Group #34202523 — subgroups

Subgroups — #34202523

Order 20224.

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

normal subgroups only

This group has 248 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	C₁₆×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#23762757	C₂	1	2	yes	maximal, complemented
3	#23761472	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761446	C₂	1	2	yes	maximal, complemented
5	C₇₉×#2246	C₂	1	1	yes	maximal, characteristic
6	#19098760	—	2	2	no	—
7	C₈×Dic₁₅₈	C₂²	1	1	yes	characteristic
8	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
9	Dic₇₉⋊C₁₆	—	2	1	no	—
10	C₁₆×Dic₇₉	—	2	1	no	—
11	#19098316	C₂²	1	2	yes	—
12	#19098307	—	2	2	no	—
13	C₂×D₁₆×C₇₉	—	2	1	no	—
14	C₂×D₁₆×C₇₉	—	2	1	no	—
15	#493×C₇₉	C₂²	1	2	yes	—
16	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#11929612	—	4	2	no	—
18	C₄×Dic₁₅₈	D₄	1	1	yes	characteristic
19	C₂×C₇₉⋊C₁₆	—	2	1	no	—
20	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
21	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
22	Dic₇₉⋊C₈	—	2	1	no	—
23	#14579082	—	4	2	no	—
24	C₈×Dic₇₉	—	2	1	no	—
25	C₂×D₈×C₇₉	—	2	2	no	—
26	C₂×D₈×C₇₉	—	2	2	no	—
27	#127×C₇₉	—	2	2	no	—
28	D₁₆×C₇₉	—	4	2	no	—
29	D₁₆×C₇₉	—	4	2	no	—
30	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₅₈	D₈	1	1	yes	characteristic
34	#11929612	—	8	4	no	—
35	C₂×C₇₉⋊C₈	—	2	1	no	—
36	C₂×C₇₉⋊C₈	—	4	2	no	—
37	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
38	Dic₇₉⋊C₄	—	2	1	no	—
39	C₄×Dic₇₉	—	2	1	no	—
40	C₇₉⋊C₁₆	—	4	1	no	—
41	C₂×D₄×C₇₉	—	4	2	no	—
42	C₂×D₄×C₇₉	—	4	2	no	—
43	C₂×D₄×C₇₉	—	4	2	no	—
44	D₈×C₇₉	—	4	2	no	—
45	D₈×C₇₉	—	4	2	no	—
46	D₈×C₇₉	—	8	2	no	—
47	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
49	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₁₆×C₇₉	—	2	1	no	—
51	C₁₆×C₇₉	SD₁₆	1	2	yes	complemented
52	C₂³×C₇₉	—	8	2	no	—
53	D₄×C₇₉	—	8	2	no	—
54	D₄×C₇₉	—	8	2	no	—
55	D₄×C₇₉	—	8	2	no	—
56	D₄×C₇₉	—	8	2	no	—
57	D₄×C₇₉	—	16	2	no	—
58	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
61	C₂×Dic₇₉	—	2	1	no	—
62	C₂×Dic₇₉	—	2	1	no	—
63	Dic₁₅₈	—	2	1	no	—
64	Dic₁₅₈	D₁₆	1	2	yes	complemented
65	C₈×C₇₉	—	2	1	no	—
66	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
67	C₈×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₇₉⋊C₈	—	4	1	no	—
69	C₇₉⋊C₈	—	8	2	no	—
70	C₂²×C₇₉	—	16	2	no	—
71	C₂²×C₇₉	—	16	2	no	—
72	C₂²×C₇₉	—	16	2	no	—
73	C₂²×C₇₉	#466	1	1	yes	characteristic
74	C₄×C₇₉	—	2	1	no	—
75	C₄×C₇₉	#460	1	1	yes	characteristic
76	C₄×C₇₉	#472	1	1	yes	characteristic
77	C₄×C₇₉	C₂×D₁₆	1	1	yes	characteristic
78	C₄×C₇₉	#508	1	1	yes	characteristic
79	Dic₇₉	—	2	2	no	—
80	Dic₇₉	—	4	1	no	—
81	#12439	—	79	1	no	maximal, Sylow
82	C₂×C₇₉	—	32	2	no	—
83	C₂×C₇₉	#1668	1	1	yes	characteristic
84	C₂×C₇₉	#2215	1	1	yes	characteristic
85	C₂×C₇₉	#2221	1	1	yes	characteristic
86	#2246	D₇₉	1	1	yes	characteristic
87	Q₈×C₁₆	—	79	1	no	—
88	#1668	—	79	2	no	—
89	#1371	—	79	1	no	—
90	#1361	—	79	2	no	—
91	C₇₉	#12439	1	1	yes	characteristic, complemented, Sylow
92	C₂×D₁₆	—	2	1	no	—
93	C₂×D₁₆	—	2	1	no	—
94	#493	C₂×D₇₉	1	2	yes	—
95	#460	—	158	2	no	—
96	C₈×Q₈	—	79	1	no	—
97	C₄⋊C₁₆	—	79	1	no	—
98	C₄⋊C₁₆	—	158	1	no	—
99	C₄×C₁₆	—	158	1	no	—
100	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
101	#335	—	79	2	no	—
102	#331	—	158	2	no	—
103	C₂×SD₁₆	—	316	2	no	—
104	C₂×D₈	—	2	2	no	—
105	C₂×D₈	—	2	2	no	—
106	#127	—	2	2	no	—
107	C₄×Q₈	—	79	1	no	—
108	D₁₆	—	4	2	no	—
109	D₁₆	—	4	2	no	—
110	C₂×C₁₆	—	158	1	no	—
111	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
112	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
113	C₄⋊C₈	—	79	1	no	—
114	C₄⋊C₈	—	158	1	no	—
115	C₄⋊C₈	—	158	2	no	—
116	#102	—	316	2	no	—
117	C₄×C₈	—	158	1	no	—
118	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
119	C₂×Q₈	—	79	1	no	—
120	C₂×D₄	—	4	2	no	—
121	C₂×D₄	—	4	2	no	—
122	C₂×D₄	—	4	2	no	—
123	SD₁₆	—	632	4	no	—
124	D₈	—	4	2	no	—
125	D₈	—	4	2	no	—
126	D₈	—	8	2	no	—
127	C₂×C₈	—	316	2	no	—
128	C₂×C₈	—	158	1	no	—
129	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
130	C₂×C₈	#11929623	1	1	yes	characteristic
131	C₄⋊C₄	—	79	1	no	—
132	C₄⋊C₄	—	158	1	no	—
133	C₄²	—	158	1	no	—
134	C₄²	D₄×D₇₉	1	1	yes	characteristic
135	C₁₆	—	2	1	no	—
136	C₁₆	—	316	1	no	—
137	C₁₆	#11929612	1	2	yes	complemented
138	C₂³	—	8	2	no	—
139	Q₈	—	79	2	no	—
140	Q₈	—	158	1	no	—
141	D₄	—	8	2	no	—
142	D₄	—	8	2	no	—
143	D₄	—	8	2	no	—
144	D₄	—	8	2	no	—
145	D₄	—	16	2	no	—
146	C₂×C₄	—	158	1	no	—
147	C₂×C₄	—	158	1	no	—
148	C₂×C₄	#14579132	1	1	yes	characteristic
149	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
150	C₂×C₄	#14579180	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	316	1	no	—
153	C₈	—	632	2	no	—
154	C₈	C₂×#11929612	1	1	yes	characteristic
155	C₈	#14579187	1	1	yes	characteristic
156	C₂²	—	16	2	no	—
157	C₂²	—	16	2	no	—
158	C₂²	—	16	2	no	—
159	C₂²	#19098796	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	316	1	no	—
162	C₄	—	158	2	no	—
163	C₄	#19098760	1	1	yes	characteristic
164	C₄	#19098830	1	1	yes	characteristic
165	C₄	D₁₆×D₇₉	1	1	yes	characteristic
166	C₄	#19098992	1	1	yes	characteristic
167	C₂	—	32	2	no	—
168	C₂	#23762757	1	1	yes	characteristic
169	C₂	#23765483	1	1	yes	characteristic
170	C₂	#23765517	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₅₈

C₂

yes

maximal, characteristic, complemented

#23762757

C₂

yes

maximal, complemented

#23761472

C₂

yes

maximal, characteristic, complemented

#23761446

C₂

yes

maximal, complemented

C₇₉×#2246

C₂

yes

maximal, characteristic

#19098760

—

C₈×Dic₁₅₈

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₁₆)

C₂²

yes

characteristic

Dic₇₉⋊C₁₆

—

C₁₆×Dic₇₉

—

#19098316

C₂²

yes

—

#19098307

—

C₂×D₁₆×C₇₉

—

C₂×D₁₆×C₇₉

—

#493×C₇₉

C₂²

yes

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×#11929612

—

C₄×Dic₁₅₈

D₄

yes

characteristic

C₂×C₇₉⋊C₁₆

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

characteristic

Dic₇₉⋊C₈

—

#14579082

—

C₈×Dic₇₉

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

#127×C₇₉

—

D₁₆×C₇₉

—

D₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

D₈

yes

characteristic

#11929612

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

D₈

yes

characteristic

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

D₈×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

complemented

C₂³×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

#135

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

D₁₆

yes

complemented

C₈×C₇₉

—

C₈×C₇₉

C₂×SD₁₆

yes

characteristic

C₈×C₇₉

AGL(1,ℤ/8)

yes

characteristic

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#466

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₂×D₁₆

yes

characteristic

C₄×C₇₉

#508

yes

characteristic

Dic₇₉

—

Dic₇₉

—

#12439

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

yes

characteristic

Q₈×C₁₆

—

#1668

—

#1371

—

#1361

—

C₇₉

#12439

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₂×D₁₆

—

#493

C₂×D₇₉

yes

—

#460

—

158

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

158

—

C₄×C₁₆

—

158

—

100

C₄×C₁₆

C₂×D₇₉

yes

characteristic

101

#335

—

102

#331

—

158

—

103

C₂×SD₁₆

—

316

—

104

C₂×D₈

—

105

C₂×D₈

—

106

#127

—

107

C₄×Q₈

—

108

D₁₆

—

109

D₁₆

—

110

C₂×C₁₆

—

158

—

111

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

112

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

—

158

—

115

C₄⋊C₈

—

158

—

116

#102

—

316

—

117

C₄×C₈

—

158

—

118

C₄×C₈

C₂²×D₇₉

yes

characteristic

119

C₂×Q₈

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

—

123

SD₁₆

—

632

—

124

D₈

—

125

D₈

—

126

D₈

—

127

C₂×C₈

—

316

—

128

C₂×C₈

—

158

—

129

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

130

C₂×C₈

#11929623

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

158

—

133

C₄²

—

158

—

134

C₄²

D₄×D₇₉

yes

characteristic

135

C₁₆

—

136

C₁₆

—

316

—

137

C₁₆

#11929612

yes

complemented

138

C₂³

—

139

Q₈

—

140

Q₈

—

158

—

141

D₄

—

142

D₄

—

143

D₄

—

144

D₄

—

145

D₄

—

146

C₂×C₄

—

158

—

147

C₂×C₄

—

158

—

148

C₂×C₄

#14579132

yes

characteristic

149

C₂×C₄

D₈×D₇₉

yes

characteristic

150

C₂×C₄

#14579180

yes

characteristic

151

C₈

—

152

C₈

—

316

—

153

C₈

—

632

—

154

C₈

C₂×#11929612

yes

characteristic

155

C₈

#14579187

yes

characteristic

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#19098796

yes

characteristic

160

C₄

—

161

C₄

—

316

—

162

C₄

—

158

—

163

C₄

#19098760

yes

characteristic

164

C₄

#19098830

yes

characteristic

165

C₄

D₁₆×D₇₉

yes

characteristic

166

C₄

#19098992

yes

characteristic

167

C₂

—

168

C₂

#23762757

yes

characteristic

169

C₂

#23765483

yes

characteristic

170

C₂

#23765517

yes

characteristic

	subgroup	G/H
center	C₂²	#19098796
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting, prime radical	C₇₉×#2246	C₂
socle	C₂²×C₇₉	#466
hypercenter	#2246	D₇₉
hypocenter	C₇₉	#12439
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

subgroup

G/H

center

C₂²

#19098796

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting, prime radical

C₇₉×#2246

C₂

C₂²×C₇₉

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

N	H ≅ G/N	type
1. C₁₆×Dic₁₅₈	C₂	semidirect
2. #23762757	C₂	semidirect
3. #23761472	C₂	semidirect
4. #23761446	C₂	semidirect
51. C₁₆×C₇₉	SD₁₆	semidirect
64. Dic₁₅₈	D₁₆	semidirect
91. C₇₉	#12439	semidirect
137. C₁₆	#11929612	semidirect

H ≅ G/N

type

1. C₁₆×Dic₁₅₈

C₂

semidirect

2. #23762757

C₂

semidirect

3. #23761472

C₂

semidirect

4. #23761446

C₂

semidirect

51. C₁₆×C₇₉

SD₁₆

semidirect

64. Dic₁₅₈

D₁₆

semidirect

91. C₇₉

#12439

semidirect

137. C₁₆

#11929612

semidirect

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

subgroup	index
C₁₆×Dic₁₅₈	2
#23762757	2
#23761472	2
#23761446	2
C₇₉×#2246	2
#12439	79

subgroup

index

C₁₆×Dic₁₅₈

C₇₉×#2246

Subgroups — #34202523

Conjugacy classes of subgroups

Complements

Maximal subgroups