Group #34206803 — subgroups

Subgroups — #34206803

Order 20224.

334 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 135 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	#23772334	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098339	C₂	1	1	yes	maximal, characteristic
3	#23761490	C₂	1	4	yes	maximal
4	C₇₉×#2150	C₂	1	1	yes	maximal, characteristic
5	#19099120	C₄	1	1	yes	characteristic
6	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
7	#19098693	—	2	2	no	—
8	C₈×Dic₁₅₈	—	2	2	no	—
9	#19098589	C₄	1	2	yes	—
10	#19098587	C₄	1	1	yes	characteristic
11	#19098518	C₂²	1	2	yes	—
12	#19098339	—	2	2	no	—
13	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₇₉	—	2	2	no	—
16	C₂×#11929608	C₂×C₄	1	2	yes	—
17	#14579161	C₈	1	2	yes	—
18	#14579153	C₈	1	2	yes	—
19	#14579140	—	2	2	no	—
20	C₄×Dic₁₅₈	—	2	2	no	—
21	#14579120	C₂×C₄	1	2	yes	—
22	C₂×C₇₉⋊C₁₆	—	2	2	no	—
23	C₂×C₇₉⋊C₁₆	—	2	2	no	—
24	C₂×C₇₉⋊C₁₆	Q₈	1	4	yes	complemented
25	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
26	Dic₇₉⋊C₈	—	2	4	no	—
27	#14579071	—	2	2	no	—
28	C₈×Dic₇₉	—	2	2	no	—
29	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
32	M₃₂×C₇₉	—	2	4	no	—
33	C₂×C₁₆×C₇₉	—	2	2	no	—
34	C₄×C₈×C₇₉	D₄	1	2	yes	—
35	C₄×C₈×C₇₉	D₄	1	2	yes	—
36	C₂×Dic₁₅₈	—	2	2	no	—
37	#11929608	—	2	4	no	—
38	C₂×C₇₉⋊C₈	—	2	2	no	—
39	C₄⋊Dic₇₉	—	2	2	no	—
40	C₄⋊Dic₇₉	C₂×C₈	1	4	yes	—
41	Dic₇₉⋊C₄	—	2	4	no	—
42	C₂²⋊Dic₇₉	—	2	2	no	—
43	C₄×Dic₇₉	—	2	2	no	—
44	C₇₉⋊C₁₆	—	2	4	no	—
45	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
46	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₉	—	2	2	no	—
48	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₇₉	C₂×D₄	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	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
56	C₁₆×C₇₉	—	4	2	no	—
57	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₇₉	—	4	2	no	—
59	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
60	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₉	#131	1	2	yes	—
65	C₂×Dic₇₉	—	2	4	no	—
66	Dic₁₅₈	—	4	4	no	—
67	C₈×C₇₉	—	2	2	no	—
68	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
69	C₈×C₇₉	#124	1	1	yes	characteristic
70	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
71	C₇₉⋊C₈	—	4	2	no	—
72	C₂²×C₇₉	—	2	2	no	—
73	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
74	C₂²×C₇₉	#446	1	2	yes	—
75	C₄×C₇₉	—	2	2	no	—
76	C₄×C₇₉	—	2	2	no	—
77	C₄×C₇₉	#391	1	1	yes	characteristic
78	C₄×C₇₉	#408	1	1	yes	characteristic
79	C₄×C₇₉	C₈×Q₈	1	2	yes	—
80	Dic₇₉	—	4	4	no	—
81	#13267	—	79	1	no	maximal, Sylow
82	C₂×C₇₉	—	2	2	no	—
83	C₂×C₇₉	#1984	1	1	yes	characteristic
84	C₂×C₇₉	#2117	1	2	yes	—
85	#2960	—	79	1	no	—
86	#2150	D₇₉	1	1	yes	characteristic
87	C₂×#346	—	79	1	no	—
88	#1379	—	79	4	no	—
89	C₇₉	#13267	1	1	yes	characteristic, complemented, Sylow
90	#533	—	79	1	no	—
91	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
92	C₂²×C₁₆	—	79	2	no	—
93	#446	—	158	2	no	—
94	C₈×Q₈	—	158	2	no	—
95	#424	—	79	2	no	—
96	#423	—	79	1	no	—
97	#405	—	79	2	no	—
98	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
99	C₄⋊C₁₆	—	2	2	no	—
100	#346	—	158	2	no	—
101	C₂×M₁₆	—	79	2	no	—
102	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
104	C₄⋊Q₈	—	79	2	no	—
105	#125	—	79	2	no	—
106	C₂²⋊Q₈	—	158	2	no	—
107	C₄×Q₈	—	158	2	no	—
108	#117	—	79	2	no	—
109	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
110	M₃₂	—	2	4	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	79	4	no	—
113	C₂×C₁₆	—	158	2	no	—
114	C₂×C₁₆	—	158	2	no	—
115	C₄⋊C₈	—	158	4	no	—
116	C₄⋊C₈	—	158	2	no	—
117	#97	—	158	2	no	—
118	C₄×C₈	—	158	2	no	—
119	C₄×C₈	D₃₁₆	1	2	yes	—
120	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
121	C₂×Q₈	—	158	2	no	—
122	C₂²×C₄	#11929607	1	2	yes	—
123	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
124	M₁₆	—	158	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	158	2	no	—
127	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
128	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
129	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
130	C₂×C₈	Q₈×D₇₉	1	2	yes	—
131	C₂×C₈	#11929623	1	2	yes	—
132	C₄⋊C₄	—	79	4	no	—
133	C₄⋊C₄	—	158	4	no	—
134	C₄⋊C₄	—	158	2	no	—
135	C₂²⋊C₄	—	158	2	no	—
136	C₄²	—	158	2	no	—
137	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
138	C₄²	C₈×D₇₉	1	1	yes	characteristic
139	C₄²	C₈×D₇₉	1	1	yes	characteristic
140	C₁₆	—	4	2	no	—
141	C₁₆	—	158	4	no	—
142	C₂³	C₂×#11929607	1	1	yes	characteristic
143	Q₈	—	316	4	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	158	4	no	—
146	C₂×C₄	(C₂×D₇₉)⋊C₈	1	2	yes	—
147	C₂×C₄	(C₂×D₇₉)⋊C₈	1	2	yes	—
148	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	#14579126	1	2	yes	—
150	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
151	C₂×C₄	#14579169	1	2	yes	—
152	C₈	—	2	2	no	—
153	C₈	—	316	2	no	—
154	C₈	#14579141	1	2	yes	—
155	C₈	#14579148	1	1	yes	characteristic
156	C₈	#14579159	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	C₂×(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
159	C₂²	#19098690	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	316	4	no	—
163	C₄	#19098468	1	1	yes	characteristic
164	C₄	#19098531	1	1	yes	characteristic
165	C₄	#19098687	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#23764335	1	1	yes	characteristic
168	C₂	#23765029	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23772334

C₂

yes

maximal, characteristic

C₂×#19098339

C₂

yes

maximal, characteristic

#23761490

C₂

yes

maximal

C₇₉×#2150

C₂

yes

maximal, characteristic

#19099120

C₄

yes

characteristic

C₂²×C₇₉⋊C₁₆

C₂²

yes

—

#19098693

—

C₈×Dic₁₅₈

—

#19098589

C₄

yes

—

#19098587

C₄

yes

characteristic

#19098518

C₂²

yes

—

#19098339

—

C₂×M₃₂×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₉

—

C₂×#11929608

C₂×C₄

yes

—

#14579161

C₈

yes

—

#14579153

C₈

yes

—

#14579140

—

C₄×Dic₁₅₈

—

#14579120

C₂×C₄

yes

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

Q₈

yes

complemented

C₄⋊(C₇₉⋊C₈)

—

Dic₇₉⋊C₈

—

#14579071

—

C₈×Dic₇₉

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₂×Dic₁₅₈

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂×C₈

yes

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

M₁₆

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂×D₄

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

C₂×C₈

yes

characteristic

C₄²×C₇₉

C₂×C₈

yes

characteristic

C₁₆×C₇₉

—

C₂³×C₇₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₈

yes

—

C₂×C₄×C₇₉

C₂²⋊C₈

yes

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×Q₈

yes

—

C₂×C₄×C₇₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₉

#131

yes

—

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂²⋊Q₈

yes

—

C₈×C₇₉

#124

yes

characteristic

C₈×C₇₉

C₄⋊Q₈

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₇₉

#446

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₈×Q₈

yes

—

Dic₇₉

—

#13267

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1984

yes

characteristic

C₂×C₇₉

#2117

yes

—

#2960

—

#2150

D₇₉

yes

characteristic

C₂×#346

—

#1379

—

C₇₉

#13267

yes

characteristic, complemented, Sylow

#533

—

C₂×M₃₂

C₂×D₇₉

yes

—

C₂²×C₁₆

—

#446

—

158

—

C₈×Q₈

—

158

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₄⋊C₁₆

—

100

#346

—

158

—

101

C₂×M₁₆

—

102

C₂²×C₈

C₄×D₇₉

yes

characteristic

103

C₂²×C₈

C₂²×D₇₉

yes

characteristic

104

C₄⋊Q₈

—

105

#125

—

106

C₂²⋊Q₈

—

158

—

107

C₄×Q₈

—

158

—

108

#117

—

109

C₂×C₄²

C₄×D₇₉

yes

characteristic

110

M₃₂

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

158

—

114

C₂×C₁₆

—

158

—

115

C₄⋊C₈

—

158

—

116

C₄⋊C₈

—

158

—

117

#97

—

158

—

118

C₄×C₈

—

158

—

119

C₄×C₈

D₃₁₆

yes

—

120

C₄×C₈

C₂²⋊D₇₉

yes

—

121

C₂×Q₈

—

158

—

122

C₂²×C₄

#11929607

yes

—

123

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

124

M₁₆

—

158

—

125

C₂×C₈

—

126

C₂×C₈

—

158

—

127

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

128

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

129

C₂×C₈

C₂×D₃₁₆

yes

characteristic

130

C₂×C₈

Q₈×D₇₉

yes

—

131

C₂×C₈

#11929623

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

158

—

134

C₄⋊C₄

—

158

—

135

C₂²⋊C₄

—

158

—

136

C₄²

—

158

—

137

C₄²

(C₂×D₇₉)⋊C₄

yes

—

138

C₄²

C₈×D₇₉

yes

characteristic

139

C₄²

C₈×D₇₉

yes

characteristic

140

C₁₆

—

141

C₁₆

—

158

—

142

C₂³

C₂×#11929607

yes

characteristic

143

Q₈

—

316

—

144

C₂×C₄

—

145

C₂×C₄

—

158

—

146

C₂×C₄

(C₂×D₇₉)⋊C₈

yes

—

147

C₂×C₄

(C₂×D₇₉)⋊C₈

yes

—

148

C₂×C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

149

C₂×C₄

#14579126

yes

—

150

C₂×C₄

C₂×C₈×D₇₉

yes

characteristic

151

C₂×C₄

#14579169

yes

—

152

C₈

—

153

C₈

—

316

—

154

C₈

#14579141

yes

—

155

C₈

#14579148

yes

characteristic

156

C₈

#14579159

yes

characteristic

157

C₂²

—

158

C₂²

C₂×(C₂×D₇₉)⋊C₈

yes

characteristic

159

C₂²

#19098690

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

316

—

163

C₄

#19098468

yes

characteristic

164

C₄

#19098531

yes

characteristic

165

C₄

#19098687

yes

—

166

C₂

—

167

C₂

#23764335

yes

characteristic

168

C₂

#23765029

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2150	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₈
prime radical	C₂³×C₇₉	C₂×M₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2150	D₇₉
hypocenter	C₇₉	#13267
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#2150

C₂

socle

C₂²×C₇₉

C₂×C₂²⋊C₈

prime radical

C₂³×C₇₉

C₂×M₁₆

cosocle

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
24. C₂×C₇₉⋊C₁₆	Q₈	semidirect
89. C₇₉	#13267	semidirect

H ≅ G/N

type

24. C₂×C₇₉⋊C₁₆

Q₈

semidirect

89. C₇₉

#13267

semidirect

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

subgroup	index
#23772334	2
C₂×#19098339	2
#23761490	2
C₇₉×#2150	2
#13267	79

subgroup

index

C₇₉×#2150

Subgroups — #34206803

Conjugacy classes of subgroups

Complements

Maximal subgroups