Group #14583527 — subgroups

Subgroups — #14583527

Order 2752.

206 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 111 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	#13063325	C₂	1	2	yes	maximal, complemented
2	#13063324	C₂	1	2	yes	maximal, complemented
3	#13063317	C₂	1	2	yes	maximal, complemented
4	#13063316	C₂	1	2	yes	maximal, complemented
5	#13063302	C₂	1	4	yes	maximal
6	#13063284	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic
8	#117×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₄×C₄₃	C₂²	1	1	yes	characteristic
10	C₂²×Dic₄₃	C₂²	1	2	yes	—
11	C₂×Dic₈₆	C₂²	1	4	yes	—
12	C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
15	C₄²×C₄₃	C₂²	1	2	yes	—
16	C₂²⋊Dic₄₃	C₂²	1	2	yes	—
17	C₄⋊Dic₄₃	C₂²	1	2	yes	—
18	C₄⋊Dic₄₃	C₂²	1	2	yes	—
19	C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
20	C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
21	C₄⋊Dic₄₃	C₂²	1	4	yes	—
22	Dic₄₃⋊C₄	C₂²	1	4	yes	—
23	Dic₄₃⋊C₄	C₂²	1	4	yes	—
24	C₄×Dic₄₃	C₂²	1	2	yes	—
25	C₂³×C₄₃	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₄₃	Q₈	1	4	yes	complemented
27	C₂×C₄×C₄₃	C₂³	1	2	yes	—
28	C₂×C₄×C₄₃	C₂³	1	2	yes	—
29	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₄₃	—	4	2	no	—
32	C₂×Dic₄₃	C₂³	1	2	yes	—
33	C₂×Dic₄₃	C₂³	1	2	yes	—
34	C₂×Dic₄₃	C₂³	1	4	yes	—
35	Dic₈₆	—	4	4	no	—
36	C₂²×C₄₃	—	2	1	no	—
37	C₂²×C₄₃	—	2	1	no	—
38	C₂²×C₄₃	C₂×Q₈	1	2	yes	—
39	C₂²×C₄₃	C₂⁴	1	1	yes	characteristic
40	C₄×C₄₃	—	2	2	no	—
41	C₄×C₄₃	—	2	2	no	—
42	C₄×C₄₃	C₂×Q₈	1	2	yes	—
43	C₄×C₄₃	C₂×Q₈	1	2	yes	—
44	Dic₄₃	—	2	2	no	—
45	Dic₄₃	—	2	4	no	—
46	Dic₄₃	—	2	2	no	—
47	C₂×C₄₃	—	2	2	no	—
48	C₂×C₄₃	C₂²×Q₈	1	1	yes	characteristic
49	C₂×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
50	C₂×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
51	#544	—	43	1	no	maximal, Sylow
52	C₄₃	#544	1	1	yes	characteristic, complemented, Sylow
53	C₄⋊Q₈	—	43	2	no	—
54	C₄⋊Q₈	—	43	2	no	—
55	#125	—	43	2	no	—
56	#125	—	43	2	no	—
57	C₂²⋊Q₈	—	43	4	no	—
58	#117	—	43	1	no	—
59	#117	D₄₃	1	1	yes	characteristic
60	C₂×C₄⋊C₄	—	43	1	no	—
61	C₂×Q₈	—	43	4	no	—
62	C₂²×C₄	—	43	2	no	—
63	C₂²×C₄	C₂×D₄₃	1	1	yes	characteristic
64	C₄⋊C₄	—	43	2	no	—
65	C₄⋊C₄	—	43	2	no	—
66	C₄⋊C₄	—	43	1	no	—
67	C₄⋊C₄	—	43	1	no	—
68	C₄⋊C₄	—	43	4	no	—
69	C₄⋊C₄	—	43	4	no	—
70	C₄⋊C₄	—	43	4	no	—
71	C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
72	C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
73	C₂²⋊C₄	—	43	2	no	—
74	C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
75	C₄²	—	43	2	no	—
76	C₄²	C₂×D₄₃	1	2	yes	—
77	C₂³	C₂²×D₄₃	1	1	yes	characteristic
78	Q₈	—	172	4	no	—
79	C₂×C₄	—	43	2	no	—
80	C₂×C₄	—	43	2	no	—
81	C₂×C₄	—	43	4	no	—
82	C₂×C₄	—	172	2	no	—
83	C₂×C₄	Dic₈₆	1	4	yes	complemented
84	C₂×C₄	C₂²×D₄₃	1	2	yes	—
85	C₂×C₄	C₂²×D₄₃	1	2	yes	—
86	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
87	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
88	C₂²	—	2	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	C₂×Dic₈₆	1	2	yes	—
91	C₂²	C₂³×D₄₃	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	86	4	no	—
95	C₄	—	86	2	no	—
96	C₄	—	86	2	no	—
97	C₄	C₂×Dic₈₆	1	2	yes	—
98	C₄	C₂×Dic₈₆	1	2	yes	—
99	C₂	—	2	2	no	—
100	C₂	C₂²×Dic₈₆	1	1	yes	characteristic
101	C₂	#13063371	1	1	yes	characteristic
102	C₂	#13063372	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13063325

C₂

yes

maximal, complemented

#13063324

C₂

yes

maximal, complemented

#13063317

C₂

yes

maximal, complemented

#13063316

C₂

yes

maximal, complemented

#13063302

C₂

yes

maximal

#13063284

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₄₃

C₂

yes

maximal, characteristic

#117×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₄×C₄₃

C₂²

yes

characteristic

C₂²×Dic₄₃

C₂²

yes

—

C₂×Dic₈₆

C₂²

yes

—

C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂²⋊C₄×C₄₃

C₂²

yes

—

C₄²×C₄₃

C₂²

yes

—

C₂²⋊Dic₄₃

C₂²

yes

—

C₄⋊Dic₄₃

C₂²

yes

—

C₄⋊Dic₄₃

C₂²

yes

—

C₄⋊Dic₄₃

C₂²

yes

characteristic

C₄⋊Dic₄₃

C₂²

yes

characteristic

C₄⋊Dic₄₃

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

C₄×Dic₄₃

C₂²

yes

—

C₂³×C₄₃

C₂³

yes

characteristic

C₂×C₄×C₄₃

Q₈

yes

complemented

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

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

—

Dic₈₆

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×Q₈

yes

—

C₂²×C₄₃

C₂⁴

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×Q₈

yes

—

C₄×C₄₃

C₂×Q₈

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

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

—

#117

—

#117

D₄₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄₃

yes

characteristic

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

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₄₃

yes

—

C₄²

—

C₄²

C₂×D₄₃

yes

—

C₂³

C₂²×D₄₃

yes

characteristic

Q₈

—

172

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

172

—

C₂×C₄

Dic₈₆

yes

complemented

C₂×C₄

C₂²×D₄₃

yes

—

C₂×C₄

C₂²×D₄₃

yes

—

C₂×C₄

C₂²×D₄₃

yes

characteristic

C₂×C₄

C₂²×D₄₃

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂×Dic₈₆

yes

—

C₂²

C₂³×D₄₃

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

C₂×Dic₈₆

yes

—

C₄

C₂×Dic₈₆

yes

—

C₂

—

100

C₂

C₂²×Dic₈₆

yes

characteristic

101

C₂

#13063371

yes

characteristic

102

C₂

#13063372

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₃
commutator, socle, cosocle	C₂²×C₄₃	C₂⁴
Fitting	#117×C₄₃	C₂
prime radical	C₂³×C₄₃	C₂³
hypercenter	#117	D₄₃
hypocenter	C₄₃	#544
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₃

commutator, socle, cosocle

C₂²×C₄₃

C₂⁴

Fitting

#117×C₄₃

C₂

prime radical

C₂³×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. #13063325	C₂	semidirect
2. #13063324	C₂	semidirect
3. #13063317	C₂	semidirect
4. #13063316	C₂	semidirect
26. C₂×C₄×C₄₃	Q₈	semidirect
52. C₄₃	#544	semidirect
83. C₂×C₄	Dic₈₆	semidirect

H ≅ G/N

type

1. #13063325

C₂

semidirect

2. #13063324

C₂

semidirect

3. #13063317

C₂

semidirect

4. #13063316

C₂

semidirect

26. C₂×C₄×C₄₃

Q₈

semidirect

52. C₄₃

#544

semidirect

83. C₂×C₄

Dic₈₆

semidirect

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

subgroup	index
#13063325	2
#13063324	2
#13063317	2
#13063316	2
#13063302	2
#13063284	2
C₂×C₄⋊Dic₄₃	2
#117×C₄₃	2
#544	43

subgroup

index

C₂×C₄⋊Dic₄₃

#117×C₄₃

#544

Subgroups — #14583527

Conjugacy classes of subgroups

Complements

Maximal subgroups