Group #23764158 — subgroups

Subgroups — #23764158

Order 10112.

256 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14579094	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#19098433	C₂	1	1	yes	maximal, characteristic
4	#19098325	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₉	—	2	1	no	—
8	C₂×C₄⋊Dic₇₉	—	2	1	no	—
9	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
10	#14579094	—	2	4	no	—
11	C₄⋊(C₇₉⋊C₈)	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₇₉	—	2	1	no	—
15	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
16	#95×C₇₉	—	2	1	no	—
17	C₂²×Dic₇₉	—	2	2	no	—
18	C₂×C₇₉⋊C₈	—	2	2	no	—
19	C₂×C₇₉⋊C₈	—	2	2	no	—
20	C₂×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
21	C₄⋊Dic₇₉	—	2	2	no	—
22	C₄⋊Dic₇₉	—	2	2	no	—
23	C₄⋊Dic₇₉	—	4	2	no	—
24	C₄⋊Dic₇₉	—	4	2	no	—
25	C₂²×C₄×C₇₉	—	2	1	no	—
26	C₂²×C₄×C₇₉	—	2	1	no	—
27	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₇₉	—	2	1	no	—
31	C₄⋊C₄×C₇₉	—	2	1	no	—
32	C₄⋊C₄×C₇₉	D₄	1	2	yes	—
33	C₄⋊C₄×C₇₉	Q₈	1	2	yes	—
34	C₄²×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	—	2	2	no	—
41	C₂×C₄×C₇₉	—	2	2	no	—
42	C₂×C₄×C₇₉	—	4	1	no	—
43	C₂×C₄×C₇₉	—	4	1	no	—
44	C₂×C₄×C₇₉	—	4	1	no	—
45	C₂×C₄×C₇₉	—	4	1	no	—
46	C₂×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
48	C₂×C₄×C₇₉	D₈	1	2	yes	—
49	C₂×C₄×C₇₉	Q₁₆	1	2	yes	—
50	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
56	C₂×Dic₇₉	—	4	2	no	—
57	C₂×Dic₇₉	—	4	2	no	—
58	C₂×Dic₇₉	—	4	2	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×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	—	4	2	no	—
69	C₄×C₇₉	—	4	2	no	—
70	C₄×C₇₉	#107	1	4	yes	complemented
71	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
72	C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
73	C₄×C₇₉	#125	1	1	yes	characteristic
74	C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
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	—	79	1	no	—
87	#384	—	79	1	no	—
88	C₄×C₄⋊C₄	D₇₉	1	1	yes	characteristic
89	#340	—	79	2	no	—
90	C₂²×C₈	—	79	2	no	—
91	C₂×C₄⋊C₄	—	79	2	no	—
92	C₂×C₄⋊C₄	—	158	1	no	—
93	C₂×C₄⋊C₄	—	158	1	no	—
94	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
95	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
96	C₂×C₄²	—	2	1	no	—
97	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
98	#107	—	158	4	no	—
99	C₄⋊C₈	—	79	4	no	—
100	#95	—	2	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	158	2	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	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	316	2	no	—
113	C₄⋊C₄	—	316	2	no	—
114	C₄⋊C₄	—	158	2	no	—
115	C₄⋊C₄	—	158	2	no	—
116	C₄⋊C₄	Dic₁₅₈	1	2	yes	—
117	C₄⋊C₄	D₃₁₆	1	2	yes	—
118	C₄²	—	4	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	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	316	2	no	—
131	C₂×C₄	—	316	2	no	—
132	C₂×C₄	—	316	2	no	—
133	C₂×C₄	Dic₇₉⋊C₄	1	2	yes	—
134	C₂×C₄	Dic₇₉⋊C₄	1	2	yes	—
135	C₂×C₄	#11929609	1	2	yes	—
136	C₂×C₄	#11929614	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
138	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
139	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
140	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
141	C₂×C₄	#11929624	1	1	yes	characteristic
142	C₂×C₄	#11929624	1	1	yes	characteristic
143	C₈	—	158	4	no	—
144	C₂²	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
145	C₂²	#14579135	1	1	yes	characteristic
146	C₂²	#14579140	1	1	yes	characteristic
147	C₂²	C₂×#11929609	1	1	yes	characteristic
148	C₂²	C₂×#11929614	1	1	yes	characteristic
149	C₂²	#14579186	1	1	yes	characteristic
150	C₂²	#14579191	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	632	2	no	—
155	C₄	#14579094	1	4	yes	complemented
156	C₄	C₄×D₃₁₆	1	1	yes	characteristic
157	C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
158	C₄	#14579153	1	1	yes	characteristic
159	C₄	#14579161	1	1	yes	characteristic
160	C₂	#19098461	1	1	yes	characteristic
161	C₂	C₂×#14579094	1	1	yes	characteristic
162	C₂	#19098606	1	1	yes	characteristic
163	C₂	#19098758	1	1	yes	characteristic
164	C₂	#19098773	1	1	yes	characteristic
165	C₂	#19098843	1	1	yes	characteristic
166	C₂	#19098858	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579094

C₂

yes

maximal

C₂×C₄⋊(C₇₉⋊C₈)

C₂

yes

maximal, characteristic

#19098433

C₂

yes

maximal, characteristic

#19098325

C₂

yes

maximal

C₄×C₄⋊C₄×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

#14579094

—

C₄⋊(C₇₉⋊C₈)

C₄

yes

complemented

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

#95×C₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

—

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

D₄

yes

—

C₄⋊C₄×C₇₉

Q₈

yes

—

C₄²×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

—

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

—

C₇₉⋊C₈

—

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

—

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

—

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

—

#384

—

C₄×C₄⋊C₄

D₇₉

yes

characteristic

#340

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

#107

—

158

—

C₄⋊C₈

—

100

#95

—

101

C₂²×C₄

—

102

C₂²×C₄

—

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

—

316

—

113

C₄⋊C₄

—

316

—

114

C₄⋊C₄

—

158

—

115

C₄⋊C₄

—

158

—

116

C₄⋊C₄

Dic₁₅₈

yes

—

117

C₄⋊C₄

D₃₁₆

yes

—

118

C₄²

—

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

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

316

—

131

C₂×C₄

—

316

—

132

C₂×C₄

—

316

—

133

C₂×C₄

Dic₇₉⋊C₄

yes

—

134

C₂×C₄

Dic₇₉⋊C₄

yes

—

135

C₂×C₄

#11929609

yes

—

136

C₂×C₄

#11929614

yes

—

137

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

138

C₂×C₄

C₂×D₃₁₆

yes

characteristic

139

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

140

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

141

C₂×C₄

#11929624

yes

characteristic

142

C₂×C₄

#11929624

yes

characteristic

143

C₈

—

158

—

144

C₂²

C₂×Dic₇₉⋊C₄

yes

characteristic

145

C₂²

#14579135

yes

characteristic

146

C₂²

#14579140

yes

characteristic

147

C₂²

C₂×#11929609

yes

characteristic

148

C₂²

C₂×#11929614

yes

characteristic

149

C₂²

#14579186

yes

characteristic

150

C₂²

#14579191

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

632

—

155

C₄

#14579094

yes

complemented

156

C₄

C₄×D₃₁₆

yes

characteristic

157

C₄

C₄×Dic₁₅₈

yes

characteristic

158

C₄

#14579153

yes

characteristic

159

C₄

#14579161

yes

characteristic

160

C₂

#19098461

yes

characteristic

161

C₂

C₂×#14579094

yes

characteristic

162

C₂

#19098606

yes

characteristic

163

C₂

#19098758

yes

characteristic

164

C₂

#19098773

yes

characteristic

165

C₂

#19098843

yes

characteristic

166

C₂

#19098858

yes

characteristic

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

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₄×C₄⋊C₄×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

cosocle

C₂²×C₄×C₇₉

C₂³

hypercenter

C₄×C₄⋊C₄

D₇₉

hypocenter

C₇₉

#1947

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
11. C₄⋊(C₇₉⋊C₈)	C₄	semidirect
70. C₄×C₇₉	#107	semidirect
84. C₇₉	#1947	semidirect
155. C₄	#14579094	semidirect

H ≅ G/N

type

11. C₄⋊(C₇₉⋊C₈)

C₄

semidirect

70. C₄×C₇₉

#107

semidirect

84. C₇₉

#1947

semidirect

155. C₄

#14579094

semidirect

Subgroups — #23764158

Conjugacy classes of subgroups

Complements