Group #34187733 — subgroups

Subgroups — #34187733

Order 20224.

478 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 159 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	#23763572	C₂	1	1	yes	maximal, characteristic
2	#23762072	C₂	1	4	yes	maximal
3	#23761683	C₂	1	1	yes	maximal, characteristic
4	C₂×#494×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#19098616	C₂²	1	2	yes	—
6	C₂×#14579097	—	2	2	no	—
7	#19098587	C₂²	1	2	yes	—
8	#19098309	—	2	8	no	—
9	#19098284	—	2	1	no	—
10	C₈×C₇₉⋊C₈	—	2	1	no	—
11	C₂²×Q₁₆×C₇₉	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
13	#494×C₇₉	—	2	1	no	—
14	#494×C₇₉	—	2	1	no	—
15	#494×C₇₉	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#11929608	—	2	2	no	—
18	#14579097	—	4	4	no	—
19	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
20	C₄⋊(C₇₉⋊C₈)	D₄	1	4	yes	—
21	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
22	C₄×C₇₉⋊C₈	—	2	2	no	—
23	C₂²×Q₈×C₇₉	—	2	2	no	—
24	C₂×Q₁₆×C₇₉	—	2	4	no	—
25	C₂×Q₁₆×C₇₉	—	2	4	no	—
26	C₂×Q₁₆×C₇₉	—	4	4	no	—
27	C₂×Q₁₆×C₇₉	—	4	4	no	—
28	C₂²×C₈×C₇₉	D₄	1	2	yes	—
29	C₄⋊Q₈×C₇₉	—	2	4	no	—
30	C₄⋊Q₈×C₇₉	—	2	2	no	—
31	C₄⋊Q₈×C₇₉	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₇₉	—	2	4	no	—
33	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
36	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
37	#11929608	—	4	4	no	—
38	C₂×C₇₉⋊C₈	—	2	4	no	—
39	C₂×C₇₉⋊C₈	—	4	2	no	—
40	C₂×Q₈×C₇₉	—	2	4	no	—
41	C₂×Q₈×C₇₉	—	4	2	no	—
42	C₂×Q₈×C₇₉	—	4	4	no	—
43	C₂×Q₈×C₇₉	—	4	4	no	—
44	C₂²×C₄×C₇₉	—	4	2	no	—
45	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	#41	1	2	yes	—
47	Q₁₆×C₇₉	—	4	8	no	—
48	Q₁₆×C₇₉	—	4	8	no	—
49	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₇₉	D₈	1	4	yes	—
52	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
53	C₄⋊C₄×C₇₉	—	2	8	no	—
54	C₄⋊C₄×C₇₉	—	4	4	no	—
55	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₇₉	C₂×D₄	1	2	yes	—
58	C₂³×C₇₉	#124	1	1	yes	characteristic
59	Q₈×C₇₉	—	4	8	no	—
60	Q₈×C₇₉	—	8	2	no	—
61	Q₈×C₇₉	—	8	2	no	—
62	C₂×C₄×C₇₉	—	2	4	no	—
63	C₂×C₄×C₇₉	—	4	4	no	—
64	C₂×C₄×C₇₉	—	8	4	no	—
65	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₇₉	#121	1	2	yes	—
68	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
69	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
71	C₈×C₇₉	#102	1	4	yes	—
72	C₈×C₇₉	#102	1	4	yes	—
73	C₇₉⋊C₈	—	4	4	no	—
74	C₇₉⋊C₈	—	8	2	no	—
75	C₂²×C₇₉	—	2	2	no	—
76	C₂²×C₇₉	#390	1	1	yes	characteristic
77	C₂²×C₇₉	#486	1	2	yes	—
78	C₄×C₇₉	—	2	2	no	—
79	C₄×C₇₉	—	8	4	no	—
80	C₄×C₇₉	C₂×#102	1	2	yes	—
81	C₄×C₇₉	#473	1	4	yes	—
82	C₄×C₇₉	#492	1	1	yes	characteristic
83	C₄×C₇₉	#493	1	1	yes	characteristic
84	#9772	—	79	1	no	maximal, Sylow
85	C₂×C₇₉	—	2	2	no	—
86	C₂×C₇₉	#1957	1	1	yes	characteristic
87	C₂×C₇₉	#1979	1	2	yes	—
88	C₂×#494	D₇₉	1	1	yes	characteristic
89	#1848	—	79	1	no	—
90	#1546	—	79	4	no	—
91	#1450	—	79	1	no	—
92	C₇₉	#9772	1	1	yes	characteristic, complemented, Sylow
93	C₂²×Q₁₆	—	2	2	no	—
94	C₂×C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
95	#494	—	2	1	no	—
96	#494	—	2	1	no	—
97	#494	Dic₇₉	1	4	yes	—
98	#431	—	79	2	no	—
99	C₂×#108	—	158	2	no	—
100	#423	—	79	2	no	—
101	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
102	#332	—	158	8	no	—
103	#322	—	158	1	no	—
104	C₈²	—	158	1	no	—
105	C₂²×Q₈	—	2	2	no	—
106	C₂×Q₁₆	—	2	4	no	—
107	C₂×Q₁₆	—	2	4	no	—
108	C₂×Q₁₆	—	4	4	no	—
109	C₂×Q₁₆	—	4	4	no	—
110	C₂×M₁₆	—	158	2	no	—
111	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
112	C₄⋊Q₈	—	2	4	no	—
113	C₄⋊Q₈	—	2	2	no	—
114	C₄⋊Q₈	C₂×Dic₇₉	1	4	yes	—
115	C₂×C₄⋊C₄	—	2	4	no	—
116	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
117	#108	—	316	4	no	—
118	C₄⋊C₈	—	79	4	no	—
119	C₄⋊C₈	—	158	4	no	—
120	C₂²⋊C₈	—	158	2	no	—
121	C₄×C₈	—	158	2	no	—
122	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
123	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
124	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
125	C₂×Q₈	—	2	4	no	—
126	C₂×Q₈	—	4	2	no	—
127	C₂×Q₈	—	4	4	no	—
128	C₂×Q₈	—	4	4	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
131	C₂²×C₄	#11929623	1	2	yes	—
132	Q₁₆	—	4	8	no	—
133	Q₁₆	—	4	8	no	—
134	M₁₆	—	316	4	no	—
135	C₂×C₈	—	316	2	no	—
136	C₂×C₈	—	158	4	no	—
137	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
138	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
139	C₂×C₈	#11929609	1	4	yes	—
140	C₂×C₈	#11929612	1	4	yes	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
144	C₄²	D₄×D₇₉	1	2	yes	—
145	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
146	C₂³	#14579148	1	1	yes	characteristic
147	Q₈	—	4	8	no	—
148	Q₈	—	8	2	no	—
149	Q₈	—	8	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
154	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
155	C₂×C₄	#14579132	1	2	yes	—
156	C₂×C₄	#14579157	1	1	yes	characteristic
157	C₂×C₄	C₂×#11929609	1	2	yes	—
158	C₂×C₄	C₂×#11929612	1	2	yes	—
159	C₈	—	316	4	no	—
160	C₈	—	632	2	no	—
161	C₈	#14579082	1	4	yes	—
162	C₈	#14579082	1	4	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	#19098466	1	1	yes	characteristic
165	C₂²	#19098909	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	8	4	no	—
168	C₄	C₂×#14579082	1	2	yes	—
169	C₄	#19098835	1	4	yes	—
170	C₄	#19098935	1	1	yes	characteristic
171	C₄	#19098939	1	1	yes	characteristic
172	C₂	—	2	2	no	—
173	C₂	#23764207	1	1	yes	characteristic
174	C₂	#23764316	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763572

C₂

yes

maximal, characteristic

#23762072

C₂

yes

maximal

#23761683

C₂

yes

maximal, characteristic

C₂×#494×C₇₉

C₂

yes

maximal, characteristic

#19098616

C₂²

yes

—

C₂×#14579097

—

#19098587

C₂²

yes

—

#19098309

—

#19098284

—

C₈×C₇₉⋊C₈

—

C₂²×Q₁₆×C₇₉

—

C₂×C₄⋊Q₈×C₇₉

C₂²

yes

—

#494×C₇₉

—

#494×C₇₉

—

#494×C₇₉

C₄

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

#14579097

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₂²×Q₈×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

#124

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₈×C₇₉

#102

yes

—

C₈×C₇₉

#102

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#390

yes

characteristic

C₂²×C₇₉

#486

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×#102

yes

—

C₄×C₇₉

#473

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

#1957

yes

characteristic

C₂×C₇₉

#1979

yes

—

C₂×#494

D₇₉

yes

—

—

—

C₇₉

#9772

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₇₉

yes

—

#494

—

#494

—

#494

Dic₇₉

yes

—

#431

—

C₂×#108

—

158

—

100

#423

—

101

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

102

#332

—

158

—

103

#322

—

158

—

104

C₈²

—

158

—

105

C₂²×Q₈

—

106

C₂×Q₁₆

—

107

C₂×Q₁₆

—

108

C₂×Q₁₆

—

109

C₂×Q₁₆

—

110

C₂×M₁₆

—

158

—

111

C₂²×C₈

C₂²⋊D₇₉

yes

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

C₂×Dic₇₉

yes

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

C₂²×D₇₉

yes

characteristic

117

#108

—

316

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

158

—

120

C₂²⋊C₈

—

158

—

121

C₄×C₈

—

158

—

122

C₄×C₈

C₂×Dic₇₉

yes

—

123

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

124

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

131

C₂²×C₄

#11929623

yes

—

132

Q₁₆

—

133

Q₁₆

—

134

M₁₆

—

316

—

135

C₂×C₈

—

316

—

136

C₂×C₈

—

158

—

137

C₂×C₈

C₂²⋊Dic₇₉

yes

—

138

C₂×C₈

C₂²⋊Dic₇₉

yes

—

139

C₂×C₈

#11929609

yes

—

140

C₂×C₈

#11929612

yes

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄²

C₂²×Dic₇₉

yes

characteristic

144

C₄²

D₄×D₇₉

yes

—

145

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

146

C₂³

#14579148

yes

characteristic

147

Q₈

—

148

Q₈

—

149

Q₈

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

154

C₂×C₄

D₄×Dic₇₉

yes

—

155

C₂×C₄

#14579132

yes

—

156

C₂×C₄

#14579157

yes

characteristic

157

C₂×C₄

C₂×#11929609

yes

—

158

C₂×C₄

C₂×#11929612

yes

—

159

C₈

—

316

—

160

C₈

—

632

—

161

C₈

#14579082

yes

—

162

C₈

#14579082

yes

—

163

C₂²

—

164

C₂²

#19098466

yes

characteristic

165

C₂²

#19098909

yes

—

166

C₄

—

167

C₄

—

168

C₄

C₂×#14579082

yes

—

169

C₄

#19098835

yes

—

170

C₄

#19098935

yes

characteristic

171

C₄

#19098939

yes

characteristic

172

C₂

—

173

C₂

#23764207

yes

characteristic

174

C₂

#23764316

yes

—

	subgroup	G/H
center	C₂²	#19098466
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#494×C₇₉	C₂
socle	C₂²×C₇₉	#390
prime radical	C₂³×C₇₉	#124
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#494	D₇₉
hypocenter	C₇₉	#9772
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂²

#19098466

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₂×#494×C₇₉

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
92. C₇₉	#9772	semidirect

H ≅ G/N

type

92. C₇₉

#9772

semidirect

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

subgroup	index
#23763572	2
#23762072	2
#23761683	2
C₂×#494×C₇₉	2
#9772	79

subgroup

index

C₂×#494×C₇₉

Subgroups — #34187733

Conjugacy classes of subgroups

Complements

Maximal subgroups