Group #14583528 — subgroups

Subgroups — #14583528

Order 2752.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#13063323	C₂	1	2	yes	maximal, complemented
3	#13063315	C₂	1	2	yes	maximal, complemented
4	#13063315	C₂	1	2	yes	maximal, complemented
5	#13063304	C₂	1	1	yes	maximal, characteristic
6	#13063302	C₂	1	2	yes	maximal
7	#13063281	C₂	1	2	yes	maximal
8	C₂×Dic₄₃⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂²⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
10	C₂×Q₈×C₄₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₄₃	C₂²	1	1	yes	characteristic
12	C₂²×Dic₄₃	C₂²	1	1	yes	characteristic
13	C₂²×Dic₄₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₈₆	C₂²	1	2	yes	—
15	C₂×Dic₈₆	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
19	C₂²⋊Dic₄₃	C₂²	1	2	yes	—
20	C₄⋊Dic₄₃	C₂²	1	2	yes	—
21	C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
22	Dic₄₃⋊C₄	C₂²	1	2	yes	—
23	Dic₄₃⋊C₄	C₂²	1	2	yes	—
24	Dic₄₃⋊C₄	C₂²	1	2	yes	—
25	Dic₄₃⋊C₄	C₂²	1	2	yes	—
26	Dic₄₃⋊C₄	C₂²	1	2	yes	—
27	Dic₄₃⋊C₄	C₂²	1	2	yes	—
28	Dic₄₃⋊C₄	C₂²	1	2	yes	—
29	C₄×Dic₄₃	C₂²	1	2	yes	—
30	C₄×Dic₄₃	C₂²	1	2	yes	—
31	C₂³×C₄₃	C₂³	1	1	yes	characteristic
32	Q₈×C₄₃	—	4	1	no	—
33	C₂×C₄×C₄₃	—	4	1	no	—
34	C₂×C₄×C₄₃	C₂³	1	2	yes	—
35	C₂×C₄×C₄₃	C₂³	1	2	yes	—
36	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
38	C₂×Dic₄₃	—	4	1	no	—
39	C₂×Dic₄₃	Q₈	1	4	yes	complemented
40	C₂×Dic₄₃	C₂³	1	2	yes	—
41	C₂×Dic₄₃	C₂³	1	2	yes	—
42	C₂×Dic₄₃	C₂³	1	2	yes	—
43	C₂×Dic₄₃	C₂³	1	2	yes	—
44	Dic₈₆	—	4	1	no	—
45	Dic₈₆	—	4	2	no	—
46	C₂²×C₄₃	—	2	1	no	—
47	C₂²×C₄₃	—	2	1	no	—
48	C₂²×C₄₃	C₂×Q₈	1	2	yes	—
49	C₂²×C₄₃	C₂⁴	1	1	yes	characteristic
50	C₄×C₄₃	—	2	1	no	—
51	C₄×C₄₃	—	2	2	no	—
52	C₄×C₄₃	—	2	1	no	—
53	C₄×C₄₃	—	2	2	no	—
54	Dic₄₃	—	2	2	no	—
55	Dic₄₃	—	2	2	no	—
56	Dic₄₃	—	2	2	no	—
57	Dic₄₃	C₂×Q₈	1	4	yes	—
58	C₂×C₄₃	—	2	2	no	—
59	C₂×C₄₃	C₂²×Q₈	1	1	yes	characteristic
60	C₂×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
61	C₂×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
62	#544	—	43	1	no	maximal, Sylow
63	C₄₃	#544	1	1	yes	characteristic, complemented, Sylow
64	C₄⋊Q₈	—	43	2	no	—
65	C₄⋊Q₈	—	43	2	no	—
66	#125	—	43	2	no	—
67	#125	—	43	2	no	—
68	C₂²⋊Q₈	—	43	1	no	—
69	C₂²⋊Q₈	—	43	2	no	—
70	C₂²⋊Q₈	D₄₃	1	1	yes	characteristic
71	#117	—	43	2	no	—
72	C₂×C₄⋊C₄	—	43	1	no	—
73	C₂×Q₈	—	43	1	no	—
74	C₂×Q₈	—	43	2	no	—
75	C₂×Q₈	C₂×D₄₃	1	1	yes	characteristic
76	C₂²×C₄	—	43	1	no	—
77	C₂²×C₄	—	43	1	no	—
78	C₂²×C₄	C₂×D₄₃	1	1	yes	characteristic
79	C₄⋊C₄	—	43	2	no	—
80	C₄⋊C₄	—	43	2	no	—
81	C₄⋊C₄	—	43	2	no	—
82	C₄⋊C₄	—	43	2	no	—
83	C₄⋊C₄	—	43	2	no	—
84	C₄⋊C₄	—	43	1	no	—
85	C₄⋊C₄	—	43	2	no	—
86	C₄⋊C₄	—	43	2	no	—
87	C₄⋊C₄	—	43	2	no	—
88	C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
89	C₄⋊C₄	C₂×D₄₃	1	2	yes	—
90	C₂²⋊C₄	—	43	2	no	—
91	C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
92	C₄²	—	43	2	no	—
93	C₄²	—	43	2	no	—
94	C₂³	C₂²×D₄₃	1	1	yes	characteristic
95	Q₈	—	4	1	no	—
96	Q₈	—	172	1	no	—
97	Q₈	—	172	2	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	43	4	no	—
100	C₂×C₄	—	43	2	no	—
101	C₂×C₄	—	43	2	no	—
102	C₂×C₄	—	43	2	no	—
103	C₂×C₄	—	43	2	no	—
104	C₂×C₄	—	172	1	no	—
105	C₂×C₄	C₂²×D₄₃	1	2	yes	—
106	C₂×C₄	C₂²×D₄₃	1	2	yes	—
107	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
108	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	Q₈×D₄₃	1	2	yes	—
112	C₂²	C₂³×D₄₃	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	43	4	no	—
118	C₄	—	86	2	no	—
119	C₄	—	86	2	no	—
120	C₄	—	86	2	no	—
121	C₂	—	2	2	no	—
122	C₂	C₂×Q₈×D₄₃	1	1	yes	characteristic
123	C₂	#13063370	1	1	yes	characteristic
124	C₂	#13063372	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13063324

C₂

yes

maximal, complemented

#13063323

C₂

yes

maximal, complemented

#13063315

C₂

yes

maximal, complemented

#13063315

C₂

yes

maximal, complemented

#13063304

C₂

yes

maximal, characteristic

#13063302

C₂

yes

maximal

#13063281

C₂

yes

maximal

C₂×Dic₄₃⋊C₄

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₄₃

C₂

yes

maximal, characteristic

C₂×Q₈×C₄₃

C₂²

yes

characteristic

C₂²×C₄×C₄₃

C₂²

yes

characteristic

C₂²×Dic₄₃

C₂²

yes

characteristic

C₂²×Dic₄₃

C₂²

yes

characteristic

C₂×Dic₈₆

C₂²

yes

—

C₂×Dic₈₆

C₂²

yes

characteristic

C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₄⋊C₄×C₄₃

C₂²

yes

—

C₂²⋊C₄×C₄₃

C₂²

yes

—

C₂²⋊Dic₄₃

C₂²

yes

—

C₄⋊Dic₄₃

C₂²

yes

—

C₄⋊Dic₄₃

C₂²

yes

characteristic

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

C₄×Dic₄₃

C₂²

yes

—

C₄×Dic₄₃

C₂²

yes

—

C₂³×C₄₃

C₂³

yes

characteristic

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂³

yes

—

C₂×C₄×C₄₃

C₂³

yes

—

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

Q₈

yes

complemented

C₂×Dic₄₃

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

—

Dic₈₆

—

Dic₈₆

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×Q₈

yes

—

C₂²×C₄₃

C₂⁴

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

C₂×Q₈

yes

—

C₂×C₄₃

—

C₂×C₄₃

C₂²×Q₈

yes

characteristic

C₂×C₄₃

H(5,𝔽₂)

yes

characteristic

C₂×C₄₃

2₋¹⁺⁴

yes

—

C₄₃

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₄₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄₃

yes

characteristic

C₄⋊C₄

C₂×D₄₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₄₃

yes

—

C₄²

—

C₄²

—

C₂³

C₂²×D₄₃

yes

characteristic

Q₈

—

Q₈

—

172

—

Q₈

—

172

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

172

—

105

C₂×C₄

C₂²×D₄₃

yes

—

106

C₂×C₄

C₂²×D₄₃

yes

—

107

C₂×C₄

C₂²×D₄₃

yes

characteristic

108

C₂×C₄

C₂²×D₄₃

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

Q₈×D₄₃

yes

—

112

C₂²

C₂³×D₄₃

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₂

—

122

C₂

C₂×Q₈×D₄₃

yes

characteristic

123

C₂

#13063370

yes

characteristic

124

C₂

#13063372

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₃
commutator, socle, cosocle	C₂²×C₄₃	C₂⁴
Fitting	C₂²⋊Q₈×C₄₃	C₂
prime radical	C₂³×C₄₃	C₂³
hypercenter	C₂²⋊Q₈	D₄₃
hypocenter	C₄₃	#544
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₃

commutator, socle, cosocle

C₂²×C₄₃

C₂⁴

Fitting

C₂²⋊Q₈×C₄₃

C₂

prime radical

C₂³×C₄₃

C₂³

hypercenter

C₂²⋊Q₈

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #13063324	C₂	semidirect
2. #13063323	C₂	semidirect
3. #13063315	C₂	semidirect
4. #13063315	C₂	semidirect
39. C₂×Dic₄₃	Q₈	semidirect
63. C₄₃	#544	semidirect

H ≅ G/N

type

1. #13063324

C₂

semidirect

2. #13063323

C₂

semidirect

3. #13063315

C₂

semidirect

4. #13063315

C₂

semidirect

39. C₂×Dic₄₃

Q₈

semidirect

63. C₄₃

#544

semidirect

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

subgroup	index
#13063324	2
#13063323	2
#13063315	2
#13063304	2
#13063302	2
#13063281	2
C₂×Dic₄₃⋊C₄	2
C₂²⋊Q₈×C₄₃	2
#544	43

subgroup

index

C₂×Dic₄₃⋊C₄

C₂²⋊Q₈×C₄₃

#544

Subgroups — #14583528

Conjugacy classes of subgroups

Complements

Maximal subgroups