Group #34175372 — subgroups

Subgroups — #34175372

Order 20224.

198 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 91 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	#23762163	C₂	1	1	yes	maximal, characteristic
2	#23762156	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
4	#19098620	—	2	1	no	—
5	#19098587	—	2	1	no	—
6	#19098316	C₄	1	2	yes	—
7	#19098314	—	2	1	no	—
8	#19098314	—	2	1	no	—
9	#19098314	C₄	1	2	yes	—
10	C₂²×C₁₆×C₇₉	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₇₉	—	2	1	no	—
13	C₄⋊C₁₆×C₇₉	—	2	1	no	—
14	C₂×#11929608	—	4	1	no	—
15	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
16	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
17	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
18	C₂²⋊(C₇₉⋊C₈)	—	4	1	no	—
19	#14579072	—	4	1	no	—
20	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₉	—	2	2	no	—
24	C₂×C₁₆×C₇₉	—	4	2	no	—
25	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
29	#11929608	—	8	2	no	—
30	C₂×C₇₉⋊C₈	—	4	2	no	—
31	C₂×C₇₉⋊C₈	—	4	2	no	—
32	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
35	C₂×C₈×C₇₉	—	2	1	no	—
36	C₂×C₈×C₇₉	—	2	1	no	—
37	C₂×C₈×C₇₉	—	4	1	no	—
38	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₇₉	D₈	1	1	yes	characteristic
45	C₄²×C₇₉	Q₁₆	1	1	yes	characteristic
46	C₁₆×C₇₉	—	4	2	no	—
47	C₂³×C₇₉	#106	1	1	yes	characteristic
48	C₂×C₄×C₇₉	—	2	2	no	—
49	C₂×C₄×C₇₉	—	2	1	no	—
50	C₂×C₄×C₇₉	—	2	1	no	—
51	C₂×C₄×C₇₉	—	2	2	no	—
52	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
53	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
56	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
58	C₈×C₇₉	—	2	1	no	—
59	C₈×C₇₉	—	2	1	no	—
60	C₈×C₇₉	—	2	1	no	—
61	C₈×C₇₉	#100	1	1	yes	characteristic
62	C₈×C₇₉	#101	1	1	yes	characteristic
63	C₇₉⋊C₈	—	8	2	no	—
64	C₇₉⋊C₈	—	8	2	no	—
65	C₂²×C₇₉	—	2	1	no	—
66	C₂²×C₇₉	—	2	1	no	—
67	C₂²×C₇₉	#340	1	1	yes	characteristic
68	C₂²×C₇₉	#365	1	2	yes	—
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	#338	1	1	yes	characteristic
74	C₄×C₇₉	#343	1	1	yes	characteristic
75	C₄×C₇₉	#359	1	2	yes	—
76	#7243	—	79	1	no	maximal, Sylow
77	C₂×C₇₉	—	2	2	no	—
78	C₂×C₇₉	#1292	1	1	yes	characteristic
79	C₂×C₇₉	#1382	1	1	yes	characteristic
80	C₂×C₇₉	#1383	1	1	yes	characteristic
81	C₂×C₄⋊C₁₆	D₇₉	1	1	yes	characteristic
82	#1568	—	79	1	no	—
83	#1566	—	79	1	no	—
84	C₇₉	#7243	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₁₆	Dic₇₉	1	2	yes	—
86	#432	—	158	1	no	—
87	#423	—	158	1	no	—
88	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
89	C₄⋊C₁₆	—	2	1	no	—
90	C₄⋊C₁₆	—	2	1	no	—
91	#335	—	79	2	no	—
92	#334	—	79	2	no	—
93	#334	—	158	1	no	—
94	#334	—	158	1	no	—
95	C₂×M₁₆	—	316	1	no	—
96	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
97	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
98	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	4	2	no	—
101	C₄⋊C₈	—	316	1	no	—
102	C₄⋊C₈	—	158	2	no	—
103	C₄⋊C₈	—	158	2	no	—
104	C₂²⋊C₈	—	316	1	no	—
105	#97	—	316	1	no	—
106	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
107	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
108	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
109	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
110	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
111	C₂²×C₄	#11929613	1	1	yes	characteristic
112	C₂²×C₄	#11929613	1	1	yes	characteristic
113	M₁₆	—	632	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	316	2	no	—
118	C₂×C₈	—	316	2	no	—
119	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
120	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
121	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
122	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
123	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
124	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
125	C₄²	D₆₃₂	1	1	yes	characteristic
126	C₄²	Dic₃₁₆	1	1	yes	characteristic
127	C₁₆	—	4	2	no	—
128	C₂³	#14579093	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
134	C₂×C₄	#14579083	1	1	yes	characteristic
135	C₂×C₄	#14579083	1	1	yes	characteristic
136	C₂×C₄	#14579086	1	1	yes	characteristic
137	C₂×C₄	#14579086	1	1	yes	characteristic
138	C₂×C₄	#14579095	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	632	2	no	—
143	C₈	—	632	2	no	—
144	C₈	#14579078	1	1	yes	characteristic
145	C₈	#14579080	1	1	yes	characteristic
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	#19098326	1	1	yes	characteristic
149	C₂²	#19098386	1	2	yes	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	C₇₉⋊#338	1	1	yes	characteristic
155	C₄	#19098332	1	1	yes	characteristic
156	C₄	#19098371	1	2	yes	—
157	C₂	—	2	2	no	—
158	C₂	#23761275	1	1	yes	characteristic
159	C₂	#23761496	1	1	yes	characteristic
160	C₂	#23761499	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23762163

C₂

yes

maximal, characteristic

#23762156

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆×C₇₉

C₂

yes

maximal, characteristic

#19098620

—

#19098587

—

#19098316

C₄

yes

—

#19098314

—

#19098314

—

#19098314

C₄

yes

—

C₂²×C₁₆×C₇₉

C₄

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₉

—

C₄⋊C₁₆×C₇₉

—

C₂×#11929608

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

Q₈

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

SD₁₆

yes

characteristic

C₂²×C₄×C₇₉

SD₁₆

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄²

yes

characteristic

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

C₄²×C₇₉

D₈

yes

characteristic

C₄²×C₇₉

Q₁₆

yes

characteristic

C₁₆×C₇₉

—

C₂³×C₇₉

#106

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#340

yes

characteristic

C₂²×C₇₉

#365

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

#7243

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₄⋊C₁₆

D₇₉

yes

characteristic

#1568

—

#1566

—

C₇₉

#7243

yes

characteristic, complemented, Sylow

C₂²×C₁₆

Dic₇₉

yes

—

#432

—

158

—

#423

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#335

—

#334

—

#334

—

158

—

#334

—

158

—

C₂×M₁₆

—

316

—

C₂²×C₈

Dic₁₅₈

yes

characteristic

C₂²×C₈

C₂×Dic₇₉

yes

characteristic

C₂×C₄²

D₃₁₆

yes

characteristic

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₄⋊C₈

—

316

—

102

C₄⋊C₈

—

158

—

103

C₄⋊C₈

—

158

—

104

C₂²⋊C₈

—

316

—

105

#97

—

316

—

106

C₄×C₈

C₄×D₇₉

yes

characteristic

107

C₄×C₈

C₄×D₇₉

yes

characteristic

108

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

109

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

110

C₂²×C₄

C₄⋊Dic₇₉

yes

characteristic

111

C₂²×C₄

#11929613

yes

characteristic

112

C₂²×C₄

#11929613

yes

characteristic

113

M₁₆

—

632

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

316

—

118

C₂×C₈

—

316

—

119

C₂×C₈

C₄×Dic₇₉

yes

characteristic

120

C₂×C₈

C₂²⋊Dic₇₉

yes

characteristic

121

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

122

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

123

C₄²

(C₂×D₇₉)⋊C₄

yes

characteristic

124

C₄²

(C₂×D₇₉)⋊C₄

yes

characteristic

125

C₄²

D₆₃₂

yes

characteristic

126

C₄²

Dic₃₁₆

yes

characteristic

127

C₁₆

—

128

C₂³

#14579093

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₇₉⋊#95

yes

characteristic

134

C₂×C₄

#14579083

yes

characteristic

135

C₂×C₄

#14579083

yes

characteristic

136

C₂×C₄

#14579086

yes

characteristic

137

C₂×C₄

#14579086

yes

characteristic

138

C₂×C₄

#14579095

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

632

—

143

C₈

—

632

—

144

C₈

#14579078

yes

characteristic

145

C₈

#14579080

yes

characteristic

146

C₂²

—

147

C₂²

—

148

C₂²

#19098326

yes

characteristic

149

C₂²

#19098386

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

C₇₉⋊#338

yes

characteristic

155

C₄

#19098332

yes

characteristic

156

C₄

#19098371

yes

—

157

C₂

—

158

C₂

#23761275

yes

characteristic

159

C₂

#23761496

yes

characteristic

160

C₂

#23761499

yes

characteristic

	subgroup	G/H
center	C₂²	#19098326
commutator	C₂×C₈×C₇₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₂×C₄⋊C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	#340
prime radical	C₂³×C₇₉	#106
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₇₉
hypocenter	C₇₉	#7243
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂²

#19098326

commutator

C₂×C₈×C₇₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₇₉

Fitting

C₂×C₄⋊C₁₆×C₇₉

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄×C₈×C₇₉

C₂²

hypercenter

C₂×C₄⋊C₁₆

D₇₉

hypocenter

C₇₉

#7243

LCS

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
84. C₇₉	#7243	semidirect

H ≅ G/N

type

84. C₇₉

#7243

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23762163	2
#23762156	2
C₂×C₄⋊C₁₆×C₇₉	2
#7243	79

subgroup

index

#23762163

#23762156

C₂×C₄⋊C₁₆×C₇₉

#7243

Subgroups — #34175372

Conjugacy classes of subgroups

Complements

Maximal subgroups