Group #34208852 — subgroups

Subgroups — #34208852

Order 20224.

190 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 67 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×#19098403	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#23765258	C₂	1	1	yes	maximal, characteristic
4	#23761599	C₂	1	1	yes	maximal, characteristic
5	#23761582	C₂	1	1	yes	maximal, characteristic
6	C₃₂×Dic₇₉	C₂	1	1	yes	maximal, characteristic
7	C₇₉×#1416	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₆₃₂	C₂²	1	1	yes	characteristic
9	Q₁₆×Dic₇₉	—	2	1	no	—
10	#19098657	—	2	1	no	—
11	#19098403	C₄	1	4	yes	complemented
12	C₂×C₇₉⋊C₃₂	C₂²	1	1	yes	characteristic
13	#19098389	C₂²	1	1	yes	characteristic
14	#19098367	—	2	1	no	—
15	#19098366	—	2	1	no	—
16	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
17	C₂×Q₃₂×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×C₃₂×C₇₉	C₂²	1	1	yes	characteristic
19	#366×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×Dic₃₁₆	—	2	1	no	—
21	Q₈×Dic₇₉	—	4	1	no	—
22	#14579126	—	4	1	no	—
23	Dic₆₃₂	—	2	1	no	—
24	Dic₆₃₂	C₂×C₄	1	2	yes	—
25	C₂×C₇₉⋊C₁₆	D₄	1	1	yes	characteristic
26	#14579095	—	2	1	no	—
27	#14579085	—	4	1	no	—
28	#14579084	—	4	1	no	—
29	C₈×Dic₇₉	D₄	1	1	yes	characteristic
30	C₇₉⋊C₃₂	C₂×C₄	1	2	yes	—
31	C₂×Q₁₆×C₇₉	—	2	1	no	—
32	Q₃₂×C₇₉	—	2	1	no	—
33	Q₃₂×C₇₉	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
35	#107×C₇₉	—	2	1	no	—
36	C₃₂×C₇₉	—	2	1	no	—
37	C₂×Dic₁₅₈	—	4	1	no	—
38	Dic₃₁₆	—	2	2	no	—
39	Dic₃₁₆	—	4	1	no	—
40	C₂×C₇₉⋊C₈	D₈	1	1	yes	characteristic
41	C₄⋊Dic₇₉	—	4	1	no	—
42	C₄⋊Dic₇₉	—	8	1	no	—
43	Dic₇₉⋊C₄	—	8	1	no	—
44	C₄×Dic₇₉	—	8	1	no	—
45	C₄×Dic₇₉	—	8	1	no	—
46	C₄×Dic₇₉	D₈	1	1	yes	characteristic
47	C₇₉⋊C₁₆	—	2	1	no	—
48	C₂×Q₈×C₇₉	—	4	1	no	—
49	Q₁₆×C₇₉	—	2	2	no	—
50	Q₁₆×C₇₉	—	4	1	no	—
51	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₄⋊C₄×C₇₉	—	4	1	no	—
53	C₁₆×C₇₉	C₂²×C₄	1	1	yes	characteristic
54	C₁₆×C₇₉	#41	1	1	yes	characteristic
55	Q₈×C₇₉	—	4	2	no	—
56	Q₈×C₇₉	—	8	1	no	—
57	C₂×C₄×C₇₉	—	8	1	no	—
58	C₂×C₄×C₇₉	—	8	1	no	—
59	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
60	C₂×Dic₇₉	—	8	1	no	—
61	C₂×Dic₇₉	—	8	1	no	—
62	C₂×Dic₇₉	D₁₆	1	1	yes	characteristic
63	C₂×Dic₇₉	D₁₆	1	1	yes	characteristic
64	Dic₁₅₈	—	4	2	no	—
65	Dic₁₅₈	—	8	1	no	—
66	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
67	C₈×C₇₉	#135	1	1	yes	characteristic
68	C₇₉⋊C₈	—	2	1	no	—
69	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
70	C₄×C₇₉	—	8	2	no	—
71	C₄×C₇₉	—	16	1	no	—
72	C₄×C₇₉	C₄×D₈	1	1	yes	characteristic
73	C₄×C₇₉	#508	1	1	yes	characteristic
74	Dic₇₉	—	2	1	no	—
75	Dic₇₉	—	8	2	no	—
76	Dic₇₉	—	16	1	no	—
77	Dic₇₉	Q₆₄	1	2	yes	complemented
78	C₄×Q₆₄	—	79	1	no	maximal, Sylow
79	C₂×C₇₉	C₄×D₁₆	1	1	yes	characteristic
80	C₂×C₇₉	C₂×Q₆₄	1	1	yes	characteristic
81	C₂×C₇₉	#2262	1	1	yes	characteristic
82	C₂×Q₆₄	—	79	1	no	—
83	C₄×Q₃₂	—	79	1	no	—
84	C₄×Q₃₂	—	79	1	no	—
85	#1423	—	79	1	no	—
86	#1416	—	79	1	no	—
87	#1416	D₇₉	1	1	yes	characteristic
88	C₄×C₃₂	—	79	1	no	—
89	C₇₉	C₄×Q₆₄	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₃₂	—	79	1	no	—
91	C₂×Q₃₂	C₂×D₇₉	1	1	yes	characteristic
92	C₄×Q₁₆	—	158	1	no	—
93	C₄×Q₁₆	—	158	1	no	—
94	Q₆₄	—	79	4	no	—
95	C₂×C₃₂	—	79	1	no	—
96	C₂×C₃₂	C₂×D₇₉	1	1	yes	characteristic
97	#366	—	79	1	no	—
98	#366	C₂×D₇₉	1	1	yes	characteristic
99	#358	—	158	1	no	—
100	#358	—	158	1	no	—
101	C₄×C₁₆	—	79	1	no	—
102	C₂×Q₁₆	—	2	1	no	—
103	C₂×Q₁₆	—	158	1	no	—
104	C₄×Q₈	—	316	1	no	—
105	C₄×Q₈	—	316	1	no	—
106	Q₃₂	—	2	1	no	—
107	Q₃₂	—	79	2	no	—
108	Q₃₂	—	158	1	no	—
109	Q₃₂	C₄×D₇₉	1	2	yes	—
110	C₂×C₁₆	—	79	1	no	—
111	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
112	#107	—	2	1	no	—
113	#107	—	158	1	no	—
114	#103	—	316	1	no	—
115	#103	—	316	1	no	—
116	C₄×C₈	—	79	1	no	—
117	C₃₂	—	2	1	no	—
118	C₃₂	—	79	2	no	—
119	C₂×Q₈	—	4	1	no	—
120	C₂×Q₈	—	316	1	no	—
121	Q₁₆	—	2	2	no	—
122	Q₁₆	—	4	1	no	—
123	Q₁₆	—	316	1	no	—
124	Q₁₆	—	158	2	no	—
125	C₂×C₈	—	79	1	no	—
126	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	—	316	1	no	—
129	C₄⋊C₄	—	632	1	no	—
130	C₄⋊C₄	—	632	1	no	—
131	C₄²	—	79	1	no	—
132	C₄²	—	632	1	no	—
133	C₄²	—	632	1	no	—
134	C₁₆	—	158	1	no	—
135	C₁₆	C₂×C₄×D₇₉	1	1	yes	characteristic
136	C₁₆	#11929623	1	1	yes	characteristic
137	Q₈	—	4	2	no	—
138	Q₈	—	8	1	no	—
139	Q₈	—	316	2	no	—
140	Q₈	—	632	1	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	79	1	no	—
144	C₂×C₄	—	79	1	no	—
145	C₂×C₄	—	632	1	no	—
146	C₂×C₄	—	632	1	no	—
147	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
148	C₈	—	158	1	no	—
149	C₈	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
150	C₈	#14579182	1	1	yes	characteristic
151	C₂²	D₁₆×D₇₉	1	1	yes	characteristic
152	C₄	—	8	2	no	—
153	C₄	—	16	1	no	—
154	C₄	—	79	2	no	—
155	C₄	—	632	2	no	—
156	C₄	—	158	1	no	—
157	C₄	—	1264	1	no	—
158	C₄	#19098643	1	1	yes	characteristic
159	C₄	#19098994	1	1	yes	characteristic
160	C₂	#23765244	1	1	yes	characteristic
161	C₂	Q₆₄×D₇₉	1	1	yes	characteristic
162	C₂	#23765708	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098403

C₂

yes

maximal, characteristic

Q₃₂×Dic₇₉

C₂

yes

maximal, characteristic

#23765258

C₂

yes

maximal, characteristic

#23761599

C₂

yes

maximal, characteristic

#23761582

C₂

yes

maximal, characteristic

C₃₂×Dic₇₉

C₂

yes

maximal, characteristic

C₇₉×#1416

C₂

yes

maximal, characteristic

C₂×Dic₆₃₂

C₂²

yes

characteristic

Q₁₆×Dic₇₉

—

#19098657

—

#19098403

C₄

yes

complemented

C₂×C₇₉⋊C₃₂

C₂²

yes

characteristic

#19098389

C₂²

yes

characteristic

#19098367

—

#19098366

—

C₁₆×Dic₇₉

C₂²

yes

characteristic

C₂×Q₃₂×C₇₉

C₂²

yes

characteristic

C₂×C₃₂×C₇₉

C₂²

yes

characteristic

#366×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

—

Q₈×Dic₇₉

—

#14579126

—

Dic₆₃₂

—

Dic₆₃₂

C₂×C₄

yes

—

C₂×C₇₉⋊C₁₆

D₄

yes

—

—

—

C₈×Dic₇₉

D₄

yes

characteristic

C₇₉⋊C₃₂

C₂×C₄

yes

—

C₂×Q₁₆×C₇₉

—

Q₃₂×C₇₉

—

Q₃₂×C₇₉

C₂×C₄

yes

—

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

#107×C₇₉

—

C₃₂×C₇₉

—

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

—

C₂×C₇₉⋊C₈

D₈

yes

characteristic

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

D₈

yes

characteristic

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₁₆×C₇₉

C₂²×C₄

yes

characteristic

C₁₆×C₇₉

#41

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

D₁₆

yes

characteristic

C₂×Dic₇₉

D₁₆

yes

characteristic

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

C₄×D₄

yes

characteristic

C₈×C₇₉

#135

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

C₂×D₁₆

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₄×D₈

yes

characteristic

C₄×C₇₉

#508

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

Q₆₄

yes

complemented

C₄×Q₆₄

—

maximal, Sylow

C₂×C₇₉

C₄×D₁₆

yes

characteristic

C₂×C₇₉

C₂×Q₆₄

yes

characteristic

C₂×C₇₉

#2262

yes

characteristic

C₂×Q₆₄

—

C₄×Q₃₂

—

C₄×Q₃₂

—

#1423

—

#1416

—

#1416

D₇₉

yes

characteristic

C₄×C₃₂

—

C₇₉

C₄×Q₆₄

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

C₂×D₇₉

yes

characteristic

C₄×Q₁₆

—

158

—

C₄×Q₁₆

—

158

—

Q₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₉

yes

characteristic

#366

—

#366

C₂×D₇₉

yes

characteristic

#358

—

158

—

100

#358

—

158

—

101

C₄×C₁₆

—

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

158

—

104

C₄×Q₈

—

316

—

105

C₄×Q₈

—

316

—

106

Q₃₂

—

107

Q₃₂

—

108

Q₃₂

—

158

—

109

Q₃₂

C₄×D₇₉

yes

—

110

C₂×C₁₆

—

111

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

112

#107

—

113

#107

—

158

—

114

#103

—

316

—

115

#103

—

316

—

116

C₄×C₈

—

117

C₃₂

—

118

C₃₂

—

119

C₂×Q₈

—

120

C₂×Q₈

—

316

—

121

Q₁₆

—

122

Q₁₆

—

123

Q₁₆

—

316

—

124

Q₁₆

—

158

—

125

C₂×C₈

—

126

C₂×C₈

D₄×D₇₉

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

316

—

129

C₄⋊C₄

—

632

—

130

C₄⋊C₄

—

632

—

131

C₄²

—

132

C₄²

—

632

—

133

C₄²

—

632

—

134

C₁₆

—

158

—

135

C₁₆

C₂×C₄×D₇₉

yes

characteristic

136

C₁₆

#11929623

yes

characteristic

137

Q₈

—

138

Q₈

—

139

Q₈

—

316

—

140

Q₈

—

632

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

632

—

146

C₂×C₄

—

632

—

147

C₂×C₄

D₈×D₇₉

yes

characteristic

148

C₈

—

158

—

149

C₈

(C₂²⋊D₇₉)⋊C₄

yes

characteristic

150

C₈

#14579182

yes

characteristic

151

C₂²

D₁₆×D₇₉

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

632

—

156

C₄

—

158

—

157

C₄

—

1264

—

158

C₄

#19098643

yes

characteristic

159

C₄

#19098994

yes

characteristic

160

C₂

#23765244

yes

characteristic

161

C₂

Q₆₄×D₇₉

yes

characteristic

162

C₂

#23765708

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₇₉
commutator	C₁₆×C₇₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₇₉×#1416	C₂
socle, prime radical	C₂²×C₇₉	C₂×D₁₆
cosocle	C₂×C₁₆×C₇₉	C₂³
hypercenter	#1416	D₇₉
hypocenter	C₇₉	C₄×Q₆₄
LCS	G ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

subgroup

G/H

center

C₂²

D₁₆×D₇₉

commutator

C₁₆×C₇₉

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₇₉

Fitting

C₇₉×#1416

C₂

socle, prime radical

C₂²×C₇₉

C₂×D₁₆

cosocle

C₂×C₁₆×C₇₉

C₂³

hypercenter

#1416

D₇₉

hypocenter

C₇₉

C₄×Q₆₄

LCS

G ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

N	H ≅ G/N	type
11. #19098403	C₄	semidirect
77. Dic₇₉	Q₆₄	semidirect
89. C₇₉	C₄×Q₆₄	semidirect

H ≅ G/N

type

11. #19098403

C₄

semidirect

77. Dic₇₉

Q₆₄

semidirect

89. C₇₉

C₄×Q₆₄

semidirect

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

subgroup	index
C₂×#19098403	2
Q₃₂×Dic₇₉	2
#23765258	2
#23761599	2
#23761582	2
C₃₂×Dic₇₉	2
C₇₉×#1416	2
C₄×Q₆₄	79

subgroup

index

Q₃₂×Dic₇₉

C₃₂×Dic₇₉

C₇₉×#1416

C₄×Q₆₄

Subgroups — #34208852

Conjugacy classes of subgroups

Complements

Maximal subgroups