Group #28360940 — subgroups

Subgroups — #28360940

Order 13952.

192 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 57 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	#20294990	C₂	1	2	yes	maximal
2	#381×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#13182013	—	2	2	no	—
4	C₂²⋊#11724784	—	2	2	no	—
5	C₂²⋊#11724784	C₄	1	4	yes	complemented
6	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₄×C₁₀₉	C₄	1	2	yes	—
8	#95×C₁₀₉	—	2	2	no	—
9	#13182013	—	4	4	no	—
10	C₂×#11724784	—	2	4	no	—
11	C₂⁴×C₁₀₉	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₁₀₉	—	2	2	no	—
13	C₂²×C₄×C₁₀₉	—	4	2	no	—
14	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
17	C₂²×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
18	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
19	C₂³×C₁₀₉	—	2	2	no	—
20	C₂³×C₁₀₉	—	4	2	no	—
21	C₂³×C₁₀₉	C₄²	1	1	yes	characteristic
22	C₂³×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₀₉	—	2	4	no	—
25	C₂×C₄×C₁₀₉	—	4	2	no	—
26	C₂×C₄×C₁₀₉	—	4	1	no	—
27	C₂×C₄×C₁₀₉	—	4	1	no	—
28	C₂×C₄×C₁₀₉	—	4	4	no	—
29	C₂×C₄×C₁₀₉	—	4	2	no	—
30	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
31	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
32	#11724784	—	4	4	no	—
33	C₂²×C₁₀₉	—	2	2	no	—
34	C₂²×C₁₀₉	—	2	1	no	—
35	C₂²×C₁₀₉	—	2	1	no	—
36	C₂²×C₁₀₉	—	2	2	no	—
37	C₂²×C₁₀₉	—	2	2	no	—
38	C₂²×C₁₀₉	—	4	4	no	—
39	C₂²×C₁₀₉	#95	1	1	yes	characteristic
40	C₂²×C₁₀₉	#99	1	2	yes	—
41	C₄×C₁₀₉	—	2	2	no	—
42	C₄×C₁₀₉	—	2	2	no	—
43	C₄×C₁₀₉	—	8	2	no	—
44	C₂×C₁₀₉	—	2	2	no	—
45	C₂×C₁₀₉	—	4	1	no	—
46	C₂×C₁₀₉	—	4	1	no	—
47	C₂×C₁₀₉	#342	1	1	yes	characteristic
48	C₂×C₁₀₉	#344	1	2	yes	—
49	#1306	—	109	1	no	maximal, Sylow
50	C₁₀₉	#1306	1	1	yes	characteristic, complemented, Sylow
51	#407	—	109	2	no	—
52	#381	D₁₀₉	1	1	yes	characteristic
53	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
54	C₂×M₁₆	—	218	2	no	—
55	C₂×C₂²⋊C₄	Dic₁₀₉	1	2	yes	—
56	C₂²⋊C₈	—	109	4	no	—
57	C₂²⋊C₈	—	218	2	no	—
58	#95	—	2	2	no	—
59	C₂⁴	C₂×Dic₁₀₉	1	1	yes	characteristic
60	C₂²×C₄	—	2	2	no	—
61	C₂²×C₄	—	4	2	no	—
62	C₂²×C₄	Dic₂₁₈	1	1	yes	characteristic
63	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
64	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
65	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
66	M₁₆	—	436	4	no	—
67	C₂×C₈	—	218	4	no	—
68	C₂²⋊C₄	—	4	2	no	—
69	C₂³	—	2	2	no	—
70	C₂³	—	4	2	no	—
71	C₂³	C₄×Dic₁₀₉	1	1	yes	characteristic
72	C₂³	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
73	C₂³	C₄⋊Dic₁₀₉	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	#13182006	1	2	yes	—
81	C₂×C₄	#13182008	1	2	yes	—
82	C₈	—	436	4	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	2	1	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	4	4	no	—
89	C₂²	#16812606	1	1	yes	characteristic
90	C₂²	#16812614	1	2	yes	—
91	C₄	—	2	2	no	—
92	C₄	—	2	2	no	—
93	C₄	—	8	2	no	—
94	C₂	—	2	2	no	—
95	C₂	—	4	1	no	—
96	C₂	—	4	1	no	—
97	C₂	#20294793	1	1	yes	characteristic
98	C₂	#20294796	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20294990

C₂

yes

maximal

#381×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

—

C₂²⋊#11724784

—

C₂²⋊#11724784

C₄

yes

complemented

C₂³×C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₁₀₉

C₄

yes

—

#95×C₁₀₉

—

#13182013

—

C₂×#11724784

—

C₂⁴×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

Q₈

yes

characteristic

C₂²⋊C₄×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₄²

yes

characteristic

C₂³×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂³×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#95

yes

characteristic

C₂²×C₁₀₉

#99

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#342

yes

characteristic

C₂×C₁₀₉

#344

yes

—

#1306

—

109

maximal, Sylow

C₁₀₉

#1306

yes

characteristic, complemented, Sylow

#407

—

109

—

#381

D₁₀₉

yes

characteristic

C₂³×C₄

C₂×D₁₀₉

yes

characteristic

C₂×M₁₆

—

218

—

C₂×C₂²⋊C₄

Dic₁₀₉

yes

—

C₂²⋊C₈

—

109

—

C₂²⋊C₈

—

218

—

#95

—

C₂⁴

C₂×Dic₁₀₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₂₁₈

yes

characteristic

C₂²×C₄

C₄×D₁₀₉

yes

—

C₂²×C₄

D₄₃₆

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

M₁₆

—

436

—

C₂×C₈

—

218

—

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

C₄×Dic₁₀₉

yes

characteristic

C₂³

C₂²⋊Dic₁₀₉

yes

characteristic

C₂³

C₄⋊Dic₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#13182006

yes

—

C₂×C₄

#13182008

yes

—

C₈

—

436

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#16812606

yes

characteristic

C₂²

#16812614

yes

—

C₄

—

C₄

—

C₄

—

C₂

—

C₂

—

C₂

—

C₂

#20294793

yes

characteristic

C₂

#20294796

yes

—

	subgroup	G/H
center	C₂²	#16812606
commutator	C₂³×C₁₀₉	C₄²
Frattini	C₂³×C₄	C₂×D₁₀₉
Fitting	#381×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#95
prime radical	C₂⁴×C₁₀₉	C₂×C₄
cosocle	C₂³×C₄×C₁₀₉	C₂²
hypercenter	#381	D₁₀₉
hypocenter	C₁₀₉	#1306
LCS	G ▷ C₂³×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #381

subgroup

G/H

center

C₂²

#16812606

commutator

C₂³×C₁₀₉

C₄²

Frattini

C₂³×C₄

C₂×D₁₀₉

Fitting

#381×C₁₀₉

C₂

socle

C₂²×C₁₀₉

#95

prime radical

C₂⁴×C₁₀₉

C₂×C₄

cosocle

C₂³×C₄×C₁₀₉

C₂²

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #381

N	H ≅ G/N	type
5. C₂²⋊#11724784	C₄	semidirect
50. C₁₀₉	#1306	semidirect

H ≅ G/N

type

5. C₂²⋊#11724784

C₄

semidirect

50. C₁₀₉

#1306

semidirect

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

subgroup	index
#20294990	2
#381×C₁₀₉	2
#1306	109

subgroup

index

#20294990

#381×C₁₀₉

#1306

109

Subgroups — #28360940

Conjugacy classes of subgroups

Complements

Maximal subgroups