Group #28364814 — subgroups

Subgroups — #28364814

Order 13952.

152 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×#13182004	C₂	1	1	yes	maximal, characteristic
2	#20295064	C₂	1	1	yes	maximal, characteristic
3	#20294853	C₂	1	2	yes	maximal, complemented
4	#20294851	C₂	1	2	yes	maximal, complemented
5	#427×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#16812658	—	2	1	no	—
7	C₂×#13182004	C₄	1	4	yes	complemented
8	C₂×#13182004	C₂²	1	1	yes	characteristic
9	C₂×#13182004	C₂²	1	1	yes	characteristic
10	#16812633	C₂²	1	2	yes	—
11	#16812631	—	2	1	no	—
12	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	#117×C₁₀₉	—	2	1	no	—
14	#107×C₁₀₉	C₂²	1	2	yes	—
15	#106×C₁₀₉	—	2	1	no	—
16	C₄⋊Dic₁₀₉	—	2	2	no	—
17	C₂²⋊Dic₁₀₉	—	4	1	no	—
18	C₄×Dic₁₀₉	—	4	1	no	—
19	#13182004	C₂×C₄	1	2	yes	—
20	#13182004	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
22	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
24	C₂×C₈×C₁₀₉	Q₈	1	2	yes	—
25	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₁₀₉	—	2	2	no	—
27	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
28	C₄²×C₁₀₉	—	4	1	no	—
29	C₂³×C₁₀₉	D₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	—	4	2	no	—
31	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
32	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	Q₁₆	1	2	yes	—
34	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₁₀₉	—	4	2	no	—
36	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
37	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₁₀₉	—	2	2	no	—
40	C₂²×C₁₀₉	#107	1	2	yes	—
41	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
42	C₄×C₁₀₉	—	8	2	no	—
43	C₄×C₁₀₉	#107	1	2	yes	—
44	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
46	Dic₁₀₉	—	8	2	no	—
47	C₂×C₁₀₉	—	2	2	no	—
48	C₂×C₁₀₉	C₂×#107	1	1	yes	characteristic
49	C₂×C₁₀₉	#508	1	2	yes	—
50	#2158	—	109	1	no	maximal, Sylow
51	C₁₀₉	#2158	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₁₆	—	109	1	no	—
53	#427	—	109	1	no	—
54	#427	D₁₀₉	1	1	yes	characteristic
55	#367	—	109	2	no	—
56	#366	—	109	2	no	—
57	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
58	#117	—	2	1	no	—
59	#117	—	218	1	no	—
60	C₂×C₁₆	—	109	4	no	—
61	C₂×C₁₆	—	109	1	no	—
62	C₂×C₁₆	—	109	1	no	—
63	#107	—	109	2	no	—
64	#107	C₂×D₁₀₉	1	2	yes	—
65	#106	—	2	1	no	—
66	#106	—	218	1	no	—
67	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
68	C₂×C₈	Dic₂₁₈	1	2	yes	—
69	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
70	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
71	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	218	2	no	—
74	C₂²⋊C₄	—	4	1	no	—
75	C₂²⋊C₄	—	436	1	no	—
76	C₄²	—	4	1	no	—
77	C₄²	—	436	1	no	—
78	C₁₆	—	109	2	no	—
79	C₁₆	—	109	2	no	—
80	C₂³	#13182014	1	1	yes	characteristic
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	436	2	no	—
83	C₂×C₄	#13182008	1	2	yes	—
84	C₂×C₄	#13182014	1	1	yes	characteristic
85	C₂×C₄	#13182019	1	2	yes	—
86	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
87	C₈	#13182008	1	2	yes	—
88	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
89	C₈	C₂×Dic₂₁₈	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	#16812632	1	2	yes	—
92	C₂²	C₂×#13182014	1	1	yes	characteristic
93	C₄	—	8	2	no	—
94	C₄	—	872	2	no	—
95	C₄	#16812632	1	2	yes	—
96	C₄	C₂×#13182008	1	1	yes	characteristic
97	C₄	C₂×#13182019	1	1	yes	characteristic
98	C₂	—	2	2	no	—
99	C₂	C₂×#16812632	1	1	yes	characteristic
100	C₂	#20295456	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13182004

C₂

yes

maximal, characteristic

#20295064

C₂

yes

maximal, characteristic

#20294853

C₂

yes

maximal, complemented

#20294851

C₂

yes

maximal, complemented

#427×C₁₀₉

C₂

yes

maximal, characteristic

#16812658

—

C₂×#13182004

C₄

yes

complemented

C₂×#13182004

C₂²

yes

characteristic

C₂×#13182004

C₂²

yes

characteristic

#16812633

C₂²

yes

—

#16812631

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

—

#107×C₁₀₉

C₂²

yes

—

#106×C₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

#13182004

C₂×C₄

yes

—

#13182004

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

Q₈

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

D₈

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

D₈

yes

characteristic

C₂×C₄×C₁₀₉

Q₁₆

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

C₈×C₁₀₉

C₂²×C₄

yes

characteristic

C₈×C₁₀₉

C₂×Q₈

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#107

yes

—

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

#107

yes

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂×Q₁₆

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#107

yes

characteristic

C₂×C₁₀₉

#508

yes

—

#2158

—

109

maximal, Sylow

C₁₀₉

#2158

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

109

—

#427

—

109

—

#427

D₁₀₉

yes

characteristic

#367

—

109

—

#366

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

—

#117

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

#107

—

109

—

#107

C₂×D₁₀₉

yes

—

#106

—

#106

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₈

Dic₂₁₈

yes

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

C₂²⋊C₄

—

436

—

C₄²

—

C₄²

—

436

—

C₁₆

—

109

—

C₁₆

—

109

—

C₂³

#13182014

yes

characteristic

C₂×C₄

—

C₂×C₄

—

436

—

C₂×C₄

#13182008

yes

—

C₂×C₄

#13182014

yes

characteristic

C₂×C₄

#13182019

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₈

#13182008

yes

—

C₈

C₂×C₄×D₁₀₉

yes

characteristic

C₈

C₂×Dic₂₁₈

yes

characteristic

C₂²

—

C₂²

#16812632

yes

—

C₂²

C₂×#13182014

yes

characteristic

C₄

—

C₄

—

872

—

C₄

#16812632

yes

—

C₄

yes

C₄

yes

C₂

—

C₂

C₂×#16812632

yes

characteristic

100

C₂

#20295456

yes

—

	subgroup	G/H
center	C₂×C₄	#13182014
commutator	C₈×C₁₀₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#427×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₈
prime radical	C₂³×C₁₀₉	D₈
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	#427	D₁₀₉
hypocenter	C₁₀₉	#2158
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₂×C₄

#13182014

commutator

C₈×C₁₀₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

#427×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₈

prime radical

C₂³×C₁₀₉

D₈

cosocle

C₂×C₈×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
3. #20294853	C₂	semidirect
4. #20294851	C₂	semidirect
7. C₂×#13182004	C₄	semidirect
51. C₁₀₉	#2158	semidirect

H ≅ G/N

type

3. #20294853

C₂

semidirect

4. #20294851

C₂

semidirect

7. C₂×#13182004

C₄

semidirect

51. C₁₀₉

#2158

semidirect

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

subgroup	index
C₂²×#13182004	2
#20295064	2
#20294853	2
#20294851	2
#427×C₁₀₉	2
#2158	109

subgroup

index

#427×C₁₀₉

109

Subgroups — #28364814

Conjugacy classes of subgroups

Complements

Maximal subgroups