Group #25017161 — subgroups

Subgroups — #25017161

Order 11008.

186 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 105 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	#19136889	C₂	1	2	yes	maximal, complemented
2	#19132793	C₂	1	1	yes	maximal, characteristic, complemented
3	#19132737	C₂	1	2	yes	maximal, complemented
4	C₄×C₄₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#2170	C₂	1	1	yes	maximal, characteristic
6	#14582733	—	2	2	no	—
7	#14582733	C₂²	1	2	yes	—
8	C₂×C₄₃⋊C₃₂	—	2	2	no	—
9	C₂×C₄₃⋊C₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₄₃⋊C₃₂	C₂²	1	1	yes	characteristic
11	C₄₃×#504	C₂²	1	2	yes	—
12	C₄₃×#443	C₄	1	1	yes	characteristic
13	C₄₃×#364	C₄	1	1	yes	characteristic
14	C₄₃×#350	C₄	1	2	yes	—
15	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃⋊C₃₂	—	2	2	no	—
17	C₄₃⋊C₃₂	—	2	1	no	—
18	C₄₃⋊C₃₂	D₄	1	1	yes	characteristic
19	C₄₃⋊C₃₂	D₄	1	1	yes	characteristic
20	#135×C₄₃	C₈	1	1	yes	characteristic
21	#131×C₄₃	C₂×C₄	1	2	yes	—
22	M₃₂×C₄₃	—	2	2	no	—
23	M₃₂×C₄₃	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₄₃	—	2	2	no	—
25	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
27	#108×C₄₃	C₈	1	1	yes	characteristic
28	#104×C₄₃	C₈	1	2	yes	—
29	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
30	#41×C₄₃	C₂×C₈	1	2	yes	—
31	Q₁₆×C₄₃	C₁₆	1	1	yes	characteristic
32	SD₁₆×C₄₃	C₁₆	1	2	yes	—
33	D₈×C₄₃	C₁₆	1	1	yes	characteristic
34	M₁₆×C₄₃	—	2	2	no	—
35	M₁₆×C₄₃	C₂×C₈	1	2	yes	—
36	C₂×C₈×C₄₃	—	2	2	no	—
37	C₂×C₈×C₄₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
40	C₁₆×C₄₃	—	2	2	no	—
41	C₁₆×C₄₃	—	2	1	no	—
42	C₁₆×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₄₃	#41	1	1	yes	characteristic
44	Q₈×C₄₃	C₂×C₁₆	1	2	yes	—
45	D₄×C₄₃	—	2	2	no	—
46	D₄×C₄₃	C₂×C₁₆	1	2	yes	—
47	C₂×C₄×C₄₃	—	2	1	no	—
48	C₂×C₄×C₄₃	—	2	2	no	—
49	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₄₃	—	2	2	no	—
51	C₈×C₄₃	C₂×C₁₆	1	1	yes	characteristic
52	C₈×C₄₃	C₂×C₁₆	1	1	yes	characteristic
53	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
54	C₈×C₄₃	#131	1	1	yes	characteristic
55	#13647	—	43	1	no	maximal, Sylow
56	C₂²×C₄₃	—	2	2	no	—
57	C₂²×C₄₃	#504	1	1	yes	characteristic
58	C₄×C₄₃	—	2	2	no	—
59	C₄×C₄₃	—	2	2	no	—
60	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
61	C₄×C₄₃	C₂²×C₁₆	1	1	yes	characteristic
62	#2258	—	43	2	no	—
63	#2170	D₄₃	1	1	yes	characteristic
64	#1422	—	43	1	no	—
65	#1401	—	43	2	no	—
66	C₄×C₃₂	—	43	1	no	—
67	C₂×C₄₃	—	2	1	no	—
68	C₂×C₄₃	—	4	2	no	—
69	C₂×C₄₃	D₄×C₁₆	1	1	yes	characteristic
70	#504	C₂×D₄₃	1	2	yes	—
71	#443	Dic₄₃	1	1	yes	characteristic
72	M₆₄	—	43	2	no	—
73	M₆₄	—	86	2	no	—
74	C₂×C₃₂	—	43	1	no	—
75	C₂×C₃₂	—	43	1	no	—
76	C₂×C₃₂	—	86	2	no	—
77	#364	Dic₄₃	1	1	yes	characteristic
78	#350	Dic₄₃	1	2	yes	—
79	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
80	C₄₃	#13647	1	1	yes	characteristic, complemented, Sylow
81	#135	C₄₃⋊C₈	1	1	yes	characteristic
82	#131	C₂×Dic₄₃	1	2	yes	—
83	M₃₂	—	2	2	no	—
84	M₃₂	C₂×Dic₄₃	1	2	yes	—
85	C₂×C₁₆	—	2	2	no	—
86	C₂×C₁₆	C₂×Dic₄₃	1	1	yes	characteristic
87	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
88	#108	C₄₃⋊C₈	1	1	yes	characteristic
89	#104	C₄₃⋊C₈	1	2	yes	—
90	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
91	C₃₂	—	43	1	no	—
92	C₃₂	—	43	1	no	—
93	C₃₂	—	86	2	no	—
94	C₃₂	—	86	1	no	—
95	#41	C₂×C₄₃⋊C₈	1	2	yes	—
96	Q₁₆	C₄₃⋊C₁₆	1	1	yes	characteristic
97	SD₁₆	C₄₃⋊C₁₆	1	2	yes	—
98	D₈	C₄₃⋊C₁₆	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	C₂×C₄₃⋊C₈	1	2	yes	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	C₂×C₄₃⋊C₈	1	1	yes	characteristic
103	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
104	C₄²	C₂×C₄₃⋊C₈	1	1	yes	characteristic
105	C₁₆	—	2	2	no	—
106	C₁₆	—	2	1	no	—
107	C₁₆	D₄×D₄₃	1	1	yes	characteristic
108	C₁₆	#10621900	1	1	yes	characteristic
109	Q₈	C₂×C₄₃⋊C₁₆	1	2	yes	—
110	D₄	—	2	2	no	—
111	D₄	C₂×C₄₃⋊C₁₆	1	2	yes	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
115	C₈	—	2	2	no	—
116	C₈	C₂×C₄₃⋊C₁₆	1	1	yes	characteristic
117	C₈	C₂×C₄₃⋊C₁₆	1	1	yes	characteristic
118	C₈	D₄×Dic₄₃	1	1	yes	characteristic
119	C₈	#13063332	1	1	yes	characteristic
120	C₂²	—	2	2	no	—
121	C₂²	#14583316	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	D₄×C₄₃⋊C₈	1	1	yes	characteristic
125	C₄	C₂²×C₄₃⋊C₁₆	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	4	2	no	—
128	C₂	D₄×C₄₃⋊C₁₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136889

C₂

yes

maximal, complemented

#19132793

C₂

yes

maximal, characteristic, complemented

#19132737

C₂

yes

maximal, complemented

C₄×C₄₃⋊C₃₂

C₂

yes

maximal, characteristic, complemented

C₄₃×#2170

C₂

yes

maximal, characteristic

#14582733

—

#14582733

C₂²

yes

—

C₂×C₄₃⋊C₃₂

—

C₂×C₄₃⋊C₃₂

C₂²

yes

characteristic

C₂×C₄₃⋊C₃₂

C₂²

yes

characteristic

C₄₃×#504

C₂²

yes

—

C₄₃×#443

C₄

yes

characteristic

C₄₃×#364

C₄

yes

characteristic

C₄₃×#350

C₄

yes

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₄₃⋊C₃₂

—

C₄₃⋊C₃₂

—

C₄₃⋊C₃₂

D₄

yes

characteristic

C₄₃⋊C₃₂

D₄

yes

characteristic

#135×C₄₃

C₈

yes

characteristic

#131×C₄₃

C₂×C₄

yes

—

M₃₂×C₄₃

—

M₃₂×C₄₃

C₂×C₄

yes

—

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₃

C₂³

yes

characteristic

#108×C₄₃

C₈

yes

characteristic

#104×C₄₃

C₈

yes

—

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

#41×C₄₃

C₂×C₈

yes

—

Q₁₆×C₄₃

C₁₆

yes

characteristic

SD₁₆×C₄₃

C₁₆

yes

—

D₈×C₄₃

C₁₆

yes

characteristic

M₁₆×C₄₃

—

M₁₆×C₄₃

C₂×C₈

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×C₈

yes

characteristic

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×C₈

yes

characteristic

C₁₆×C₄₃

—

C₁₆×C₄₃

—

C₁₆×C₄₃

C₂×D₄

yes

characteristic

C₁₆×C₄₃

#41

yes

characteristic

Q₈×C₄₃

C₂×C₁₆

yes

—

D₄×C₄₃

—

D₄×C₄₃

C₂×C₁₆

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²×C₈

yes

characteristic

C₈×C₄₃

—

C₈×C₄₃

C₂×C₁₆

yes

characteristic

C₈×C₄₃

C₂×C₁₆

yes

characteristic

C₈×C₄₃

C₄×D₄

yes

C₈×C₄₃

yes

—

C₂²×C₄₃

—

C₂²×C₄₃

#504

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₈×D₄

yes

characteristic

C₄×C₄₃

C₂²×C₁₆

yes

characteristic

#2258

—

#2170

D₄₃

yes

characteristic

#1422

—

#1401

—

C₄×C₃₂

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

D₄×C₁₆

yes

characteristic

#504

C₂×D₄₃

yes

—

#443

Dic₄₃

yes

characteristic

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

#364

Dic₄₃

yes

characteristic

#350

Dic₄₃

yes

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₄₃

#13647

yes

characteristic, complemented, Sylow

#135

C₄₃⋊C₈

yes

characteristic

#131

C₂×Dic₄₃

yes

—

M₃₂

—

M₃₂

C₂×Dic₄₃

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₄₃

yes

characteristic

C₂×C₁₆

C₂²×D₄₃

yes

characteristic

#108

C₄₃⋊C₈

yes

characteristic

#104

C₄₃⋊C₈

yes

—

C₄×C₈

C₂×Dic₄₃

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

C₃₂

—

#41

C₂×C₄₃⋊C₈

yes

—

Q₁₆

C₄₃⋊C₁₆

yes

characteristic

SD₁₆

C₄₃⋊C₁₆

yes

—

D₈

C₄₃⋊C₁₆

yes

characteristic

M₁₆

—

100

M₁₆

C₂×C₄₃⋊C₈

yes

—

101

C₂×C₈

—

102

C₂×C₈

C₂×C₄₃⋊C₈

yes

characteristic

103

C₂×C₈

C₂²×Dic₄₃

yes

characteristic

104

C₄²

C₂×C₄₃⋊C₈

yes

characteristic

105

C₁₆

—

106

C₁₆

—

107

C₁₆

D₄×D₄₃

yes

characteristic

108

C₁₆

#10621900

yes

characteristic

109

Q₈

C₂×C₄₃⋊C₁₆

yes

—

110

D₄

—

111

D₄

C₂×C₄₃⋊C₁₆

yes

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²×C₄₃⋊C₈

yes

characteristic

115

C₈

—

116

C₈

C₂×C₄₃⋊C₁₆

yes

characteristic

117

C₈

C₂×C₄₃⋊C₁₆

yes

characteristic

118

C₈

D₄×Dic₄₃

yes

characteristic

119

C₈

#13063332

yes

characteristic

120

C₂²

—

121

C₂²

#14583316

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

D₄×C₄₃⋊C₈

yes

characteristic

125

C₄

C₂²×C₄₃⋊C₁₆

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

D₄×C₄₃⋊C₁₆

yes

characteristic

	subgroup	G/H
center	C₁₆	D₄×D₄₃
commutator	C₄×C₄₃	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₄₃
Fitting	C₄₃×#2170	C₂
socle	C₂×C₄₃	D₄×C₁₆
prime radical	#135×C₄₃	C₈
cosocle	C₂×C₁₆×C₄₃	C₂³
hypercenter	#2170	D₄₃
hypocenter	C₄₃	#13647
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #2170

subgroup

G/H

center

C₁₆

D₄×D₄₃

commutator

C₄×C₄₃

C₂²×C₁₆

Frattini

C₂×C₁₆

C₂²×D₄₃

Fitting

C₄₃×#2170

C₂

socle

C₂×C₄₃

D₄×C₁₆

prime radical

#135×C₄₃

C₈

cosocle

C₂×C₁₆×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #2170

N	H ≅ G/N	type
1. #19136889	C₂	semidirect
2. #19132793	C₂	semidirect
3. #19132737	C₂	semidirect
4. C₄×C₄₃⋊C₃₂	C₂	semidirect
80. C₄₃	#13647	semidirect

H ≅ G/N

type

1. #19136889

C₂

semidirect

2. #19132793

C₂

semidirect

3. #19132737

C₂

semidirect

4. C₄×C₄₃⋊C₃₂

C₂

semidirect

80. C₄₃

#13647

semidirect

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

subgroup	index
#19136889	2
#19132793	2
#19132737	2
C₄×C₄₃⋊C₃₂	2
C₄₃×#2170	2
#13647	43

subgroup

index

#19136889

#19132793

#19132737

C₄×C₄₃⋊C₃₂

C₄₃×#2170

#13647

Subgroups — #25017161

Conjugacy classes of subgroups

Complements

Maximal subgroups