Group #28379383 — subgroups

Subgroups — #28379383

Order 13952.

152 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 64 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	#20296036	C₂	1	2	yes	maximal
2	#20295953	C₂	1	2	yes	maximal, complemented
3	#20295952	C₂	1	2	yes	maximal, complemented
4	C₂×C₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#16812778	—	2	2	no	—
6	#16812768	C₂²	1	4	yes	—
7	#16812767	—	2	2	no	—
8	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#13182004	C₄	1	1	yes	characteristic, complemented
10	C₁₆×D₁₀₉	C₄	1	1	yes	characteristic, complemented
11	C₁₆×D₁₀₉	C₄	1	1	yes	characteristic, complemented
12	C₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
13	C₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
15	C₄⋊#89137	—	2	4	no	—
16	C₂²⋊#89137	—	4	2	no	—
17	C₄×#89137	—	4	2	no	—
18	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
19	C₂×#11724784	Q₈	1	1	yes	characteristic
20	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₈×D₁₀₉	D₄	1	1	yes	characteristic
22	C₈×D₁₀₉	Q₈	1	1	yes	characteristic
23	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
24	#13182004	C₂×C₄	1	1	yes	characteristic
25	#13182004	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×D₁₀₉	D₈	1	1	yes	characteristic
30	C₂×#89137	—	4	4	no	—
31	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×Dic₁₀₉	Q₁₆	1	1	yes	characteristic
33	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
34	C₄×D₁₀₉	D₈	1	1	yes	characteristic
35	C₄×D₁₀₉	Q₁₆	1	1	yes	characteristic
36	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
38	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	#11724784	C₄⋊C₄	1	1	yes	characteristic
40	#11724784	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₁₀₉	#107	1	1	yes	characteristic
42	C₂×D₁₀₉	—	2	2	no	—
43	C₂×D₁₀₉	#107	1	1	yes	characteristic
44	C₂×D₁₀₉	C₂×D₈	1	1	yes	characteristic
45	#89137	—	8	4	no	—
46	C₄×C₁₀₉	#107	1	1	yes	characteristic
47	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
48	Dic₁₀₉	#107	1	1	yes	characteristic
49	Dic₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₁₀₉	—	2	1	no	—
51	C₂×C₁₀₉	C₂×#107	1	1	yes	characteristic
52	D₁₀₉	—	2	1	no	—
53	D₁₀₉	#508	1	2	yes	—
54	#2158	—	109	1	no	maximal, Sylow
55	C₁₀₉	#2158	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₁₆	—	109	1	no	—
57	#427	—	109	2	no	—
58	#367	—	109	2	no	—
59	#366	—	109	2	no	—
60	C₂²×C₈	—	109	1	no	—
61	#117	—	218	2	no	—
62	C₂×C₁₆	—	109	1	no	—
63	C₂×C₁₆	—	109	1	no	—
64	C₂×C₁₆	—	109	1	no	—
65	C₂×C₁₆	—	109	1	no	—
66	C₂×C₁₆	—	109	1	no	—
67	C₂×C₁₆	#89137	1	1	yes	characteristic, complemented
68	#107	—	109	4	no	—
69	#106	—	218	2	no	—
70	C₂²×C₄	—	109	1	no	—
71	C₂×C₈	—	109	1	no	—
72	C₂×C₈	—	109	1	no	—
73	C₂×C₈	—	109	1	no	—
74	C₂×C₈	—	109	1	no	—
75	C₂×C₈	—	109	1	no	—
76	C₂×C₈	C₂×#89137	1	1	yes	characteristic
77	C₄⋊C₄	—	218	4	no	—
78	C₂²⋊C₄	—	436	2	no	—
79	C₄²	—	436	2	no	—
80	C₁₆	—	109	1	no	—
81	C₁₆	—	109	1	no	—
82	C₁₆	C₂×#89137	1	1	yes	characteristic
83	C₁₆	C₂×#89137	1	1	yes	characteristic
84	C₂³	—	109	1	no	—
85	C₂×C₄	—	109	1	no	—
86	C₂×C₄	—	109	1	no	—
87	C₂×C₄	—	109	1	no	—
88	C₂×C₄	—	109	1	no	—
89	C₂×C₄	—	109	1	no	—
90	C₂×C₄	—	436	4	no	—
91	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
92	C₈	—	109	1	no	—
93	C₈	—	109	1	no	—
94	C₈	C₄⋊#89137	1	1	yes	characteristic
95	C₈	C₂²×#89137	1	1	yes	characteristic
96	C₂²	—	109	1	no	—
97	C₂²	—	109	1	no	—
98	C₂²	—	218	2	no	—
99	C₂²	#16812768	1	1	yes	characteristic
100	C₄	—	872	4	no	—
101	C₄	—	109	1	no	—
102	C₄	—	109	1	no	—
103	C₄	#16812768	1	1	yes	characteristic
104	C₄	C₂×C₄⋊#89137	1	1	yes	characteristic
105	C₂	—	2	1	no	—
106	C₂	—	109	2	no	—
107	C₂	—	218	1	no	—
108	C₂	C₂×#16812768	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296036

C₂

yes

maximal

#20295953

C₂

yes

maximal, complemented

#20295952

C₂

yes

maximal, complemented

C₂×C₁₆×D₁₀₉

C₂

yes

maximal, characteristic

#16812778

—

#16812768

C₂²

yes

—

#16812767

—

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₂×#13182004

C₄

yes

characteristic, complemented

C₁₆×D₁₀₉

C₄

yes

characteristic, complemented

C₁₆×D₁₀₉

C₄

yes

characteristic, complemented

C₁₆×D₁₀₉

C₂²

yes

characteristic

C₁₆×D₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₄

yes

characteristic, complemented

C₄⋊#89137

—

C₂²⋊#89137

—

C₄×#89137

—

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×#11724784

Q₈

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

C₈×D₁₀₉

Q₈

yes

characteristic

C₈×D₁₀₉

C₂³

yes

characteristic

#13182004

C₂×C₄

yes

characteristic

#13182004

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

D₈

yes

characteristic

C₂×#89137

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

Q₁₆

yes

characteristic

C₄×D₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

D₈

yes

characteristic

C₄×D₁₀₉

Q₁₆

yes

characteristic

C₄×D₁₀₉

C₂×D₄

yes

characteristic

C₈×C₁₀₉

C₄⋊C₄

yes

characteristic

C₈×C₁₀₉

C₂²×C₄

yes

characteristic

#11724784

C₄⋊C₄

yes

characteristic

#11724784

C₂×Q₈

yes

characteristic

C₂²×C₁₀₉

#107

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

#107

yes

characteristic

C₂×D₁₀₉

C₂×D₈

yes

characteristic

#89137

—

C₄×C₁₀₉

#107

yes

characteristic

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

Dic₁₀₉

#107

yes

characteristic

Dic₁₀₉

C₂×Q₁₆

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#107

yes

characteristic

D₁₀₉

—

D₁₀₉

#508

yes

—

#2158

—

109

maximal, Sylow

C₁₀₉

#2158

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

109

—

#427

—

109

—

#367

—

109

—

#366

—

109

—

C₂²×C₈

—

109

—

#117

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

#89137

yes

characteristic, complemented

#107

—

109

—

#106

—

218

—

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₂×#89137

yes

characteristic

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

436

—

C₄²

—

436

—

C₁₆

—

109

—

C₁₆

—

109

—

C₁₆

yes

C₁₆

yes

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

C₄⋊#89137

yes

characteristic

C₈

—

109

—

C₈

—

109

—

C₈

C₄⋊#89137

yes

characteristic

C₈

C₂²×#89137

yes

characteristic

C₂²

—

109

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

#16812768

yes

characteristic

100

C₄

—

872

—

101

C₄

—

109

—

102

C₄

—

109

—

103

C₄

#16812768

yes

characteristic

104

C₄

C₂×C₄⋊#89137

yes

characteristic

105

C₂

—

106

C₂

—

109

—

107

C₂

—

218

—

108

C₂

C₂×#16812768

yes

characteristic

	subgroup	G/H
center	C₄	#16812768
commutator	C₈×C₁₀₉	C₂²×C₄
Frattini	C₈	C₂²×#89137
Fitting	C₂×C₁₆×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×#107
prime radical	C₂²×D₁₀₉	D₈
cosocle	C₈×D₁₀₉	C₂³
hypercenter	C₂×C₁₆	#89137
hypocenter	C₁₀₉	#2158
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

subgroup

G/H

center

C₄

#16812768

commutator

C₈×C₁₀₉

C₂²×C₄

Frattini

C₈

C₂²×#89137

Fitting

C₂×C₁₆×C₁₀₉

C₄

socle

C₂×C₁₀₉

C₂×#107

prime radical

C₂²×D₁₀₉

D₈

cosocle

C₈×D₁₀₉

C₂³

C₂×C₁₆

C₁₀₉

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

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

N	H ≅ G/N	type
2. #20295953	C₂	semidirect
3. #20295952	C₂	semidirect
9. C₂×#13182004	C₄	semidirect
10. C₁₆×D₁₀₉	C₄	semidirect
11. C₁₆×D₁₀₉	C₄	semidirect
14. C₂×C₁₆×C₁₀₉	C₄	semidirect
55. C₁₀₉	#2158	semidirect
67. C₂×C₁₆	#89137	semidirect

H ≅ G/N

type

2. #20295953

C₂

semidirect

3. #20295952

C₂

semidirect

9. C₂×#13182004

C₄

semidirect

10. C₁₆×D₁₀₉

C₄

semidirect

11. C₁₆×D₁₀₉

C₄

semidirect

14. C₂×C₁₆×C₁₀₉

C₄

semidirect

55. C₁₀₉

#2158

semidirect

67. C₂×C₁₆

#89137

semidirect

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

subgroup	index
#20296036	2
#20295953	2
#20295952	2
C₂×C₁₆×D₁₀₉	2
#2158	109

subgroup

index

C₂×C₁₆×D₁₀₉

109

Subgroups — #28379383

Conjugacy classes of subgroups

Complements

Maximal subgroups