Group #28361148 — subgroups

Subgroups — #28361148

Order 13952.

128 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂×#16812639	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812635	C₂	1	1	yes	maximal, characteristic
3	#427×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182013	—	2	1	no	—
5	#16812639	—	2	1	no	—
6	#16812639	C₄	1	2	yes	complemented
7	C₂×#13182004	—	2	1	no	—
8	#16812635	—	2	1	no	—
9	#16812635	C₄	1	2	yes	complemented
10	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	#117×C₁₀₉	—	2	1	no	—
12	#107×C₁₀₉	—	2	1	no	—
13	#106×C₁₀₉	C₄	1	2	yes	—
14	#13182013	—	2	2	no	—
15	#13182013	—	4	1	no	—
16	C₂×#11724784	—	4	1	no	—
17	#13182004	—	2	2	no	—
18	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
24	C₂×C₈×C₁₀₉	Q₈	1	1	yes	characteristic
25	C₄⋊C₄×C₁₀₉	—	2	2	no	—
26	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
27	C₄²×C₁₀₉	—	4	1	no	—
28	C₂³×C₁₀₉	SD₁₆	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	—	4	2	no	—
30	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	Q₁₆	1	1	yes	characteristic
36	C₈×C₁₀₉	C₄²	1	1	yes	characteristic
37	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
39	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
40	#11724784	—	4	2	no	—
41	C₂²×C₁₀₉	—	4	1	no	—
42	C₂²×C₁₀₉	#102	1	1	yes	characteristic
43	C₂²×C₁₀₉	#103	1	1	yes	characteristic
44	C₂²×C₁₀₉	#106	1	1	yes	characteristic
45	C₄×C₁₀₉	—	8	2	no	—
46	C₄×C₁₀₉	#95	1	1	yes	characteristic
47	C₄×C₁₀₉	#102	1	1	yes	characteristic
48	C₄×C₁₀₉	#103	1	1	yes	characteristic
49	C₄×C₁₀₉	#107	1	1	yes	characteristic
50	C₂×C₁₀₉	—	2	1	no	—
51	C₂×C₁₀₉	—	2	1	no	—
52	C₂×C₁₀₉	—	2	1	no	—
53	C₂×C₁₀₉	#340	1	1	yes	characteristic
54	#1389	—	109	1	no	maximal, Sylow
55	C₁₀₉	#1389	1	1	yes	characteristic, complemented, Sylow
56	C₂×M₃₂	—	109	1	no	—
57	C₂×#108	—	109	1	no	—
58	#427	D₁₀₉	1	1	yes	characteristic
59	C₂×M₁₆	—	218	1	no	—
60	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
61	#117	—	2	1	no	—
62	M₃₂	—	109	2	no	—
63	M₃₂	—	218	1	no	—
64	C₂×C₁₆	—	218	1	no	—
65	#108	—	109	2	no	—
66	#108	—	218	1	no	—
67	#107	—	2	1	no	—
68	#106	Dic₁₀₉	1	2	yes	—
69	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
70	M₁₆	—	436	1	no	—
71	M₁₆	—	218	2	no	—
72	C₂×C₈	—	436	1	no	—
73	C₂×C₈	Dic₂₁₈	1	1	yes	characteristic
74	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
75	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
76	C₂×C₈	D₄₃₆	1	1	yes	characteristic
77	C₂×C₈	C₂×Dic₁₀₉	1	1	yes	characteristic
78	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
79	C₄⋊C₄	—	2	2	no	—
80	C₂²⋊C₄	—	4	1	no	—
81	C₄²	—	4	1	no	—
82	C₁₆	—	218	2	no	—
83	C₂³	#13182017	1	1	yes	characteristic
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	#13182006	1	1	yes	characteristic
86	C₂×C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
87	C₂×C₄	#13182008	1	1	yes	characteristic
88	C₂×C₄	#13182014	1	1	yes	characteristic
89	C₂×C₄	#13182016	1	1	yes	characteristic
90	C₂×C₄	#13182019	1	1	yes	characteristic
91	C₈	—	436	2	no	—
92	C₈	C₄×Dic₁₀₉	1	1	yes	characteristic
93	C₈	#13182006	1	1	yes	characteristic
94	C₈	#13182008	1	1	yes	characteristic
95	C₈	C₄⋊Dic₁₀₉	1	1	yes	characteristic
96	C₂²	—	4	1	no	—
97	C₂²	#16812620	1	1	yes	characteristic
98	C₂²	#16812622	1	1	yes	characteristic
99	C₂²	#16812630	1	1	yes	characteristic
100	C₄	—	8	2	no	—
101	C₄	#16812606	1	1	yes	characteristic
102	C₄	#16812619	1	1	yes	characteristic
103	C₄	#16812623	1	1	yes	characteristic
104	C₄	#16812632	1	1	yes	characteristic
105	C₂	—	2	1	no	—
106	C₂	—	2	1	no	—
107	C₂	—	2	1	no	—
108	C₂	#20294788	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812639

C₂

yes

maximal, characteristic

C₂×#16812635

C₂

yes

maximal, characteristic

#427×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

—

#16812639

—

#16812639

C₄

yes

—

—

C₄

yes

complemented

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

—

#107×C₁₀₉

—

#106×C₁₀₉

C₄

yes

—

—

—

—

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

Q₈

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

SD₁₆

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

D₈

yes

characteristic

C₂×C₄×C₁₀₉

SD₁₆

yes

characteristic

C₂×C₄×C₁₀₉

Q₁₆

yes

characteristic

C₈×C₁₀₉

C₄²

yes

characteristic

C₈×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₈×C₁₀₉

C₄⋊C₄

yes

characteristic

C₈×C₁₀₉

C₄⋊C₄

yes

characteristic

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#340

yes

characteristic

#1389

—

109

maximal, Sylow

C₁₀₉

#1389

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

C₂×#108

—

109

—

#427

D₁₀₉

yes

characteristic

C₂×M₁₆

—

218

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

—

M₃₂

—

109

—

M₃₂

—

218

—

C₂×C₁₆

—

218

—

#108

—

109

—

#108

—

218

—

#107

—

#106

Dic₁₀₉

yes

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

M₁₆

—

436

—

M₁₆

—

218

—

C₂×C₈

—

436

—

C₂×C₈

Dic₂₁₈

yes

characteristic

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

D₄₃₆

yes

characteristic

C₂×C₈

C₂×Dic₁₀₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

218

—

C₂³

#13182017

yes

characteristic

C₂×C₄

—

C₂×C₄

#13182006

yes

characteristic

C₂×C₄

C₂²⋊Dic₁₀₉

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

436

—

C₈

C₄×Dic₁₀₉

yes

C₈

yes

C₈

yes

C₈

C₄⋊Dic₁₀₉

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

100

C₄

—

101

C₄

#16812606

yes

characteristic

102

C₄

#16812619

yes

characteristic

103

C₄

#16812623

yes

characteristic

104

C₄

#16812632

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

108

C₂

#20294788

yes

characteristic

	subgroup	G/H
center	C₄	#16812619
commutator	C₈×C₁₀₉	C₄²
Frattini	C₂²×C₈	C₂×D₁₀₉
Fitting	#427×C₁₀₉	C₂
socle	C₂×C₁₀₉	#340
prime radical	C₂³×C₁₀₉	SD₁₆
cosocle	C₂²×C₈×C₁₀₉	C₂²
hypercenter	#427	D₁₀₉
hypocenter	C₁₀₉	#1389
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₄

#16812619

commutator

C₈×C₁₀₉

C₄²

Frattini

C₂²×C₈

C₂×D₁₀₉

Fitting

#427×C₁₀₉

C₂

socle

C₂×C₁₀₉

#340

prime radical

C₂³×C₁₀₉

SD₁₆

cosocle

C₂²×C₈×C₁₀₉

C₂²

D₁₀₉

C₁₀₉

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
6. #16812639	C₄	semidirect
9. #16812635	C₄	semidirect
55. C₁₀₉	#1389	semidirect

H ≅ G/N

type

6. #16812639

C₄

semidirect

9. #16812635

C₄

semidirect

55. C₁₀₉

#1389

semidirect

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

subgroup	index
C₂×#16812639	2
C₂×#16812635	2
#427×C₁₀₉	2
#1389	109

subgroup

index

C₂×#16812639

C₂×#16812635

#427×C₁₀₉

#1389

109

Subgroups — #28361148

Conjugacy classes of subgroups

Complements

Maximal subgroups