Group #24983481 — subgroups

Subgroups — #24983481

Order 11008.

210 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 119 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×#14582720	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#13063265	—	2	2	no	—
4	#14582720	—	2	2	no	—
5	#14582720	C₄	1	4	yes	—
6	C₂²×C₁₆×C₄₃	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₄₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₄₃	—	2	2	no	—
10	#13063265	—	2	4	no	—
11	#13063265	—	4	2	no	—
12	C₂×C₄₃⋊C₁₆	—	4	2	no	—
13	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
15	C₂×C₄²×C₄₃	Q₈	1	1	yes	characteristic
16	C₂×C₁₆×C₄₃	—	2	1	no	—
17	C₂×C₁₆×C₄₃	—	2	1	no	—
18	C₂×C₁₆×C₄₃	—	2	1	no	—
19	C₂×C₁₆×C₄₃	—	2	1	no	—
20	C₂×C₁₆×C₄₃	C₈	1	2	yes	—
21	C₂×C₁₆×C₄₃	C₈	1	2	yes	—
22	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₄₃	D₄	1	2	yes	—
24	C₂²×C₄×C₄₃	—	2	1	no	—
25	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
27	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
28	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₄₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₄₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₄₃	M₁₆	1	1	yes	characteristic
32	C₂×C₈×C₄₃	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
34	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
35	C₂×C₈×C₄₃	Q₁₆	1	1	yes	characteristic
36	C₄²×C₄₃	—	2	1	no	—
37	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
38	C₁₆×C₄₃	—	2	2	no	—
39	C₁₆×C₄₃	—	2	2	no	—
40	C₄₃⋊C₁₆	—	4	4	no	—
41	C₂³×C₄₃	#108	1	1	yes	characteristic
42	C₂×C₄×C₄₃	—	2	4	no	—
43	C₂×C₄×C₄₃	—	2	2	no	—
44	C₂×C₄×C₄₃	#95	1	1	yes	characteristic
45	C₂×C₄×C₄₃	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₄₃	C₄⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
48	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
49	C₂×C₄×C₄₃	#108	1	1	yes	characteristic
50	C₈×C₄₃	C₄×C₈	1	1	yes	characteristic
51	C₈×C₄₃	#97	1	1	yes	characteristic
52	C₈×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
53	C₈×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
54	C₈×C₄₃	#102	1	2	yes	—
55	C₈×C₄₃	#103	1	2	yes	—
56	#7160	—	43	1	no	maximal, Sylow
57	C₂²×C₄₃	—	2	2	no	—
58	C₂²×C₄₃	#334	1	1	yes	characteristic
59	C₂²×C₄₃	#335	1	1	yes	characteristic
60	C₂²×C₄₃	#341	1	1	yes	characteristic
61	C₄×C₄₃	—	2	4	no	—
62	C₄×C₄₃	#334	1	1	yes	characteristic
63	C₄×C₄₃	#335	1	1	yes	characteristic
64	C₄×C₄₃	#336	1	1	yes	characteristic
65	C₄×C₄₃	#340	1	1	yes	characteristic
66	C₂×#364	—	43	2	no	—
67	C₂×C₄×C₁₆	D₄₃	1	1	yes	characteristic
68	C₂×C₄₃	—	2	1	no	—
69	C₂×C₄₃	—	2	1	no	—
70	C₂×C₄₃	#1277	1	1	yes	characteristic
71	C₂×C₄₃	#1373	1	2	yes	—
72	C₂×M₃₂	—	86	2	no	—
73	C₂²×C₁₆	Dic₄₃	1	1	yes	characteristic
74	C₂²×C₁₆	Dic₄₃	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
76	#364	—	43	4	no	—
77	#364	—	86	2	no	—
78	C₄×C₁₆	—	2	2	no	—
79	C₄₃	#7160	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
81	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
82	C₂×C₄²	Dic₈₆	1	1	yes	characteristic
83	M₃₂	—	86	4	no	—
84	M₃₂	—	172	2	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	—	2	1	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	—	172	2	no	—
90	C₂×C₁₆	C₄₃⋊C₈	1	2	yes	—
91	C₂×C₁₆	C₄₃⋊C₈	1	2	yes	—
92	C₄×C₈	C₄×D₄₃	1	2	yes	—
93	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
96	C₂×C₈	#10621873	1	2	yes	—
97	C₂×C₈	D₃₄₄	1	1	yes	characteristic
98	C₂×C₈	Dic₁₇₂	1	1	yes	characteristic
99	C₂×C₈	C₂×C₄₃⋊C₈	1	1	yes	characteristic
100	C₂×C₈	C₂×C₄₃⋊C₈	1	1	yes	characteristic
101	C₂×C₈	#10621877	1	1	yes	characteristic
102	C₂×C₈	#10621877	1	1	yes	characteristic
103	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
104	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
105	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
106	C₄²	—	2	1	no	—
107	C₄²	Dic₄₃⋊C₄	1	2	yes	—
108	C₁₆	—	2	2	no	—
109	C₁₆	—	2	2	no	—
110	C₁₆	—	172	4	no	—
111	C₂³	#13063261	1	1	yes	characteristic
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₄₃⋊#95	1	1	yes	characteristic
115	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
116	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
117	C₂×C₄	#13063257	1	1	yes	characteristic
118	C₂×C₄	#13063259	1	1	yes	characteristic
119	C₂×C₄	#13063261	1	1	yes	characteristic
120	C₈	C₄×C₄₃⋊C₈	1	1	yes	characteristic
121	C₈	#13063236	1	1	yes	characteristic
122	C₈	C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
123	C₈	C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
124	C₈	#13063247	1	2	yes	—
125	C₈	#13063250	1	2	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	#14582650	1	1	yes	characteristic
128	C₂²	#14582652	1	1	yes	characteristic
129	C₂²	#14582664	1	1	yes	characteristic
130	C₄	—	2	4	no	—
131	C₄	#14582650	1	1	yes	characteristic
132	C₄	#14582652	1	1	yes	characteristic
133	C₄	#14582654	1	1	yes	characteristic
134	C₄	#14582662	1	1	yes	characteristic
135	C₂	—	2	1	no	—
136	C₂	—	2	1	no	—
137	C₂	#19132430	1	1	yes	characteristic
138	C₂	#19132671	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582720

C₂

yes

maximal

C₂×C₄×C₁₆×C₄₃

C₂

yes

maximal, characteristic

C₂×#13063265

—

#14582720

—

#14582720

C₄

yes

—

C₂²×C₁₆×C₄₃

C₄

yes

characteristic

C₂²×C₁₆×C₄₃

C₄

yes

characteristic

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄×C₁₆×C₄₃

—

#13063265

—

#13063265

—

C₂×C₄₃⋊C₁₆

—

C₂²×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

Q₈

yes

characteristic

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

C₈

yes

—

C₂×C₁₆×C₄₃

C₈

yes

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₃

C₄²

yes

characteristic

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×C₈

yes

characteristic

C₂×C₈×C₄₃

C₂×C₈

yes

characteristic

C₂×C₈×C₄₃

M₁₆

yes

characteristic

C₂×C₈×C₄₃

M₁₆

yes

characteristic

C₂×C₈×C₄₃

D₈

yes

characteristic

C₂×C₈×C₄₃

SD₁₆

yes

—

C₂×C₈×C₄₃

Q₁₆

yes

characteristic

C₄²×C₄₃

—

C₄²×C₄₃

C₄⋊C₄

yes

—

C₁₆×C₄₃

—

C₁₆×C₄₃

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

#108

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#95

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₈×C₄₃

C₄×C₈

yes

characteristic

C₈×C₄₃

#97

yes

characteristic

C₈×C₄₃

C₂²⋊C₈

yes

characteristic

C₈×C₄₃

C₂²⋊C₈

yes

characteristic

C₈×C₄₃

#102

yes

—

C₈×C₄₃

#103

yes

—

#7160

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₂×C₄×C₁₆

D₄₃

yes

characteristic

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#1277

yes

characteristic

C₂×C₄₃

#1373

yes

—

C₂×M₃₂

—

C₂²×C₁₆

Dic₄₃

yes

characteristic

C₂²×C₁₆

Dic₄₃

yes

characteristic

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

#364

—

#364

—

C₄×C₁₆

—

C₄₃

#7160

yes

characteristic, complemented, Sylow

C₂²×C₈

D₁₇₂

yes

characteristic

C₂²×C₈

C₂×Dic₄₃

yes

characteristic

C₂×C₄²

Dic₈₆

yes

characteristic

M₃₂

—

M₃₂

—

172

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

172

—

C₂×C₁₆

C₄₃⋊C₈

yes

—

C₂×C₁₆

C₄₃⋊C₈

yes

—

C₄×C₈

C₄×D₄₃

yes

—

C₄×C₈

C₂²⋊D₄₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₄⋊Dic₄₃

yes

characteristic

C₂×C₈

#10621873

yes

—

C₂×C₈

D₃₄₄

yes

characteristic

C₂×C₈

Dic₁₇₂

yes

characteristic

C₂×C₈

C₂×C₄₃⋊C₈

yes

characteristic

100

C₂×C₈

C₂×C₄₃⋊C₈

yes

characteristic

101

C₂×C₈

#10621877

yes

characteristic

102

C₂×C₈

#10621877

yes

characteristic

103

C₂×C₈

C₄×Dic₄₃

yes

characteristic

104

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

105

C₂×C₈

C₂²⋊Dic₄₃

yes

characteristic

106

C₄²

—

107

C₄²

Dic₄₃⋊C₄

yes

—

108

C₁₆

—

109

C₁₆

—

110

C₁₆

—

172

—

111

C₂³

#13063261

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₄₃⋊#95

yes

characteristic

115

C₂×C₄

C₄⋊(C₄₃⋊C₈)

yes

characteristic

116

C₂×C₄

C₄⋊(C₄₃⋊C₈)

yes

characteristic

117

C₂×C₄

#13063257

yes

characteristic

118

C₂×C₄

#13063259

yes

characteristic

119

C₂×C₄

#13063261

yes

characteristic

120

C₈

C₄×C₄₃⋊C₈

yes

characteristic

121

C₈

#13063236

yes

characteristic

122

C₈

C₂²⋊(C₄₃⋊C₈)

yes

characteristic

123

C₈

C₂²⋊(C₄₃⋊C₈)

yes

characteristic

124

C₈

#13063247

yes

—

125

C₈

#13063250

yes

—

126

C₂²

—

127

C₂²

#14582650

yes

characteristic

128

C₂²

#14582652

yes

characteristic

129

C₂²

#14582664

yes

characteristic

130

C₄

—

131

C₄

#14582650

yes

characteristic

132

C₄

#14582652

yes

characteristic

133

C₄

#14582654

yes

characteristic

134

C₄

#14582662

yes

characteristic

135

C₂

—

136

C₂

—

137

C₂

#19132430

yes

characteristic

138

C₂

#19132671

yes

—

	subgroup	G/H
center	C₂×C₈	D₃₄₄
commutator	C₈×C₄₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₃
Fitting	C₂×C₄×C₁₆×C₄₃	C₂
socle	C₂²×C₄₃	#335
prime radical	C₂³×C₄₃	#108
cosocle	C₂×C₄×C₈×C₄₃	C₂²
hypercenter	C₂×C₄×C₁₆	D₄₃
hypocenter	C₄₃	#7160
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂×C₈

D₃₄₄

commutator

C₈×C₄₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₄₃

Fitting

C₂×C₄×C₁₆×C₄₃

C₂

C₂²×C₄₃

C₂³×C₄₃

C₂×C₄×C₈×C₄₃

C₂²

hypercenter

C₂×C₄×C₁₆

D₄₃

hypocenter

C₄₃

#7160

LCS

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

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

N	H ≅ G/N	type
79. C₄₃	#7160	semidirect

H ≅ G/N

type

79. C₄₃

#7160

semidirect

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

subgroup	index
C₂×#14582720	2
C₂×C₄×C₁₆×C₄₃	2
#7160	43

subgroup

index

C₂×#14582720

C₂×C₄×C₁₆×C₄₃

#7160

Subgroups — #24983481

Conjugacy classes of subgroups

Complements

Maximal subgroups