Group #24983467 — subgroups

Subgroups — #24983467

Order 11008.

242 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 151 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×#14582652	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
4	#14582652	—	2	2	no	—
5	#14582652	C₄	1	4	yes	—
6	C₂²×C₁₆×C₄₃	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
8	C₄×C₁₆×C₄₃	—	2	2	no	—
9	C₂²×C₄₃⋊C₈	—	4	2	no	—
10	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
11	C₄⋊(C₄₃⋊C₈)	—	4	2	no	—
12	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₄₃	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₄₃	—	2	2	no	—
16	C₂×C₁₆×C₄₃	—	2	2	no	—
17	C₂×C₁₆×C₄₃	C₈	1	4	yes	complemented
18	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₄₃	D₄	1	2	yes	—
20	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	Q₁₆	1	1	yes	characteristic
23	C₂×C₈×C₄₃	—	2	2	no	—
24	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
25	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₄₃	M₁₆	1	2	yes	—
29	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
30	C₄²×C₄₃	SD₁₆	1	2	yes	—
31	C₂×C₄₃⋊C₈	—	4	4	no	—
32	C₂×C₄₃⋊C₈	—	8	2	no	—
33	C₂×C₄₃⋊C₈	—	8	2	no	—
34	C₁₆×C₄₃	—	2	4	no	—
35	C₂³×C₄₃	#107	1	1	yes	characteristic
36	C₂×C₄×C₄₃	—	2	2	no	—
37	C₂×C₄×C₄₃	#95	1	1	yes	characteristic
38	C₂×C₄×C₄₃	#102	1	2	yes	—
39	C₂×C₄×C₄₃	#103	1	2	yes	—
40	C₂×C₄×C₄₃	C₄⋊C₈	1	2	yes	—
41	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
42	C₂×C₄×C₄₃	#108	1	2	yes	—
43	C₂×C₄×C₄₃	D₁₆	1	1	yes	characteristic
44	C₂×C₄×C₄₃	SD₃₂	1	2	yes	—
45	C₂×C₄×C₄₃	Q₃₂	1	1	yes	characteristic
46	C₈×C₄₃	—	2	2	no	—
47	C₈×C₄₃	C₄×C₈	1	1	yes	characteristic
48	C₈×C₄₃	#97	1	1	yes	characteristic
49	C₈×C₄₃	C₂²⋊C₈	1	2	yes	—
50	C₄₃⋊C₈	—	8	4	no	—
51	#7154	—	43	1	no	maximal, Sylow
52	C₂²×C₄₃	#335	1	2	yes	—
53	C₂²×C₄₃	#340	1	1	yes	characteristic
54	C₂²×C₄₃	#366	1	1	yes	characteristic
55	C₂²×C₄₃	#367	1	1	yes	characteristic
56	C₂²×C₄₃	#368	1	2	yes	—
57	C₄×C₄₃	—	2	2	no	—
58	C₄×C₄₃	#334	1	2	yes	—
59	C₄×C₄₃	#336	1	1	yes	characteristic
60	C₄×C₄₃	#341	1	1	yes	characteristic
61	C₄×C₄₃	#357	1	2	yes	—
62	C₄×C₄₃	#358	1	2	yes	—
63	C₂×C₄×C₁₆	D₄₃	1	1	yes	characteristic
64	C₂×#335	—	43	2	no	—
65	C₂×C₄₃	#1277	1	1	yes	characteristic
66	C₂×C₄₃	#1371	1	2	yes	—
67	C₂×C₄₃	#1372	1	2	yes	—
68	C₂×C₄₃	#1380	1	1	yes	characteristic
69	C₂×C₄₃	#1381	1	1	yes	characteristic
70	C₂²×C₁₆	Dic₄₃	1	2	yes	—
71	C₂×C₄⋊C₈	—	86	2	no	—
72	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
73	C₄×C₁₆	—	2	2	no	—
74	#335	—	43	4	no	—
75	#335	—	86	2	no	—
76	C₄₃	#7154	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₈	—	172	2	no	—
78	C₂²×C₈	Dic₈₆	1	1	yes	characteristic
79	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
80	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
81	C₂×C₁₆	—	2	2	no	—
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	C₄₃⋊C₈	1	4	yes	complemented
84	C₄⋊C₈	—	86	4	no	—
85	C₄⋊C₈	—	172	2	no	—
86	C₄×C₈	C₄×D₄₃	1	2	yes	—
87	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
88	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
89	C₂²×C₄	Dic₁₇₂	1	1	yes	characteristic
90	C₂²×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	344	2	no	—
93	C₂×C₈	—	344	2	no	—
94	C₂×C₈	—	172	4	no	—
95	C₂×C₈	C₂×C₄₃⋊C₈	1	2	yes	—
96	C₂×C₈	#10621877	1	2	yes	—
97	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
98	C₂×C₈	Dic₄₃⋊C₄	1	2	yes	—
99	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
100	C₄²	#10621873	1	2	yes	—
101	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
102	C₁₆	—	2	4	no	—
103	C₂³	#13063259	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₄₃⋊#95	1	1	yes	characteristic
106	C₂×C₄	#13063247	1	2	yes	—
107	C₂×C₄	#13063250	1	2	yes	—
108	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	2	yes	—
109	C₂×C₄	#13063257	1	1	yes	characteristic
110	C₂×C₄	#13063261	1	2	yes	—
111	C₂×C₄	D₆₈₈	1	1	yes	characteristic
112	C₂×C₄	#13063270	1	2	yes	—
113	C₂×C₄	Dic₃₄₄	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	344	4	no	—
116	C₈	C₄×C₄₃⋊C₈	1	1	yes	characteristic
117	C₈	#13063236	1	1	yes	characteristic
118	C₈	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
119	C₂²	#14582652	1	2	yes	—
120	C₂²	#14582662	1	1	yes	characteristic
121	C₂²	#14582725	1	1	yes	characteristic
122	C₂²	#14582727	1	1	yes	characteristic
123	C₂²	#14582729	1	2	yes	—
124	C₄	—	2	2	no	—
125	C₄	#14582650	1	2	yes	—
126	C₄	#14582654	1	1	yes	characteristic
127	C₄	#14582664	1	1	yes	characteristic
128	C₄	#14582701	1	2	yes	—
129	C₄	#14582704	1	2	yes	—
130	C₂	#19132430	1	1	yes	characteristic
131	C₂	#19132667	1	2	yes	—
132	C₂	#19132669	1	2	yes	—
133	C₂	#19132687	1	1	yes	characteristic
134	C₂	#19132689	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582652

C₂

yes

maximal

C₂×C₄×C₁₆×C₄₃

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₃⋊C₈)

—

#14582652

—

#14582652

C₄

yes

—

C₂²×C₁₆×C₄₃

C₄

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄×C₁₆×C₄₃

—

C₂²×C₄₃⋊C₈

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₂²×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₃

Q₈

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

C₈

yes

complemented

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

D₈

yes

characteristic

C₂²×C₄×C₄₃

Q₁₆

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄²

yes

characteristic

C₂×C₈×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₂×C₈×C₄₃

C₂×C₈

yes

—

C₂×C₈×C₄₃

M₁₆

yes

—

C₄²×C₄₃

C₂²⋊C₄

yes

—

C₄²×C₄₃

SD₁₆

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₁₆×C₄₃

—

C₂³×C₄₃

#107

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#95

yes

characteristic

C₂×C₄×C₄₃

#102

yes

—

C₂×C₄×C₄₃

#103

yes

—

C₂×C₄×C₄₃

C₄⋊C₈

yes

—

C₂×C₄×C₄₃

#106

yes

characteristic

C₂×C₄×C₄₃

#108

yes

—

C₂×C₄×C₄₃

D₁₆

yes

characteristic

C₂×C₄×C₄₃

SD₃₂

yes

—

C₂×C₄×C₄₃

Q₃₂

yes

characteristic

C₈×C₄₃

—

C₈×C₄₃

C₄×C₈

yes

characteristic

C₈×C₄₃

#97

yes

characteristic

C₈×C₄₃

C₂²⋊C₈

yes

—

C₄₃⋊C₈

—

#7154

—

maximal, Sylow

C₂²×C₄₃

#335

yes

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₄×C₄₃

—

C₄×C₄₃

#334

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₄×C₄₃

#358

yes

—

C₂×C₄×C₁₆

D₄₃

yes

characteristic

C₂×#335

—

C₂×C₄₃

#1277

yes

characteristic

C₂×C₄₃

#1371

yes

—

C₂×C₄₃

#1372

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂²×C₁₆

Dic₄₃

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

C₄×C₁₆

—

#335

—

#335

—

C₄₃

#7154

yes

characteristic, complemented, Sylow

C₂²×C₈

—

172

—

C₂²×C₈

Dic₈₆

yes

characteristic

C₂²×C₈

C₂×Dic₄₃

yes

characteristic

C₂×C₄²

D₁₇₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄₃⋊C₈

yes

complemented

C₄⋊C₈

—

C₄⋊C₈

—

172

—

C₄×C₈

C₄×D₄₃

yes

—

C₄×C₈

C₂²⋊D₄₃

yes

—

C₂²×C₄

D₃₄₄

yes

characteristic

C₂²×C₄

Dic₁₇₂

yes

characteristic

C₂²×C₄

C₄⋊Dic₄₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

344

—

C₂×C₈

—

344

—

C₂×C₈

—

172

—

C₂×C₈

C₂×C₄₃⋊C₈

yes

—

C₂×C₈

#10621877

yes

—

C₂×C₈

C₄×Dic₄₃

yes

characteristic

C₂×C₈

Dic₄₃⋊C₄

yes

—

C₂×C₈

C₂²⋊Dic₄₃

yes

characteristic

100

C₄²

#10621873

yes

—

101

C₄²

(C₂×D₄₃)⋊C₄

yes

—

102

C₁₆

—

103

C₂³

#13063259

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

C₄₃⋊#95

yes

characteristic

106

C₂×C₄

#13063247

yes

—

107

C₂×C₄

#13063250

yes

—

108

C₂×C₄

C₄⋊(C₄₃⋊C₈)

yes

—

109

C₂×C₄

#13063257

yes

characteristic

110

C₂×C₄

#13063261

yes

—

111

C₂×C₄

D₆₈₈

yes

characteristic

112

C₂×C₄

#13063270

yes

—

113

C₂×C₄

Dic₃₄₄

yes

characteristic

114

C₈

—

115

C₈

—

344

—

116

C₈

C₄×C₄₃⋊C₈

yes

characteristic

117

C₈

#13063236

yes

characteristic

118

C₈

C₂²⋊(C₄₃⋊C₈)

yes

—

119

C₂²

#14582652

yes

—

120

C₂²

#14582662

yes

characteristic

121

C₂²

#14582725

yes

characteristic

122

C₂²

#14582727

yes

characteristic

123

C₂²

#14582729

yes

—

124

C₄

—

125

C₄

#14582650

yes

—

126

C₄

#14582654

yes

characteristic

127

C₄

#14582664

yes

characteristic

128

C₄

#14582701

yes

—

129

C₄

#14582704

yes

—

130

C₂

#19132430

yes

characteristic

131

C₂

#19132667

yes

—

132

C₂

#19132669

yes

—

133

C₂

#19132687

yes

characteristic

134

C₂

#19132689

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	D₃₄₄
commutator	C₈×C₄₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₃
Fitting	C₂×C₄×C₁₆×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	#107
cosocle	C₂×C₄×C₈×C₄₃	C₂²
hypercenter	C₂×C₄×C₁₆	D₄₃
hypocenter	C₄₃	#7154
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂²×C₄

D₃₄₄

commutator

C₈×C₄₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₄₃

Fitting

C₂×C₄×C₁₆×C₄₃

C₂

socle, prime radical

C₂³×C₄₃

#107

cosocle

C₂×C₄×C₈×C₄₃

C₂²

hypercenter

C₂×C₄×C₁₆

D₄₃

hypocenter

C₄₃

#7154

LCS

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
17. C₂×C₁₆×C₄₃	C₈	semidirect
76. C₄₃	#7154	semidirect
83. C₂×C₁₆	C₄₃⋊C₈	semidirect

H ≅ G/N

type

17. C₂×C₁₆×C₄₃

C₈

semidirect

76. C₄₃

#7154

semidirect

83. C₂×C₁₆

C₄₃⋊C₈

semidirect

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

subgroup	index
C₂×#14582652	2
C₂×C₄×C₁₆×C₄₃	2
#7154	43

subgroup

index

C₂×#14582652

C₂×C₄×C₁₆×C₄₃

#7154

Subgroups — #24983467

Conjugacy classes of subgroups

Complements

Maximal subgroups