Group #14583653 — subgroups

Subgroups — #14583653

Order 2752.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	#13063324	C₂	1	2	yes	maximal, complemented
2	#13063320	C₂	1	2	yes	maximal, complemented
3	#13063320	C₂	1	2	yes	maximal, complemented
4	#13063317	C₂	1	1	yes	maximal, characteristic, complemented
5	#13063313	C₂	1	1	yes	maximal, characteristic
6	#13063310	C₂	1	2	yes	maximal
7	#13063305	C₂	1	2	yes	maximal, complemented
8	#13063303	C₂	1	2	yes	maximal, complemented
9	#125×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₁₇₂	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄₃	C₂²	1	2	yes	—
12	C₂×Dic₈₆	C₂²	1	2	yes	—
13	C₂×Dic₈₆	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
16	C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
17	C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
18	C₄²×C₄₃	C₂²	1	1	yes	characteristic
19	(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
20	(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
24	C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₄₃	C₂²	1	2	yes	—
26	C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
27	Dic₄₃⋊C₄	C₂²	1	2	yes	—
28	Dic₄₃⋊C₄	C₂²	1	2	yes	—
29	Dic₄₃⋊C₄	C₂²	1	2	yes	—
30	C₄×Dic₄₃	C₂²	1	2	yes	—
31	C₂²×D₄₃	C₂³	1	2	yes	—
32	C₂×C₄×C₄₃	C₂³	1	2	yes	—
33	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₄₃	C₂³	1	2	yes	—
36	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
37	C₂×Dic₄₃	C₂³	1	2	yes	—
38	C₂×Dic₄₃	C₂³	1	2	yes	—
39	C₂×Dic₄₃	C₂³	1	2	yes	—
40	D₁₇₂	—	4	1	no	—
41	C₄×D₄₃	—	4	2	no	—
42	Dic₈₆	—	4	1	no	—
43	Dic₈₆	—	4	2	no	—
44	C₂²×C₄₃	C₂⁴	1	1	yes	characteristic
45	C₂×D₄₃	—	2	2	no	—
46	C₂×D₄₃	—	2	2	no	—
47	C₂×D₄₃	—	2	2	no	—
48	C₄×C₄₃	—	2	2	no	—
49	C₄×C₄₃	—	2	1	no	—
50	C₄×C₄₃	—	2	2	no	—
51	C₄×C₄₃	—	2	1	no	—
52	C₄×C₄₃	—	2	1	no	—
53	Dic₄₃	—	2	2	no	—
54	Dic₄₃	—	2	2	no	—
55	Dic₄₃	—	2	2	no	—
56	C₂×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
58	C₂×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
59	D₄₃	—	4	2	no	—
60	#563	—	43	1	no	maximal, Sylow
61	C₄₃	#563	1	1	yes	characteristic, complemented, Sylow
62	C₄⋊Q₈	—	43	2	no	—
63	#126	—	43	2	no	—
64	#126	—	43	2	no	—
65	#125	—	43	1	no	—
66	#125	D₄₃	1	1	yes	characteristic, complemented
67	#124	—	43	1	no	—
68	#123	—	43	2	no	—
69	C₂²⋊Q₈	—	43	2	no	—
70	C₂²⋊Q₈	—	43	2	no	—
71	C₂×Q₈	—	43	2	no	—
72	C₂×Q₈	—	43	1	no	—
73	C₂×D₄	—	43	1	no	—
74	C₂²×C₄	—	43	2	no	—
75	C₄⋊C₄	—	43	1	no	—
76	C₄⋊C₄	—	43	2	no	—
77	C₄⋊C₄	—	43	2	no	—
78	C₄⋊C₄	—	43	2	no	—
79	C₄⋊C₄	—	43	2	no	—
80	C₄⋊C₄	—	43	1	no	—
81	C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
82	C₄⋊C₄	C₂×D₄₃	1	2	yes	—
83	C₄⋊C₄	C₂×D₄₃	1	2	yes	—
84	C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
85	C₂²⋊C₄	—	43	2	no	—
86	C₂²⋊C₄	—	43	2	no	—
87	C₂²⋊C₄	—	43	2	no	—
88	C₂²⋊C₄	—	43	2	no	—
89	C₂²⋊C₄	—	43	2	no	—
90	C₄²	—	43	2	no	—
91	C₄²	C₂×D₄₃	1	1	yes	characteristic
92	C₂³	—	43	2	no	—
93	Q₈	—	172	2	no	—
94	Q₈	—	172	1	no	—
95	D₄	—	172	1	no	—
96	C₂×C₄	—	43	2	no	—
97	C₂×C₄	—	43	2	no	—
98	C₂×C₄	—	43	2	no	—
99	C₂×C₄	—	172	2	no	—
100	C₂×C₄	C₂²×D₄₃	1	2	yes	—
101	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
103	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₄₃	1	2	yes	—
105	C₂²	—	86	2	no	—
106	C₂²	—	86	2	no	—
107	C₂²	—	86	2	no	—
108	C₂²	C₂³×D₄₃	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	86	2	no	—
115	C₄	—	86	2	no	—
116	C₄	—	86	2	no	—
117	C₂	—	172	2	no	—
118	C₂	#13063371	1	1	yes	characteristic
119	C₂	#13063372	1	1	yes	characteristic
120	C₂	#13063373	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13063324

C₂

yes

maximal, complemented

#13063320

C₂

yes

maximal, complemented

#13063320

C₂

yes

maximal, complemented

#13063317

C₂

yes

maximal, characteristic, complemented

#13063313

C₂

yes

maximal, characteristic

#13063310

C₂

yes

maximal

#13063305

C₂

yes

maximal, complemented

#13063303

C₂

yes

maximal, complemented

#125×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×D₁₇₂

C₂²

yes

characteristic

C₂×C₄×D₄₃

C₂²

yes

—

C₂×Dic₈₆

C₂²

yes

—

C₂×Dic₈₆

C₂²

yes

characteristic

C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₄⋊C₄×C₄₃

C₂²

yes

—

C₄⋊C₄×C₄₃

C₂²

yes

—

C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₄²×C₄₃

C₂²

yes

characteristic

(C₂×D₄₃)⋊C₄

C₂²

yes

—

(C₂×D₄₃)⋊C₄

C₂²

yes

—

(C₂×D₄₃)⋊C₄

C₂²

yes

—

(C₂×D₄₃)⋊C₄

C₂²

yes

—

(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₄₃

C₂²

yes

characteristic

C₄⋊Dic₄₃

C₂²

yes

—

C₄⋊Dic₄₃

C₂²

yes

characteristic

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

C₄×Dic₄₃

C₂²

yes

—

C₂²×D₄₃

C₂³

yes

—

C₂×C₄×C₄₃

C₂³

yes

—

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₄×C₄₃

C₂³

yes

—

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂×Dic₄₃

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

—

D₁₇₂

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

C₂²×C₄₃

C₂⁴

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

C₂×C₄₃

H(5,𝔽₂)

yes

characteristic

C₂×C₄₃

2₋¹⁺⁴

yes

characteristic

C₂×C₄₃

2₋¹⁺⁴

yes

characteristic

D₄₃

—

#563

—

maximal, Sylow

C₄₃

#563

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

—

—

D₄₃

yes

characteristic, complemented

#124

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄₃

yes

characteristic

C₄⋊C₄

C₂×D₄₃

yes

—

C₄⋊C₄

C₂×D₄₃

yes

—

C₄⋊C₄

C₂×D₄₃

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₄₃

yes

characteristic

C₂³

—

Q₈

—

172

—

Q₈

—

172

—

D₄

—

172

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

172

—

100

C₂×C₄

C₂²×D₄₃

yes

—

101

C₂×C₄

C₂²×D₄₃

yes

characteristic

102

C₂×C₄

C₂²×D₄₃

yes

characteristic

103

C₂×C₄

C₂²×D₄₃

yes

characteristic

104

C₂×C₄

C₂²×D₄₃

yes

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₂³×D₄₃

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₂

—

172

—

118

C₂

#13063371

yes

characteristic

119

C₂

#13063372

yes

characteristic

120

C₂

#13063373

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₃
commutator, socle, cosocle	C₂²×C₄₃	C₂⁴
Fitting	#125×C₄₃	C₂
prime radical	C₂×D₁₇₂	C₂²
hypercenter	#125	D₄₃
hypocenter	C₄₃	#563
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₃

commutator, socle, cosocle

C₂²×C₄₃

C₂⁴

Fitting

#125×C₄₃

C₂

prime radical

C₂×D₁₇₂

C₂²

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #13063324	C₂	semidirect
2. #13063320	C₂	semidirect
3. #13063320	C₂	semidirect
4. #13063317	C₂	semidirect
7. #13063305	C₂	semidirect
8. #13063303	C₂	semidirect
9. #125×C₄₃	C₂	semidirect
61. C₄₃	#563	semidirect
66. #125	D₄₃	semidirect

H ≅ G/N

type

1. #13063324

C₂

semidirect

2. #13063320

C₂

semidirect

3. #13063320

C₂

semidirect

4. #13063317

C₂

semidirect

7. #13063305

C₂

semidirect

8. #13063303

C₂

semidirect

9. #125×C₄₃

C₂

semidirect

61. C₄₃

#563

semidirect

66. #125

D₄₃

semidirect

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

subgroup	index
#13063324	2
#13063320	2
#13063317	2
#13063313	2
#13063310	2
#13063305	2
#13063303	2
#125×C₄₃	2
#563	43

subgroup

index

#125×C₄₃

Subgroups — #14583653

Conjugacy classes of subgroups

Complements

Maximal subgroups