Group #23775822 — subgroups

Subgroups — #23775822

Order 10112.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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	#19099279	C₂	1	2	yes	maximal, complemented
2	#19098890	C₂	1	4	yes	maximal, complemented
3	#19098850	C₂	1	2	yes	maximal, complemented
4	#19098800	C₂	1	2	yes	maximal, complemented
5	C₄×D₆₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×D₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#19098600	C₂	1	2	yes	maximal, complemented
8	#500×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₇₉	C₂²	1	2	yes	—
11	#14579150	—	2	4	no	—
12	#14579136	—	2	2	no	—
13	#14579136	C₂²	1	4	yes	—
14	C₄×D₃₁₆	C₂²	1	2	yes	—
15	C₄×D₃₁₆	C₂²	1	1	yes	characteristic
16	#14579119	—	2	2	no	—
17	#14579119	C₂²	1	4	yes	—
18	#14579095	C₂²	1	1	yes	characteristic
19	#14579092	C₂²	1	4	yes	—
20	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
21	#14579083	C₂²	1	2	yes	—
22	#14579081	C₂²	1	4	yes	—
23	(C₂×D₇₉)⋊C₈	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
25	#107×C₇₉	C₂²	1	2	yes	—
26	#107×C₇₉	C₂²	1	2	yes	—
27	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×D₃₁₆	—	2	4	no	—
29	C₂×D₃₁₆	D₄	1	1	yes	characteristic
30	C₂×D₃₁₆	C₂³	1	2	yes	—
31	C₂×C₄×D₇₉	—	2	2	no	—
32	C₂×C₄×D₇₉	C₂³	1	2	yes	—
33	D₆₃₂	—	2	2	no	—
34	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
35	C₈×D₇₉	—	2	4	no	—
36	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
37	C₄⋊Dic₇₉	C₂³	1	2	yes	—
38	(C₂×D₇₉)⋊C₄	—	2	4	no	—
39	(C₂×D₇₉)⋊C₄	—	2	4	no	—
40	(C₂×D₇₉)⋊C₄	—	2	2	no	—
41	(C₂×D₇₉)⋊C₄	D₄	1	2	yes	—
42	C₄×Dic₇₉	—	2	4	no	—
43	C₂×Q₈×C₇₉	—	2	2	no	—
44	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₇₉	—	2	2	no	—
47	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
48	C₄²×C₇₉	C₂³	1	1	yes	characteristic
49	C₂²×D₇₉	—	2	2	no	—
50	C₂²×D₇₉	C₂×D₄	1	2	yes	—
51	Q₈×C₇₉	—	8	2	no	—
52	C₂×C₄×C₇₉	—	2	4	no	—
53	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₇₉	—	2	2	no	—
57	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
58	D₃₁₆	—	2	2	no	—
59	D₃₁₆	—	2	2	no	—
60	D₃₁₆	—	2	2	no	—
61	D₃₁₆	—	8	4	no	—
62	C₄×D₇₉	—	2	2	no	—
63	C₄×D₇₉	—	2	2	no	—
64	C₄×D₇₉	—	4	4	no	—
65	C₈×C₇₉	—	2	1	no	—
66	C₈×C₇₉	#41	1	2	yes	—
67	C₇₉⋊C₈	—	2	2	no	—
68	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
69	C₂×D₇₉	—	2	2	no	—
70	C₂×D₇₉	—	2	2	no	—
71	C₂×D₇₉	—	2	2	no	—
72	C₂×D₇₉	—	4	2	no	—
73	C₂×D₇₉	—	4	2	no	—
74	C₂×D₇₉	—	4	2	no	—
75	C₄×C₇₉	—	2	1	no	—
76	C₄×C₇₉	—	4	4	no	—
77	C₄×C₇₉	#135	1	2	yes	—
78	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
79	C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
80	Dic₇₉	—	2	2	no	—
81	Dic₇₉	—	4	2	no	—
82	C₂×C₇₉	#547	1	1	yes	characteristic
83	C₂×C₇₉	C₂×#135	1	1	yes	characteristic
84	C₂×C₇₉	#576	1	1	yes	characteristic
85	D₇₉	—	4	2	no	—
86	D₇₉	—	8	2	no	—
87	#3341	—	79	1	no	maximal, Sylow
88	C₇₉	#3341	1	1	yes	characteristic, complemented, Sylow
89	#555	—	79	2	no	—
90	#500	D₇₉	1	1	yes	characteristic, complemented
91	#482	—	79	4	no	—
92	#476	—	79	2	no	—
93	#466	—	79	2	no	—
94	C₄×D₈	—	79	1	no	—
95	C₈×D₄	—	79	1	no	—
96	#427	—	79	2	no	—
97	C₂×D₈	—	79	1	no	—
98	C₂²×C₈	—	79	2	no	—
99	C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
100	#124	—	158	4	no	—
101	#121	—	79	4	no	—
102	#121	—	158	2	no	—
103	C₄×D₄	—	79	1	no	—
104	C₄×D₄	—	79	2	no	—
105	#117	—	79	4	no	—
106	#117	—	158	2	no	—
107	#107	—	79	1	no	—
108	#107	C₂×D₇₉	1	2	yes	—
109	#107	C₂×D₇₉	1	2	yes	—
110	#106	—	79	4	no	—
111	C₄⋊C₈	—	79	1	no	—
112	#102	—	79	2	no	—
113	#102	—	79	4	no	—
114	C₂²⋊C₈	—	79	2	no	—
115	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
116	C₂×Q₈	—	2	2	no	—
117	C₂×D₄	—	79	1	no	—
118	C₂×D₄	—	79	2	no	—
119	C₂×D₄	—	158	4	no	—
120	C₂²×C₄	—	79	2	no	—
121	C₂²×C₄	—	158	2	no	—
122	D₈	—	158	2	no	—
123	C₂×C₈	—	79	2	no	—
124	C₂×C₈	—	158	4	no	—
125	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
126	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	79	1	no	—
129	C₄⋊C₄	—	79	2	no	—
130	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
131	C₂²⋊C₄	—	79	2	no	—
132	C₂²⋊C₄	—	158	4	no	—
133	C₂²⋊C₄	—	158	4	no	—
134	C₂²⋊C₄	—	158	2	no	—
135	C₄²	—	158	4	no	—
136	C₄²	C₂²×D₇₉	1	1	yes	characteristic
137	C₂³	—	79	2	no	—
138	C₂³	—	158	2	no	—
139	Q₈	—	8	2	no	—
140	D₄	—	632	4	no	—
141	D₄	—	158	2	no	—
142	D₄	—	158	2	no	—
143	D₄	—	158	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	316	4	no	—
146	C₂×C₄	—	79	2	no	—
147	C₂×C₄	—	158	2	no	—
148	C₂×C₄	—	158	2	no	—
149	C₂×C₄	—	158	2	no	—
150	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
151	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
152	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	158	2	no	—
155	C₈	#11929625	1	2	yes	—
156	C₂²	—	316	2	no	—
157	C₂²	—	316	2	no	—
158	C₂²	—	316	2	no	—
159	C₂²	—	158	2	no	—
160	C₂²	—	158	2	no	—
161	C₂²	—	158	2	no	—
162	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	4	no	—
165	C₄	—	316	2	no	—
166	C₄	—	158	2	no	—
167	C₄	#14579184	1	2	yes	—
168	C₄	C₂×#11929625	1	1	yes	characteristic
169	C₄	#14579209	1	1	yes	characteristic
170	C₂	—	316	2	no	—
171	C₂	—	632	2	no	—
172	C₂	#19099216	1	1	yes	characteristic
173	C₂	C₂×#14579184	1	1	yes	characteristic
174	C₂	#19099364	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099279

C₂

yes

maximal, complemented

#19098890

C₂

yes

maximal, complemented

#19098850

C₂

yes

maximal, complemented

#19098800

C₂

yes

maximal, complemented

C₄×D₆₃₂

C₂

yes

maximal, characteristic, complemented

C₈×D₃₁₆

C₂

yes

maximal, characteristic, complemented

#19098600

C₂

yes

maximal, complemented

#500×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×D₆₃₂

C₂²

yes

characteristic

C₂×C₈×D₇₉

C₂²

yes

—

#14579150

—

#14579136

—

#14579136

C₂²

yes

—

C₄×D₃₁₆

C₂²

yes

—

C₄×D₃₁₆

C₂²

yes

characteristic

#14579119

—

#14579119

C₂²

yes

—

#14579095

C₂²

yes

characteristic

#14579092

C₂²

yes

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

#14579083

C₂²

yes

—

#14579081

C₂²

yes

—

(C₂×D₇₉)⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₇₉

C₂²

yes

—

#107×C₇₉

C₂²

yes

—

#107×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×D₃₁₆

—

C₂×D₃₁₆

D₄

yes

characteristic

C₂×D₃₁₆

C₂³

yes

—

C₂×C₄×D₇₉

—

C₂×C₄×D₇₉

C₂³

yes

—

D₆₃₂

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₈×D₇₉

—

C₄⋊Dic₇₉

D₄

yes

characteristic

C₄⋊Dic₇₉

C₂³

yes

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

D₄

yes

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

C₂×D₄

yes

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₄

yes

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

C₄×D₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#41

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#135

yes

—

C₄×C₇₉

C₂×#41

yes

characteristic

C₄×C₇₉

2₋¹⁺⁴

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

D₇₉

—

#3341

—

maximal, Sylow

C₇₉

#3341

yes

characteristic, complemented, Sylow

#555

—

#500

D₇₉

yes

characteristic, complemented

#482

—

#476

—

#466

—

C₄×D₈

—

C₈×D₄

—

#427

—

C₂×D₈

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₇₉

yes

—

100

#124

—

158

—

101

#121

—

102

#121

—

158

—

103

C₄×D₄

—

104

C₄×D₄

—

105

#117

—

106

#117

—

158

—

107

#107

—

108

#107

C₂×D₇₉

yes

—

109

#107

C₂×D₇₉

yes

—

110

#106

—

111

C₄⋊C₈

—

112

#102

—

113

#102

—

114

C₂²⋊C₈

—

115

C₄×C₈

C₂×D₇₉

yes

characteristic

116

C₂×Q₈

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

158

—

120

C₂²×C₄

—

121

C₂²×C₄

—

158

—

122

D₈

—

158

—

123

C₂×C₈

—

124

C₂×C₈

—

158

—

125

C₂×C₈

C₂²×D₇₉

yes

characteristic

126

C₂×C₈

C₂²×D₇₉

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

C₂²×D₇₉

yes

—

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

158

—

133

C₂²⋊C₄

—

158

—

134

C₂²⋊C₄

—

158

—

135

C₄²

—

158

—

136

C₄²

C₂²×D₇₉

yes

characteristic

137

C₂³

—

138

C₂³

—

158

—

139

Q₈

—

140

D₄

—

632

—

141

D₄

—

158

—

142

D₄

—

158

—

143

D₄

—

158

—

144

C₂×C₄

—

145

C₂×C₄

—

316

—

146

C₂×C₄

—

147

C₂×C₄

—

158

—

148

C₂×C₄

—

158

—

149

C₂×C₄

—

158

—

150

C₂×C₄

D₄×D₇₉

yes

characteristic

151

C₂×C₄

D₄×D₇₉

yes

characteristic

152

C₂×C₄

C₂³×D₇₉

yes

characteristic

153

C₈

—

154

C₈

—

158

—

155

C₈

#11929625

yes

—

156

C₂²

—

316

—

157

C₂²

—

316

—

158

C₂²

—

316

—

159

C₂²

—

158

—

160

C₂²

—

158

—

161

C₂²

—

158

—

162

C₂²

C₂×D₄×D₇₉

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

316

—

166

C₄

—

158

—

167

C₄

#14579184

yes

—

168

C₄

C₂×#11929625

yes

characteristic

169

C₄

#14579209

yes

characteristic

170

C₂

—

316

—

171

C₂

—

632

—

172

C₂

#19099216

yes

characteristic

173

C₂

C₂×#14579184

yes

characteristic

174

C₂

#19099364

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#500×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
hypercenter	#500	D₇₉
hypocenter	C₇₉	#3341
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#500×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
1. #19099279	C₂	semidirect
2. #19098890	C₂	semidirect
3. #19098850	C₂	semidirect
4. #19098800	C₂	semidirect
5. C₄×D₆₃₂	C₂	semidirect
6. C₈×D₃₁₆	C₂	semidirect
7. #19098600	C₂	semidirect
8. #500×C₇₉	C₂	semidirect
88. C₇₉	#3341	semidirect
90. #500	D₇₉	semidirect

H ≅ G/N

type

1. #19099279

C₂

semidirect

2. #19098890

C₂

semidirect

3. #19098850

C₂

semidirect

4. #19098800

C₂

semidirect

5. C₄×D₆₃₂

C₂

semidirect

6. C₈×D₃₁₆

C₂

semidirect

7. #19098600

C₂

semidirect

8. #500×C₇₉

C₂

semidirect

88. C₇₉

#3341

semidirect

90. #500

D₇₉

semidirect

Subgroups — #23775822

Conjugacy classes of subgroups

Complements