Group #34202531 — subgroups

Subgroups — #34202531

Order 20224.

216 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 75 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	#23762770	C₂	1	2	yes	maximal
3	#23761472	C₂	1	1	yes	maximal, characteristic
4	#23761450	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₇₉	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	#19098848	—	2	2	no	—
10	#19098316	C₂²	1	2	yes	—
11	#19098310	—	2	2	no	—
12	C₈×Q₈×C₇₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₇₉	—	2	1	no	—
14	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₇₉	—	2	1	no	—
16	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₁₆	—	2	2	no	—
18	C₂×Dic₃₁₆	—	2	2	no	—
19	#14579161	—	2	2	no	—
20	Dic₆₃₂	—	4	2	no	—
21	Dic₆₃₂	—	4	2	no	—
22	#14579092	—	4	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	#14579085	—	4	2	no	—
25	C₄×Q₈×C₇₉	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₉	—	2	1	no	—
27	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₇₉	—	2	1	no	—
30	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₇₉	—	2	1	no	—
32	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₅₈	—	4	2	no	—
34	Dic₃₁₆	—	4	2	no	—
35	Dic₃₁₆	—	4	2	no	—
36	Dic₃₁₆	—	8	2	no	—
37	C₂×C₇₉⋊C₈	—	4	2	no	—
38	C₄⋊Dic₇₉	—	4	2	no	—
39	C₄⋊Dic₇₉	—	4	2	no	—
40	C₂×Q₈×C₇₉	D₈	1	1	yes	characteristic
41	C₂×C₈×C₇₉	—	2	1	no	—
42	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
44	C₄⋊C₄×C₇₉	—	2	1	no	—
45	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
46	C₄²×C₇₉	—	2	1	no	—
47	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₇₉	—	2	1	no	—
49	C₁₆×C₇₉	—	4	1	no	—
50	C₁₆×C₇₉	SD₁₆	1	2	yes	—
51	Q₈×C₇₉	—	2	1	no	—
52	Q₈×C₇₉	Q₃₂	1	2	yes	complemented
53	C₂×C₄×C₇₉	—	2	1	no	—
54	C₂×C₄×C₇₉	—	2	1	no	—
55	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
56	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
58	C₂×Dic₇₉	—	8	2	no	—
59	Dic₁₅₈	—	8	2	no	—
60	Dic₁₅₈	—	8	2	no	—
61	C₈×C₇₉	—	2	1	no	—
62	C₈×C₇₉	—	4	1	no	—
63	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
64	C₈×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₇₉⋊C₈	—	8	2	no	—
66	C₂²×C₇₉	#466	1	1	yes	characteristic
67	C₄×C₇₉	—	2	2	no	—
68	C₄×C₇₉	—	2	1	no	—
69	C₄×C₇₉	—	4	1	no	—
70	C₄×C₇₉	#460	1	1	yes	characteristic
71	C₄×C₇₉	#472	1	1	yes	characteristic
72	C₄×C₇₉	C₂×Q₃₂	1	1	yes	characteristic
73	C₄×C₇₉	#508	1	1	yes	characteristic
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	D₇₉	1	1	yes	characteristic
81	#1670	—	79	2	no	—
82	#1371	—	79	1	no	—
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	C₂×D₇₉	1	1	yes	characteristic
90	C₄⋊C₁₆	—	2	1	no	—
91	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
92	C₄×C₁₆	—	2	1	no	—
93	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
94	#335	—	79	2	no	—
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	D₃₁₆	1	1	yes	characteristic
100	Q₃₂	—	316	2	no	—
101	Q₃₂	—	316	2	no	—
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
104	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
105	#106	—	316	2	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	158	2	no	—
108	C₄⋊C₈	D₃₁₆	1	1	yes	characteristic
109	#103	—	316	2	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
112	C₂×Q₈	—	316	2	no	—
113	C₂×Q₈	D₆₃₂	1	1	yes	characteristic
114	Q₁₆	—	316	2	no	—
115	Q₁₆	—	316	2	no	—
116	Q₁₆	—	632	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	316	2	no	—
119	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
120	C₂×C₈	#11929624	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	316	2	no	—
123	C₄⋊C₄	—	316	2	no	—
124	C₄⋊C₄	D₆₃₂	1	1	yes	characteristic
125	C₄²	—	2	1	no	—
126	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	4	1	no	—
129	C₁₆	#11929611	1	2	yes	—
130	Q₈	—	2	1	no	—
131	Q₈	—	632	2	no	—
132	Q₈	—	632	2	no	—
133	Q₈	Dic₆₃₂	1	2	yes	complemented
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	632	2	no	—
137	C₂×C₄	#14579135	1	1	yes	characteristic
138	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
139	C₂×C₄	#14579183	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	4	1	no	—
142	C₈	—	632	2	no	—
143	C₈	C₂×#11929611	1	1	yes	characteristic
144	C₈	#14579186	1	1	yes	characteristic
145	C₂²	#19098799	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	2	no	—
148	C₄	—	4	1	no	—
149	C₄	—	1264	2	no	—
150	C₄	#19098763	1	1	yes	characteristic
151	C₄	#19098833	1	1	yes	characteristic
152	C₄	C₂×Dic₆₃₂	1	1	yes	characteristic
153	C₄	#19098995	1	1	yes	characteristic
154	C₂	#23762760	1	1	yes	characteristic
155	C₂	#23765491	1	1	yes	characteristic
156	C₂	#23765525	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765653

C₂

yes

maximal, characteristic

#23762770

C₂

yes

maximal

#23761472

C₂

yes

maximal, characteristic

#23761450

C₂

yes

maximal

Q₈×C₁₆×C₇₉

C₂

yes

maximal, characteristic

C₂×Dic₆₃₂

—

C₂×Dic₆₃₂

—

#19098943

C₂²

yes

—

#19098848

—

#19098316

C₂²

yes

—

#19098310

—

C₈×Q₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₉

—

C₄⋊C₁₆×C₇₉

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

#14579161

—

Dic₆₃₂

—

Dic₆₃₂

—

#14579092

—

C₄⋊(C₇₉⋊C₈)

—

#14579085

—

C₄×Q₈×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂×Q₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

D₈

yes

characteristic

C₄²×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₁₆×C₇₉

SD₁₆

yes

—

Q₈×C₇₉

—

Q₈×C₇₉

Q₃₂

yes

complemented

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

#135

yes

characteristic

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

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

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₂×Q₃₂

yes

characteristic

C₄×C₇₉

#508

yes

characteristic

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

Q₈×C₁₆

D₇₉

yes

—

—

—

C₇₉

#12440

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

158

—

C₂×Q₃₂

—

158

—

#494

—

#475

—

158

—

C₈×Q₈

C₂×D₇₉

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₇₉

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

#335

—

#332

—

158

—

C₂×Q₁₆

—

158

—

C₂×Q₁₆

—

158

—

C₄⋊Q₈

—

158

—

C₄×Q₈

D₃₁₆

yes

characteristic

100

Q₃₂

—

316

—

101

Q₃₂

—

316

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

104

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

105

#106

—

316

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

158

—

108

C₄⋊C₈

D₃₁₆

yes

characteristic

109

#103

—

316

—

110

C₄×C₈

—

111

C₄×C₈

C₂²×D₇₉

yes

characteristic

112

C₂×Q₈

—

316

—

113

C₂×Q₈

D₆₃₂

yes

characteristic

114

Q₁₆

—

316

—

115

Q₁₆

—

316

—

116

Q₁₆

—

632

—

117

C₂×C₈

—

118

C₂×C₈

—

316

—

119

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

120

C₂×C₈

#11929624

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

316

—

123

C₄⋊C₄

—

316

—

124

C₄⋊C₄

D₆₃₂

yes

characteristic

125

C₄²

—

126

C₄²

C₂×D₃₁₆

yes

characteristic

127

C₁₆

—

128

C₁₆

—

129

C₁₆

#11929611

yes

—

130

Q₈

—

131

Q₈

—

632

—

132

Q₈

—

632

—

133

Q₈

Dic₆₃₂

yes

complemented

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

632

—

137

C₂×C₄

#14579135

yes

characteristic

138

C₂×C₄

C₂×D₆₃₂

yes

characteristic

139

C₂×C₄

#14579183

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

632

—

143

C₈

C₂×#11929611

yes

characteristic

144

C₈

#14579186

yes

characteristic

145

C₂²

#19098799

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

1264

—

150

C₄

#19098763

yes

characteristic

151

C₄

#19098833

yes

characteristic

152

C₄

C₂×Dic₆₃₂

yes

characteristic

153

C₄

#19098995

yes

characteristic

154

C₂

#23762760

yes

characteristic

155

C₂

#23765491

yes

characteristic

156

C₂

#23765525

yes

characteristic

	subgroup	G/H
center	C₂²	#19098799
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	Q₈×C₁₆×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	#466
hypercenter	Q₈×C₁₆	D₇₉
hypocenter	C₇₉	#12440
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#19098799

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

Q₈×C₁₆×C₇₉

C₂

socle, prime radical

C₂²×C₇₉

#466

hypercenter

Q₈×C₁₆

D₇₉

hypocenter

C₇₉

#12440

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
52. Q₈×C₇₉	Q₃₂	semidirect
84. C₇₉	#12440	semidirect
133. Q₈	Dic₆₃₂	semidirect

H ≅ G/N

type

52. Q₈×C₇₉

Q₃₂

semidirect

84. C₇₉

#12440

semidirect

133. Q₈

Dic₆₃₂

semidirect

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

subgroup	index
#23765653	2
#23762770	2
#23761472	2
#23761450	2
Q₈×C₁₆×C₇₉	2
#12440	79

subgroup

index

Q₈×C₁₆×C₇₉

Subgroups — #34202531

Conjugacy classes of subgroups

Complements

Maximal subgroups