Group #34204618 — subgroups

Subgroups — #34204618

Order 20224.

322 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 151 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	#23763086	C₂	1	1	yes	maximal, characteristic
3	#23761477	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14579072	C₄	1	1	yes	characteristic
7	C₂×#14579071	C₄	1	2	yes	—
8	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
9	#19098409	C₄	1	1	yes	characteristic
10	C₄⋊(C₇₉⋊C₁₆)	—	2	2	no	—
11	C₂²×C₁₆×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	C₂×C₄	1	2	yes	—
16	C₂×C₇₉⋊C₁₆	—	2	2	no	—
17	C₂×C₇₉⋊C₁₆	—	2	4	no	—
18	#14579072	—	2	2	no	—
19	#14579071	—	2	4	no	—
20	C₈×Dic₇₉	—	2	2	no	—
21	C₈×Dic₇₉	—	2	2	no	—
22	C₇₉⋊#95	C₈	1	2	yes	—
23	C₇₉⋊#95	C₈	1	2	yes	—
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	C₂×C₁₆×C₇₉	—	2	2	no	—
28	C₂×C₁₆×C₇₉	—	2	4	no	—
29	C₄×C₈×C₇₉	D₄	1	2	yes	—
30	C₄×C₈×C₇₉	Q₈	1	2	yes	—
31	C₂²×Dic₇₉	C₂×C₈	1	4	yes	—
32	C₂×C₇₉⋊C₈	—	2	2	no	—
33	C₂×C₇₉⋊C₈	—	2	2	no	—
34	C₂×C₇₉⋊C₈	M₁₆	1	4	yes	—
35	C₄×Dic₇₉	—	2	2	no	—
36	C₄×Dic₇₉	—	2	2	no	—
37	C₇₉⋊C₁₆	—	4	2	no	—
38	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
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	M₁₆	1	2	yes	—
49	C₁₆×C₇₉	—	4	2	no	—
50	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₉	—	2	2	no	—
52	C₂×C₄×C₇₉	—	2	2	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	C₂×M₁₆	1	2	yes	—
58	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
59	C₂×Dic₇₉	—	2	4	no	—
60	C₂×Dic₇₉	—	2	4	no	—
61	C₂×Dic₇₉	—	2	4	no	—
62	C₈×C₇₉	—	2	2	no	—
63	C₈×C₇₉	#124	1	1	yes	characteristic
64	C₈×C₇₉	#124	1	1	yes	characteristic
65	C₈×C₇₉	#125	1	2	yes	—
66	C₇₉⋊C₈	—	2	4	no	—
67	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
68	C₂²×C₇₉	#431	1	2	yes	—
69	C₂²×C₇₉	#504	1	2	yes	—
70	C₂²×C₇₉	#504	1	2	yes	—
71	C₄×C₇₉	—	2	2	no	—
72	C₄×C₇₉	—	2	2	no	—
73	C₄×C₇₉	#382	1	1	yes	characteristic
74	C₄×C₇₉	#423	1	1	yes	characteristic
75	C₄×C₇₉	#432	1	2	yes	—
76	Dic₇₉	—	4	4	no	—
77	#12880	—	79	1	no	maximal, Sylow
78	C₂×C₇₉	#1831	1	1	yes	characteristic
79	C₂×C₇₉	#2151	1	1	yes	characteristic
80	C₂×C₇₉	#2151	1	1	yes	characteristic
81	C₂×C₇₉	#2164	1	4	yes	—
82	C₂×C₄⋊C₁₆	—	79	1	no	—
83	C₂×C₄⋊C₁₆	D₇₉	1	1	yes	characteristic
84	#1726	—	79	1	no	—
85	#1374	—	79	4	no	—
86	C₇₉	#12880	1	1	yes	characteristic, complemented, Sylow
87	C₂²×C₁₆	—	79	2	no	—
88	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
89	C₂×#97	—	79	2	no	—
90	C₂×#97	—	79	1	no	—
91	C₂×C₄×C₈	—	79	2	no	—
92	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
93	#376	—	79	1	no	—
94	C₄⋊C₁₆	—	2	2	no	—
95	C₄⋊C₁₆	—	158	2	no	—
96	C₂²×C₈	—	79	2	no	—
97	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
98	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
99	C₂×C₄²	—	79	2	no	—
100	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	2	4	no	—
103	C₂×C₁₆	—	158	2	no	—
104	C₂×C₁₆	—	158	4	no	—
105	#97	—	158	4	no	—
106	#97	—	158	2	no	—
107	C₄×C₈	—	158	2	no	—
108	C₄×C₈	—	158	2	no	—
109	C₄×C₈	Dic₁₅₈	1	2	yes	—
110	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
111	#95	—	79	2	no	—
112	#95	—	79	2	no	—
113	C₂²×C₄	—	79	4	no	—
114	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
115	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
116	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	79	4	no	—
119	C₂×C₈	—	158	2	no	—
120	C₂×C₈	—	158	2	no	—
121	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
122	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
123	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
124	C₂×C₈	#11929623	1	2	yes	—
125	C₂×C₈	#11929625	1	2	yes	—
126	C₄²	—	158	2	no	—
127	C₄²	—	158	2	no	—
128	C₄²	Dic₇₉⋊C₄	1	2	yes	—
129	C₄²	#11929607	1	2	yes	—
130	C₁₆	—	4	2	no	—
131	C₁₆	—	316	2	no	—
132	C₂³	C₂×C₈×D₇₉	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	158	4	no	—
136	C₂×C₄	—	158	4	no	—
137	C₂×C₄	—	158	4	no	—
138	C₂×C₄	Dic₇₉⋊C₈	1	2	yes	—
139	C₂×C₄	Dic₇₉⋊C₈	1	2	yes	—
140	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
141	C₂×C₄	#14579119	1	2	yes	—
142	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
143	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	158	4	no	—
146	C₈	#14579148	1	1	yes	characteristic
147	C₈	#14579150	1	1	yes	characteristic
148	C₈	#14579152	1	2	yes	—
149	C₂²	C₂×Dic₇₉⋊C₈	1	1	yes	characteristic
150	C₂²	#19098615	1	2	yes	—
151	C₂²	#19098979	1	2	yes	—
152	C₂²	#19098981	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	316	4	no	—
156	C₄	#19098428	1	1	yes	characteristic
157	C₄	#19098586	1	1	yes	characteristic
158	C₄	#19098619	1	2	yes	—
159	C₂	#23763516	1	1	yes	characteristic
160	C₂	#23765158	1	1	yes	characteristic
161	C₂	#23765160	1	1	yes	characteristic
162	C₂	#23765206	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₇₉⋊C₁₆)

C₂

yes

maximal, characteristic

#23763086

C₂

yes

maximal, characteristic

#23761477

C₂

yes

maximal

C₂×C₄⋊C₁₆×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₁₆

C₂²

yes

—

C₂×#14579072

C₄

yes

characteristic

C₂×#14579071

C₄

yes

—

C₂×C₈×Dic₇₉

C₂²

yes

—

#19098409

C₄

yes

characteristic

C₄⋊(C₇₉⋊C₁₆)

—

C₂²×C₁₆×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₉

—

C₂²×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₇₉

C₂×C₄

yes

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

#14579072

—

#14579071

—

C₈×Dic₇₉

—

C₈×Dic₇₉

—

C₇₉⋊#95

C₈

yes

—

C₇₉⋊#95

C₈

yes

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

C₂²×Dic₇₉

C₂×C₈

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

M₁₆

yes

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₉

C₂×C₈

yes

characteristic

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

M₁₆

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₉

—

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

C₂×M₁₆

yes

—

C₂×C₄×C₇₉

C₂×M₁₆

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₈×C₇₉

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₇₉

#431

yes

—

C₂²×C₇₉

#504

yes

—

C₂²×C₇₉

#504

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

D₇₉

yes

characteristic

#1726

—

#1374

—

C₇₉

#12880

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₉

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#376

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

158

—

C₂²×C₈

—

C₂²×C₈

C₄×D₇₉

yes

characteristic

C₂²×C₈

C₂²×D₇₉

yes

characteristic

C₂×C₄²

—

100

C₂×C₄²

C₄×D₇₉

yes

characteristic

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

158

—

104

C₂×C₁₆

—

158

—

105

#97

—

158

—

106

#97

—

158

—

107

C₄×C₈

—

158

—

108

C₄×C₈

—

158

—

109

C₄×C₈

Dic₁₅₈

yes

—

110

C₄×C₈

C₂²⋊D₇₉

yes

—

111

#95

—

112

#95

—

113

C₂²×C₄

—

114

C₂²×C₄

C₈×D₇₉

yes

characteristic

115

C₂²×C₄

C₈×D₇₉

yes

characteristic

116

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

158

—

120

C₂×C₈

—

158

—

121

C₂×C₈

Dic₇₉⋊C₄

yes

—

122

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

123

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

124

C₂×C₈

#11929623

yes

—

125

C₂×C₈

#11929625

yes

—

126

C₄²

—

158

—

127

C₄²

—

158

—

128

C₄²

Dic₇₉⋊C₄

yes

—

129

C₄²

#11929607

yes

—

130

C₁₆

—

131

C₁₆

—

316

—

132

C₂³

C₂×C₈×D₇₉

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

158

—

136

C₂×C₄

—

158

—

137

C₂×C₄

—

158

—

138

C₂×C₄

Dic₇₉⋊C₈

yes

—

139

C₂×C₄

Dic₇₉⋊C₈

yes

—

140

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

141

C₂×C₄

#14579119

yes

—

142

C₂×C₄

M₁₆×D₇₉

yes

—

143

C₂×C₄

C₂×#11929607

yes

characteristic

144

C₈

—

145

C₈

—

158

—

146

C₈

#14579148

yes

characteristic

147

C₈

#14579150

yes

characteristic

148

C₈

#14579152

yes

—

149

C₂²

C₂×Dic₇₉⋊C₈

yes

characteristic

150

C₂²

#19098615

yes

—

151

C₂²

#19098979

yes

—

152

C₂²

#19098981

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

316

—

156

C₄

#19098428

yes

characteristic

157

C₄

#19098586

yes

characteristic

158

C₄

#19098619

yes

—

159

C₂

#23763516

yes

characteristic

160

C₂

#23765158

yes

characteristic

161

C₂

#23765160

yes

characteristic

162

C₂

#23765206

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₉
commutator, socle, prime radical	C₂³×C₇₉	C₂²×C₈
Fitting	C₂×C₄⋊C₁₆×C₇₉	C₂
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₇₉
hypocenter	C₇₉	#12880
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₇₉

commutator, socle, prime radical

C₂³×C₇₉

C₂²×C₈

Fitting

C₂×C₄⋊C₁₆×C₇₉

C₂

cosocle

C₂²×C₈×C₇₉

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
86. C₇₉	#12880	semidirect

H ≅ G/N

type

86. C₇₉

#12880

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
#23763086	2
#23761477	2
C₂×C₄⋊C₁₆×C₇₉	2
#12880	79

subgroup

index

C₂×C₄⋊(C₇₉⋊C₁₆)

#23763086

#23761477

C₂×C₄⋊C₁₆×C₇₉

#12880

Subgroups — #34204618

Conjugacy classes of subgroups

Complements

Maximal subgroups