Group #28360922 — subgroups

Subgroups — #28360922

Order 13952.

150 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 65 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	#20295079	C₂	1	1	yes	maximal, characteristic
2	#20295050	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182013	C₄	1	2	yes	—
5	C₄⋊#11724784	—	2	1	no	—
6	C₄⋊#11724784	—	2	1	no	—
7	C₄⋊#11724784	C₄	1	2	yes	—
8	C₂²⋊#11724784	—	2	1	no	—
9	C₄×#11724784	—	2	1	no	—
10	C₂×M₁₆×C₁₀₉	—	2	1	no	—
11	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
12	#97×C₁₀₉	C₄	1	2	yes	—
13	C₄×C₈×C₁₀₉	—	2	1	no	—
14	#13182013	—	4	2	no	—
15	C₂×#11724784	—	2	2	no	—
16	C₂×#11724784	—	2	2	no	—
17	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
20	M₁₆×C₁₀₉	—	2	4	no	—
21	C₂×C₈×C₁₀₉	—	2	2	no	—
22	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
25	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
26	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	—	2	1	no	—
28	C₂×C₄×C₁₀₉	—	2	1	no	—
29	C₂×C₄×C₁₀₉	—	2	2	no	—
30	C₂×C₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
37	C₈×C₁₀₉	—	2	4	no	—
38	#11724784	—	4	2	no	—
39	#11724784	—	4	2	no	—
40	C₂²×C₁₀₉	—	2	1	no	—
41	C₂²×C₁₀₉	—	2	1	no	—
42	C₂²×C₁₀₉	—	2	1	no	—
43	C₂²×C₁₀₉	#95	1	1	yes	characteristic
44	C₄×C₁₀₉	—	2	1	no	—
45	C₄×C₁₀₉	—	2	1	no	—
46	C₄×C₁₀₉	—	2	1	no	—
47	C₄×C₁₀₉	#100	1	1	yes	characteristic
48	C₄×C₁₀₉	#101	1	1	yes	characteristic
49	C₄×C₁₀₉	#104	1	2	yes	—
50	C₄×C₁₀₉	#108	1	2	yes	—
51	C₂×C₁₀₉	—	4	1	no	—
52	C₂×C₁₀₉	#339	1	1	yes	characteristic
53	C₂×C₁₀₉	#341	1	1	yes	characteristic
54	C₂×C₁₀₉	#343	1	1	yes	characteristic
55	#1299	—	109	1	no	maximal, Sylow
56	C₁₀₉	#1299	1	1	yes	characteristic, complemented, Sylow
57	#431	—	109	1	no	—
58	#423	—	109	1	no	—
59	C₄×M₁₆	D₁₀₉	1	1	yes	characteristic
60	C₂×M₁₆	—	2	1	no	—
61	C₂×M₁₆	—	109	2	no	—
62	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
63	C₄⋊C₈	—	109	2	no	—
64	C₄⋊C₈	—	218	1	no	—
65	C₄⋊C₈	—	218	1	no	—
66	C₂²⋊C₈	—	218	1	no	—
67	#97	Dic₁₀₉	1	2	yes	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	218	1	no	—
70	C₂²×C₄	Dic₂₁₈	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
72	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
73	M₁₆	—	2	4	no	—
74	M₁₆	—	436	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	218	2	no	—
77	C₂×C₈	—	218	2	no	—
78	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
79	C₄²	D₄₃₆	1	1	yes	characteristic
80	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
81	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
82	C₂³	#13182008	1	1	yes	characteristic
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₄×Dic₁₀₉	1	1	yes	characteristic
88	C₂×C₄	#13182006	1	1	yes	characteristic
89	C₂×C₄	#13182006	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
91	C₂×C₄	#13182008	1	1	yes	characteristic
92	C₂×C₄	C₄⋊Dic₁₀₉	1	1	yes	characteristic
93	C₈	—	2	4	no	—
94	C₈	—	436	2	no	—
95	C₈	—	436	2	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	—	2	1	no	—
99	C₂²	#16812606	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	#16812615	1	1	yes	characteristic
104	C₄	#16812617	1	1	yes	characteristic
105	C₄	#16812627	1	2	yes	—
106	C₄	#16812634	1	2	yes	—
107	C₂	—	4	1	no	—
108	C₂	#20294786	1	1	yes	characteristic
109	C₂	#20294790	1	1	yes	characteristic
110	C₂	#20294794	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295079

C₂

yes

maximal, characteristic

#20295050

C₂

yes

maximal, characteristic

C₄×M₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₄

yes

—

C₄⋊#11724784

—

C₄⋊#11724784

—

C₄⋊#11724784

C₄

yes

—

C₂²⋊#11724784

—

C₄×#11724784

—

C₂×M₁₆×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#97×C₁₀₉

C₄

yes

—

C₄×C₈×C₁₀₉

—

#13182013

—

C₂×#11724784

—

C₂×#11724784

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

Q₈

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₂³×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₈×C₁₀₉

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#95

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

C₄×C₁₀₉

#108

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1299

yes

characteristic, complemented, Sylow

#431

—

109

—

#423

—

109

—

C₄×M₁₆

D₁₀₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

C₂²⋊C₈

—

218

—

#97

Dic₁₀₉

yes

—

C₄×C₈

—

C₄×C₈

—

218

—

C₂²×C₄

Dic₂₁₈

yes

characteristic

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

436

—

C₂×C₈

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

C₄×D₁₀₉

yes

characteristic

C₄²

D₄₃₆

yes

characteristic

C₄²

C₂×Dic₁₀₉

yes

characteristic

C₄²

C₂²⋊D₁₀₉

yes

characteristic

C₂³

#13182008

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₀₉

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₁₀₉

yes

characteristic

C₂×C₄

#13182008

yes

characteristic

C₂×C₄

C₄⋊Dic₁₀₉

yes

characteristic

C₈

—

C₈

—

436

—

C₈

—

436

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#16812606

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

#16812615

yes

characteristic

104

C₄

#16812617

yes

characteristic

105

C₄

#16812627

yes

—

106

C₄

#16812634

yes

—

107

C₂

—

108

C₂

#20294786

yes

characteristic

109

C₂

#20294790

yes

characteristic

110

C₂

#20294794

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13182006
commutator	C₂×C₄×C₁₀₉	C₄²
Frattini	C₂×C₄²	C₂×D₁₀₉
Fitting	C₄×M₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#95
prime radical	C₂³×C₁₀₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₀₉	C₂²
hypercenter	C₄×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#1299
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

#13182006

commutator

C₂×C₄×C₁₀₉

C₄²

Frattini

C₂×C₄²

C₂×D₁₀₉

Fitting

C₄×M₁₆×C₁₀₉

C₂

socle

C₂²×C₁₀₉

#95

prime radical

C₂³×C₁₀₉

C₄⋊C₄

cosocle

C₂×C₄²×C₁₀₉

C₂²

hypercenter

C₄×M₁₆

D₁₀₉

hypocenter

C₁₀₉

#1299

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
56. C₁₀₉	#1299	semidirect

H ≅ G/N

type

56. C₁₀₉

#1299

semidirect

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

subgroup	index
#20295079	2
#20295050	2
C₄×M₁₆×C₁₀₉	2
#1299	109

subgroup

index

#20295079

#20295050

C₄×M₁₆×C₁₀₉

#1299

109

Subgroups — #28360922

Conjugacy classes of subgroups

Complements

Maximal subgroups