Group #24985964 — subgroups

Subgroups — #24985964

Order 11008.

518 conjugacy classes of subgroups; only the 291 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 291 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	#19134340	C₂	1	2	yes	maximal
2	C₄×#13063235	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582621	C₂	1	2	yes	maximal
4	C₂×#14582620	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063235	C₄	1	4	yes	—
7	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
9	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
10	#14582653	C₄	1	8	yes	—
11	#14582621	C₄	1	8	yes	complemented
12	#14582620	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
15	C₈²×C₄₃	C₄	1	4	yes	complemented
16	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
17	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₄₃	C₂×C₄	1	4	yes	—
20	#13063235	—	2	8	no	—
21	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
22	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
23	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
27	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
33	C₂²×Dic₄₃	C₄²	1	4	yes	—
34	C₂×C₈×C₄₃	—	2	8	no	—
35	C₂×C₈×C₄₃	C₄²	1	8	yes	—
36	C₂×C₈×C₄₃	C₄²	1	4	yes	—
37	C₂×C₈×C₄₃	M₁₆	1	8	yes	—
38	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
39	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
41	C₄×Dic₄₃	—	2	4	no	—
42	C₄×Dic₄₃	—	2	8	no	—
43	C₂×C₄₃⋊C₈	—	2	4	no	—
44	C₂×C₄₃⋊C₈	—	2	8	no	—
45	C₂×C₄₃⋊C₈	C₄²	1	4	yes	—
46	C₂×C₄₃⋊C₈	C₄²	1	8	yes	—
47	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
50	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
52	C₂×C₄×C₄₃	C₂×M₁₆	1	4	yes	—
53	C₂×C₄×C₄₃	#131	1	8	yes	—
54	C₂×Dic₄₃	—	2	4	no	—
55	C₂×Dic₄₃	—	2	8	no	—
56	C₈×C₄₃	—	2	8	no	—
57	C₈×C₄₃	#97	1	8	yes	—
58	C₄₃⋊C₈	—	2	4	no	—
59	C₄₃⋊C₈	—	2	8	no	—
60	#7844	—	43	1	no	maximal, Sylow
61	C₂²×C₄₃	C₄³	1	1	yes	characteristic
62	C₂²×C₄₃	C₄×M₁₆	1	2	yes	—
63	C₂²×C₄₃	#405	1	4	yes	—
64	C₄×C₄₃	C₂×#97	1	2	yes	—
65	C₄×C₄₃	C₄×M₁₆	1	2	yes	—
66	C₄×C₄₃	#405	1	4	yes	—
67	C₄×C₄₃	#405	1	4	yes	—
68	Dic₄₃	—	4	4	no	—
69	#1744	—	43	2	no	—
70	C₄×#97	—	43	1	no	—
71	C₂×#322	—	43	1	no	—
72	C₂×#322	—	43	2	no	—
73	C₂×C₈²	D₄₃	1	1	yes	characteristic
74	C₂×C₄₃	#1453	1	4	yes	—
75	C₂×C₄₃	C₄×#97	1	1	yes	characteristic
76	C₂×C₄₃	#1726	1	2	yes	—
77	C₂×#97	—	43	4	no	—
78	C₂×C₄×C₈	—	43	1	no	—
79	C₂×C₄×C₈	—	43	2	no	—
80	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
81	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
82	C₄³	—	43	1	no	—
83	#336	—	43	8	no	—
84	#322	—	43	4	no	—
85	#322	—	43	8	no	—
86	C₈²	Dic₄₃	1	4	yes	complemented
87	C₄₃	#7844	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	43	4	no	—
89	C₂²×C₈	—	43	2	no	—
90	C₂²×C₈	C₄×D₄₃	1	4	yes	—
91	C₂²×C₈	C₄×D₄₃	1	2	yes	—
92	C₂×C₄²	—	43	4	no	—
93	C₂×C₄²	—	43	2	no	—
94	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
95	#97	—	86	8	no	—
96	C₄×C₈	—	43	4	no	—
97	C₄×C₈	—	43	4	no	—
98	C₄×C₈	—	43	4	no	—
99	C₄×C₈	C₄×D₄₃	1	4	yes	—
100	C₄×C₈	C₄×D₄₃	1	2	yes	—
101	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
102	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
103	C₂²×C₄	—	43	4	no	—
104	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
106	C₂×C₈	—	2	8	no	—
107	C₂×C₈	—	43	4	no	—
108	C₂×C₈	—	43	8	no	—
109	C₂×C₈	—	86	4	no	—
110	C₂×C₈	—	86	8	no	—
111	C₂×C₈	#10621872	1	8	yes	—
112	C₂×C₈	C₄×Dic₄₃	1	8	yes	—
113	C₂×C₈	C₄×Dic₄₃	1	4	yes	—
114	C₄²	—	86	4	no	—
115	C₄²	—	86	8	no	—
116	C₄²	C₂×C₄×D₄₃	1	2	yes	—
117	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
118	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
119	C₂³	C₄²×D₄₃	1	1	yes	characteristic
120	C₂×C₄	—	86	4	no	—
121	C₂×C₄	—	86	8	no	—
122	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
123	C₂×C₄	C₄²×D₄₃	1	2	yes	—
124	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
126	C₂×C₄	C₂×#10621872	1	4	yes	—
127	C₂×C₄	#13063331	1	8	yes	—
128	C₈	—	2	8	no	—
129	C₈	—	86	4	no	—
130	C₈	—	86	8	no	—
131	C₈	#13063235	1	8	yes	—
132	C₂²	C₄²×Dic₄₃	1	1	yes	characteristic
133	C₂²	C₄×#10621872	1	2	yes	—
134	C₂²	#14582853	1	4	yes	—
135	C₄	—	172	4	no	—
136	C₄	C₂×#13063235	1	2	yes	—
137	C₄	C₄×#10621872	1	2	yes	—
138	C₄	#14582853	1	4	yes	—
139	C₄	#14582854	1	4	yes	—
140	C₂	#19132889	1	4	yes	—
141	C₂	C₄×#13063235	1	1	yes	characteristic
142	C₂	#19134281	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19134340

C₂

yes

maximal

C₄×#13063235

C₂

yes

maximal, characteristic

C₂×#14582621

C₂

yes

maximal

C₂×#14582620

C₂

yes

maximal, characteristic

C₂×C₈²×C₄₃

C₂

yes

maximal, characteristic

C₂×#13063235

C₄

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

C₄²×Dic₄₃

C₂²

yes

characteristic

#14582653

C₄

yes

—

#14582621

C₄

yes

complemented

#14582620

C₄

yes

complemented

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₃

C₂²

yes

—

C₈²×C₄₃

C₄

yes

complemented

C₂²×C₄₃⋊C₈

C₂×C₄

yes

—

C₂²×C₄₃⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₄₃

C₂×C₄

yes

—

C₂×C₄×Dic₄₃

C₂×C₄

yes

—

#13063235

—

C₄×C₄₃⋊C₈

C₂×C₄

yes

—

C₄×C₄₃⋊C₈

C₂×C₄

yes

—

C₄×C₄₃⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₄₃

C₂×C₄

yes

—

C₂²×C₈×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂²×C₄

yes

—

C₂²×Dic₄₃

C₄²

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄²

yes

—

C₂×C₈×C₄₃

C₄²

yes

—

C₂×C₈×C₄₃

M₁₆

yes

—

C₄²×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₄²

yes

—

C₂×C₄₃⋊C₈

C₄²

yes

—

C₂³×C₄₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₃

C₂×C₄²

yes

—

C₂×C₄×C₄₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₃

C₂×C₄²

yes

—

C₂×C₄×C₄₃

C₂×M₁₆

yes

—

C₂×C₄×C₄₃

#131

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#97

yes

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#7844

—

maximal, Sylow

C₂²×C₄₃

C₄³

yes

characteristic

C₂²×C₄₃

C₄×M₁₆

yes

—

C₂²×C₄₃

#405

yes

—

C₄×C₄₃

C₂×#97

yes

—

C₄×C₄₃

C₄×M₁₆

yes

—

C₄×C₄₃

#405

yes

—

C₄×C₄₃

#405

yes

—

Dic₄₃

—

#1744

—

C₄×#97

—

C₂×#322

—

C₂×#322

—

C₂×C₈²

D₄₃

yes

characteristic

C₂×C₄₃

#1453

yes

—

C₂×C₄₃

C₄×#97

yes

characteristic

C₂×C₄₃

#1726

yes

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

C₂×C₄×C₈

C₂×D₄₃

yes

—

C₄³

—

#336

—

#322

—

#322

—

C₈²

Dic₄₃

yes

complemented

C₄₃

#7844

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄×D₄₃

yes

—

C₂²×C₈

C₄×D₄₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₄₃

yes

characteristic

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₄₃

yes

—

100

C₄×C₈

C₄×D₄₃

yes

—

101

C₄×C₈

C₂×Dic₄₃

yes

—

102

C₄×C₈

C₂×Dic₄₃

yes

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

105

C₂²×C₄

C₂×C₄×D₄₃

yes

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

#10621872

yes

—

112

C₂×C₈

C₄×Dic₄₃

yes

—

113

C₂×C₈

C₄×Dic₄₃

yes

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₂×C₄×D₄₃

yes

—

117

C₄²

C₂×C₄×D₄₃

yes

characteristic

118

C₄²

C₂²×Dic₄₃

yes

characteristic

119

C₂³

C₄²×D₄₃

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₄²×D₄₃

yes

characteristic

123

C₂×C₄

C₄²×D₄₃

yes

—

124

C₂×C₄

C₂×C₄×Dic₄₃

yes

characteristic

125

C₂×C₄

C₂×C₄×Dic₄₃

yes

—

126

C₂×C₄

C₂×#10621872

yes

—

127

C₂×C₄

#13063331

yes

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

#13063235

yes

—

132

C₂²

C₄²×Dic₄₃

yes

characteristic

133

C₂²

C₄×#10621872

yes

—

134

C₂²

#14582853

yes

—

135

C₄

—

172

—

136

C₄

C₂×#13063235

yes

—

137

C₄

C₄×#10621872

yes

—

138

C₄

#14582853

yes

—

139

C₄

#14582854

yes

—

140

C₂

#19132889

yes

—

141

C₂

C₄×#13063235

yes

characteristic

142

C₂

#19134281

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₃
commutator	C₂²×C₄₃	C₄³
Fitting	C₂×C₈²×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₄²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×C₈²	D₄₃
hypocenter	C₄₃	#7844
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₄₃

commutator

C₂²×C₄₃

C₄³

Fitting

C₂×C₈²×C₄₃

C₂

socle, prime radical

C₂³×C₄₃

C₂×C₄²

cosocle

C₂×C₄²×C₄₃

C₂³

hypercenter

C₂×C₈²

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
11. #14582621	C₄	semidirect
12. #14582620	C₄	semidirect
15. C₈²×C₄₃	C₄	semidirect
86. C₈²	Dic₄₃	semidirect
87. C₄₃	#7844	semidirect

H ≅ G/N

type

11. #14582621

C₄

semidirect

12. #14582620

C₄

semidirect

15. C₈²×C₄₃

C₄

semidirect

86. C₈²

Dic₄₃

semidirect

87. C₄₃

#7844

semidirect

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

subgroup	index
#19134340	2
C₄×#13063235	2
C₂×#14582621	2
C₂×#14582620	2
C₂×C₈²×C₄₃	2
#7844	43

subgroup

index

C₂×C₈²×C₄₃

Subgroups — #24985964

Conjugacy classes of subgroups

Complements

Maximal subgroups