Group #34204629 — subgroups

Subgroups — #34204629

Order 20224.

342 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 143 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	#23765156	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#23761481	C₂	1	4	yes	maximal
4	C₇₉×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579101	C₂²	1	2	yes	—
6	C₂×C₄×C₇₉⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₇₉	C₄	1	2	yes	—
8	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
9	C₄²×Dic₇₉	C₄	1	1	yes	characteristic
10	C₄⋊(C₇₉⋊C₁₆)	—	2	2	no	—
11	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₇₉	—	2	2	no	—
14	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₇₉	—	2	2	no	—
16	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
17	#14579101	—	2	4	no	—
18	C₂×C₇₉⋊C₁₆	—	2	2	no	—
19	C₄×C₇₉⋊C₈	C₈	1	2	yes	—
20	C₄×C₇₉⋊C₈	C₈	1	2	yes	—
21	C₈×Dic₇₉	—	2	2	no	—
22	C₈×Dic₇₉	—	2	2	no	—
23	C₈×Dic₇₉	—	2	4	no	—
24	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₇₉	—	2	4	no	—
28	C₂×C₁₆×C₇₉	—	2	2	no	—
29	C₄×C₈×C₇₉	D₄	1	2	yes	—
30	C₄×C₈×C₇₉	Q₈	1	2	yes	—
31	C₂²×Dic₇₉	—	2	2	no	—
32	C₂×C₇₉⋊C₈	—	2	2	no	—
33	C₂×C₇₉⋊C₈	—	2	2	no	—
34	C₂×C₇₉⋊C₈	C₂×C₈	1	4	yes	—
35	C₄×Dic₇₉	—	2	8	no	—
36	C₄×Dic₇₉	—	2	2	no	—
37	C₄×Dic₇₉	M₁₆	1	4	yes	—
38	C₇₉⋊C₁₆	—	4	2	no	—
39	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
40	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	—	2	2	no	—
42	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₇₉	#41	1	2	yes	—
46	C₂×C₈×C₇₉	#41	1	2	yes	—
47	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
48	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
50	C₁₆×C₇₉	—	4	2	no	—
51	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
52	C₂×C₄×C₇₉	—	2	4	no	—
53	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₉	#117	1	2	yes	—
57	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
59	C₂×Dic₇₉	—	2	8	no	—
60	C₂×Dic₇₉	—	2	4	no	—
61	C₈×C₇₉	—	2	2	no	—
62	C₈×C₇₉	#124	1	1	yes	characteristic
63	C₈×C₇₉	#124	1	1	yes	characteristic
64	C₈×C₇₉	#125	1	2	yes	—
65	C₇₉⋊C₈	—	2	4	no	—
66	C₂²×C₇₉	—	2	2	no	—
67	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
68	C₂²×C₇₉	#432	1	2	yes	—
69	C₄×C₇₉	—	2	2	no	—
70	C₄×C₇₉	#364	1	4	yes	—
71	C₄×C₇₉	#382	1	1	yes	characteristic
72	C₄×C₇₉	#423	1	1	yes	characteristic
73	C₄×C₇₉	#431	1	2	yes	—
74	Dic₇₉	—	2	8	no	—
75	#12883	—	79	1	no	maximal, Sylow
76	C₂×C₇₉	—	2	2	no	—
77	C₂×C₇₉	#1831	1	1	yes	characteristic
78	C₂×C₇₉	C₂×#364	1	2	yes	—
79	#2150	—	79	1	no	—
80	#2150	D₇₉	1	1	yes	characteristic
81	C₄²×C₈	—	79	1	no	—
82	#1376	—	79	4	no	—
83	C₇₉	#12883	1	1	yes	characteristic, complemented, Sylow
84	C₂×M₃₂	—	79	2	no	—
85	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
86	C₂×C₄×C₈	—	79	1	no	—
87	C₂×C₄×C₈	—	79	2	no	—
88	C₂×C₄×C₈	—	79	2	no	—
89	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
90	C₄³	—	79	1	no	—
91	C₄⋊C₁₆	—	2	2	no	—
92	C₄⋊C₁₆	—	158	2	no	—
93	C₂²×C₈	—	79	2	no	—
94	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
96	C₂×C₄²	—	79	2	no	—
97	C₂×C₄²	—	158	2	no	—
98	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
99	M₃₂	—	2	4	no	—
100	M₃₂	—	158	4	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	158	2	no	—
103	C₄×C₈	—	79	2	no	—
104	C₄×C₈	—	79	2	no	—
105	C₄×C₈	—	158	4	no	—
106	C₄×C₈	—	158	2	no	—
107	C₄×C₈	—	158	2	no	—
108	C₄×C₈	Dic₁₅₈	1	2	yes	—
109	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
110	C₂²×C₄	—	158	2	no	—
111	C₂²×C₄	#11929607	1	2	yes	—
112	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	79	4	no	—
115	C₂×C₈	—	158	2	no	—
116	C₂×C₈	—	158	2	no	—
117	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
118	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
119	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
120	C₂×C₈	#11929623	1	2	yes	—
121	C₂×C₈	#11929625	1	2	yes	—
122	C₄²	—	79	4	no	—
123	C₄²	—	158	8	no	—
124	C₄²	—	158	2	no	—
125	C₄²	Dic₇₉⋊C₄	1	2	yes	—
126	C₄²	C₈×D₇₉	1	1	yes	characteristic
127	C₄²	C₈×D₇₉	1	1	yes	characteristic
128	C₁₆	—	4	2	no	—
129	C₁₆	—	316	2	no	—
130	C₂³	C₂×#11929607	1	1	yes	characteristic
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	158	8	no	—
133	C₂×C₄	—	158	4	no	—
134	C₂×C₄	Dic₇₉⋊C₈	1	2	yes	—
135	C₂×C₄	Dic₇₉⋊C₈	1	2	yes	—
136	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
137	C₂×C₄	#14579119	1	2	yes	—
138	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
139	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
140	C₈	—	2	2	no	—
141	C₈	—	158	4	no	—
142	C₈	#14579148	1	1	yes	characteristic
143	C₈	#14579150	1	1	yes	characteristic
144	C₈	#14579152	1	2	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	C₂×Dic₇₉⋊C₈	1	1	yes	characteristic
147	C₂²	#19098619	1	2	yes	—
148	C₄	—	2	2	no	—
149	C₄	—	158	8	no	—
150	C₄	#19098383	1	4	yes	—
151	C₄	#19098428	1	1	yes	characteristic
152	C₄	#19098586	1	1	yes	characteristic
153	C₄	#19098615	1	2	yes	—
154	C₂	—	2	2	no	—
155	C₂	#23763516	1	1	yes	characteristic
156	C₂	C₂×#19098383	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765156

C₂

yes

maximal, characteristic

C₄×C₈×Dic₇₉

C₂

yes

maximal, characteristic

#23761481

C₂

yes

maximal

C₇₉×#2150

C₂

yes

maximal, characteristic

C₂×#14579101

C₂²

yes

—

C₂×C₄×C₇₉⋊C₈

C₄

yes

characteristic

C₂×C₈×Dic₇₉

C₄

yes

—

C₂×C₈×Dic₇₉

C₂²

yes

—

C₄²×Dic₇₉

C₄

yes

characteristic

C₄⋊(C₇₉⋊C₁₆)

—

C₂×M₃₂×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₉

—

C₂²×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₇₉

—

C₂×C₄×Dic₇₉

C₂×C₄

yes

—

#14579101

—

C₂×C₇₉⋊C₁₆

—

C₄×C₇₉⋊C₈

C₈

yes

—

C₄×C₇₉⋊C₈

C₈

yes

—

C₈×Dic₇₉

—

C₈×Dic₇₉

—

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

Q₈

yes

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×C₈

yes

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

M₁₆

yes

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

M₁₆

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₄²×C₇₉

C₄⋊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

C₄⋊C₈

yes

—

C₂×C₄×C₇₉

C₄⋊C₈

yes

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#117

yes

—

C₂×C₄×C₇₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₉

C₂×M₁₆

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₈×C₇₉

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₇₉

#432

yes

—

C₄×C₇₉

—

C₄×C₇₉

#364

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

#12883

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1831

yes

characteristic

C₂×C₇₉

C₂×#364

yes

—

#2150

—

#2150

D₇₉

yes

characteristic

C₄²×C₈

—

#1376

—

C₇₉

#12883

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₇₉

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₄³

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

158

—

C₂²×C₈

—

C₂²×C₈

C₄×D₇₉

yes

characteristic

C₂²×C₈

C₂²×D₇₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

158

—

C₂×C₄²

C₄×D₇₉

yes

characteristic

M₃₂

—

100

M₃₂

—

158

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

158

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

158

—

106

C₄×C₈

—

158

—

107

C₄×C₈

—

158

—

108

C₄×C₈

Dic₁₅₈

yes

—

109

C₄×C₈

C₂²⋊D₇₉

yes

—

110

C₂²×C₄

—

158

—

111

C₂²×C₄

#11929607

yes

—

112

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

158

—

116

C₂×C₈

—

158

—

117

C₂×C₈

Dic₇₉⋊C₄

yes

—

118

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

119

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

120

C₂×C₈

#11929623

yes

—

121

C₂×C₈

#11929625

yes

—

122

C₄²

—

123

C₄²

—

158

—

124

C₄²

—

158

—

125

C₄²

Dic₇₉⋊C₄

yes

—

126

C₄²

C₈×D₇₉

yes

characteristic

127

C₄²

C₈×D₇₉

yes

characteristic

128

C₁₆

—

129

C₁₆

—

316

—

130

C₂³

C₂×#11929607

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

158

—

133

C₂×C₄

—

158

—

134

C₂×C₄

Dic₇₉⋊C₈

yes

—

135

C₂×C₄

Dic₇₉⋊C₈

yes

—

136

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

137

C₂×C₄

#14579119

yes

—

138

C₂×C₄

C₂×C₈×D₇₉

yes

characteristic

139

C₂×C₄

M₁₆×D₇₉

yes

—

140

C₈

—

141

C₈

—

158

—

142

C₈

#14579148

yes

characteristic

143

C₈

#14579150

yes

characteristic

144

C₈

#14579152

yes

—

145

C₂²

—

146

C₂²

C₂×Dic₇₉⋊C₈

yes

characteristic

147

C₂²

#19098619

yes

—

148

C₄

—

149

C₄

—

158

—

150

C₄

#19098383

yes

—

151

C₄

#19098428

yes

characteristic

152

C₄

#19098586

yes

characteristic

153

C₄

#19098615

yes

—

154

C₂

—

155

C₂

#23763516

yes

characteristic

156

C₂

C₂×#19098383

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×C₄⋊C₈
prime radical	C₂³×C₇₉	C₂×M₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2150	D₇₉
hypocenter	C₇₉	#12883
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×C₄⋊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
83. C₇₉	#12883	semidirect

H ≅ G/N

type

83. C₇₉

#12883

semidirect

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

subgroup	index
#23765156	2
C₄×C₈×Dic₇₉	2
#23761481	2
C₇₉×#2150	2
#12883	79

subgroup

index

#23765156

C₄×C₈×Dic₇₉

#23761481

C₇₉×#2150

#12883

Subgroups — #34204629

Conjugacy classes of subgroups

Complements

Maximal subgroups