D₄	C₂²⋊D₇₉	1	1	yes	characteristic
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	2	1	no	—
119	Q₁₆	—	316	4	no	—
120	C₂×C₈	—	158	1	no	—
121	C₂×C₈	—	158	1	no	—
122	C₂×C₈	—	158	1	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	158	1	no	—
129	C₄⋊C₄	—	158	1	no	—
130	C₄⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
131	C₄⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
132	C₂²⋊C₄	—	2	2	no	—
133	C₂²⋊C₄	—	2	1	no	—
134	C₄²	—	2	1	no	—
135	C₄²	C₂²×D₇₉	1	1	yes	characteristic
136	C₂³	—	2	1	no	—
137	Q₈	—	2	1	no	—
138	Q₈	—	2	1	no	—
139	Q₈	—	4	2	no	—
140	Q₈	—	316	2	no	—
141	D₄	—	2	1	no	—
142	D₄	#11929614	1	2	yes	complemented
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	316	1	no	—
151	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
152	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
153	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
154	C₈	—	316	1	no	—
155	C₈	—	316	1	no	—
156	C₈	—	158	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	4	1	no	—
159	C₂²	—	4	1	no	—
160	C₂²	#14579130	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	4	1	no	—
163	C₄	—	4	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	4	2	no	—
166	C₄	—	632	1	no	—
167	C₄	C₂×#11929614	1	1	yes	characteristic
168	C₄	#14579181	1	1	yes	characteristic
169	C₄	#14579191	1	1	yes	characteristic
170	C₄	#14579192	1	1	yes	characteristic
171	C₂	—	4	2	no	—
172	C₂	#19098719	1	1	yes	characteristic
173	C₂	#19098725	1	1	yes	characteristic
174	C₂	#19098731	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098942

C₂

yes

maximal, characteristic, complemented

#19098858

C₂

yes

maximal, characteristic, complemented

#19098853

C₂

yes

maximal, characteristic

#19098312

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₇₉⋊C₈)

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₇₉⋊C₈)

C₂

yes

maximal, characteristic

#554×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929614

—

#14579161

C₂²

yes

characteristic

#14579094

—

#14579092

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

#14579085

—

#14579082

—

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

#125×C₇₉

—

C₂²⋊Q₈×C₇₉

—

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

C₂²

yes

characteristic

C₄×D₄×C₇₉

—

C₄×D₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

—

C₂×Dic₁₅₈

—

#11929614

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂×Q₈×C₇₉

D₄

yes

characteristic

C₂×Q₈×C₇₉

D₄

yes

—

C₂×D₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

Q₁₆

yes

complemented

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×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

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

Dic₁₅₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#120

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×Q₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#554

D₇₉

yes

—

—

—

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₂×Q₁₆

—

158

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₉

yes

characteristic

#125

—

100

C₂²⋊Q₈

—

101

C₂²⋊Q₈

—

102

C₄×Q₈

C₂×D₇₉

yes

characteristic

103

C₄×D₄

—

104

C₄×D₄

C₂×D₇₉

yes

characteristic

105

C₂×C₄⋊C₄

—

106

#107

—

158

—

107

#106

—

158

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

#103

—

158

—

111

#102

—

158

—

112

C₄×C₈

—

113

C₂×Q₈

—

158

—

114

C₂×Q₈

C₂²⋊D₇₉

yes

characteristic

115

C₂×Q₈

C₂²⋊D₇₉

yes

—

116

C₂×D₄

C₂²⋊D₇₉

yes

characteristic

117

C₂²×C₄

—

118

C₂²×C₄

—

119

Q₁₆

—

316

—

120

C₂×C₈

—

158

—

121

C₂×C₈

—

158

—

122

C₂×C₈

—

158

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

158

—

129

C₄⋊C₄

—

158

—

130

C₄⋊C₄

C₂²⋊D₇₉

yes

characteristic

131

C₄⋊C₄

C₂²⋊D₇₉

yes

characteristic

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

—

135

C₄²

C₂²×D₇₉

yes

characteristic

136

C₂³

—

137

Q₈

—

138

Q₈

—

139

Q₈

—

140

Q₈

—

316

—

141

D₄

—

142

D₄

#11929614

yes

complemented

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

316

—

151

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

152

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

153

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

154

C₈

—

316

—

155

C₈

—

316

—

156

C₈

—

158

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

#14579130

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

632

—

167

C₄

C₂×#11929614

yes

characteristic

168

C₄

#14579181

yes

characteristic

169

C₄

#14579191

yes

characteristic

170

C₄

#14579192

yes

characteristic

171

C₂

—

172

C₂

#19098719

yes

characteristic

173

C₂

#19098725

yes

characteristic

174

C₂

#19098731

yes

characteristic

	subgroup	G/H
center	C₂²	#14579130
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	#554×C₇₉	C₂
socle	C₂²×C₇₉	#120
prime radical	C₂×D₄×C₇₉	D₄
hypercenter	#554	D₇₉
hypocenter	C₇₉	#1632
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #554

subgroup

G/H

center

C₂²

#14579130

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

#554×C₇₉

C₂

socle

C₂²×C₇₉

#120

prime radical

C₂×D₄×C₇₉

D₄

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #554

N	H ≅ G/N	type
1. #19098942	C₂	semidirect
2. #19098858	C₂	semidirect
4. #19098312	C₂	semidirect
5. Q₈⋊(C₇₉⋊C₈)	C₂	semidirect
53. D₄×C₇₉	Q₁₆	semidirect
88. C₇₉	#1632	semidirect
142. D₄	#11929614	semidirect

H ≅ G/N

type

1. #19098942

C₂

semidirect

2. #19098858

C₂

semidirect

4. #19098312

C₂

semidirect

5. Q₈⋊(C₇₉⋊C₈)

C₂

semidirect

53. D₄×C₇₉

Q₁₆

semidirect

88. C₇₉

#1632

semidirect

142. D₄

#11929614

semidirect

Subgroups — #23762534

Conjugacy classes of subgroups

Complements