Group #34206862 — subgroups

Subgroups — #34206862

Order 20224.

334 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 131 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×#19098965	C₂	1	1	yes	maximal, characteristic
2	#23765156	C₂	1	1	yes	maximal, characteristic
3	#23761510	C₂	1	4	yes	maximal
4	C₄×#108×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579153	—	2	1	no	—
6	C₂×#14579101	C₂²	1	2	yes	—
7	#19098965	—	2	2	no	—
8	#19098965	C₄	1	4	yes	—
9	C₂×#14579095	C₂²	1	2	yes	—
10	C₂×#14579093	—	2	1	no	—
11	C₄⋊(C₇₉⋊C₁₆)	—	2	2	no	—
12	C₂×#108×C₇₉	C₂²	1	2	yes	—
13	C₄×M₁₆×C₇₉	—	2	1	no	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	#341×C₇₉	—	2	1	no	—
16	#14579153	—	2	4	no	—
17	#14579153	—	4	1	no	—
18	#14579153	—	4	1	no	—
19	C₂×C₄⋊Dic₇₉	—	2	2	no	—
20	C₂×Dic₇₉⋊C₄	—	4	2	no	—
21	#14579101	—	2	4	no	—
22	C₂×C₇₉⋊C₁₆	—	2	2	no	—
23	#14579095	—	2	2	no	—
24	#14579095	C₂×C₄	1	4	yes	—
25	#14579093	—	2	4	no	—
26	C₂×M₁₆×C₇₉	—	2	2	no	—
27	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
30	#108×C₇₉	—	2	2	no	—
31	#108×C₇₉	Q₈	1	4	yes	—
32	#97×C₇₉	—	4	1	no	—
33	C₄×C₈×C₇₉	—	4	1	no	—
34	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₉	D₄	1	2	yes	—
36	C₂²×Dic₇₉	—	4	2	no	—
37	C₄⋊Dic₇₉	—	2	4	no	—
38	C₄⋊Dic₇₉	—	4	2	no	—
39	Dic₇₉⋊C₄	—	4	4	no	—
40	Dic₇₉⋊C₄	—	4	4	no	—
41	C₇₉⋊C₁₆	—	4	2	no	—
42	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
43	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₇₉	—	2	4	no	—
45	M₁₆×C₇₉	—	4	2	no	—
46	C₂×C₈×C₇₉	—	2	2	no	—
47	C₂×C₈×C₇₉	—	4	2	no	—
48	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₇₉	#41	1	2	yes	—
53	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₇₉	D₈	1	1	yes	characteristic
55	C₄²×C₇₉	D₈	1	1	yes	characteristic
56	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₇₉	—	4	2	no	—
58	C₂×C₄×C₇₉	#102	1	2	yes	—
59	C₂×C₄×C₇₉	#102	1	2	yes	—
60	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
62	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₇₉	#137	1	2	yes	—
64	C₂×Dic₇₉	—	4	4	no	—
65	C₂×Dic₇₉	—	8	4	no	—
66	C₈×C₇₉	—	2	2	no	—
67	C₈×C₇₉	—	4	4	no	—
68	C₈×C₇₉	C₄×Q₈	1	2	yes	—
69	C₈×C₇₉	#124	1	1	yes	characteristic
70	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
71	C₂²×C₇₉	—	2	2	no	—
72	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
73	C₂²×C₇₉	#475	1	2	yes	—
74	C₄×C₇₉	—	2	2	no	—
75	C₄×C₇₉	—	2	2	no	—
76	C₄×C₇₉	#391	1	1	yes	characteristic
77	C₄×C₇₉	#419	1	1	yes	characteristic
78	C₄×C₇₉	#474	1	2	yes	—
79	Dic₇₉	—	8	4	no	—
80	#13280	—	79	1	no	maximal, Sylow
81	C₂×C₇₉	—	2	2	no	—
82	C₂×C₇₉	#1988	1	1	yes	characteristic
83	C₂×C₇₉	#2148	1	2	yes	—
84	C₂×#499	—	79	1	no	—
85	#2150	—	79	1	no	—
86	C₄×#108	D₇₉	1	1	yes	characteristic
87	#1387	—	79	4	no	—
88	C₇₉	#13280	1	1	yes	characteristic, complemented, Sylow
89	C₂×#125	—	158	1	no	—
90	C₂×M₃₂	—	79	2	no	—
91	#499	—	79	4	no	—
92	#499	—	158	2	no	—
93	C₂×#108	C₂×D₇₉	1	2	yes	—
94	C₂×#107	—	79	2	no	—
95	C₂×#106	—	158	1	no	—
96	C₄×M₁₆	—	2	1	no	—
97	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
98	C₄⋊C₁₆	—	158	2	no	—
99	#341	—	2	1	no	—
100	C₂×M₁₆	—	2	2	no	—
101	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
103	#125	—	316	1	no	—
104	#125	—	316	1	no	—
105	#125	—	158	4	no	—
106	C₂×C₄⋊C₄	—	316	2	no	—
107	C₂×C₄⋊C₄	—	158	2	no	—
108	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
109	M₃₂	—	158	4	no	—
110	C₂×C₁₆	—	158	2	no	—
111	#108	—	2	2	no	—
112	#108	Dic₁₅₈	1	4	yes	—
113	#107	—	79	4	no	—
114	#107	—	158	2	no	—
115	#106	—	158	4	no	—
116	#97	—	4	1	no	—
117	C₄×C₈	—	4	1	no	—
118	C₄×C₈	C₄×D₇₉	1	2	yes	—
119	C₄×C₈	D₃₁₆	1	2	yes	—
120	C₂²×C₄	—	316	2	no	—
121	C₂²×C₄	#11929611	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
123	M₁₆	—	2	4	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
128	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
129	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
131	C₂×C₈	#11929624	1	2	yes	—
132	C₄⋊C₄	—	316	4	no	—
133	C₄⋊C₄	—	316	4	no	—
134	C₄⋊C₄	—	316	2	no	—
135	C₄⋊C₄	—	158	4	no	—
136	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
137	C₄²	#11929609	1	1	yes	characteristic
138	C₄²	#11929609	1	1	yes	characteristic
139	C₁₆	—	316	2	no	—
140	C₂³	C₂×#11929611	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	316	4	no	—
143	C₂×C₄	—	632	4	no	—
144	C₂×C₄	#14579081	1	2	yes	—
145	C₂×C₄	#14579081	1	2	yes	—
146	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
147	C₂×C₄	#14579140	1	2	yes	—
148	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
149	C₂×C₄	#14579191	1	2	yes	—
150	C₈	—	2	2	no	—
151	C₈	—	4	4	no	—
152	C₈	C₄×Dic₁₅₈	1	2	yes	—
153	C₈	#14579149	1	1	yes	characteristic
154	C₈	#14579161	1	1	yes	characteristic
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#14579081	1	1	yes	characteristic
157	C₂²	#19098848	1	2	yes	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	632	4	no	—
161	C₄	#19098469	1	1	yes	characteristic
162	C₄	#19098571	1	1	yes	characteristic
163	C₄	#19098843	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	#23764352	1	1	yes	characteristic
166	C₂	#23765147	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098965

C₂

yes

maximal, characteristic

#23765156

C₂

yes

maximal, characteristic

#23761510

C₂

yes

maximal

C₄×#108×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579153

—

C₂×#14579101

C₂²

yes

—

#19098965

—

#19098965

C₄

yes

—

C₂×#14579095

C₂²

yes

—

C₂×#14579093

—

C₄⋊(C₇₉⋊C₁₆)

—

C₂×#108×C₇₉

C₂²

yes

—

C₄×M₁₆×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#341×C₇₉

—

#14579153

—

#14579153

—

#14579153

—

C₂×C₄⋊Dic₇₉

—

C₂×Dic₇₉⋊C₄

—

#14579101

—

C₂×C₇₉⋊C₁₆

—

#14579095

—

#14579095

C₂×C₄

yes

—

#14579093

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

#108×C₇₉

—

#108×C₇₉

Q₈

yes

—

#97×C₇₉

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

SD₁₆

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₄²×C₇₉

C₂²⋊C₄

yes

—

C₄²×C₇₉

D₈

yes

characteristic

C₄²×C₇₉

D₈

yes

characteristic

C₂³×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#102

yes

—

C₂×C₄×C₇₉

#102

yes

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

#137

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₄×Q₈

yes

—

C₈×C₇₉

#124

yes

characteristic

C₈×C₇₉

C₄⋊Q₈

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

C₂×#102

yes

characteristic

C₂²×C₇₉

#475

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

#13280

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1988

yes

characteristic

C₂×C₇₉

#2148

yes

—

C₂×#499

—

#2150

—

C₄×#108

D₇₉

yes

characteristic

#1387

—

C₇₉

#13280

yes

characteristic, complemented, Sylow

C₂×#125

—

158

—

C₂×M₃₂

—

#499

—

#499

—

158

—

C₂×#108

C₂×D₇₉

yes

—

C₂×#107

—

C₂×#106

—

158

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₄⋊C₁₆

—

158

—

#341

—

100

C₂×M₁₆

—

101

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

102

C₂²×C₈

C₂²×D₇₉

yes

characteristic

103

#125

—

316

—

104

#125

—

316

—

105

#125

—

158

—

106

C₂×C₄⋊C₄

—

316

—

107

C₂×C₄⋊C₄

—

158

—

108

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

109

M₃₂

—

158

—

110

C₂×C₁₆

—

158

—

111

#108

—

112

#108

Dic₁₅₈

yes

—

113

#107

—

114

#107

—

158

—

115

#106

—

158

—

116

#97

—

117

C₄×C₈

—

118

C₄×C₈

C₄×D₇₉

yes

—

119

C₄×C₈

D₃₁₆

yes

—

120

C₂²×C₄

—

316

—

121

C₂²×C₄

#11929611

yes

—

122

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

128

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

129

C₂×C₈

C₂×D₃₁₆

yes

characteristic

130

C₂×C₈

C₂×Dic₁₅₈

yes

—

131

C₂×C₈

#11929624

yes

—

132

C₄⋊C₄

—

316

—

133

C₄⋊C₄

—

316

—

134

C₄⋊C₄

—

316

—

135

C₄⋊C₄

—

158

—

136

C₄²

(C₂×D₇₉)⋊C₄

yes

—

137

C₄²

#11929609

yes

characteristic

138

C₄²

#11929609

yes

characteristic

139

C₁₆

—

316

—

140

C₂³

C₂×#11929611

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

316

—

143

C₂×C₄

—

632

—

144

C₂×C₄

#14579081

yes

—

145

C₂×C₄

#14579081

yes

—

146

C₂×C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

147

C₂×C₄

#14579140

yes

—

148

C₂×C₄

C₂×#11929609

yes

characteristic

149

C₂×C₄

#14579191

yes

—

150

C₈

—

151

C₈

—

152

C₈

C₄×Dic₁₅₈

yes

—

153

C₈

#14579149

yes

characteristic

154

C₈

#14579161

yes

characteristic

155

C₂²

—

156

C₂²

C₂×#14579081

yes

characteristic

157

C₂²

#19098848

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

632

—

161

C₄

#19098469

yes

characteristic

162

C₄

#19098571

yes

characteristic

163

C₄

#19098843

yes

—

164

C₂

—

165

C₂

#23764352

yes

characteristic

166

C₂

#23765147

yes

—

	subgroup	G/H
center	C₂²	C₂×#14579081
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₄×#108×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#102
prime radical	C₂³×C₇₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₄×#108	D₇₉
hypocenter	C₇₉	#13280
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

subgroup

G/H

center

C₂²

C₂×#14579081

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₄×#108×C₇₉

C₂

socle

C₂²×C₇₉

C₂×#102

prime radical

C₂³×C₇₉

C₂×SD₁₆

cosocle

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

N	H ≅ G/N	type
88. C₇₉	#13280	semidirect

H ≅ G/N

type

88. C₇₉

#13280

semidirect

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

subgroup	index
C₂×#19098965	2
#23765156	2
#23761510	2
C₄×#108×C₇₉	2
#13280	79

subgroup

index

C₄×#108×C₇₉

Subgroups — #34206862

Conjugacy classes of subgroups

Complements

Maximal subgroups