Group #34206867 — subgroups

Subgroups — #34206867

Order 20224.

366 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 167 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×C₄×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#23763564	C₂	1	1	yes	maximal, characteristic
3	#23761491	C₂	1	4	yes	maximal
4	C₂×#499×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14579095	C₂²	1	2	yes	—
7	C₂×#14579093	—	2	1	no	—
8	#19098429	—	2	1	no	—
9	C₄×C₇₉⋊C₁₆	—	2	2	no	—
10	C₂×#125×C₇₉	—	2	1	no	—
11	#499×C₇₉	—	2	2	no	—
12	#499×C₇₉	C₄	1	4	yes	complemented
13	C₂×#107×C₇₉	C₂²	1	2	yes	—
14	C₂×#106×C₇₉	—	2	1	no	—
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₇₉	—	2	2	no	—
17	C₂×C₇₉⋊C₁₆	—	2	2	no	—
18	C₂×C₇₉⋊C₁₆	—	2	2	no	—
19	C₂×C₇₉⋊C₁₆	Q₈	1	4	yes	—
20	#14579095	—	2	4	no	—
21	#14579093	—	4	2	no	—
22	C₇₉⋊#95	—	4	2	no	—
23	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
25	#125×C₇₉	—	2	4	no	—
26	#125×C₇₉	—	4	2	no	—
27	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₇₉	—	4	2	no	—
29	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
30	#107×C₇₉	—	2	2	no	—
31	#107×C₇₉	C₂×C₄	1	4	yes	—
32	#106×C₇₉	—	2	4	no	—
33	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₇₉	D₄	1	2	yes	—
35	C₂²×Dic₇₉	—	4	2	no	—
36	C₄⋊Dic₇₉	—	4	2	no	—
37	C₄⋊Dic₇₉	—	4	2	no	—
38	C₇₉⋊C₁₆	—	2	4	no	—
39	C₂²×C₄×C₇₉	—	4	2	no	—
40	C₂²×C₄×C₇₉	D₈	1	2	yes	—
41	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
44	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
45	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₇₉	#41	1	2	yes	—
49	C₄⋊C₄×C₇₉	—	2	4	no	—
50	C₄⋊C₄×C₇₉	—	4	8	no	—
51	C₄⋊C₄×C₇₉	—	4	2	no	—
52	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₇₉	SD₁₆	1	2	yes	—
54	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₇₉	—	2	4	no	—
56	C₂×C₄×C₇₉	—	4	4	no	—
57	C₂×C₄×C₇₉	—	8	4	no	—
58	C₂×C₄×C₇₉	#102	1	4	yes	—
59	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×Dic₇₉	—	8	2	no	—
66	C₂×Dic₇₉	—	8	4	no	—
67	C₈×C₇₉	#103	1	4	yes	—
68	C₈×C₇₉	C₄×Q₈	1	2	yes	—
69	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
70	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
71	C₂²×C₇₉	#486	1	1	yes	characteristic
72	C₂²×C₇₉	#500	1	1	yes	characteristic
73	C₂²×C₇₉	#508	1	4	yes	—
74	C₄×C₇₉	—	2	4	no	—
75	C₄×C₇₉	—	8	4	no	—
76	C₄×C₇₉	#391	1	1	yes	characteristic
77	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
78	C₄×C₇₉	#487	1	1	yes	characteristic
79	C₄×C₇₉	#498	1	1	yes	characteristic
80	Dic₇₉	—	16	2	no	—
81	#13281	—	79	1	no	maximal, Sylow
82	C₂×C₇₉	#1990	1	1	yes	characteristic
83	C₂×C₇₉	#2138	1	2	yes	—
84	C₂×C₇₉	#2242	1	2	yes	—
85	C₂×C₇₉	#2253	1	2	yes	—
86	C₂×#499	D₇₉	1	1	yes	characteristic
87	C₂×C₄×C₁₆	—	79	1	no	—
88	#1846	—	79	1	no	—
89	#1380	—	79	4	no	—
90	C₇₉	#13281	1	1	yes	characteristic, complemented, Sylow
91	C₂×#125	—	2	1	no	—
92	C₂²×C₁₆	—	79	2	no	—
93	#499	—	2	2	no	—
94	#499	Dic₇₉	1	4	yes	complemented
95	C₂×#107	—	79	2	no	—
96	C₂×#107	C₂×D₇₉	1	2	yes	—
97	C₂×#106	—	2	1	no	—
98	C₂×#106	—	158	1	no	—
99	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
100	#382	—	158	1	no	—
101	C₄×C₁₆	—	158	2	no	—
102	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
104	#125	—	2	4	no	—
105	#125	—	4	2	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	—	4	2	no	—
108	C₂×C₄⋊C₄	—	158	2	no	—
109	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
110	C₂×C₁₆	—	79	4	no	—
111	C₂×C₁₆	—	158	2	no	—
112	C₂×C₁₆	—	158	2	no	—
113	#107	—	2	2	no	—
114	#107	—	158	4	no	—
115	#107	C₂×Dic₇₉	1	4	yes	—
116	#106	—	2	4	no	—
117	#106	—	316	2	no	—
118	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
119	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
120	#95	—	316	2	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	316	2	no	—
123	C₂²×C₄	#11929609	1	2	yes	—
124	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
126	C₂×C₈	#11929611	1	2	yes	—
127	C₂×C₈	#11929614	1	2	yes	—
128	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
129	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
130	C₂×C₈	Q₈×D₇₉	1	2	yes	—
131	C₂×C₈	#11929625	1	2	yes	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	4	8	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	316	2	no	—
136	C₄⋊C₄	—	316	2	no	—
137	C₄²	C₂²⋊Dic₇₉	1	2	yes	—
138	C₄²	#11929612	1	2	yes	—
139	C₁₆	—	158	4	no	—
140	C₂³	C₂×#11929609	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	—	632	2	no	—
145	C₂×C₄	—	632	4	no	—
146	C₂×C₄	#14579082	1	4	yes	—
147	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
148	C₂×C₄	#14579150	1	1	yes	characteristic
149	C₂×C₄	#14579159	1	1	yes	characteristic
150	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
151	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
152	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
153	C₈	#14579085	1	4	yes	—
154	C₈	Q₈×Dic₇₉	1	2	yes	—
155	C₈	#14579137	1	2	yes	—
156	C₂²	C₂×#14579082	1	1	yes	characteristic
157	C₂²	#19098911	1	1	yes	characteristic
158	C₂²	#19098966	1	1	yes	characteristic
159	C₂²	#19098993	1	4	yes	—
160	C₄	—	2	4	no	—
161	C₄	—	8	4	no	—
162	C₄	—	1264	2	no	—
163	C₄	#19098470	1	1	yes	characteristic
164	C₄	C₂×#14579085	1	1	yes	characteristic
165	C₄	#19098915	1	1	yes	characteristic
166	C₄	#19098959	1	1	yes	characteristic
167	C₂	#23764362	1	1	yes	characteristic
168	C₂	#23765107	1	2	yes	—
169	C₂	#23765634	1	2	yes	—
170	C₂	#23765676	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₇₉⋊C₁₆

C₂

yes

maximal, characteristic

#23763564

C₂

yes

maximal, characteristic

#23761491

C₂

yes

maximal

C₂×#499×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₁₆

C₂²

yes

—

C₂×#14579095

C₂²

yes

—

C₂×#14579093

—

#19098429

—

C₄×C₇₉⋊C₁₆

—

C₂×#125×C₇₉

—

#499×C₇₉

—

#499×C₇₉

C₄

yes

complemented

C₂×#107×C₇₉

C₂²

yes

—

C₂×#106×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

Q₈

yes

—

#14579095

—

#14579093

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

#125×C₇₉

—

#125×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

D₄

yes

characteristic

#107×C₇₉

—

#107×C₇₉

C₂×C₄

yes

—

#106×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₈

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

Q₁₆

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

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²⋊C₄

yes

—

C₄²×C₇₉

SD₁₆

yes

—

C₂³×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#102

yes

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

#103

yes

—

C₈×C₇₉

C₄×Q₈

yes

—

C₈×C₇₉

C₂²⋊Q₈

yes

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#2242

yes

—

C₂×C₇₉

#2253

yes

—

C₂×#499

D₇₉

yes

characteristic

C₂×C₄×C₁₆

—

#1846

—

#1380

—

C₇₉

#13281

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂²×C₁₆

—

#499

—

#499

Dic₇₉

yes

complemented

C₂×#107

—

C₂×#107

C₂×D₇₉

yes

—

C₂×#106

—

C₂×#106

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

100

#382

—

158

—

101

C₄×C₁₆

—

158

—

102

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

103

C₂²×C₈

C₂²×D₇₉

yes

characteristic

104

#125

—

105

#125

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

158

—

109

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

110

C₂×C₁₆

—

111

C₂×C₁₆

—

158

—

112

C₂×C₁₆

—

158

—

113

#107

—

114

#107

—

158

—

115

#107

C₂×Dic₇₉

yes

—

116

#106

—

117

#106

—

316

—

118

C₄×C₈

C₂×Dic₇₉

yes

—

119

C₄×C₈

C₂²⋊D₇₉

yes

—

120

#95

—

316

—

121

C₂²×C₄

—

122

C₂²×C₄

—

316

—

123

C₂²×C₄

#11929609

yes

—

124

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

125

C₂×C₈

C₂²⋊Dic₇₉

yes

—

126

C₂×C₈

#11929611

yes

—

127

C₂×C₈

#11929614

yes

—

128

C₂×C₈

C₂²×Dic₇₉

yes

characteristic

129

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

130

C₂×C₈

Q₈×D₇₉

yes

—

131

C₂×C₈

#11929625

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

316

—

136

C₄⋊C₄

—

316

—

137

C₄²

C₂²⋊Dic₇₉

yes

—

138

C₄²

#11929612

yes

—

139

C₁₆

—

158

—

140

C₂³

C₂×#11929609

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

632

—

145

C₂×C₄

—

632

—

146

C₂×C₄

#14579082

yes

—

147

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

148

C₂×C₄

#14579150

yes

characteristic

149

C₂×C₄

#14579159

yes

characteristic

150

C₂×C₄

C₂×#11929611

yes

characteristic

151

C₂×C₄

C₂×#11929612

yes

characteristic

152

C₂×C₄

C₂×#11929614

yes

characteristic

153

C₈

#14579085

yes

—

154

C₈

Q₈×Dic₇₉

yes

—

155

C₈

#14579137

yes

—

156

C₂²

C₂×#14579082

yes

characteristic

157

C₂²

#19098911

yes

characteristic

158

C₂²

#19098966

yes

characteristic

159

C₂²

#19098993

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

1264

—

163

C₄

#19098470

yes

characteristic

164

C₄

C₂×#14579085

yes

characteristic

165

C₄

#19098915

yes

characteristic

166

C₄

#19098959

yes

characteristic

167

C₂

#23764362

yes

characteristic

168

C₂

#23765107

yes

—

169

C₂

#23765634

yes

—

170

C₂

#23765676

yes

—

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#499×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#499	D₇₉
hypocenter	C₇₉	#13281
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂³

C₂×#11929609

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₂×#499×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₈

cosocle

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

N	H ≅ G/N	type
12. #499×C₇₉	C₄	semidirect
90. C₇₉	#13281	semidirect
94. #499	Dic₇₉	semidirect

H ≅ G/N

type

12. #499×C₇₉

C₄

semidirect

90. C₇₉

#13281

semidirect

94. #499

Dic₇₉

semidirect

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

subgroup	index
C₂×C₄×C₇₉⋊C₁₆	2
#23763564	2
#23761491	2
C₂×#499×C₇₉	2
#13281	79

subgroup

index

C₂×C₄×C₇₉⋊C₁₆

#23763564

#23761491

C₂×#499×C₇₉

#13281

Subgroups — #34206867

Conjugacy classes of subgroups

Complements

Maximal subgroups