Group #25017079 — subgroups

Subgroups — #25017079

Order 11008.

218 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 127 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×#14582733	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	Dic₄₃⋊C₃₂	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₃×M₆₄	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₃⋊C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₄₃	C₄	1	1	yes	characteristic
7	#14582733	—	2	2	no	—
8	C₂×C₄₃⋊C₃₂	C₂²	1	2	yes	—
9	C₁₆×Dic₄₃	C₄	1	2	yes	—
10	C₁₆×Dic₄₃	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
12	C₄₃×M₆₄	—	2	2	no	—
13	C₂×C₃₂×C₄₃	C₂²	1	2	yes	—
14	C₂²×C₄₃⋊C₈	C₈	1	1	yes	characteristic
15	C₂×C₄×Dic₄₃	C₈	1	1	yes	characteristic
16	C₂×C₄₃⋊C₁₆	—	2	2	no	—
17	C₂×C₄₃⋊C₁₆	C₂×C₄	1	2	yes	—
18	C₈×Dic₄₃	C₈	1	2	yes	—
19	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
20	C₄₃⋊C₃₂	—	2	2	no	—
21	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
24	C₂×C₁₆×C₄₃	Q₈	1	2	yes	—
25	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
26	C₃₂×C₄₃	—	2	2	no	—
27	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
28	C₂²×Dic₄₃	C₁₆	1	2	yes	—
29	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₄₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₄₃	M₁₆	1	2	yes	—
32	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
33	C₄×Dic₄₃	C₁₆	1	1	yes	characteristic
34	C₄×Dic₄₃	C₁₆	1	1	yes	characteristic
35	C₄×Dic₄₃	C₂×C₈	1	2	yes	—
36	C₂×C₄₃⋊C₈	—	2	2	no	—
37	C₂×C₄₃⋊C₈	C₂×C₈	1	2	yes	—
38	C₁₆×C₄₃	C₄⋊C₄	1	2	yes	—
39	C₁₆×C₄₃	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₄₃	C₂×Q₈	1	1	yes	characteristic
41	C₄₃⋊C₁₆	—	2	2	no	—
42	C₂³×C₄₃	C₂×C₁₆	1	1	yes	characteristic
43	C₂×C₄×C₄₃	C₄⋊C₈	1	2	yes	—
44	C₂×C₄×C₄₃	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₄₃	M₃₂	1	2	yes	—
46	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
47	C₂×Dic₄₃	—	2	4	no	—
48	C₂×Dic₄₃	C₂×C₁₆	1	2	yes	—
49	C₂×Dic₄₃	C₂×C₁₆	1	2	yes	—
50	C₈×C₄₃	C₄⋊C₈	1	2	yes	—
51	C₈×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₈×C₄₃	C₂×M₁₆	1	1	yes	characteristic
53	C₄₃⋊C₈	—	2	2	no	—
54	#13627	—	43	1	no	maximal, Sylow
55	C₂²×C₄₃	—	2	2	no	—
56	C₂²×C₄₃	C₄⋊C₁₆	1	2	yes	—
57	C₂²×C₄₃	C₂²×C₁₆	1	1	yes	characteristic
58	C₄×C₄₃	C₄⋊C₁₆	1	2	yes	—
59	C₄×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
60	C₄×C₄₃	C₂×M₃₂	1	1	yes	characteristic
61	Dic₄₃	—	2	2	no	—
62	Dic₄₃	M₆₄	1	4	yes	complemented
63	C₂×M₆₄	—	43	1	no	—
64	C₂×M₆₄	D₄₃	1	1	yes	characteristic
65	C₂×C₄×C₁₆	—	43	1	no	—
66	C₄⋊C₃₂	—	43	4	no	—
67	C₂×C₄₃	—	2	2	no	—
68	C₂×C₄₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
69	C₂×C₄₃	C₂×M₆₄	1	2	yes	—
70	C₂²×C₁₆	—	43	1	no	—
71	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
72	C₂×C₄×C₈	—	43	1	no	—
73	M₆₄	—	2	2	no	—
74	M₆₄	—	86	2	no	—
75	C₂×C₃₂	—	43	2	no	—
76	C₂×C₃₂	C₂×D₄₃	1	2	yes	—
77	C₄×C₁₆	—	43	2	no	—
78	C₄×C₁₆	—	43	2	no	—
79	C₄₃	#13627	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	—	43	1	no	—
81	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
82	C₂×C₄²	—	43	1	no	—
83	C₂×C₁₆	—	43	2	no	—
84	C₂×C₁₆	—	86	2	no	—
85	C₂×C₁₆	Dic₈₆	1	2	yes	—
86	C₂×C₁₆	C₄×D₄₃	1	1	yes	characteristic
87	C₂×C₁₆	C₂²⋊D₄₃	1	2	yes	—
88	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
89	C₄×C₈	—	43	2	no	—
90	C₄×C₈	—	43	2	no	—
91	C₃₂	—	2	2	no	—
92	C₃₂	—	86	2	no	—
93	C₂²×C₄	—	43	2	no	—
94	C₂²×C₄	C₈×D₄₃	1	1	yes	characteristic
95	C₂×C₈	—	43	2	no	—
96	C₂×C₈	—	86	2	no	—
97	C₂×C₈	C₈×D₄₃	1	1	yes	characteristic
98	C₂×C₈	#10621872	1	2	yes	—
99	C₂×C₈	Dic₄₃⋊C₄	1	2	yes	—
100	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
101	C₄²	—	43	1	no	—
102	C₄²	—	43	1	no	—
103	C₄²	—	43	2	no	—
104	C₁₆	—	86	2	no	—
105	C₁₆	Dic₄₃⋊C₄	1	2	yes	—
106	C₁₆	C₂×Dic₈₆	1	1	yes	characteristic
107	C₁₆	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
108	C₂³	C₁₆×D₄₃	1	1	yes	characteristic
109	C₂×C₄	—	43	2	no	—
110	C₂×C₄	—	43	2	no	—
111	C₂×C₄	—	86	4	no	—
112	C₂×C₄	Dic₄₃⋊C₈	1	2	yes	—
113	C₂×C₄	C₁₆×D₄₃	1	1	yes	characteristic
114	C₂×C₄	#13063264	1	2	yes	—
115	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
116	C₈	—	86	2	no	—
117	C₈	Dic₄₃⋊C₈	1	2	yes	—
118	C₈	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
119	C₈	C₂×#10621872	1	1	yes	characteristic
120	C₂²	—	2	2	no	—
121	C₂²	Dic₄₃⋊C₁₆	1	2	yes	—
122	C₂²	C₂×C₁₆×D₄₃	1	1	yes	characteristic
123	C₄	—	43	4	no	—
124	C₄	—	86	2	no	—
125	C₄	Dic₄₃⋊C₁₆	1	2	yes	—
126	C₄	C₂×Dic₄₃⋊C₈	1	1	yes	characteristic
127	C₄	C₂×#13063264	1	1	yes	characteristic
128	C₂	—	2	2	no	—
129	C₂	C₂×Dic₄₃⋊C₁₆	1	1	yes	characteristic
130	C₂	M₆₄×D₄₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582733

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₄₃

C₂

yes

maximal, characteristic

Dic₄₃⋊C₃₂

C₂

yes

maximal, complemented

C₂×C₄₃×M₆₄

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₁₆

C₄

yes

characteristic

C₂×C₈×Dic₄₃

C₄

yes

characteristic

#14582733

—

C₂×C₄₃⋊C₃₂

C₂²

yes

—

C₁₆×Dic₄₃

C₄

yes

—

C₁₆×Dic₄₃

C₂²

yes

—

C₂²×C₁₆×C₄₃

C₂²

yes

characteristic

C₄₃×M₆₄

—

C₂×C₃₂×C₄₃

C₂²

yes

—

C₂²×C₄₃⋊C₈

C₈

yes

characteristic

C₂×C₄×Dic₄₃

C₈

yes

characteristic

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

C₂×C₄

yes

—

C₈×Dic₄₃

C₈

yes

—

C₈×Dic₄₃

C₂×C₄

yes

—

C₄₃⋊C₃₂

—

C₂²×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₃

D₄

yes

—

C₂×C₁₆×C₄₃

Q₈

yes

—

C₂×C₁₆×C₄₃

C₂³

yes

characteristic

C₃₂×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₈

yes

characteristic

C₂²×Dic₄₃

C₁₆

yes

—

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₂×C₈×C₄₃

C₂×C₈

yes

characteristic

C₂×C₈×C₄₃

M₁₆

yes

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₄×Dic₄₃

C₁₆

yes

characteristic

C₄×Dic₄₃

C₁₆

yes

characteristic

C₄×Dic₄₃

C₂×C₈

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×C₈

yes

—

C₁₆×C₄₃

C₄⋊C₄

yes

—

C₁₆×C₄₃

C₂×D₄

yes

characteristic

C₁₆×C₄₃

C₂×Q₈

yes

characteristic

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₈

yes

—

C₂×C₄×C₄₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₄₃

M₃₂

yes

—

C₂×C₄×C₄₃

C₂²×C₈

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×C₁₆

yes

—

C₂×Dic₄₃

C₂×C₁₆

yes

—

C₈×C₄₃

C₄⋊C₈

yes

—

C₈×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₄₃

C₂×M₁₆

yes

characteristic

C₄₃⋊C₈

—

#13627

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

C₄⋊C₁₆

yes

—

C₂²×C₄₃

C₂²×C₁₆

yes

characteristic

C₄×C₄₃

C₄⋊C₁₆

yes

—

C₄×C₄₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₄₃

C₂×M₃₂

yes

characteristic

Dic₄₃

—

Dic₄₃

M₆₄

yes

complemented

C₂×M₆₄

—

C₂×M₆₄

D₄₃

yes

characteristic

C₂×C₄×C₁₆

—

C₄⋊C₃₂

—

C₂×C₄₃

—

C₂×C₄₃

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₄₃

C₂×M₆₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₄₃

yes

characteristic

C₂×C₄×C₈

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₄₃

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄₃

#13627

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₄₃

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Dic₈₆

yes

—

C₂×C₁₆

C₄×D₄₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₄₃

yes

—

C₂×C₁₆

C₂²×D₄₃

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₂²×C₄

—

C₂²×C₄

C₈×D₄₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×D₄₃

yes

characteristic

C₂×C₈

#10621872

yes

—

C₂×C₈

Dic₄₃⋊C₄

yes

—

100

C₂×C₈

C₂×C₄×D₄₃

yes

characteristic

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₁₆

—

105

C₁₆

Dic₄₃⋊C₄

yes

—

106

C₁₆

C₂×Dic₈₆

yes

characteristic

107

C₁₆

C₂×C₂²⋊D₄₃

yes

characteristic

108

C₂³

C₁₆×D₄₃

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

Dic₄₃⋊C₈

yes

—

113

C₂×C₄

C₁₆×D₄₃

yes

characteristic

114

C₂×C₄

#13063264

yes

—

115

C₂×C₄

C₂×C₈×D₄₃

yes

characteristic

116

C₈

—

117

C₈

Dic₄₃⋊C₈

yes

—

118

C₈

C₂×Dic₄₃⋊C₄

yes

characteristic

119

C₈

C₂×#10621872

yes

characteristic

120

C₂²

—

121

C₂²

Dic₄₃⋊C₁₆

yes

—

122

C₂²

C₂×C₁₆×D₄₃

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

Dic₄₃⋊C₁₆

yes

—

126

C₄

C₂×Dic₄₃⋊C₈

yes

characteristic

127

C₄

C₂×#13063264

yes

characteristic

128

C₂

—

129

C₂

C₂×Dic₄₃⋊C₁₆

yes

characteristic

130

C₂

M₆₄×D₄₃

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₄₃
commutator, socle	C₂²×C₄₃	C₂²×C₁₆
Fitting	C₂×C₄₃×M₆₄	C₂
prime radical	C₂³×C₄₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₄₃	C₂³
hypercenter	C₂×M₆₄	D₄₃
hypocenter	C₄₃	#13627
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₄₃

commutator, socle

C₂²×C₄₃

C₂²×C₁₆

Fitting

C₂×C₄₃×M₆₄

C₂

prime radical

C₂³×C₄₃

C₂×C₁₆

cosocle

C₂×C₁₆×C₄₃

C₂³

hypercenter

C₂×M₆₄

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

N	H ≅ G/N	type
3. Dic₄₃⋊C₃₂	C₂	semidirect
62. Dic₄₃	M₆₄	semidirect
79. C₄₃	#13627	semidirect

H ≅ G/N

type

3. Dic₄₃⋊C₃₂

C₂

semidirect

62. Dic₄₃

M₆₄

semidirect

79. C₄₃

#13627

semidirect

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

subgroup	index
C₂×#14582733	2
C₂×C₁₆×Dic₄₃	2
Dic₄₃⋊C₃₂	2
C₂×C₄₃×M₆₄	2
#13627	43

subgroup

index

C₂×#14582733

C₂×C₁₆×Dic₄₃

Dic₄₃⋊C₃₂

C₂×C₄₃×M₆₄

#13627

Subgroups — #25017079

Conjugacy classes of subgroups

Complements

Maximal subgroups