Group #25005634 — subgroups

Subgroups — #25005634

Order 11008.

260 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 163 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	#19136394	C₂	1	2	yes	maximal
2	#19132513	C₂	1	2	yes	maximal, complemented
3	#19132512	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1450	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₄₃⋊C₁₆)	C₄	1	4	yes	—
7	C₂²⋊(C₄₃⋊C₁₆)	C₄	1	4	yes	—
8	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
9	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
10	C₄₃×#431	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
12	C₄₃×#322	C₄	1	2	yes	—
13	C₈²×C₄₃	C₂²	1	1	yes	characteristic
14	C₈²×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	—
16	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	—
17	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
19	C₄⋊C₈×C₄₃	C₈	1	4	yes	—
20	C₂²⋊C₈×C₄₃	C₈	1	4	yes	—
21	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
27	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₄₃	—	2	4	no	—
29	C₂×C₈×C₄₃	C₄²	1	2	yes	—
30	C₂×C₈×C₄₃	C₂×C₈	1	4	yes	—
31	C₂×C₈×C₄₃	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
33	C₄²×C₄₃	C₄²	1	1	yes	characteristic
34	C₄²×C₄₃	C₂×C₈	1	2	yes	—
35	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
36	C₄₃⋊C₁₆	—	2	4	no	—
37	C₄₃⋊C₁₆	—	2	4	no	—
38	C₂³×C₄₃	C₄×C₈	1	1	yes	characteristic
39	C₂×C₄×C₄₃	—	2	2	no	—
40	C₂×C₄×C₄₃	—	2	4	no	—
41	C₂×C₄×C₄₃	C₄×C₈	1	2	yes	—
42	C₂×C₄×C₄₃	C₄×C₈	1	1	yes	characteristic
43	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
44	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
45	C₈×C₄₃	—	2	2	no	—
46	C₈×C₄₃	—	2	4	no	—
47	C₈×C₄₃	—	2	4	no	—
48	C₈×C₄₃	#131	1	4	yes	—
49	#11412	—	43	1	no	maximal, Sylow
50	C₂²×C₄₃	—	2	2	no	—
51	C₂²×C₄₃	—	2	1	no	—
52	C₂²×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
53	C₄×C₄₃	—	2	2	no	—
54	C₄×C₄₃	—	2	1	no	—
55	C₄×C₄₃	#405	1	2	yes	—
56	C₄×C₄₃	#504	1	4	yes	—
57	#2162	—	43	2	no	—
58	#1450	D₄₃	1	1	yes	characteristic
59	#1312	—	43	2	no	—
60	#1311	—	43	1	no	—
61	C₈×C₁₆	—	43	1	no	—
62	C₂×C₄₃	—	4	1	no	—
63	C₂×C₄₃	#1450	1	1	yes	characteristic
64	C₂×C₄₃	#2108	1	2	yes	—
65	#431	Dic₄₃	1	2	yes	—
66	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
67	C₄⋊C₁₆	—	43	4	no	—
68	C₂²⋊C₁₆	—	43	4	no	—
69	C₄×C₁₆	—	43	2	no	—
70	C₄×C₁₆	—	43	2	no	—
71	#322	Dic₄₃	1	2	yes	—
72	C₈²	C₂×D₄₃	1	1	yes	characteristic
73	C₈²	C₂×D₄₃	1	1	yes	characteristic
74	C₄₃	#11412	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	C₄×D₄₃	1	2	yes	—
76	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
77	C₂×C₁₆	—	43	4	no	—
78	C₂×C₁₆	—	43	4	no	—
79	C₄⋊C₈	C₄₃⋊C₈	1	4	yes	—
80	C₂²⋊C₈	C₄₃⋊C₈	1	4	yes	—
81	C₄×C₈	C₄×D₄₃	1	2	yes	—
82	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
83	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
84	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
85	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
86	C₂²×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
87	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	C₂×C₄₃⋊C₈	1	4	yes	—
90	C₂×C₈	C₂×C₄₃⋊C₈	1	4	yes	—
91	C₂×C₈	C₄×Dic₄₃	1	2	yes	—
92	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
93	C₄²	C₂×C₄₃⋊C₈	1	2	yes	—
94	C₄²	C₄×Dic₄₃	1	1	yes	characteristic
95	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
96	C₁₆	—	86	4	no	—
97	C₁₆	—	86	4	no	—
98	C₂³	C₄×C₄₃⋊C₈	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	C₄×C₄₃⋊C₈	1	1	yes	characteristic
102	C₂×C₄	C₄×C₄₃⋊C₈	1	2	yes	—
103	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×C₄₃⋊C₈	1	2	yes	—
105	C₈	—	2	4	no	—
106	C₈	—	2	4	no	—
107	C₈	—	2	2	no	—
108	C₈	#13063331	1	4	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	—	2	1	no	—
111	C₂²	C₂×C₄×C₄₃⋊C₈	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	1	no	—
114	C₄	#14582854	1	2	yes	—
115	C₄	#14583316	1	4	yes	—
116	C₂	—	4	1	no	—
117	C₂	#19132878	1	1	yes	characteristic
118	C₂	#19136192	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19136394

C₂

yes

maximal

#19132513

C₂

yes

maximal, complemented

#19132512

C₂

yes

maximal, characteristic, complemented

C₈×C₄₃⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₄₃×#1450

C₂

yes

maximal, characteristic

C₄⋊(C₄₃⋊C₁₆)

C₄

yes

—

C₂²⋊(C₄₃⋊C₁₆)

C₄

yes

—

C₄×C₄₃⋊C₁₆

C₂²

yes

—

C₄×C₄₃⋊C₁₆

C₂²

yes

—

C₄₃×#431

C₄

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#322

C₄

yes

—

C₈²×C₄₃

C₂²

yes

characteristic

C₈²×C₄₃

C₂²

yes

characteristic

C₂×C₄₃⋊C₁₆

C₂×C₄

yes

—

C₂×C₄₃⋊C₁₆

C₂×C₄

yes

—

C₂²×C₈×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₄₃

C₈

yes

—

C₂²⋊C₈×C₄₃

C₈

yes

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₄²

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₈

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄²

yes

—

C₂×C₈×C₄₃

C₂×C₈

yes

—

C₂×C₈×C₄₃

C₂×C₈

yes

—

C₂×C₈×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

C₄²

yes

characteristic

C₄²×C₄₃

C₂×C₈

yes

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄₃⋊C₁₆

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₄×C₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×C₈

yes

—

C₂×C₄×C₄₃

C₄×C₈

yes

characteristic

C₂×C₄×C₄₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₃

C₂²×C₈

yes

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#131

yes

—

#11412

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₄×C₈

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#405

yes

—

C₄×C₄₃

#504

yes

—

#2162

—

#1450

D₄₃

yes

characteristic

#1312

—

#1311

—

C₈×C₁₆

—

C₂×C₄₃

—

C₂×C₄₃

#1450

yes

characteristic

C₂×C₄₃

#2108

yes

—

#431

Dic₄₃

yes

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#322

Dic₄₃

yes

—

C₈²

C₂×D₄₃

yes

characteristic

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#11412

yes

characteristic, complemented, Sylow

C₂²×C₈

C₄×D₄₃

yes

—

C₂×C₄²

C₂×Dic₄₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

C₄₃⋊C₈

yes

—

C₂²⋊C₈

C₄₃⋊C₈

yes

—

C₄×C₈

C₄×D₄₃

yes

—

C₄×C₈

C₂×Dic₄₃

yes

—

C₄×C₈

C₂×Dic₄₃

yes

—

C₄×C₈

C₂×Dic₄₃

yes

characteristic

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂²×C₄

C₂×C₄₃⋊C₈

yes

—

C₂²×C₄

C₄×Dic₄₃

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂×C₄₃⋊C₈

yes

—

C₂×C₈

C₂×C₄₃⋊C₈

yes

—

C₂×C₈

C₄×Dic₄₃

yes

—

C₂×C₈

C₂×C₄×D₄₃

yes

—

C₄²

C₂×C₄₃⋊C₈

yes

—

C₄²

C₄×Dic₄₃

yes

characteristic

C₄²

C₂²×Dic₄₃

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

C₄×C₄₃⋊C₈

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄×C₄₃⋊C₈

yes

characteristic

102

C₂×C₄

C₄×C₄₃⋊C₈

yes

—

103

C₂×C₄

C₂×C₄×Dic₄₃

yes

characteristic

104

C₂×C₄

C₂²×C₄₃⋊C₈

yes

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

#13063331

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂×C₄×C₄₃⋊C₈

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

#14582854

yes

—

115

C₄

#14583316

yes

—

116

C₂

—

117

C₂

#19132878

yes

characteristic

118

C₂

#19136192

yes

—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₄₃
commutator, socle	C₂²×C₄₃	C₂×C₄×C₈
Fitting	C₄₃×#1450	C₂
prime radical	C₂³×C₄₃	C₄×C₈
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	#1450	D₄₃
hypocenter	C₄₃	#11412
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄×C₈ ◁ #1450

subgroup

G/H

center, Frattini

C₄×C₈

C₂²×D₄₃

commutator, socle

C₂²×C₄₃

C₂×C₄×C₈

Fitting

C₄₃×#1450

C₂

prime radical

C₂³×C₄₃

C₄×C₈

cosocle

C₄×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₄×C₈ ◁ #1450

N	H ≅ G/N	type
2. #19132513	C₂	semidirect
3. #19132512	C₂	semidirect
4. C₈×C₄₃⋊C₁₆	C₂	semidirect
74. C₄₃	#11412	semidirect

H ≅ G/N

type

2. #19132513

C₂

semidirect

3. #19132512

C₂

semidirect

4. C₈×C₄₃⋊C₁₆

C₂

semidirect

74. C₄₃

#11412

semidirect

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

subgroup	index
#19136394	2
#19132513	2
#19132512	2
C₈×C₄₃⋊C₁₆	2
C₄₃×#1450	2
#11412	43

subgroup

index

#19136394

#19132513

#19132512

C₈×C₄₃⋊C₁₆

C₄₃×#1450

#11412

Subgroups — #25005634

Conjugacy classes of subgroups

Complements

Maximal subgroups