Group #27203413 — subgroups

Subgroups — #27203413

Order 13184.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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	#20267730	C₂	1	3	yes	maximal
2	#20267724	C₂	1	6	yes	maximal
3	#20267711	C₂	1	3	yes	maximal
4	#20267659	C₂	1	2	yes	maximal
5	#384×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₀₃	C₂²	1	3	yes	—
8	C₂×Dic₁₀₃⋊C₄	C₂²	1	6	yes	—
9	C₁₀₃⋊#95	C₂²	1	6	yes	—
10	C₁₀₃⋊#95	C₂²	1	6	yes	—
11	C₁₀₃⋊#95	C₂²	1	3	yes	—
12	C₁₀₃⋊#95	C₂²	1	3	yes	—
13	C₂×C₄⋊C₄×C₁₀₃	C₂²	1	3	yes	—
14	C₂×C₄⋊C₄×C₁₀₃	C₂²	1	3	yes	—
15	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₀₆	—	4	3	no	—
17	C₂²×Dic₁₀₃	C₂³	1	2	yes	—
18	C₂²×Dic₁₀₃	C₂³	1	6	yes	—
19	C₄⋊Dic₁₀₃	—	4	3	no	—
20	Dic₁₀₃⋊C₄	—	4	6	no	—
21	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₃	C₂³	1	3	yes	—
23	C₂²×C₄×C₁₀₃	C₂³	1	3	yes	—
24	C₄⋊C₄×C₁₀₃	—	4	3	no	—
25	C₄⋊C₄×C₁₀₃	—	4	3	no	—
26	C₄²×C₁₀₃	Q₈	1	4	yes	complemented
27	C₂³×C₁₀₃	C₂⁴	1	1	yes	characteristic
28	C₂×C₄×C₁₀₃	—	2	3	no	—
29	C₂×C₄×C₁₀₃	—	2	3	no	—
30	C₂×C₄×C₁₀₃	—	2	6	no	—
31	C₂×C₄×C₁₀₃	—	2	3	no	—
32	C₂×C₄×C₁₀₃	—	2	3	no	—
33	C₂×C₄×C₁₀₃	C₂×Q₈	1	6	yes	—
34	C₂×Dic₁₀₃	—	2	6	no	—
35	C₂×Dic₁₀₃	—	2	6	no	—
36	C₂×Dic₁₀₃	—	2	12	no	—
37	Dic₂₀₆	—	4	4	no	—
38	C₂²×C₁₀₃	C₂²×Q₈	1	1	yes	characteristic
39	C₂²×C₁₀₃	H(5,𝔽₂)	1	3	yes	—
40	C₂²×C₁₀₃	2₋¹⁺⁴	1	3	yes	—
41	C₄×C₁₀₃	—	2	6	no	—
42	C₄×C₁₀₃	—	4	1	no	—
43	C₄×C₁₀₃	—	4	3	no	—
44	Dic₁₀₃	—	4	2	no	—
45	Dic₁₀₃	—	4	6	no	—
46	C₂×C₁₀₃	#544	1	3	yes	—
47	C₂×C₁₀₃	#560	1	1	yes	characteristic
48	C₂×C₁₀₃	#563	1	3	yes	—
49	#2843	—	103	1	no	maximal, Sylow
50	C₁₀₃	#2843	1	1	yes	characteristic, complemented, Sylow
51	#400	—	103	3	no	—
52	#398	—	103	6	no	—
53	#395	—	103	3	no	—
54	#384	D₁₀₃	1	1	yes	characteristic
55	#383	—	103	2	no	—
56	C₂²×Q₈	—	103	1	no	—
57	C₂×C₄⋊C₄	—	103	6	no	—
58	C₂×C₄⋊C₄	—	103	3	no	—
59	C₂×C₄⋊C₄	C₂×D₁₀₃	1	3	yes	—
60	C₂×C₄⋊C₄	C₂×D₁₀₃	1	3	yes	—
61	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
62	#95	—	103	6	no	—
63	#95	—	103	6	no	—
64	#95	—	103	3	no	—
65	#95	—	103	3	no	—
66	C₂×Q₈	—	412	3	no	—
67	C₂²×C₄	—	103	6	no	—
68	C₂²×C₄	—	103	2	no	—
69	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₁₀₃	1	3	yes	—
71	C₂²×C₄	C₂²×D₁₀₃	1	3	yes	—
72	C₄⋊C₄	—	4	3	no	—
73	C₄⋊C₄	—	4	3	no	—
74	C₄⋊C₄	—	412	6	no	—
75	C₄⋊C₄	—	412	3	no	—
76	C₄²	Dic₂₀₆	1	4	yes	complemented
77	C₂³	C₂³×D₁₀₃	1	1	yes	characteristic
78	Q₈	—	412	4	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	—	2	3	no	—
84	C₂×C₄	—	206	6	no	—
85	C₂×C₄	—	206	6	no	—
86	C₂×C₄	—	206	12	no	—
87	C₂×C₄	C₂×Dic₂₀₆	1	6	yes	—
88	C₂²	C₂²×Dic₂₀₆	1	1	yes	characteristic
89	C₂²	#15693066	1	3	yes	—
90	C₂²	#15693069	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	4	1	no	—
93	C₄	—	4	3	no	—
94	C₄	—	412	2	no	—
95	C₄	—	412	6	no	—
96	C₂	#20268422	1	3	yes	—
97	C₂	#20268528	1	1	yes	characteristic
98	C₂	#20268540	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20267730

C₂

yes

maximal

#20267724

C₂

yes

maximal

#20267711

C₂

yes

maximal

#20267659

C₂

yes

maximal

#384×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₆

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₃

C₂²

yes

—

C₂×Dic₁₀₃⋊C₄

C₂²

yes

—

C₁₀₃⋊#95

C₂²

yes

—

C₁₀₃⋊#95

C₂²

yes

—

C₁₀₃⋊#95

C₂²

yes

—

C₁₀₃⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₃

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₃

C₂²

yes

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₂×Dic₂₀₆

—

C₂²×Dic₁₀₃

C₂³

yes

—

C₂²×Dic₁₀₃

C₂³

yes

—

C₄⋊Dic₁₀₃

—

Dic₁₀₃⋊C₄

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₂²×C₄×C₁₀₃

C₂³

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄²×C₁₀₃

Q₈

yes

complemented

C₂³×C₁₀₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂×Q₈

yes

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

Dic₂₀₆

—

C₂²×C₁₀₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₀₃

H(5,𝔽₂)

yes

—

C₂²×C₁₀₃

2₋¹⁺⁴

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

#544

yes

—

C₂×C₁₀₃

#560

yes

characteristic

C₂×C₁₀₃

#563

yes

—

#2843

—

103

maximal, Sylow

C₁₀₃

#2843

yes

characteristic, complemented, Sylow

#400

—

103

—

#398

—

103

—

#395

—

103

—

#384

D₁₀₃

yes

characteristic

#383

—

103

—

C₂²×Q₈

—

103

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

C₂×D₁₀₃

yes

—

C₂×C₄⋊C₄

C₂×D₁₀₃

yes

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

#95

—

103

—

#95

—

103

—

#95

—

103

—

#95

—

103

—

C₂×Q₈

—

412

—

C₂²×C₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

C₂²×C₄

C₂²×D₁₀₃

yes

—

C₂²×C₄

C₂²×D₁₀₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

412

—

C₄⋊C₄

—

412

—

C₄²

Dic₂₀₆

yes

complemented

C₂³

C₂³×D₁₀₃

yes

characteristic

Q₈

—

412

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

206

—

C₂×C₄

—

206

—

C₂×C₄

—

206

—

C₂×C₄

C₂×Dic₂₀₆

yes

—

C₂²

C₂²×Dic₂₀₆

yes

characteristic

C₂²

#15693066

yes

—

C₂²

#15693069

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

412

—

C₄

—

412

—

C₂

#20268422

yes

—

C₂

#20268528

yes

characteristic

C₂

#20268540

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₃
commutator, socle, prime radical, cosocle	C₂³×C₁₀₃	C₂⁴
Fitting	#384×C₁₀₃	C₂
hypercenter	#384	D₁₀₃
hypocenter	C₁₀₃	#2843
LCS	G ▷ C₂³×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₃

commutator, socle, prime radical, cosocle

C₂³×C₁₀₃

C₂⁴

Fitting

#384×C₁₀₃

C₂

D₁₀₃

C₁₀₃

G ▷ C₂³×C₁₀₃ ▷ C₁₀₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
26. C₄²×C₁₀₃	Q₈	semidirect
50. C₁₀₃	#2843	semidirect
76. C₄²	Dic₂₀₆	semidirect

H ≅ G/N

type

26. C₄²×C₁₀₃

Q₈

semidirect

50. C₁₀₃

#2843

semidirect

76. C₄²

Dic₂₀₆

semidirect

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

subgroup	index
#20267730	2
#20267724	2
#20267711	2
#20267659	2
#384×C₁₀₃	2
#2843	103

subgroup

index

#384×C₁₀₃

103

Subgroups — #27203413

Conjugacy classes of subgroups

Complements

Maximal subgroups