Group #23764161 — subgroups

Subgroups — #23764161

Order 10112.

256 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 121 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×#14579092	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
4	#19098325	C₂	1	2	yes	maximal
5	#384×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₇₉	—	2	1	no	—
10	#14579092	—	2	4	no	—
11	C₄⋊(C₇₉⋊C₈)	C₄	1	4	yes	complemented
12	C₇₉⋊#95	—	2	1	no	—
13	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
15	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₇₉	—	2	1	no	—
18	C₂²×Dic₇₉	—	2	1	no	—
19	C₂×C₇₉⋊C₈	—	2	2	no	—
20	C₂×C₇₉⋊C₈	—	2	2	no	—
21	C₂×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
22	C₄⋊Dic₇₉	—	2	1	no	—
23	C₄⋊Dic₇₉	—	2	1	no	—
24	C₄⋊Dic₇₉	D₄	1	2	yes	—
25	C₄⋊Dic₇₉	Q₈	1	2	yes	—
26	C₄×Dic₇₉	—	4	2	no	—
27	C₂²×C₄×C₇₉	—	2	2	no	—
28	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₇₉	—	2	2	no	—
32	C₄⋊C₄×C₇₉	—	2	2	no	—
33	C₄⋊C₄×C₇₉	—	4	2	no	—
34	C₄⋊C₄×C₇₉	—	4	2	no	—
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	—	4	2	no	—
41	C₂×C₄×C₇₉	—	4	2	no	—
42	C₂×C₄×C₇₉	—	4	2	no	—
43	C₂×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₇₉	SD₁₆	1	2	yes	—
46	C₂×C₄×C₇₉	SD₁₆	1	2	yes	—
47	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
52	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
53	C₂×Dic₇₉	—	2	2	no	—
54	C₂×Dic₇₉	—	2	2	no	—
55	C₂×Dic₇₉	—	4	1	no	—
56	C₂×Dic₇₉	—	4	1	no	—
57	C₂×Dic₇₉	—	4	1	no	—
58	C₂×Dic₇₉	—	4	1	no	—
59	C₇₉⋊C₈	—	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×SD₁₆	1	1	yes	characteristic
64	C₂²×C₇₉	C₂×SD₁₆	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	—	8	2	no	—
69	C₄×C₇₉	#106	1	4	yes	complemented
70	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
71	C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
72	C₄×C₇₉	#125	1	1	yes	characteristic
73	C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
74	Dic₇₉	—	4	2	no	—
75	Dic₇₉	—	4	2	no	—
76	C₂×C₇₉	#389	1	1	yes	characteristic
77	C₂×C₇₉	C₂×#106	1	1	yes	characteristic
78	C₂×C₇₉	#428	1	1	yes	characteristic
79	C₂×C₇₉	#460	1	1	yes	characteristic
80	C₂×C₇₉	#461	1	1	yes	characteristic
81	C₂×C₇₉	#475	1	1	yes	characteristic
82	C₂×C₇₉	#476	1	1	yes	characteristic
83	#1948	—	79	1	no	maximal, Sylow
84	C₇₉	#1948	1	1	yes	characteristic, complemented, Sylow
85	C₂×#106	—	79	2	no	—
86	C₂×C₄⋊C₈	—	79	1	no	—
87	#384	D₇₉	1	1	yes	characteristic
88	C₄×C₄⋊C₄	—	79	1	no	—
89	#340	—	79	2	no	—
90	C₂²×C₈	—	79	2	no	—
91	C₂×C₄⋊C₄	—	2	1	no	—
92	C₂×C₄⋊C₄	—	2	1	no	—
93	C₂×C₄⋊C₄	—	79	1	no	—
94	C₂×C₄⋊C₄	—	79	1	no	—
95	C₂×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
96	C₂×C₄²	—	158	1	no	—
97	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
98	#106	—	158	4	no	—
99	C₄⋊C₈	—	79	4	no	—
100	#95	—	158	1	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	158	1	no	—
103	C₂²×C₄	—	158	1	no	—
104	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
107	C₂×C₈	—	79	4	no	—
108	C₂×C₈	—	158	2	no	—
109	C₂×C₈	—	158	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	79	2	no	—
115	C₄⋊C₄	—	79	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	C₄×D₇₉	1	2	yes	—
121	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	316	1	no	—
128	C₂×C₄	—	316	1	no	—
129	C₂×C₄	—	316	1	no	—
130	C₂×C₄	—	316	1	no	—
131	C₂×C₄	—	158	2	no	—
132	C₂×C₄	—	158	2	no	—
133	C₂×C₄	Dic₇₉⋊C₄	1	2	yes	—
134	C₂×C₄	Dic₇₉⋊C₄	1	2	yes	—
135	C₂×C₄	#11929611	1	2	yes	—
136	C₂×C₄	#11929612	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₇₉	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	—	158	4	no	—
144	C₂²	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
145	C₂²	#14579132	1	1	yes	characteristic
146	C₂²	#14579137	1	1	yes	characteristic
147	C₂²	C₂×#11929611	1	1	yes	characteristic
148	C₂²	C₂×#11929612	1	1	yes	characteristic
149	C₂²	#14579187	1	1	yes	characteristic
150	C₂²	#14579192	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	8	2	no	—
153	C₄	—	316	2	no	—
154	C₄	—	316	2	no	—
155	C₄	#14579092	1	4	yes	complemented
156	C₄	#14579123	1	1	yes	characteristic
157	C₄	#14579126	1	1	yes	characteristic
158	C₄	#14579152	1	1	yes	characteristic
159	C₄	#14579160	1	1	yes	characteristic
160	C₂	#19098459	1	1	yes	characteristic
161	C₂	C₂×#14579092	1	1	yes	characteristic
162	C₂	#19098603	1	1	yes	characteristic
163	C₂	#19098760	1	1	yes	characteristic
164	C₂	#19098765	1	1	yes	characteristic
165	C₂	#19098845	1	1	yes	characteristic
166	C₂	#19098850	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579092

C₂

yes

maximal

C₂×C₄⋊(C₇₉⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₇₉

C₂

yes

maximal, characteristic

#19098325

C₂

yes

maximal

#384×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄×Dic₇₉

—

#14579092

—

C₄⋊(C₇₉⋊C₈)

C₄

yes

complemented

C₇₉⋊#95

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

—

C₄⋊Dic₇₉

Q₈

yes

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×C₄

yes

—

C₄²×C₇₉

Q₈

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₄⋊C₄

yes

—

C₂×C₄×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

SD₁₆

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

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₉

#121

yes

characteristic

C₂²×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₉

C₂×SD₁₆

yes

characteristic

C₂²×C₇₉

C₂×SD₁₆

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#106

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

—

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×#106

—

C₂×C₄⋊C₈

—

#384

D₇₉

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₄²

—

158

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

#106

—

158

—

C₄⋊C₈

—

100

#95

—

158

—

101

C₂²×C₄

—

102

C₂²×C₄

—

158

—

103

C₂²×C₄

—

158

—

104

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

105

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

106

C₂²×C₄

C₂²×D₇₉

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

158

—

109

C₂×C₈

—

158

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

158

—

117

C₄⋊C₄

—

158

—

118

C₄²

—

316

—

119

C₄²

Dic₁₅₈

yes

—

120

C₄²

C₄×D₇₉

yes

—

121

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

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

Dic₇₉⋊C₄

yes

—

134

C₂×C₄

Dic₇₉⋊C₄

yes

—

135

C₂×C₄

#11929611

yes

—

136

C₂×C₄

#11929612

yes

—

137

C₂×C₄

C₂×C₄×D₇₉

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

—

158

—

144

C₂²

C₂×Dic₇₉⋊C₄

yes

characteristic

145

C₂²

#14579132

yes

characteristic

146

C₂²

#14579137

yes

characteristic

147

C₂²

C₂×#11929611

yes

characteristic

148

C₂²

C₂×#11929612

yes

characteristic

149

C₂²

#14579187

yes

characteristic

150

C₂²

#14579192

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

316

—

154

C₄

—

316

—

155

C₄

#14579092

yes

complemented

156

C₄

#14579123

yes

characteristic

157

C₄

#14579126

yes

characteristic

158

C₄

#14579152

yes

characteristic

159

C₄

#14579160

yes

characteristic

160

C₂

#19098459

yes

characteristic

161

C₂

C₂×#14579092

yes

characteristic

162

C₂

#19098603

yes

characteristic

163

C₂

#19098760

yes

characteristic

164

C₂

#19098765

yes

characteristic

165

C₂

#19098845

yes

characteristic

166

C₂

#19098850

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#384×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#384	D₇₉
hypocenter	C₇₉	#1948
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#384×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
11. C₄⋊(C₇₉⋊C₈)	C₄	semidirect
69. C₄×C₇₉	#106	semidirect
84. C₇₉	#1948	semidirect
155. C₄	#14579092	semidirect

H ≅ G/N

type

11. C₄⋊(C₇₉⋊C₈)

C₄

semidirect

69. C₄×C₇₉

#106

semidirect

84. C₇₉

#1948

semidirect

155. C₄

#14579092

semidirect

Subgroups — #23764161

Conjugacy classes of subgroups

Complements