Group #27209292 — subgroups

Subgroups — #27209292

Order 13184.

930 conjugacy classes of subgroups; only the 451 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 451 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	#20268545	C₂	1	10	yes	maximal, complemented
2	#20268539	C₂	1	5	yes	maximal, complemented
3	#20268347	C₂	1	5	yes	maximal, complemented
4	#20268326	C₂	1	10	yes	maximal, complemented
5	#564×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#15693020	C₂²	1	10	yes	—
7	#15693016	C₂²	1	20	yes	—
8	#15693013	C₂²	1	5	yes	—
9	#15693009	C₂²	1	10	yes	—
10	#15693006	C₂²	1	20	yes	—
11	#15693001	C₂²	1	10	yes	—
12	#15692999	C₂²	1	20	yes	—
13	#15692996	C₂²	1	10	yes	—
14	#15692986	C₂²	1	10	yes	—
15	#15692983	C₂²	1	10	yes	—
16	#15692979	C₂²	1	10	yes	—
17	C₄²×D₁₀₃	C₂²	1	5	yes	—
18	#125×C₁₀₃	C₂²	1	10	yes	—
19	#125×C₁₀₃	C₂²	1	5	yes	—
20	C₂×Dic₂₀₆	C₂³	1	10	yes	—
21	C₂×D₄₁₂	C₂³	1	10	yes	—
22	C₂×C₄×D₁₀₃	C₂³	1	5	yes	—
23	C₂×C₄×D₁₀₃	C₂³	1	10	yes	—
24	C₄⋊Dic₁₀₃	C₂³	1	10	yes	—
25	Dic₁₀₃⋊C₄	C₂³	1	20	yes	—
26	(C₂×D₁₀₃)⋊C₄	C₂³	1	20	yes	—
27	(C₂×D₁₀₃)⋊C₄	C₂³	1	20	yes	—
28	C₄×Dic₁₀₃	C₂³	1	10	yes	—
29	C₄×Dic₁₀₃	C₂³	1	5	yes	—
30	C₄⋊C₄×C₁₀₃	C₂³	1	10	yes	—
31	C₄⋊C₄×C₁₀₃	C₂³	1	10	yes	—
32	C₄⋊C₄×C₁₀₃	C₂³	1	10	yes	—
33	C₄²×C₁₀₃	C₂³	1	5	yes	—
34	C₂²×D₁₀₃	C₂⁴	1	5	yes	—
35	C₂×C₄×C₁₀₃	C₂⁴	1	5	yes	—
36	C₂×C₄×C₁₀₃	C₂⁴	1	10	yes	—
37	C₂×Dic₁₀₃	C₂⁴	1	10	yes	—
38	C₂×Dic₁₀₃	C₂⁴	1	1	yes	characteristic
39	D₄₁₂	—	4	10	no	—
40	C₄×D₁₀₃	—	2	10	no	—
41	C₄×D₁₀₃	—	4	10	no	—
42	Dic₂₀₆	—	4	10	no	—
43	C₂²×C₁₀₃	C₂⁵	1	1	yes	characteristic
44	C₂×D₁₀₃	—	2	5	no	—
45	C₂×D₁₀₃	—	2	10	no	—
46	C₄×C₁₀₃	—	2	5	no	—
47	C₄×C₁₀₃	—	2	10	no	—
48	Dic₁₀₃	—	2	10	no	—
49	Dic₁₀₃	2₋¹⁺⁴	1	2	yes	—
50	C₂×C₁₀₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
51	C₂×C₁₀₃	#585	1	2	yes	—
52	D₁₀₃	—	4	5	no	—
53	#3531	—	103	1	no	maximal, Sylow
54	C₁₀₃	#3531	1	1	yes	characteristic, complemented, Sylow
55	#564	D₁₀₃	1	1	yes	characteristic, complemented
56	#563	—	103	10	no	—
57	#562	—	103	5	no	—
58	#533	—	103	5	no	—
59	#529	—	103	10	no	—
60	C₄⋊Q₈	—	103	10	no	—
61	#126	—	103	20	no	—
62	#125	—	103	5	no	—
63	#125	C₂×D₁₀₃	1	10	yes	—
64	#125	C₂×D₁₀₃	1	5	yes	—
65	#124	—	103	10	no	—
66	#123	—	103	20	no	—
67	C₂²⋊Q₈	—	103	10	no	—
68	C₂²⋊Q₈	—	103	20	no	—
69	#121	—	103	10	no	—
70	C₄×Q₈	—	103	10	no	—
71	C₄×D₄	—	103	10	no	—
72	#117	—	103	10	no	—
73	C₂×C₄²	—	103	5	no	—
74	C₂×Q₈	—	103	10	no	—
75	C₂×D₄	—	103	10	no	—
76	C₂²×C₄	—	103	10	no	—
77	C₂²×C₄	—	103	5	no	—
78	C₄⋊C₄	—	103	20	no	—
79	C₄⋊C₄	—	103	10	no	—
80	C₄⋊C₄	C₂²×D₁₀₃	1	10	yes	—
81	C₄⋊C₄	C₂²×D₁₀₃	1	10	yes	—
82	C₄⋊C₄	C₂²×D₁₀₃	1	10	yes	—
83	C₂²⋊C₄	—	103	20	no	—
84	C₂²⋊C₄	—	103	20	no	—
85	C₄²	—	103	10	no	—
86	C₄²	—	103	5	no	—
87	C₄²	C₂²×D₁₀₃	1	5	yes	—
88	C₂³	—	103	5	no	—
89	Q₈	—	412	10	no	—
90	D₄	—	412	10	no	—
91	C₂×C₄	—	103	1	no	—
92	C₂×C₄	—	103	10	no	—
93	C₂×C₄	—	412	10	no	—
94	C₂×C₄	—	206	10	no	—
95	C₂×C₄	C₂³×D₁₀₃	1	5	yes	—
96	C₂×C₄	C₂³×D₁₀₃	1	10	yes	—
97	C₂²	—	206	5	no	—
98	C₂²	—	206	10	no	—
99	C₂²	C₂⁴×D₁₀₃	1	1	yes	characteristic
100	C₄	—	2	5	no	—
101	C₄	—	2	10	no	—
102	C₄	—	103	2	no	—
103	C₄	—	206	10	no	—
104	C₂	—	412	5	no	—
105	C₂	2₋¹⁺⁴×D₁₀₃	1	1	yes	characteristic
106	C₂	#20268628	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20268545

C₂

yes

maximal, complemented

#20268539

C₂

yes

maximal, complemented

#20268347

C₂

yes

maximal, complemented

#20268326

C₂

yes

maximal, complemented

#564×C₁₀₃

C₂

yes

maximal, characteristic, complemented

#15693020

C₂²

yes

—

#15693016

C₂²

yes

—

#15693013

C₂²

yes

—

#15693009

C₂²

yes

—

#15693006

C₂²

yes

—

#15693001

C₂²

yes

—

#15692999

C₂²

yes

—

#15692996

C₂²

yes

—

#15692986

C₂²

yes

—

#15692983

C₂²

yes

—

#15692979

C₂²

yes

—

C₄²×D₁₀₃

C₂²

yes

—

#125×C₁₀₃

C₂²

yes

—

#125×C₁₀₃

C₂²

yes

—

C₂×Dic₂₀₆

C₂³

yes

—

C₂×D₄₁₂

C₂³

yes

—

C₂×C₄×D₁₀₃

C₂³

yes

—

C₂×C₄×D₁₀₃

C₂³

yes

—

C₄⋊Dic₁₀₃

C₂³

yes

—

Dic₁₀₃⋊C₄

C₂³

yes

—

(C₂×D₁₀₃)⋊C₄

C₂³

yes

—

(C₂×D₁₀₃)⋊C₄

C₂³

yes

—

C₄×Dic₁₀₃

C₂³

yes

—

C₄×Dic₁₀₃

C₂³

yes

—

C₄⋊C₄×C₁₀₃

C₂³

yes

—

C₄⋊C₄×C₁₀₃

C₂³

yes

—

C₄⋊C₄×C₁₀₃

C₂³

yes

—

C₄²×C₁₀₃

C₂³

yes

—

C₂²×D₁₀₃

C₂⁴

yes

—

C₂×C₄×C₁₀₃

C₂⁴

yes

—

C₂×C₄×C₁₀₃

C₂⁴

yes

—

C₂×Dic₁₀₃

C₂⁴

yes

—

C₂×Dic₁₀₃

C₂⁴

yes

characteristic

D₄₁₂

—

C₄×D₁₀₃

—

C₄×D₁₀₃

—

Dic₂₀₆

—

C₂²×C₁₀₃

C₂⁵

yes

characteristic

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

2₋¹⁺⁴

yes

—

C₂×C₁₀₃

C₂×2₋¹⁺⁴

yes

characteristic

C₂×C₁₀₃

#585

yes

—

D₁₀₃

—

#3531

—

103

maximal, Sylow

C₁₀₃

#3531

yes

characteristic, complemented, Sylow

#564

D₁₀₃

yes

characteristic, complemented

#563

—

103

—

#562

—

103

—

#533

—

103

—

#529

—

103

—

C₄⋊Q₈

—

103

—

#126

—

103

—

#125

—

103

—

#125

C₂×D₁₀₃

yes

—

#125

C₂×D₁₀₃

yes

—

#124

—

103

—

#123

—

103

—

C₂²⋊Q₈

—

103

—

C₂²⋊Q₈

—

103

—

#121

—

103

—

C₄×Q₈

—

103

—

C₄×D₄

—

103

—

#117

—

103

—

C₂×C₄²

—

103

—

C₂×Q₈

—

103

—

C₂×D₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

—

103

—

C₄⋊C₄

—

103

—

C₄⋊C₄

—

103

—

C₄⋊C₄

C₂²×D₁₀₃

yes

—

C₄⋊C₄

C₂²×D₁₀₃

yes

—

C₄⋊C₄

C₂²×D₁₀₃

yes

—

C₂²⋊C₄

—

103

—

C₂²⋊C₄

—

103

—

C₄²

—

103

—

C₄²

—

103

—

C₄²

C₂²×D₁₀₃

yes

—

C₂³

—

103

—

Q₈

—

412

—

D₄

—

412

—

C₂×C₄

—

103

—

C₂×C₄

—

103

—

C₂×C₄

—

412

—

C₂×C₄

—

206

—

C₂×C₄

C₂³×D₁₀₃

yes

—

C₂×C₄

C₂³×D₁₀₃

yes

—

C₂²

—

206

—

C₂²

—

206

—

C₂²

C₂⁴×D₁₀₃

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

—

103

C₄

—

206

—

104

C₂

—

412

—

105

C₂

2₋¹⁺⁴×D₁₀₃

yes

characteristic

106

C₂

#20268628

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₃
commutator, socle, cosocle	C₂²×C₁₀₃	C₂⁵
Fitting	#564×C₁₀₃	C₂
hypercenter	#564	D₁₀₃
hypocenter	C₁₀₃	#3531
LCS	G ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₁₀₃

commutator, socle, cosocle

C₂²×C₁₀₃

C₂⁵

Fitting

#564×C₁₀₃

C₂

D₁₀₃

C₁₀₃

G ▷ C₂²×C₁₀₃ ▷ C₁₀₃

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
1. #20268545	C₂	semidirect
2. #20268539	C₂	semidirect
3. #20268347	C₂	semidirect
4. #20268326	C₂	semidirect
5. #564×C₁₀₃	C₂	semidirect
54. C₁₀₃	#3531	semidirect
55. #564	D₁₀₃	semidirect

H ≅ G/N

type

1. #20268545

C₂

semidirect

2. #20268539

C₂

semidirect

3. #20268347

C₂

semidirect

4. #20268326

C₂

semidirect

5. #564×C₁₀₃

C₂

semidirect

54. C₁₀₃

#3531

semidirect

55. #564

D₁₀₃

semidirect

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

subgroup	index
#20268545	2
#20268539	2
#20268347	2
#20268326	2
#564×C₁₀₃	2
#3531	103

subgroup

index

#564×C₁₀₃

103

Subgroups — #27209292

Conjugacy classes of subgroups

Complements

Maximal subgroups