Group #23764159 — subgroups

Subgroups — #23764159

Order 10112.

256 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₂×#14579095	C₂	1	2	yes	maximal
2	#19098433	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
4	#19098326	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₇₉	—	2	1	no	—
7	C₂×C₄⋊Dic₇₉	—	2	1	no	—
8	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₇₉	—	2	1	no	—
12	#14579095	—	2	4	no	—
13	C₇₉⋊#95	—	2	1	no	—
14	C₂²×C₈×C₇₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₇₉	C₄	1	4	yes	complemented
17	C₂²×Dic₇₉	—	2	1	no	—
18	C₂²×Dic₇₉	—	2	1	no	—
19	C₂²×Dic₇₉	—	2	2	no	—
20	C₄⋊Dic₇₉	—	2	2	no	—
21	C₄⋊Dic₇₉	—	2	1	no	—
22	C₄⋊Dic₇₉	—	2	1	no	—
23	C₄⋊Dic₇₉	—	2	2	no	—
24	C₄⋊Dic₇₉	—	4	2	no	—
25	C₄⋊Dic₇₉	—	4	2	no	—
26	C₄⋊Dic₇₉	D₄	1	2	yes	—
27	C₄⋊Dic₇₉	Q₈	1	2	yes	—
28	C₄×Dic₇₉	—	4	2	no	—
29	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₇₉	—	2	2	no	—
33	C₂×C₈×C₇₉	—	2	2	no	—
34	C₂×C₈×C₇₉	C₂×C₄	1	4	yes	—
35	C₄²×C₇₉	C₂×C₄	1	2	yes	—
36	C₄²×C₇₉	Q₈	1	2	yes	—
37	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₉	—	2	1	no	—
39	C₂×C₄×C₇₉	—	2	1	no	—
40	C₂×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₇₉	D₈	1	2	yes	—
43	C₂×C₄×C₇₉	Q₁₆	1	2	yes	—
44	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
49	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
50	C₂×Dic₇₉	—	2	2	no	—
51	C₂×Dic₇₉	—	2	2	no	—
52	C₂×Dic₇₉	—	4	1	no	—
53	C₂×Dic₇₉	—	4	1	no	—
54	C₂×Dic₇₉	—	4	1	no	—
55	C₂×Dic₇₉	—	4	1	no	—
56	C₂×Dic₇₉	—	4	2	no	—
57	C₂×Dic₇₉	—	4	2	no	—
58	C₂×Dic₇₉	—	4	2	no	—
59	C₈×C₇₉	—	2	4	no	—
60	C₂²×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₇₉	#121	1	1	yes	characteristic
62	C₂²×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
64	C₂²×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
65	C₂²×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₇₉	#137	1	1	yes	characteristic
67	C₄×C₇₉	—	2	2	no	—
68	C₄×C₇₉	#107	1	4	yes	complemented
69	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
70	C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
71	C₄×C₇₉	#125	1	1	yes	characteristic
72	C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
73	Dic₇₉	—	4	2	no	—
74	Dic₇₉	—	4	2	no	—
75	Dic₇₉	—	8	2	no	—
76	C₂×C₇₉	#389	1	1	yes	characteristic
77	C₂×C₇₉	C₂×#107	1	1	yes	characteristic
78	C₂×C₇₉	#428	1	1	yes	characteristic
79	C₂×C₇₉	#459	1	1	yes	characteristic
80	C₂×C₇₉	#462	1	1	yes	characteristic
81	C₂×C₇₉	#474	1	1	yes	characteristic
82	C₂×C₇₉	#477	1	1	yes	characteristic
83	#1947	—	79	1	no	maximal, Sylow
84	C₇₉	#1947	1	1	yes	characteristic, complemented, Sylow
85	C₂×#107	—	79	2	no	—
86	C₂×C₄⋊C₈	D₇₉	1	1	yes	characteristic
87	#384	—	79	1	no	—
88	C₄×C₄⋊C₄	—	79	1	no	—
89	#340	—	79	2	no	—
90	C₂²×C₈	C₂×D₇₉	1	2	yes	—
91	C₂×C₄⋊C₄	—	79	1	no	—
92	C₂×C₄⋊C₄	—	79	1	no	—
93	C₂×C₄⋊C₄	—	79	2	no	—
94	C₂×C₄⋊C₄	—	158	1	no	—
95	C₂×C₄⋊C₄	—	158	1	no	—
96	C₂×C₄²	—	158	1	no	—
97	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
98	#107	—	158	4	no	—
99	C₄⋊C₈	Dic₇₉	1	4	yes	complemented
100	#95	—	158	1	no	—
101	C₂²×C₄	—	158	2	no	—
102	C₂²×C₄	—	158	1	no	—
103	C₂²×C₄	—	158	1	no	—
104	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
105	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	C₂×Dic₇₉	1	4	yes	—
110	C₄⋊C₄	—	316	2	no	—
111	C₄⋊C₄	—	316	2	no	—
112	C₄⋊C₄	—	79	2	no	—
113	C₄⋊C₄	—	79	2	no	—
114	C₄⋊C₄	—	158	2	no	—
115	C₄⋊C₄	—	158	2	no	—
116	C₄⋊C₄	—	158	1	no	—
117	C₄⋊C₄	—	158	1	no	—
118	C₄²	—	316	2	no	—
119	C₄²	Dic₁₅₈	1	2	yes	—
120	C₄²	C₂×Dic₇₉	1	2	yes	—
121	C₂³	C₂×D₃₁₆	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	316	1	no	—
125	C₂×C₄	—	316	1	no	—
126	C₂×C₄	—	316	1	no	—
127	C₂×C₄	—	316	1	no	—
128	C₂×C₄	—	316	2	no	—
129	C₂×C₄	—	316	2	no	—
130	C₂×C₄	—	316	2	no	—
131	C₂×C₄	—	158	2	no	—
132	C₂×C₄	—	158	2	no	—
133	C₂×C₄	C₄⋊Dic₇₉	1	2	yes	—
134	C₂×C₄	C₄⋊Dic₇₉	1	2	yes	—
135	C₂×C₄	D₆₃₂	1	2	yes	—
136	C₂×C₄	Dic₃₁₆	1	2	yes	—
137	C₂×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
138	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
139	C₂×C₄	Q₈×D₇₉	1	1	yes	characteristic
140	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
141	C₂×C₄	#11929623	1	1	yes	characteristic
142	C₂×C₄	#11929625	1	1	yes	characteristic
143	C₈	—	2	4	no	—
144	C₂²	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
145	C₂²	#14579136	1	1	yes	characteristic
146	C₂²	#14579141	1	1	yes	characteristic
147	C₂²	C₂×D₆₃₂	1	1	yes	characteristic
148	C₂²	C₂×Dic₃₁₆	1	1	yes	characteristic
149	C₂²	#14579189	1	1	yes	characteristic
150	C₂²	#14579194	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	316	2	no	—
153	C₄	—	316	2	no	—
154	C₄	—	632	2	no	—
155	C₄	#14579095	1	4	yes	complemented
156	C₄	D₄×Dic₇₉	1	1	yes	characteristic
157	C₄	Q₈×Dic₇₉	1	1	yes	characteristic
158	C₄	#14579152	1	1	yes	characteristic
159	C₄	#14579160	1	1	yes	characteristic
160	C₂	#19098462	1	1	yes	characteristic
161	C₂	C₂×#14579095	1	1	yes	characteristic
162	C₂	#19098605	1	1	yes	characteristic
163	C₂	#19098759	1	1	yes	characteristic
164	C₂	#19098774	1	1	yes	characteristic
165	C₂	#19098844	1	1	yes	characteristic
166	C₂	#19098859	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579095

C₂

yes

maximal

#19098433

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₇₉

C₂

yes

maximal, characteristic

#19098326

C₂

yes

maximal

C₂×C₄⋊C₈×C₇₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄×Dic₇₉

—

#14579095

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₄

yes

complemented

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

—

C₄⋊Dic₇₉

Q₈

yes

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₄²×C₇₉

C₂×C₄

yes

—

C₄²×C₇₉

Q₈

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₄⋊C₄

yes

—

C₂×C₄×C₇₉

D₈

yes

—

C₂×C₄×C₇₉

Q₁₆

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₂²×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₉

#121

yes

characteristic

C₂²×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₂²×C₇₉

C₂×Q₁₆

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

#107

yes

complemented

C₄×C₇₉

C₄×D₄

yes

characteristic

C₄×C₇₉

C₄×Q₈

yes

characteristic

C₄×C₇₉

#125

yes

characteristic

C₄×C₇₉

C₄⋊Q₈

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×C₄⋊C₈

D₇₉

yes

characteristic

#384

—

C₄×C₄⋊C₄

—

#340

—

C₂²×C₈

C₂×D₇₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄²

—

158

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

#107

—

158

—

C₄⋊C₈

Dic₇₉

yes

complemented

100

#95

—

158

—

101

C₂²×C₄

—

158

—

102

C₂²×C₄

—

158

—

103

C₂²×C₄

—

158

—

104

C₂²×C₄

D₃₁₆

yes

characteristic

105

C₂²×C₄

D₃₁₆

yes

characteristic

106

C₂²×C₄

C₂²×D₇₉

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂×Dic₇₉

yes

—

110

C₄⋊C₄

—

316

—

111

C₄⋊C₄

—

316

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

158

—

115

C₄⋊C₄

—

158

—

116

C₄⋊C₄

—

158

—

117

C₄⋊C₄

—

158

—

118

C₄²

—

316

—

119

C₄²

Dic₁₅₈

yes

—

120

C₄²

C₂×Dic₇₉

yes

—

121

C₂³

C₂×D₃₁₆

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

316

—

125

C₂×C₄

—

316

—

126

C₂×C₄

—

316

—

127

C₂×C₄

—

316

—

128

C₂×C₄

—

316

—

129

C₂×C₄

—

316

—

130

C₂×C₄

—

316

—

131

C₂×C₄

—

158

—

132

C₂×C₄

—

158

—

133

C₂×C₄

C₄⋊Dic₇₉

yes

—

134

C₂×C₄

C₄⋊Dic₇₉

yes

—

135

C₂×C₄

D₆₃₂

yes

—

136

C₂×C₄

Dic₃₁₆

yes

—

137

C₂×C₄

C₂²×Dic₇₉

yes

characteristic

138

C₂×C₄

D₄×D₇₉

yes

characteristic

139

C₂×C₄

Q₈×D₇₉

yes

characteristic

140

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

141

C₂×C₄

#11929623

yes

characteristic

142

C₂×C₄

#11929625

yes

characteristic

143

C₈

—

144

C₂²

C₂×C₄⋊Dic₇₉

yes

characteristic

145

C₂²

#14579136

yes

characteristic

146

C₂²

#14579141

yes

characteristic

147

C₂²

C₂×D₆₃₂

yes

characteristic

148

C₂²

C₂×Dic₃₁₆

yes

characteristic

149

C₂²

#14579189

yes

characteristic

150

C₂²

#14579194

yes

characteristic

151

C₄

—

152

C₄

—

316

—

153

C₄

—

316

—

154

C₄

—

632

—

155

C₄

#14579095

yes

complemented

156

C₄

D₄×Dic₇₉

yes

characteristic

157

C₄

Q₈×Dic₇₉

yes

characteristic

158

C₄

#14579152

yes

characteristic

159

C₄

#14579160

yes

characteristic

160

C₂

#19098462

yes

characteristic

161

C₂

C₂×#14579095

yes

characteristic

162

C₂

#19098605

yes

characteristic

163

C₂

#19098759

yes

characteristic

164

C₂

#19098774

yes

characteristic

165

C₂

#19098844

yes

characteristic

166

C₂

#19098859

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₂×C₄⋊C₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₇₉
hypocenter	C₇₉	#1947
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₃₁₆

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₂×C₄⋊C₈×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

cosocle

C₂²×C₄×C₇₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₇₉

hypocenter

C₇₉

#1947

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
16. C₄⋊C₈×C₇₉	C₄	semidirect
68. C₄×C₇₉	#107	semidirect
84. C₇₉	#1947	semidirect
99. C₄⋊C₈	Dic₇₉	semidirect
155. C₄	#14579095	semidirect

H ≅ G/N

type

16. C₄⋊C₈×C₇₉

C₄

semidirect

68. C₄×C₇₉

#107

semidirect

84. C₇₉

#1947

semidirect

99. C₄⋊C₈

Dic₇₉

semidirect

155. C₄

#14579095

semidirect

Subgroups — #23764159

Conjugacy classes of subgroups

Complements