Group #34209065 — subgroups

Subgroups — #34209065

Order 20224.

200 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 65 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×Dic₁₂₆₄	C₂	1	2	yes	maximal
2	#23765653	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₆₃₂	C₂	1	1	yes	maximal, characteristic
4	#23761582	C₂	1	2	yes	maximal
5	C₇₉×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₃₂	—	2	1	no	—
7	C₂×Dic₆₃₂	C₂²	1	2	yes	—
8	C₂×Dic₆₃₂	C₂²	1	1	yes	characteristic
9	#19098943	—	2	1	no	—
10	C₄×Dic₃₁₆	—	2	1	no	—
11	Dic₁₂₆₄	—	2	4	no	—
12	#19098389	C₂²	1	1	yes	characteristic
13	#19098368	—	2	1	no	—
14	C₂×C₃₂×C₇₉	C₂²	1	2	yes	—
15	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₁₆	—	2	1	no	—
17	C₂×Dic₃₁₆	—	2	2	no	—
18	C₂×Dic₃₁₆	—	4	1	no	—
19	#14579161	—	4	1	no	—
20	C₄×Dic₁₅₈	—	4	1	no	—
21	Dic₆₃₂	—	2	1	no	—
22	Dic₆₃₂	—	2	2	no	—
23	Dic₆₃₂	—	2	2	no	—
24	Dic₆₃₂	—	4	2	no	—
25	Dic₆₃₂	D₄	1	2	yes	—
26	#14579095	—	2	1	no	—
27	#14579086	—	4	1	no	—
28	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
31	C₃₂×C₇₉	—	2	2	no	—
32	C₂×Dic₁₅₈	—	4	1	no	—
33	C₂×Dic₁₅₈	—	4	2	no	—
34	Dic₃₁₆	—	2	2	no	—
35	Dic₃₁₆	—	4	2	no	—
36	Dic₃₁₆	—	4	1	no	—
37	Dic₃₁₆	—	4	2	no	—
38	Dic₃₁₆	—	8	2	no	—
39	C₄⋊Dic₇₉	—	4	1	no	—
40	C₄⋊Dic₇₉	—	8	1	no	—
41	C₄⋊Dic₇₉	—	8	1	no	—
42	Dic₇₉⋊C₄	—	8	1	no	—
43	C₄×Dic₇₉	—	8	1	no	—
44	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₇₉	D₈	1	1	yes	characteristic
47	C₁₆×C₇₉	—	2	1	no	—
48	C₁₆×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₇₉	#41	1	1	yes	characteristic
50	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
51	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₇₉	—	8	1	no	—
54	C₂×Dic₇₉	—	8	1	no	—
55	C₂×Dic₇₉	—	8	2	no	—
56	Dic₁₅₈	—	4	2	no	—
57	Dic₁₅₈	—	8	1	no	—
58	Dic₁₅₈	—	8	2	no	—
59	Dic₁₅₈	—	8	2	no	—
60	C₈×C₇₉	—	2	1	no	—
61	C₈×C₇₉	#121	1	1	yes	characteristic
62	C₈×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
64	C₄×C₇₉	—	2	1	no	—
65	C₄×C₇₉	Q₆₄	1	2	yes	complemented
66	C₄×C₇₉	#459	1	1	yes	characteristic
67	C₄×C₇₉	#509	1	1	yes	characteristic
68	Dic₇₉	—	8	2	no	—
69	Dic₇₉	—	16	2	no	—
70	Dic₇₉	—	16	1	no	—
71	#13688	—	79	1	no	maximal, Sylow
72	C₂×C₇₉	#2206	1	1	yes	characteristic
73	C₂×C₇₉	C₂×Q₆₄	1	1	yes	characteristic
74	C₂×C₇₉	#2264	1	1	yes	characteristic
75	C₂×Q₆₄	—	79	2	no	—
76	#2247	—	79	1	no	—
77	C₄×Q₃₂	—	79	1	no	—
78	C₄⋊C₃₂	D₇₉	1	1	yes	characteristic
79	#1416	—	79	2	no	—
80	C₇₉	#13688	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₃₂	—	79	1	no	—
82	C₂×Q₃₂	—	79	2	no	—
83	C₂×Q₃₂	—	158	1	no	—
84	#494	—	158	1	no	—
85	C₄×Q₁₆	—	158	1	no	—
86	Q₆₄	—	158	4	no	—
87	C₂×C₃₂	C₂×D₇₉	1	2	yes	—
88	#366	—	79	1	no	—
89	#358	—	158	1	no	—
90	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
91	C₂×Q₁₆	—	316	1	no	—
92	C₂×Q₁₆	—	158	2	no	—
93	C₂×Q₁₆	—	158	1	no	—
94	C₄⋊Q₈	—	316	1	no	—
95	C₄×Q₈	—	316	1	no	—
96	Q₃₂	—	316	2	no	—
97	Q₃₂	—	79	2	no	—
98	Q₃₂	—	158	2	no	—
99	Q₃₂	—	158	2	no	—
100	Q₃₂	—	158	1	no	—
101	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
102	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
103	#107	—	158	1	no	—
104	#103	—	316	1	no	—
105	C₄×C₈	D₃₁₆	1	1	yes	characteristic
106	C₃₂	—	2	2	no	—
107	C₂×Q₈	—	316	1	no	—
108	C₂×Q₈	—	316	2	no	—
109	Q₁₆	—	316	2	no	—
110	Q₁₆	—	316	1	no	—
111	Q₁₆	—	316	2	no	—
112	Q₁₆	—	632	2	no	—
113	Q₁₆	—	158	2	no	—
114	C₂×C₈	D₆₃₂	1	1	yes	characteristic
115	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
116	C₄⋊C₄	—	316	1	no	—
117	C₄⋊C₄	—	632	1	no	—
118	C₄⋊C₄	—	632	1	no	—
119	C₄⋊C₄	—	632	1	no	—
120	C₄²	—	632	1	no	—
121	C₄²	D₆₃₂	1	1	yes	characteristic
122	C₁₆	—	2	1	no	—
123	C₁₆	D₄×D₇₉	1	1	yes	characteristic
124	C₁₆	#11929625	1	1	yes	characteristic
125	Q₈	—	316	2	no	—
126	Q₈	—	632	1	no	—
127	Q₈	—	632	2	no	—
128	Q₈	—	632	2	no	—
129	C₂×C₄	—	632	1	no	—
130	C₂×C₄	—	632	1	no	—
131	C₂×C₄	—	632	2	no	—
132	C₂×C₄	D₁₂₆₄	1	1	yes	characteristic
133	C₂×C₄	D₁₂₆₄	1	1	yes	characteristic
134	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	#14579136	1	1	yes	characteristic
137	C₈	#14579189	1	1	yes	characteristic
138	C₂²	C₂×D₁₂₆₄	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	632	2	no	—
141	C₄	—	1264	2	no	—
142	C₄	—	1264	1	no	—
143	C₄	Dic₁₂₆₄	1	2	yes	complemented
144	C₄	#19098759	1	1	yes	characteristic
145	C₄	#19099001	1	1	yes	characteristic
146	C₂	#23765436	1	1	yes	characteristic
147	C₂	C₂×Dic₁₂₆₄	1	1	yes	characteristic
148	C₂	#23765720	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₂₆₄

C₂

yes

maximal

#23765653

C₂

yes

maximal, characteristic

C₄×Dic₆₃₂

C₂

yes

maximal, characteristic

#23761582

C₂

yes

maximal

C₇₉×C₄⋊C₃₂

C₂

yes

maximal, characteristic

C₂×Dic₆₃₂

—

C₂×Dic₆₃₂

C₂²

yes

—

C₂×Dic₆₃₂

C₂²

yes

characteristic

#19098943

—

C₄×Dic₃₁₆

—

Dic₁₂₆₄

—

#19098389

C₂²

yes

characteristic

#19098368

—

C₂×C₃₂×C₇₉

C₂²

yes

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

#14579161

—

C₄×Dic₁₅₈

—

Dic₆₃₂

—

Dic₆₃₂

—

Dic₆₃₂

—

Dic₆₃₂

—

Dic₆₃₂

D₄

yes

—

#14579095

—

#14579086

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₃₂×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

D₈

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

#41

yes

characteristic

C₂×C₄×C₇₉

D₁₆

yes

characteristic

C₂×C₄×C₇₉

D₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₂²×C₇₉

C₂×D₁₆

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

Q₆₄

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

C₂×Q₆₄

yes

characteristic

C₂×C₇₉

#2264

yes

characteristic

C₂×Q₆₄

—

#2247

—

C₄×Q₃₂

—

C₄⋊C₃₂

D₇₉

yes

characteristic

#1416

—

C₇₉

#13688

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

158

—

#494

—

158

—

C₄×Q₁₆

—

158

—

Q₆₄

—

158

—

C₂×C₃₂

C₂×D₇₉

yes

—

#366

—

#358

—

158

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

316

—

C₂×Q₁₆

—

158

—

C₂×Q₁₆

—

158

—

C₄⋊Q₈

—

316

—

C₄×Q₈

—

316

—

Q₃₂

—

316

—

Q₃₂

—

Q₃₂

—

158

—

Q₃₂

—

158

—

100

Q₃₂

—

158

—

101

C₂×C₁₆

D₃₁₆

yes

characteristic

102

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

103

#107

—

158

—

104

#103

—

316

—

105

C₄×C₈

D₃₁₆

yes

characteristic

106

C₃₂

—

107

C₂×Q₈

—

316

—

108

C₂×Q₈

—

316

—

109

Q₁₆

—

316

—

110

Q₁₆

—

316

—

111

Q₁₆

—

316

—

112

Q₁₆

—

632

—

113

Q₁₆

—

158

—

114

C₂×C₈

D₆₃₂

yes

characteristic

115

C₂×C₈

C₂×D₃₁₆

yes

characteristic

116

C₄⋊C₄

—

316

—

117

C₄⋊C₄

—

632

—

118

C₄⋊C₄

—

632

—

119

C₄⋊C₄

—

632

—

120

C₄²

—

632

—

121

C₄²

D₆₃₂

yes

characteristic

122

C₁₆

—

123

C₁₆

D₄×D₇₉

yes

characteristic

124

C₁₆

#11929625

yes

characteristic

125

Q₈

—

316

—

126

Q₈

—

632

—

127

Q₈

—

632

—

128

Q₈

—

632

—

129

C₂×C₄

—

632

—

130

C₂×C₄

—

632

—

131

C₂×C₄

—

632

—

132

C₂×C₄

D₁₂₆₄

yes

characteristic

133

C₂×C₄

D₁₂₆₄

yes

characteristic

134

C₂×C₄

C₂×D₆₃₂

yes

characteristic

135

C₈

—

136

C₈

#14579136

yes

characteristic

137

C₈

#14579189

yes

characteristic

138

C₂²

C₂×D₁₂₆₄

yes

characteristic

139

C₄

—

140

C₄

—

632

—

141

C₄

—

1264

—

142

C₄

—

1264

—

143

C₄

Dic₁₂₆₄

yes

complemented

144

C₄

#19098759

yes

characteristic

145

C₄

#19099001

yes

characteristic

146

C₂

#23765436

yes

characteristic

147

C₂

C₂×Dic₁₂₆₄

yes

characteristic

148

C₂

#23765720

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₂₆₄
commutator, cosocle	C₂×C₁₆×C₇₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₇₉×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₇₉	C₂×D₁₆
hypercenter	C₄⋊C₃₂	D₇₉
hypocenter	C₇₉	#13688
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

subgroup

G/H

center

C₂²

C₂×D₁₂₆₄

commutator, cosocle

C₂×C₁₆×C₇₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₉

Fitting

C₇₉×C₄⋊C₃₂

C₂

socle, prime radical

C₂²×C₇₉

C₂×D₁₆

hypercenter

C₄⋊C₃₂

D₇₉

hypocenter

C₇₉

#13688

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

N	H ≅ G/N	type
65. C₄×C₇₉	Q₆₄	semidirect
80. C₇₉	#13688	semidirect
143. C₄	Dic₁₂₆₄	semidirect

H ≅ G/N

type

65. C₄×C₇₉

Q₆₄

semidirect

80. C₇₉

#13688

semidirect

143. C₄

Dic₁₂₆₄

semidirect

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

subgroup	index
C₂×Dic₁₂₆₄	2
#23765653	2
C₄×Dic₆₃₂	2
#23761582	2
C₇₉×C₄⋊C₃₂	2
#13688	79

subgroup

index

C₂×Dic₁₂₆₄

#23765653

C₄×Dic₆₃₂

#23761582

C₇₉×C₄⋊C₃₂

#13688

Subgroups — #34209065

Conjugacy classes of subgroups

Complements

Maximal subgroups