Group #14583469 — subgroups

Subgroups — #14583469

Order 2752.

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

normal subgroups only

This group has 270 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	C₂×#10621900	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#13063325	C₂	1	1	yes	maximal, characteristic, complemented
4	#13063323	C₂	1	1	yes	maximal, characteristic, complemented
5	#13063312	C₂	1	1	yes	maximal, characteristic, complemented
6	#13063307	C₂	1	4	yes	maximal, complemented
7	#13063302	C₂	1	4	yes	maximal, complemented
8	#13063282	C₂	1	1	yes	maximal, characteristic, complemented
9	#124×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×C₄₃	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₄₃	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₄₃	C₂²	1	2	yes	—
13	C₂²×Dic₄₃	C₂²	1	2	yes	—
14	Q₈×D₄₃	—	2	4	no	—
15	#10621900	—	2	4	no	—
16	C₂×C₄×D₄₃	C₂²	1	2	yes	—
17	C₂×C₄×D₄₃	C₂²	1	1	yes	characteristic
18	C₂×Dic₈₆	C₂²	1	2	yes	—
19	C₂×Dic₈₆	C₂²	1	1	yes	characteristic
20	C₂×Dic₈₆	C₂²	1	1	yes	characteristic
21	C₂²⋊C₄×C₄₃	C₂²	1	4	yes	—
22	C₄²×C₄₃	C₂²	1	1	yes	characteristic
23	C₂²⋊Dic₄₃	C₂²	1	4	yes	—
24	(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
25	C₄⋊Dic₄₃	C₂²	1	4	yes	—
26	Dic₄₃⋊C₄	C₂²	1	2	yes	—
27	Dic₄₃⋊C₄	C₂²	1	4	yes	—
28	C₄×Dic₄₃	C₂²	1	1	yes	characteristic
29	C₂³×C₄₃	C₂³	1	2	yes	—
30	C₂²×D₄₃	C₂³	1	1	yes	characteristic
31	Q₈×C₄₃	—	2	2	no	—
32	D₄×C₄₃	—	2	2	no	—
33	C₂×C₄×C₄₃	C₂³	1	2	yes	—
34	C₂×C₄×C₄₃	C₂³	1	2	yes	—
35	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
36	C₂²⋊D₄₃	—	2	4	no	—
37	C₂×Dic₄₃	—	2	4	no	—
38	C₂×Dic₄₃	C₂³	1	2	yes	—
39	C₂×Dic₄₃	C₂³	1	2	yes	—
40	C₂×Dic₄₃	C₂³	1	2	yes	—
41	C₂×Dic₄₃	C₂³	1	1	yes	characteristic
42	C₄×D₄₃	—	2	4	no	—
43	C₄×D₄₃	D₄	1	4	yes	—
44	Dic₈₆	—	2	2	no	—
45	Dic₈₆	—	2	2	no	—
46	Dic₈₆	—	2	4	no	—
47	C₂²×C₄₃	—	2	2	no	—
48	C₂²×C₄₃	—	2	4	no	—
49	C₂²×C₄₃	C₂⁴	1	1	yes	characteristic
50	C₂×D₄₃	—	2	2	no	—
51	C₂×D₄₃	C₂×D₄	1	2	yes	—
52	C₄×C₄₃	—	2	2	no	—
53	C₄×C₄₃	—	2	2	no	—
54	C₄×C₄₃	C₂×D₄	1	2	yes	—
55	Dic₄₃	—	2	2	no	—
56	Dic₄₃	—	2	2	no	—
57	Dic₄₃	—	2	2	no	—
58	Dic₄₃	C₂×D₄	1	2	yes	—
59	C₂×C₄₃	—	4	2	no	—
60	C₂×C₄₃	C₂²×D₄	1	1	yes	characteristic
61	C₂×C₄₃	2₋¹⁺⁴	1	2	yes	—
62	D₄₃	—	2	2	no	—
63	#536	—	43	1	no	maximal, Sylow
64	C₄₃	#536	1	1	yes	characteristic, complemented, Sylow
65	C₂×#41	—	43	1	no	—
66	C₂²×Q₈	—	43	1	no	—
67	C₄⋊Q₈	—	43	1	no	—
68	C₄⋊Q₈	—	43	1	no	—
69	#124	—	43	1	no	—
70	#124	D₄₃	1	1	yes	characteristic, complemented
71	#123	—	43	4	no	—
72	C₂²⋊Q₈	—	43	4	no	—
73	#117	—	43	1	no	—
74	#41	—	86	4	no	—
75	C₂×Q₈	—	43	2	no	—
76	C₂×Q₈	—	43	1	no	—
77	C₂×Q₈	—	43	1	no	—
78	C₂×Q₈	—	86	4	no	—
79	C₂×Q₈	C₂×D₄₃	1	1	yes	characteristic
80	C₂×D₄	—	43	2	no	—
81	C₂×D₄	C₂×D₄₃	1	1	yes	characteristic
82	C₂²×C₄	—	43	1	no	—
83	C₂²×C₄	—	43	2	no	—
84	C₂²×C₄	—	43	2	no	—
85	C₄⋊C₄	—	43	2	no	—
86	C₄⋊C₄	—	43	4	no	—
87	C₄⋊C₄	—	43	4	no	—
88	C₂²⋊C₄	—	43	4	no	—
89	C₂²⋊C₄	—	43	2	no	—
90	C₂²⋊C₄	C₂×D₄₃	1	4	yes	—
91	C₄²	—	43	1	no	—
92	C₄²	C₂×D₄₃	1	1	yes	characteristic
93	C₂³	—	43	1	no	—
94	C₂³	C₂²×D₄₃	1	2	yes	—
95	Q₈	—	2	2	no	—
96	Q₈	—	86	4	no	—
97	Q₈	—	86	2	no	—
98	Q₈	—	86	2	no	—
99	D₄	—	2	2	no	—
100	D₄	—	86	4	no	—
101	C₂×C₄	—	43	1	no	—
102	C₂×C₄	—	43	4	no	—
103	C₂×C₄	—	43	2	no	—
104	C₂×C₄	—	43	2	no	—
105	C₂×C₄	—	43	2	no	—
106	C₂×C₄	—	86	4	no	—
107	C₂×C₄	—	86	4	no	—
108	C₂×C₄	C₂²×D₄₃	1	2	yes	—
109	C₂×C₄	C₂²×D₄₃	1	2	yes	—
110	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	43	2	no	—
114	C₂²	—	86	2	no	—
115	C₂²	C₂³×D₄₃	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	43	2	no	—
119	C₄	—	86	2	no	—
120	C₄	—	86	2	no	—
121	C₄	—	86	2	no	—
122	C₄	D₄×D₄₃	1	2	yes	—
123	C₂	—	4	2	no	—
124	C₂	—	86	2	no	—
125	C₂	C₂×D₄×D₄₃	1	1	yes	characteristic
126	C₂	#13063372	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10621900

C₂

yes

maximal, characteristic

C₂×Q₈×D₄₃

C₂

yes

maximal, characteristic, complemented

#13063325

C₂

yes

maximal, characteristic, complemented

#13063323

C₂

yes

maximal, characteristic, complemented

#13063312

C₂

yes

maximal, characteristic, complemented

#13063307

C₂

yes

maximal, complemented

#13063302

C₂

yes

maximal, complemented

#13063282

C₂

yes

maximal, characteristic, complemented

#124×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₄₃

C₂²

yes

characteristic

C₂×D₄×C₄₃

C₂²

yes

characteristic

C₂×C₂²⋊D₄₃

C₂²

yes

—

C₂²×Dic₄₃

C₂²

yes

—

Q₈×D₄₃

—

#10621900

—

C₂×C₄×D₄₃

C₂²

yes

—

C₂×C₄×D₄₃

C₂²

yes

characteristic

C₂×Dic₈₆

C₂²

yes

—

C₂×Dic₈₆

C₂²

yes

characteristic

C₂×Dic₈₆

C₂²

yes

characteristic

C₂²⋊C₄×C₄₃

C₂²

yes

—

C₄²×C₄₃

C₂²

yes

characteristic

C₂²⋊Dic₄₃

C₂²

yes

—

(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₄₃

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

C₄×Dic₄₃

C₂²

yes

characteristic

C₂³×C₄₃

C₂³

yes

—

C₂²×D₄₃

C₂³

yes

characteristic

Q₈×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

C₂³

yes

—

C₂×C₄×C₄₃

C₂³

yes

—

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

characteristic

C₄×D₄₃

—

C₄×D₄₃

D₄

yes

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂⁴

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

C₂×D₄

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×D₄

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

C₂×D₄

yes

—

C₂×C₄₃

—

C₂×C₄₃

C₂²×D₄

yes

characteristic

C₂×C₄₃

2₋¹⁺⁴

yes

—

D₄₃

—

#536

—

maximal, Sylow

C₄₃

#536

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#124

—

#124

D₄₃

yes

characteristic, complemented

#123

—

C₂²⋊Q₈

—

#117

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄₃

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂×D₄₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₄₃

yes

—

C₄²

—

C₄²

C₂×D₄₃

yes

characteristic

C₂³

—

C₂³

C₂²×D₄₃

yes

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²×D₄₃

yes

—

109

C₂×C₄

C₂²×D₄₃

yes

—

110

C₂×C₄

C₂²×D₄₃

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂³×D₄₃

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

D₄×D₄₃

yes

—

123

C₂

—

124

C₂

—

125

C₂

C₂×D₄×D₄₃

yes

characteristic

126

C₂

#13063372

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₃
commutator, socle, cosocle	C₂²×C₄₃	C₂⁴
Fitting	#124×C₄₃	C₂
prime radical	C₂×#10621900	C₂
hypercenter	#124	D₄₃
hypocenter	C₄₃	#536
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₃

commutator, socle, cosocle

C₂²×C₄₃

C₂⁴

Fitting

#124×C₄₃

C₂

C₂

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
2. C₂×Q₈×D₄₃	C₂	semidirect
3. #13063325	C₂	semidirect
4. #13063323	C₂	semidirect
5. #13063312	C₂	semidirect
6. #13063307	C₂	semidirect
7. #13063302	C₂	semidirect
8. #13063282	C₂	semidirect
9. #124×C₄₃	C₂	semidirect
64. C₄₃	#536	semidirect
70. #124	D₄₃	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₄₃

C₂

semidirect

3. #13063325

C₂

semidirect

4. #13063323

C₂

semidirect

5. #13063312

C₂

semidirect

6. #13063307

C₂

semidirect

7. #13063302

C₂

semidirect

8. #13063282

C₂

semidirect

9. #124×C₄₃

C₂

semidirect

64. C₄₃

#536

semidirect

70. #124

D₄₃

semidirect

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

subgroup	index
C₂×#10621900	2
C₂×Q₈×D₄₃	2
#13063325	2
#13063323	2
#13063312	2
#13063307	2
#13063302	2
#13063282	2
#124×C₄₃	2
#536	43

subgroup

index

C₂×Q₈×D₄₃

#124×C₄₃

Subgroups — #14583469

Conjugacy classes of subgroups

Complements

Maximal subgroups