Group #34202532 — subgroups

Subgroups — #34202532

Order 20224.

216 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 69 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	#23765653	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic
3	#23762771	C₂	1	2	yes	maximal
4	#23761448	C₂	1	2	yes	maximal
5	C₇₉×#1371	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₃₂	—	2	1	no	—
7	C₂×Dic₆₃₂	—	2	1	no	—
8	#19098943	C₂²	1	2	yes	—
9	#19098849	—	2	2	no	—
10	C₈×Dic₁₅₈	C₂²	1	1	yes	characteristic
11	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
12	Dic₇₉⋊C₁₆	—	2	1	no	—
13	C₁₆×Dic₇₉	—	2	1	no	—
14	#19098308	—	2	2	no	—
15	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	#335×C₇₉	C₂²	1	2	yes	—
17	C₂×Dic₃₁₆	—	2	2	no	—
18	C₂×Dic₃₁₆	—	2	2	no	—
19	#14579161	—	2	2	no	—
20	C₄×Dic₁₅₈	D₄	1	1	yes	characteristic
21	Dic₆₃₂	—	4	2	no	—
22	Dic₆₃₂	—	4	2	no	—
23	C₂×C₇₉⋊C₁₆	—	2	1	no	—
24	#14579093	—	4	2	no	—
25	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
26	Dic₇₉⋊C₈	—	2	1	no	—
27	#14579086	—	4	2	no	—
28	C₈×Dic₇₉	—	2	1	no	—
29	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
31	C₄⋊C₈×C₇₉	—	2	2	no	—
32	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₅₈	—	4	2	no	—
34	C₂×Dic₁₅₈	D₈	1	1	yes	characteristic
35	Dic₃₁₆	—	4	2	no	—
36	Dic₃₁₆	—	4	2	no	—
37	Dic₃₁₆	—	8	2	no	—
38	C₂×C₇₉⋊C₈	—	2	1	no	—
39	C₄⋊Dic₇₉	—	4	2	no	—
40	C₄⋊Dic₇₉	—	4	2	no	—
41	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
42	Dic₇₉⋊C₄	—	2	1	no	—
43	C₄×Dic₇₉	—	2	1	no	—
44	C₇₉⋊C₁₆	—	4	1	no	—
45	C₂×C₈×C₇₉	—	4	2	no	—
46	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
48	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₇₉	—	2	1	no	—
50	C₁₆×C₇₉	SD₁₆	1	2	yes	—
51	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
54	C₂×Dic₇₉	—	2	1	no	—
55	C₂×Dic₇₉	—	2	1	no	—
56	C₂×Dic₇₉	—	8	2	no	—
57	Dic₁₅₈	—	2	1	no	—
58	Dic₁₅₈	—	8	2	no	—
59	Dic₁₅₈	—	8	2	no	—
60	Dic₁₅₈	Q₃₂	1	2	yes	complemented
61	C₈×C₇₉	—	2	1	no	—
62	C₈×C₇₉	—	8	2	no	—
63	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
64	C₈×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₇₉⋊C₈	—	4	1	no	—
66	C₂²×C₇₉	#466	1	1	yes	characteristic
67	C₄×C₇₉	—	2	1	no	—
68	C₄×C₇₉	#460	1	1	yes	characteristic
69	C₄×C₇₉	#472	1	1	yes	characteristic
70	C₄×C₇₉	C₂×Q₃₂	1	1	yes	characteristic
71	C₄×C₇₉	#508	1	1	yes	characteristic
72	Dic₇₉	—	2	2	no	—
73	Dic₇₉	—	4	1	no	—
74	Dic₇₉	—	16	2	no	—
75	#12440	—	79	1	no	maximal, Sylow
76	C₂×C₇₉	#1668	1	1	yes	characteristic
77	C₂×C₇₉	#2216	1	1	yes	characteristic
78	C₂×C₇₉	#2222	1	1	yes	characteristic
79	#2247	—	79	1	no	—
80	Q₈×C₁₆	—	79	1	no	—
81	#1670	—	79	2	no	—
82	#1371	D₇₉	1	1	yes	characteristic
83	#1362	—	79	2	no	—
84	C₇₉	#12440	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₃₂	—	158	1	no	—
86	C₂×Q₃₂	—	158	1	no	—
87	#494	—	79	2	no	—
88	#475	—	158	2	no	—
89	C₈×Q₈	—	79	1	no	—
90	C₄⋊C₁₆	—	79	1	no	—
91	C₄⋊C₁₆	—	158	1	no	—
92	C₄×C₁₆	—	158	1	no	—
93	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
94	#335	C₂×D₇₉	1	2	yes	—
95	#332	—	158	2	no	—
96	C₂×Q₁₆	—	158	2	no	—
97	C₂×Q₁₆	—	158	2	no	—
98	C₄⋊Q₈	—	158	2	no	—
99	C₄×Q₈	—	79	1	no	—
100	Q₃₂	—	316	2	no	—
101	Q₃₂	—	316	2	no	—
102	C₂×C₁₆	—	158	1	no	—
103	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
104	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
105	#106	—	316	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	79	1	no	—
108	C₄⋊C₈	—	158	1	no	—
109	#103	—	316	2	no	—
110	C₄×C₈	—	158	1	no	—
111	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
112	C₂×Q₈	—	316	2	no	—
113	C₂×Q₈	—	79	1	no	—
114	Q₁₆	—	316	2	no	—
115	Q₁₆	—	316	2	no	—
116	Q₁₆	—	632	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	158	1	no	—
119	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
120	C₂×C₈	#11929625	1	1	yes	characteristic
121	C₄⋊C₄	—	316	2	no	—
122	C₄⋊C₄	—	316	2	no	—
123	C₄⋊C₄	—	79	1	no	—
124	C₄⋊C₄	—	158	1	no	—
125	C₄²	—	158	1	no	—
126	C₄²	D₄×D₇₉	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	316	1	no	—
129	C₁₆	#11929613	1	2	yes	—
130	Q₈	—	79	2	no	—
131	Q₈	—	632	2	no	—
132	Q₈	—	632	2	no	—
133	Q₈	—	158	1	no	—
134	C₂×C₄	—	632	2	no	—
135	C₂×C₄	—	158	1	no	—
136	C₂×C₄	—	158	1	no	—
137	C₂×C₄	#14579136	1	1	yes	characteristic
138	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
139	C₂×C₄	#14579184	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	8	2	no	—
142	C₈	—	316	1	no	—
143	C₈	C₂×#11929613	1	1	yes	characteristic
144	C₈	#14579189	1	1	yes	characteristic
145	C₂²	#19098800	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	316	1	no	—
148	C₄	—	158	2	no	—
149	C₄	—	1264	2	no	—
150	C₄	#19098764	1	1	yes	characteristic
151	C₄	#19098834	1	1	yes	characteristic
152	C₄	Q₃₂×D₇₉	1	1	yes	characteristic
153	C₄	#19098992	1	1	yes	characteristic
154	C₂	#23762761	1	1	yes	characteristic
155	C₂	#23765492	1	1	yes	characteristic
156	C₂	#23765526	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765653

C₂

yes

maximal, characteristic

C₁₆×Dic₁₅₈

C₂

yes

maximal, characteristic

#23762771

C₂

yes

maximal

#23761448

C₂

yes

maximal

C₇₉×#1371

C₂

yes

maximal, characteristic

C₂×Dic₆₃₂

—

C₂×Dic₆₃₂

—

#19098943

C₂²

yes

—

#19098849

—

C₈×Dic₁₅₈

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₁₆)

C₂²

yes

characteristic

Dic₇₉⋊C₁₆

—

C₁₆×Dic₇₉

—

#19098308

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

#335×C₇₉

C₂²

yes

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

#14579161

—

C₄×Dic₁₅₈

D₄

yes

characteristic

Dic₆₃₂

—

Dic₆₃₂

—

C₂×C₇₉⋊C₁₆

—

#14579093

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

characteristic

Dic₇₉⋊C₈

—

#14579086

—

C₈×Dic₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

D₈

yes

characteristic

Dic₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₈

yes

characteristic

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

#135

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

Q₃₂

yes

complemented

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂×SD₁₆

yes

characteristic

C₈×C₇₉

AGL(1,ℤ/8)

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

#466

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₂×Q₃₂

yes

characteristic

C₄×C₇₉

#508

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

Q₈×C₁₆

—

#1670

—

#1371

D₇₉

yes

characteristic

#1362

—

C₇₉

#12440

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

158

—

C₂×Q₃₂

—

158

—

#494

—

#475

—

158

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

158

—

C₄×C₁₆

—

158

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

#335

C₂×D₇₉

yes

—

#332

—

158

—

C₂×Q₁₆

—

158

—

C₂×Q₁₆

—

158

—

C₄⋊Q₈

—

158

—

C₄×Q₈

—

100

Q₃₂

—

316

—

101

Q₃₂

—

316

—

102

C₂×C₁₆

—

158

—

103

C₂×C₁₆

D₃₁₆

yes

characteristic

104

C₂×C₁₆

D₃₁₆

yes

characteristic

105

#106

—

316

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

158

—

109

#103

—

316

—

110

C₄×C₈

—

158

—

111

C₄×C₈

C₂²×D₇₉

yes

characteristic

112

C₂×Q₈

—

316

—

113

C₂×Q₈

—

114

Q₁₆

—

316

—

115

Q₁₆

—

316

—

116

Q₁₆

—

632

—

117

C₂×C₈

—

118

C₂×C₈

—

158

—

119

C₂×C₈

C₂×D₃₁₆

yes

characteristic

120

C₂×C₈

#11929625

yes

characteristic

121

C₄⋊C₄

—

316

—

122

C₄⋊C₄

—

316

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

158

—

125

C₄²

—

158

—

126

C₄²

D₄×D₇₉

yes

characteristic

127

C₁₆

—

128

C₁₆

—

316

—

129

C₁₆

#11929613

yes

—

130

Q₈

—

131

Q₈

—

632

—

132

Q₈

—

632

—

133

Q₈

—

158

—

134

C₂×C₄

—

632

—

135

C₂×C₄

—

158

—

136

C₂×C₄

—

158

—

137

C₂×C₄

#14579136

yes

characteristic

138

C₂×C₄

D₈×D₇₉

yes

characteristic

139

C₂×C₄

#14579184

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

316

—

143

C₈

C₂×#11929613

yes

characteristic

144

C₈

#14579189

yes

characteristic

145

C₂²

#19098800

yes

characteristic

146

C₄

—

147

C₄

—

316

—

148

C₄

—

158

—

149

C₄

—

1264

—

150

C₄

#19098764

yes

characteristic

151

C₄

#19098834

yes

characteristic

152

C₄

Q₃₂×D₇₉

yes

characteristic

153

C₄

#19098992

yes

characteristic

154

C₂

#23762761

yes

characteristic

155

C₂

#23765492

yes

characteristic

156

C₂

#23765526

yes

characteristic

	subgroup	G/H
center	C₂²	#19098800
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1371	C₂
socle, prime radical	C₂²×C₇₉	#466
hypercenter	#1371	D₇₉
hypocenter	C₇₉	#12440
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂²

#19098800

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1371

C₂

C₂²×C₇₉

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

N	H ≅ G/N	type
60. Dic₁₅₈	Q₃₂	semidirect
84. C₇₉	#12440	semidirect

H ≅ G/N

type

60. Dic₁₅₈

Q₃₂

semidirect

84. C₇₉

#12440

semidirect

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

subgroup	index
#23765653	2
C₁₆×Dic₁₅₈	2
#23762771	2
#23761448	2
C₇₉×#1371	2
#12440	79

subgroup

index

C₁₆×Dic₁₅₈

C₇₉×#1371

Subgroups — #34202532

Conjugacy classes of subgroups

Complements

Maximal subgroups