Group #24983455 — subgroups

Subgroups — #24983455

Order 11008.

266 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 103 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	#19136398	C₂	1	2	yes	maximal
2	C₄×#97×C₄₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₄₃⋊C₁₆)	C₄	1	4	yes	complemented
4	C₂²⋊(C₄₃⋊C₁₆)	—	2	2	no	—
5	#14582675	—	2	2	no	—
6	C₂×#97×C₄₃	—	2	2	no	—
7	C₂×C₄×C₈×C₄₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
9	C₄³×C₄₃	C₄	1	1	yes	characteristic
10	C₂×C₄₃⋊C₁₆	—	2	4	no	—
11	C₂²×C₈×C₄₃	—	2	1	no	—
12	C₂²×C₈×C₄₃	D₄	1	2	yes	—
13	C₂×C₄²×C₄₃	—	2	1	no	—
14	C₂×C₄²×C₄₃	—	2	1	no	—
15	C₂×C₄²×C₄₃	C₈	1	2	yes	—
16	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
17	#97×C₄₃	—	2	8	no	—
18	C₄×C₈×C₄₃	—	2	1	no	—
19	C₄×C₈×C₄₃	C₈	1	2	yes	—
20	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
22	C₄×C₈×C₄₃	Q₈	1	1	yes	characteristic
23	C₂²×C₄×C₄₃	—	2	2	no	—
24	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₄₃	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₄₃	—	2	4	no	—
28	C₂×C₈×C₄₃	—	2	2	no	—
29	C₂×C₈×C₄₃	—	2	4	no	—
30	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
31	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
32	C₄²×C₄₃	—	2	4	no	—
33	C₄²×C₄₃	—	4	2	no	—
34	C₄²×C₄₃	—	4	2	no	—
35	C₄²×C₄₃	C₄²	1	1	yes	characteristic
36	C₄²×C₄₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₄₃	M₁₆	1	1	yes	characteristic
39	C₄₃⋊C₁₆	—	4	4	no	—
40	C₂³×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
41	C₂×C₄×C₄₃	—	2	2	no	—
42	C₂×C₄×C₄₃	—	2	2	no	—
43	C₂×C₄×C₄₃	—	2	2	no	—
44	C₂×C₄×C₄₃	—	2	4	no	—
45	C₂×C₄×C₄₃	—	4	2	no	—
46	C₂×C₄×C₄₃	—	4	2	no	—
47	C₂×C₄×C₄₃	#95	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₄₃	#97	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₄⋊C₈	1	1	yes	characteristic
53	C₈×C₄₃	—	2	4	no	—
54	C₈×C₄₃	—	2	2	no	—
55	C₈×C₄₃	#104	1	4	yes	—
56	#7149	—	43	1	no	maximal, Sylow
57	C₂²×C₄₃	—	2	1	no	—
58	C₂²×C₄₃	—	2	1	no	—
59	C₂²×C₄₃	#336	1	1	yes	characteristic
60	C₂²×C₄₃	#349	1	2	yes	—
61	C₄×C₄₃	—	2	1	no	—
62	C₄×C₄₃	—	2	1	no	—
63	C₄×C₄₃	—	2	1	no	—
64	C₄×C₄₃	—	2	1	no	—
65	C₄×C₄₃	—	4	4	no	—
66	C₄×C₄₃	#339	1	2	yes	—
67	C₄×C₄₃	#347	1	2	yes	—
68	#2163	—	43	2	no	—
69	C₄×#97	D₄₃	1	1	yes	characteristic
70	C₂×C₄₃	—	2	2	no	—
71	C₂×C₄₃	#1276	1	1	yes	characteristic
72	C₂×C₄₃	#1375	1	1	yes	characteristic
73	C₂×C₄₃	#1377	1	1	yes	characteristic
74	C₂×#97	—	2	2	no	—
75	C₂×C₄×C₈	Dic₄₃	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
77	C₄³	Dic₄₃	1	1	yes	characteristic
78	C₄⋊C₁₆	—	43	4	no	—
79	C₂²⋊C₁₆	—	86	2	no	—
80	#346	—	86	2	no	—
81	C₄₃	#7149	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	—	2	1	no	—
86	C₂×C₄²	C₄₃⋊C₈	1	2	yes	—
87	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
88	C₂×C₁₆	—	86	4	no	—
89	#97	—	2	8	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	C₄₃⋊C₈	1	2	yes	—
92	C₄×C₈	Dic₈₆	1	1	yes	characteristic
93	C₄×C₈	C₄×D₄₃	1	2	yes	—
94	C₄×C₈	D₁₇₂	1	1	yes	characteristic
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
97	C₂²×C₄	#10621877	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	Dic₄₃⋊C₄	1	2	yes	—
103	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
104	C₄²	—	2	4	no	—
105	C₄²	—	4	2	no	—
106	C₄²	—	4	2	no	—
107	C₄²	C₂×C₄₃⋊C₈	1	1	yes	characteristic
108	C₄²	#10621877	1	1	yes	characteristic
109	C₄²	C₄×Dic₄₃	1	1	yes	characteristic
110	C₄²	C₄⋊Dic₄₃	1	1	yes	characteristic
111	C₁₆	—	172	4	no	—
112	C₂³	C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	C₄₃⋊#95	1	1	yes	characteristic
120	C₂×C₄	C₄×C₄₃⋊C₈	1	1	yes	characteristic
121	C₂×C₄	#13063236	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
123	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
124	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
125	C₈	—	2	4	no	—
126	C₈	—	2	2	no	—
127	C₈	#13063253	1	4	yes	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	#14582654	1	1	yes	characteristic
131	C₂²	#14582681	1	2	yes	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	4	4	no	—
137	C₄	#14582659	1	2	yes	—
138	C₄	#14582677	1	2	yes	—
139	C₂	—	2	2	no	—
140	C₂	#19132428	1	1	yes	characteristic
141	C₂	#19132675	1	1	yes	characteristic
142	C₂	#19132679	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136398

C₂

yes

maximal

C₄×#97×C₄₃

C₂

yes

maximal, characteristic

C₄⋊(C₄₃⋊C₁₆)

C₄

yes

complemented

C₂²⋊(C₄₃⋊C₁₆)

—

#14582675

—

C₂×#97×C₄₃

—

C₂×C₄×C₈×C₄₃

C₄

yes

characteristic

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄³×C₄₃

C₄

yes

characteristic

C₂×C₄₃⋊C₁₆

—

C₂²×C₈×C₄₃

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

C₈

yes

—

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

#97×C₄₃

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

C₈

yes

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

Q₈

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₃

M₁₆

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₄²

yes

characteristic

C₄²×C₄₃

C₄⋊C₄

yes

characteristic

C₄²×C₄₃

C₂×C₈

yes

characteristic

C₄²×C₄₃

M₁₆

yes

characteristic

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#95

yes

characteristic

C₂×C₄×C₄₃

C₄×C₈

yes

characteristic

C₂×C₄×C₄₃

#97

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₈

yes

characteristic

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#104

yes

—

#7149

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#336

yes

characteristic

C₂²×C₄₃

#349

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#339

yes

—

C₄×C₄₃

#347

yes

—

#2163

—

C₄×#97

D₄₃

yes

characteristic

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×#97

—

C₂×C₄×C₈

Dic₄₃

yes

characteristic

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

C₄³

Dic₄₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄₃

#7149

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₄₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₄₃⋊C₈

yes

—

C₂×C₄²

C₂×Dic₄₃

yes

characteristic

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₄×C₈

C₄₃⋊C₈

yes

—

C₄×C₈

Dic₈₆

yes

characteristic

C₄×C₈

C₄×D₄₃

yes

—

C₄×C₈

D₁₇₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×C₄₃⋊C₈

yes

characteristic

C₂²×C₄

#10621877

yes

characteristic

C₂²×C₄

C₂²⋊Dic₄₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

Dic₄₃⋊C₄

yes

—

103

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

C₂×C₄₃⋊C₈

yes

characteristic

108

C₄²

#10621877

yes

characteristic

109

C₄²

C₄×Dic₄₃

yes

characteristic

110

C₄²

C₄⋊Dic₄₃

yes

characteristic

111

C₁₆

—

172

—

112

C₂³

C₂²⋊(C₄₃⋊C₈)

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₄₃⋊#95

yes

characteristic

120

C₂×C₄

C₄×C₄₃⋊C₈

yes

characteristic

121

C₂×C₄

#13063236

yes

characteristic

122

C₂×C₄

C₂²⋊(C₄₃⋊C₈)

yes

characteristic

123

C₂×C₄

C₄⋊(C₄₃⋊C₈)

yes

characteristic

124

C₂×C₄

C₄⋊(C₄₃⋊C₈)

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

#13063253

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

#14582654

yes

characteristic

131

C₂²

#14582681

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

#14582659

yes

—

138

C₄

#14582677

yes

—

139

C₂

—

140

C₂

#19132428

yes

characteristic

141

C₂

#19132675

yes

characteristic

142

C₂

#19132679

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄₃⋊#95
commutator	C₂×C₄×C₄₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₃
Fitting	C₄×#97×C₄₃	C₂
socle	C₂²×C₄₃	#336
prime radical	C₂³×C₄₃	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₄₃	C₂²
hypercenter	C₄×#97	D₄₃
hypocenter	C₄₃	#7149
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×#97

subgroup

G/H

center

C₂×C₄

C₄₃⋊#95

commutator

C₂×C₄×C₄₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₄₃

Fitting

C₄×#97×C₄₃

C₂

socle

C₂²×C₄₃

#336

prime radical

C₂³×C₄₃

C₂²⋊C₈

cosocle

C₂×C₄×C₈×C₄₃

C₂²

hypercenter

C₄×#97

D₄₃

hypocenter

C₄₃

#7149

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×#97

N	H ≅ G/N	type
3. C₄⋊(C₄₃⋊C₁₆)	C₄	semidirect
81. C₄₃	#7149	semidirect

H ≅ G/N

type

3. C₄⋊(C₄₃⋊C₁₆)

C₄

semidirect

81. C₄₃

#7149

semidirect

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

subgroup	index
#19136398	2
C₄×#97×C₄₃	2
#7149	43

subgroup

index

#19136398

C₄×#97×C₄₃

#7149

Subgroups — #24983455

Conjugacy classes of subgroups

Complements

Maximal subgroups