Group #23772494 — subgroups

Subgroups — #23772494

Order 10112.

322 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 161 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	#19099120	C₂	1	1	yes	maximal, characteristic
2	#19098693	C₂	1	2	yes	maximal, complemented
3	#19098692	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₁₅₈	C₂	1	2	yes	maximal, complemented
5	#19098621	C₂	1	4	yes	maximal
6	#19098620	C₂	1	1	yes	maximal, characteristic
7	#431×C₇₉	C₂	1	1	yes	maximal, characteristic
8	#14579161	C₄	1	1	yes	characteristic
9	#14579161	C₄	1	1	yes	characteristic
10	#14579153	C₄	1	2	yes	—
11	#14579140	C₄	1	2	yes	—
12	#14579140	C₄	1	2	yes	—
13	C₄×Dic₁₅₈	C₂²	1	2	yes	—
14	C₄×Dic₁₅₈	C₂²	1	2	yes	—
15	#14579120	C₂²	1	2	yes	—
16	C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
17	Dic₇₉⋊C₈	C₂²	1	4	yes	—
18	Dic₇₉⋊C₈	C₂²	1	4	yes	—
19	C₂²⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
20	#14579072	C₂²	1	2	yes	—
21	#14579071	C₂²	1	4	yes	—
22	C₈×Dic₇₉	C₂²	1	4	yes	—
23	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
25	C₂²⋊C₈×C₇₉	C₂²	1	2	yes	—
26	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₅₈	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₅₈	C₂×C₄	1	2	yes	—
30	C₂×C₇₉⋊C₈	C₂³	1	4	yes	—
31	C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
32	C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
33	C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
34	C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
35	Dic₇₉⋊C₄	C₂×C₄	1	4	yes	—
36	Dic₇₉⋊C₄	C₂×C₄	1	4	yes	—
37	C₂²⋊Dic₇₉	C₂×C₄	1	4	yes	—
38	C₄×Dic₇₉	C₂³	1	4	yes	—
39	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₂³	1	2	yes	—
43	C₂×C₈×C₇₉	C₂³	1	2	yes	—
44	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₇₉	C₂³	1	1	yes	characteristic
47	C₄²×C₇₉	C₂³	1	1	yes	characteristic
48	C₂³×C₇₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	—	2	2	no	—
50	C₂×C₄×C₇₉	—	2	2	no	—
51	C₂×C₄×C₇₉	—	4	1	no	—
52	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₇₉	C₂²×C₄	1	4	yes	—
59	C₂×Dic₇₉	C₂²×C₄	1	4	yes	—
60	Dic₁₅₈	—	2	4	no	—
61	Dic₁₅₈	—	4	2	no	—
62	C₈×C₇₉	—	2	2	no	—
63	C₈×C₇₉	—	2	2	no	—
64	C₇₉⋊C₈	—	2	4	no	—
65	C₂²×C₇₉	—	2	1	no	—
66	C₂²×C₇₉	—	2	1	no	—
67	C₂²×C₇₉	—	2	1	no	—
68	C₂²×C₇₉	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₇₉	—	2	1	no	—
70	C₄×C₇₉	—	2	1	no	—
71	C₄×C₇₉	—	2	1	no	—
72	C₄×C₇₉	—	2	1	no	—
73	C₄×C₇₉	#131	1	2	yes	—
74	C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
75	Dic₇₉	—	2	4	no	—
76	Dic₇₉	—	2	4	no	—
77	C₂×C₇₉	—	4	1	no	—
78	C₂×C₇₉	#519	1	1	yes	characteristic
79	C₂×C₇₉	C₂×#131	1	1	yes	characteristic
80	C₂×C₇₉	#568	1	1	yes	characteristic
81	#2985	—	79	1	no	maximal, Sylow
82	C₇₉	#2985	1	1	yes	characteristic, complemented, Sylow
83	#533	—	79	1	no	—
84	#446	—	79	4	no	—
85	#446	—	79	2	no	—
86	C₈×Q₈	—	79	2	no	—
87	#433	—	79	4	no	—
88	#432	—	79	1	no	—
89	#431	D₇₉	1	1	yes	characteristic
90	C₄⋊Q₈	—	79	1	no	—
91	C₄⋊Q₈	—	79	1	no	—
92	#125	—	79	2	no	—
93	C₂²⋊Q₈	—	79	2	no	—
94	C₂²⋊Q₈	—	79	2	no	—
95	C₄×Q₈	—	79	2	no	—
96	C₄×Q₈	—	79	2	no	—
97	#117	—	79	2	no	—
98	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
99	C₄⋊C₈	—	79	4	no	—
100	C₄⋊C₈	—	79	4	no	—
101	C₄⋊C₈	—	79	2	no	—
102	C₄⋊C₈	C₂×D₇₉	1	2	yes	—
103	C₂²⋊C₈	—	79	2	no	—
104	C₂²⋊C₈	C₂×D₇₉	1	2	yes	—
105	#97	—	79	4	no	—
106	#97	—	79	2	no	—
107	C₄×C₈	—	79	4	no	—
108	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
109	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
110	C₂×Q₈	—	79	2	no	—
111	C₂×Q₈	—	79	2	no	—
112	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
113	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
115	C₂×C₈	—	79	4	no	—
116	C₂×C₈	C₂²×D₇₉	1	2	yes	—
117	C₂×C₈	C₂²×D₇₉	1	2	yes	—
118	C₄⋊C₄	—	79	4	no	—
119	C₄⋊C₄	—	79	4	no	—
120	C₄⋊C₄	—	79	2	no	—
121	C₄⋊C₄	—	79	2	no	—
122	C₄⋊C₄	—	79	2	no	—
123	C₄⋊C₄	—	79	2	no	—
124	C₂²⋊C₄	—	79	4	no	—
125	C₄²	—	79	4	no	—
126	C₄²	C₄×D₇₉	1	1	yes	characteristic
127	C₄²	C₄×D₇₉	1	1	yes	characteristic
128	C₄²	C₂²×D₇₉	1	1	yes	characteristic
129	C₄²	C₂²×D₇₉	1	1	yes	characteristic
130	C₂³	C₂×C₄×D₇₉	1	1	yes	characteristic
131	Q₈	—	316	2	no	—
132	Q₈	—	158	4	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	79	4	no	—
137	C₂×C₄	—	79	4	no	—
138	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	2	2	no	—
146	C₈	—	158	4	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂²×C₄×D₇₉	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	158	4	no	—
156	C₄	—	158	4	no	—
157	C₄	#14579167	1	2	yes	—
158	C₄	#14579208	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#19099040	1	1	yes	characteristic
161	C₂	C₂×#14579167	1	1	yes	characteristic
162	C₂	#19099329	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099120

C₂

yes

maximal, characteristic

#19098693

C₂

yes

maximal, complemented

#19098692

C₂

yes

maximal, complemented

C₈×Dic₁₅₈

C₂

yes

maximal, complemented

#19098621

C₂

yes

maximal

#19098620

C₂

yes

maximal, characteristic

#431×C₇₉

C₂

yes

maximal, characteristic

#14579161

C₄

yes

characteristic

#14579161

C₄

yes

characteristic

#14579153

C₄

yes

—

#14579140

C₄

yes

—

#14579140

C₄

yes

—

C₄×Dic₁₅₈

C₂²

yes

—

C₄×Dic₁₅₈

C₂²

yes

—

#14579120

C₂²

yes

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

Dic₇₉⋊C₈

C₂²

yes

—

Dic₇₉⋊C₈

C₂²

yes

—

C₂²⋊(C₇₉⋊C₈)

C₂²

yes

—

#14579072

C₂²

yes

—

#14579071

C₂²

yes

—

C₈×Dic₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

—

C₂²⋊C₈×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

C₂×C₄

yes

—

C₂×Dic₁₅₈

C₂×C₄

yes

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

Dic₇₉⋊C₄

C₂×C₄

yes

—

Dic₇₉⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₇₉

C₂×C₄

yes

—

C₄×Dic₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

—

C₂×C₈×C₇₉

C₂³

yes

—

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

C₂²×C₄

yes

—

C₂×Dic₇₉

C₂²×C₄

yes

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂³×C₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#131

yes

—

C₄×C₇₉

2₋¹⁺⁴

yes

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#533

—

#446

—

#446

—

C₈×Q₈

—

#433

—

#432

—

#431

D₇₉

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

C₂×D₇₉

yes

—

103

C₂²⋊C₈

—

104

C₂²⋊C₈

C₂×D₇₉

yes

—

105

#97

—

106

#97

—

107

C₄×C₈

—

108

C₄×C₈

C₂×D₇₉

yes

characteristic

109

C₄×C₈

C₂×D₇₉

yes

characteristic

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂²×C₄

C₄×D₇₉

yes

characteristic

113

C₂²×C₄

C₄×D₇₉

yes

characteristic

114

C₂²×C₄

C₂²×D₇₉

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

C₂²×D₇₉

yes

—

117

C₂×C₈

C₂²×D₇₉

yes

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

C₄×D₇₉

yes

characteristic

127

C₄²

C₄×D₇₉

yes

characteristic

128

C₄²

C₂²×D₇₉

yes

characteristic

129

C₄²

C₂²×D₇₉

yes

characteristic

130

C₂³

C₂×C₄×D₇₉

yes

characteristic

131

Q₈

—

316

—

132

Q₈

—

158

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

139

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

140

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

141

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

142

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

143

C₂×C₄

C₂³×D₇₉

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

158

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂²×C₄×D₇₉

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

158

—

156

C₄

—

158

—

157

C₄

#14579167

yes

—

158

C₄

#14579208

yes

—

159

C₂

—

160

C₂

#19099040

yes

characteristic

161

C₂

C₂×#14579167

yes

characteristic

162

C₂

#19099329

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₉
commutator, socle	C₂²×C₇₉	C₂³×C₄
Fitting	#431×C₇₉	C₂
prime radical	C₂³×C₇₉	C₂²×C₄
cosocle	C₂×C₄×C₇₉	C₂⁴
hypercenter	#431	D₇₉
hypocenter	C₇₉	#2985
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₇₉

commutator, socle

C₂²×C₇₉

C₂³×C₄

Fitting

#431×C₇₉

C₂

prime radical

C₂³×C₇₉

C₂²×C₄

cosocle

C₂×C₄×C₇₉

C₂⁴

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. #19098693	C₂	semidirect
3. #19098692	C₂	semidirect
4. C₈×Dic₁₅₈	C₂	semidirect
82. C₇₉	#2985	semidirect

H ≅ G/N

type

2. #19098693

C₂

semidirect

3. #19098692

C₂

semidirect

4. C₈×Dic₁₅₈

C₂

semidirect

82. C₇₉

#2985

semidirect

Subgroups — #23772494

Conjugacy classes of subgroups

Complements