Group #25005575 — subgroups

Subgroups — #25005575

Order 11008.

354 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 199 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	C₄×#13063264	C₂	1	2	yes	maximal
2	C₈²×D₄₃	C₂	1	1	yes	maximal, characteristic
3	#19132512	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×C₄₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063264	C₄	1	4	yes	—
7	C₄×C₈×D₄₃	C₄	1	2	yes	—
8	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
9	#14582674	C₄	1	4	yes	—
10	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
11	C₈×C₄₃⋊C₈	C₄	1	2	yes	—
12	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₄₃	C₂²	1	2	yes	—
14	C₈²×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₄₃	C₈	1	4	yes	—
16	C₂×C₈×D₄₃	C₂×C₄	1	2	yes	—
17	C₄²×D₄₃	C₂×C₄	1	1	yes	characteristic
18	#13063264	C₈	1	16	yes	complemented
19	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	—
20	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₈×Dic₄₃	C₈	1	4	yes	—
23	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
27	C₂×C₄×D₄₃	C₄²	1	1	yes	characteristic
28	C₂×C₄×D₄₃	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₄₃	C₂×C₈	1	4	yes	—
30	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
31	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
32	C₄×Dic₄₃	C₄²	1	1	yes	characteristic
33	C₄×Dic₄₃	C₂×C₈	1	2	yes	—
34	C₂×C₄₃⋊C₈	C₄²	1	2	yes	—
35	C₂×C₄₃⋊C₈	C₂×C₈	1	4	yes	—
36	C₈×D₄₃	—	2	8	no	—
37	C₈×D₄₃	C₂×C₈	1	8	yes	—
38	C₁₆×C₄₃	C₂×C₈	1	8	yes	complemented
39	C₄₃⋊C₁₆	C₂×C₈	1	8	yes	complemented
40	C₂²×D₄₃	C₄×C₈	1	1	yes	characteristic
41	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
42	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
43	C₂×Dic₄₃	C₄×C₈	1	2	yes	—
44	C₂×Dic₄₃	C₄×C₈	1	1	yes	characteristic
45	C₄×D₄₃	—	2	4	no	—
46	C₄×D₄₃	C₄×C₈	1	4	yes	—
47	C₈×C₄₃	—	2	4	no	—
48	C₈×C₄₃	C₂²×C₈	1	4	yes	—
49	C₄₃⋊C₈	C₄×C₈	1	4	yes	—
50	C₄₃⋊C₈	M₃₂	1	8	yes	complemented
51	C₈×M₃₂	—	43	1	no	maximal, Sylow
52	C₂²×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
53	C₂×D₄₃	—	2	2	no	—
54	C₂×D₄₃	C₈²	1	2	yes	—
55	C₄×C₄₃	C₂×C₄×C₈	1	2	yes	—
56	C₄×C₄₃	C₂×M₃₂	1	4	yes	—
57	Dic₄₃	—	2	2	no	—
58	Dic₄₃	C₈²	1	2	yes	—
59	C₄×M₃₂	—	43	2	no	—
60	C₂×C₈²	—	43	1	no	—
61	#1311	—	43	1	no	—
62	#1311	D₄₃	1	1	yes	characteristic, complemented
63	C₈×C₁₆	—	43	2	no	—
64	C₂×C₄₃	C₂×C₈²	1	1	yes	characteristic
65	C₂×C₄₃	C₄×M₃₂	1	2	yes	—
66	D₄₃	—	2	2	no	—
67	C₂×M₃₂	—	43	4	no	—
68	C₂×C₄×C₈	—	43	1	no	—
69	C₂×C₄×C₈	—	43	2	no	—
70	#346	—	43	4	no	—
71	C₄×C₁₆	—	43	2	no	—
72	C₄×C₁₆	C₂×D₄₃	1	2	yes	—
73	C₈²	—	43	1	no	—
74	C₈²	—	43	2	no	—
75	C₈²	C₂×D₄₃	1	1	yes	characteristic
76	C₄₃	C₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₈	—	43	2	no	—
78	C₂²×C₈	—	43	4	no	—
79	C₂×C₄²	—	43	1	no	—
80	M₃₂	—	43	16	no	—
81	C₂×C₁₆	—	43	4	no	—
82	C₂×C₁₆	C₄×D₄₃	1	4	yes	—
83	C₄×C₈	—	43	4	no	—
84	C₄×C₈	—	43	2	no	—
85	C₄×C₈	—	43	1	no	—
86	C₄×C₈	—	43	2	no	—
87	C₄×C₈	C₄×D₄₃	1	2	yes	—
88	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
89	C₂²×C₄	—	43	1	no	—
90	C₂²×C₄	—	43	2	no	—
91	C₂×C₈	—	43	2	no	—
92	C₂×C₈	—	43	4	no	—
93	C₂×C₈	—	43	8	no	—
94	C₂×C₈	—	86	8	no	—
95	C₂×C₈	C₈×D₄₃	1	4	yes	—
96	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
97	C₄²	—	43	1	no	—
98	C₄²	—	43	2	no	—
99	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
100	C₁₆	—	43	8	no	—
101	C₁₆	C₈×D₄₃	1	8	yes	complemented
102	C₂³	—	43	1	no	—
103	C₂×C₄	—	43	1	no	—
104	C₂×C₄	—	43	4	no	—
105	C₂×C₄	—	43	2	no	—
106	C₂×C₄	—	86	4	no	—
107	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
108	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	43	8	no	—
111	C₈	—	43	4	no	—
112	C₈	C₂×C₈×D₄₃	1	4	yes	—
113	C₂²	—	43	2	no	—
114	C₂²	—	86	2	no	—
115	C₂²	C₄×C₈×D₄₃	1	1	yes	characteristic
116	C₄	—	43	2	no	—
117	C₄	—	86	2	no	—
118	C₄	C₄×C₈×D₄₃	1	2	yes	—
119	C₄	M₃₂×D₄₃	1	4	yes	—
120	C₂	—	86	2	no	—
121	C₂	C₈²×D₄₃	1	1	yes	characteristic
122	C₂	#19136188	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13063264

C₂

yes

maximal

C₈²×D₄₃

C₂

yes

maximal, characteristic

#19132512

C₂

yes

maximal, characteristic, complemented

C₁₆×C₄₃⋊C₈

C₂

yes

maximal, complemented

C₄₃×#1311

C₂

yes

maximal, characteristic, complemented

C₂×#13063264

C₄

yes

—

C₄×C₈×D₄₃

C₄

yes

—

C₄×C₈×D₄₃

C₂²

yes

characteristic

#14582674

C₄

yes

—

C₄×C₄₃⋊C₁₆

C₂²

yes

—

C₈×C₄₃⋊C₈

C₄

yes

—

C₈×C₄₃⋊C₈

C₂²

yes

characteristic

C₄×C₁₆×C₄₃

C₂²

yes

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×C₈×D₄₃

C₈

yes

—

C₂×C₈×D₄₃

C₂×C₄

yes

—

C₄²×D₄₃

C₂×C₄

yes

characteristic

#13063264

C₈

yes

complemented

C₂×C₄₃⋊C₁₆

C₂×C₄

yes

—

C₄×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₄×C₄₃⋊C₈

C₂×C₄

yes

—

C₈×Dic₄₃

C₈

yes

—

C₈×Dic₄₃

C₂×C₄

yes

—

C₂×C₁₆×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄×D₄₃

C₄²

yes

characteristic

C₂×C₄×D₄₃

C₂×C₈

yes

—

C₂×C₈×C₄₃

C₂×C₈

yes

—

C₂×C₈×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄×Dic₄₃

C₄²

yes

characteristic

C₄×Dic₄₃

C₂×C₈

yes

—

C₂×C₄₃⋊C₈

C₄²

yes

—

C₂×C₄₃⋊C₈

C₂×C₈

yes

—

C₈×D₄₃

—

C₈×D₄₃

C₂×C₈

yes

—

C₁₆×C₄₃

C₂×C₈

yes

complemented

C₄₃⋊C₁₆

C₂×C₈

yes

complemented

C₂²×D₄₃

C₄×C₈

yes

characteristic

C₂×C₄×C₄₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₃

C₂²×C₈

yes

—

C₂×Dic₄₃

C₄×C₈

yes

—

C₂×Dic₄₃

C₄×C₈

yes

characteristic

C₄×D₄₃

—

C₄×D₄₃

C₄×C₈

yes

—

C₈×C₄₃

—

C₈×C₄₃

C₂²×C₈

yes

—

C₄₃⋊C₈

C₄×C₈

yes

—

C₄₃⋊C₈

M₃₂

yes

complemented

C₈×M₃₂

—

maximal, Sylow

C₂²×C₄₃

C₂×C₄×C₈

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

C₈²

yes

—

C₄×C₄₃

C₂×C₄×C₈

yes

—

C₄×C₄₃

C₂×M₃₂

yes

—

Dic₄₃

—

Dic₄₃

C₈²

yes

—

C₄×M₃₂

—

C₂×C₈²

—

#1311

—

#1311

D₄₃

yes

characteristic, complemented

C₈×C₁₆

—

C₂×C₄₃

C₂×C₈²

yes

characteristic

C₂×C₄₃

C₄×M₃₂

yes

—

D₄₃

—

C₂×M₃₂

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₃

yes

—

C₈²

—

C₈²

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

C₈×M₃₂

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₄₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₄₃

yes

—

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×D₄₃

yes

—

C₂×C₈

C₂×C₄×D₄₃

yes

—

C₄²

—

C₄²

—

C₄²

C₂×C₄×D₄₃

yes

characteristic

100

C₁₆

—

101

C₁₆

C₈×D₄₃

yes

complemented

102

C₂³

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄²×D₄₃

yes

characteristic

108

C₂×C₄

C₂×C₈×D₄₃

yes

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

C₂×C₈×D₄₃

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₄×C₈×D₄₃

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

C₄×C₈×D₄₃

yes

—

119

C₄

M₃₂×D₄₃

yes

—

120

C₂

—

121

C₂

C₈²×D₄₃

yes

characteristic

122

C₂

#19136188

yes

—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₄₃
commutator	C₂×C₄₃	C₂×C₈²
Fitting	C₄₃×#1311	C₂
socle	C₂²×C₄₃	C₂×C₄×C₈
prime radical	C₂²×D₄₃	C₄×C₈
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	#1311	D₄₃
hypocenter	C₄₃	C₈×M₃₂
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄×C₈ ◁ #1311

subgroup

G/H

center, Frattini

C₄×C₈

C₂²×D₄₃

commutator

C₂×C₄₃

C₂×C₈²

Fitting

C₄₃×#1311

C₂

socle

C₂²×C₄₃

C₂×C₄×C₈

prime radical

C₂²×D₄₃

C₄×C₈

cosocle

C₄×C₈×C₄₃

C₂³

hypercenter

#1311

D₄₃

hypocenter

C₄₃

C₈×M₃₂

LCS

G ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₄×C₈ ◁ #1311

N	H ≅ G/N	type
3. #19132512	C₂	semidirect
4. C₁₆×C₄₃⋊C₈	C₂	semidirect
5. C₄₃×#1311	C₂	semidirect
18. #13063264	C₈	semidirect
38. C₁₆×C₄₃	C₂×C₈	semidirect
39. C₄₃⋊C₁₆	C₂×C₈	semidirect
50. C₄₃⋊C₈	M₃₂	semidirect
62. #1311	D₄₃	semidirect
76. C₄₃	C₈×M₃₂	semidirect
101. C₁₆	C₈×D₄₃	semidirect

H ≅ G/N

type

3. #19132512

C₂

semidirect

4. C₁₆×C₄₃⋊C₈

C₂

semidirect

5. C₄₃×#1311

C₂

semidirect

18. #13063264

C₈

semidirect

38. C₁₆×C₄₃

C₂×C₈

semidirect

39. C₄₃⋊C₁₆

C₂×C₈

semidirect

50. C₄₃⋊C₈

M₃₂

semidirect

62. #1311

D₄₃

semidirect

76. C₄₃

C₈×M₃₂

semidirect

101. C₁₆

C₈×D₄₃

semidirect

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

subgroup	index
C₄×#13063264	2
C₈²×D₄₃	2
#19132512	2
C₁₆×C₄₃⋊C₈	2
C₄₃×#1311	2
C₈×M₃₂	43

subgroup

index

C₄×#13063264

C₈²×D₄₃

#19132512

C₁₆×C₄₃⋊C₈

C₄₃×#1311

C₈×M₃₂

Subgroups — #25005575

Conjugacy classes of subgroups

Complements

Maximal subgroups