Group #28363820 — subgroups

Subgroups — #28363820

Order 13952.

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

normal subgroups only

This group has 184 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	#20295074	C₂	1	2	yes	maximal
2	C₂×#16812635	C₂	1	1	yes	maximal, characteristic
3	#20294981	C₂	1	1	yes	maximal, characteristic
4	#20294797	C₂	1	2	yes	maximal
5	C₂×#108×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	2	yes	—
7	C₂×#13182013	C₂²	1	1	yes	characteristic
8	#16812658	C₂²	1	1	yes	characteristic
9	#16812635	C₄	1	4	yes	complemented
10	#16812634	—	2	4	no	—
11	#16812609	—	2	1	no	—
12	C₈×Dic₁₀₉	—	2	1	no	—
13	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	#108×C₁₀₉	—	2	1	no	—
16	#108×C₁₀₉	—	2	1	no	—
17	#13182013	—	2	1	no	—
18	#13182013	—	2	2	no	—
19	#13182013	C₂×C₄	1	4	yes	—
20	#13182013	D₄	1	2	yes	—
21	C₂×#11724784	—	2	1	no	—
22	C₂×#11724784	—	2	2	no	—
23	C₄⋊Dic₁₀₉	—	2	1	no	—
24	C₂²⋊Dic₁₀₉	Q₈	1	2	yes	—
25	C₄×Dic₁₀₉	—	2	1	no	—
26	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₀₉	—	2	2	no	—
28	M₁₆×C₁₀₉	—	2	2	no	—
29	C₂×C₈×C₁₀₉	—	2	2	no	—
30	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₀₉	Q₈	1	2	yes	—
34	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
41	C₂×Dic₁₀₉	—	2	2	no	—
42	C₈×C₁₀₉	—	4	2	no	—
43	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
44	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
45	#11724784	—	2	4	no	—
46	#11724784	—	2	2	no	—
47	C₂²×C₁₀₉	—	4	1	no	—
48	C₂²×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
49	C₂²×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
51	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
53	C₄×C₁₀₉	#121	1	1	yes	characteristic
54	C₄×C₁₀₉	#125	1	1	yes	characteristic
55	Dic₁₀₉	—	4	2	no	—
56	C₂×C₁₀₉	—	2	1	no	—
57	C₂×C₁₀₉	—	2	1	no	—
58	C₂×C₁₀₉	—	2	1	no	—
59	C₂×C₁₀₉	#389	1	1	yes	characteristic
60	#1953	—	109	1	no	maximal, Sylow
61	C₁₀₉	#1953	1	1	yes	characteristic, complemented, Sylow
62	#430	—	109	2	no	—
63	C₂×#108	—	109	1	no	—
64	C₂×#108	D₁₀₉	1	1	yes	characteristic
65	#405	—	109	1	no	—
66	#344	—	109	2	no	—
67	C₂×M₁₆	—	109	2	no	—
68	C₂×M₁₆	—	109	1	no	—
69	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
70	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
71	#117	—	109	1	no	—
72	#108	—	2	1	no	—
73	#108	—	2	1	no	—
74	#108	—	109	4	no	—
75	#108	—	218	4	no	—
76	#97	—	218	1	no	—
77	C₄×C₈	—	218	1	no	—
78	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
79	M₁₆	—	2	2	no	—
80	M₁₆	—	2	2	no	—
81	M₁₆	—	109	4	no	—
82	M₁₆	—	109	2	no	—
83	M₁₆	—	218	1	no	—
84	M₁₆	—	218	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	218	2	no	—
87	C₂×C₈	—	218	1	no	—
88	C₂×C₈	Dic₂₁₈	1	2	yes	—
89	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
90	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
91	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
92	C₄⋊C₄	—	218	1	no	—
93	C₂²⋊C₄	—	109	2	no	—
94	C₄²	—	218	1	no	—
95	C₂³	Q₈×D₁₀₉	1	1	yes	characteristic
96	C₂×C₄	—	218	2	no	—
97	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
98	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
101	C₂×C₄	#13182028	1	1	yes	characteristic
102	C₂×C₄	#13182030	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₈	—	218	4	no	—
105	C₈	—	218	2	no	—
106	C₈	#13182008	1	2	yes	—
107	C₈	#13182008	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	#16812664	1	1	yes	characteristic
110	C₂²	#16812675	1	1	yes	characteristic
111	C₂²	#16812698	1	1	yes	characteristic
112	C₄	—	436	2	no	—
113	C₄	C₂×#13182008	1	1	yes	characteristic
114	C₄	#16812661	1	1	yes	characteristic
115	C₄	#16812670	1	1	yes	characteristic
116	C₄	#16812690	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#20294922	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295074

C₂

yes

maximal

C₂×#16812635

C₂

yes

maximal, characteristic

#20294981

C₂

yes

maximal, characteristic

#20294797

C₂

yes

maximal

C₂×#108×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂×#13182013

C₂²

yes

characteristic

#16812658

C₂²

yes

characteristic

#16812635

C₄

yes

complemented

#16812634

—

#16812609

—

C₈×Dic₁₀₉

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#108×C₁₀₉

—

#108×C₁₀₉

—

#13182013

—

#13182013

—

#13182013

C₂×C₄

yes

—

#13182013

D₄

yes

—

C₂×#11724784

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

Q₈

yes

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

Q₈

yes

—

C₂³×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×Q₈

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄×D₄

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#389

yes

characteristic

#1953

—

109

maximal, Sylow

C₁₀₉

#1953

yes

characteristic, complemented, Sylow

#430

—

109

—

C₂×#108

—

109

—

C₂×#108

D₁₀₉

yes

characteristic

#405

—

109

—

#344

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

—

109

—

#108

—

#108

—

#108

—

109

—

#108

—

218

—

#97

—

218

—

C₄×C₈

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

109

—

M₁₆

—

109

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

Dic₂₁₈

yes

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

109

—

C₄²

—

218

—

C₂³

Q₈×D₁₀₉

yes

characteristic

C₂×C₄

—

218

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

D₄×D₁₀₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

100

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

101

C₂×C₄

#13182028

yes

characteristic

102

C₂×C₄

#13182030

yes

characteristic

103

C₈

—

104

C₈

—

218

—

105

C₈

—

218

—

106

C₈

#13182008

yes

—

107

C₈

#13182008

yes

—

108

C₂²

—

109

C₂²

#16812664

yes

characteristic

110

C₂²

#16812675

yes

characteristic

111

C₂²

#16812698

yes

characteristic

112

C₄

—

436

—

113

C₄

C₂×#13182008

yes

characteristic

114

C₄

#16812661

yes

characteristic

115

C₄

#16812670

yes

characteristic

116

C₄

#16812690

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#20294922

yes

characteristic

	subgroup	G/H
center	C₄	#16812670
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×#108×C₁₀₉	C₂
socle	C₂×C₁₀₉	#389
prime radical	C₂³×C₁₀₉	C₂×Q₈
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×#108	D₁₀₉
hypocenter	C₁₀₉	#1953
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#16812670

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×#108×C₁₀₉

C₂

socle

C₂×C₁₀₉

#389

prime radical

C₂³×C₁₀₉

C₂×Q₈

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #16812635	C₄	semidirect
61. C₁₀₉	#1953	semidirect

H ≅ G/N

type

9. #16812635

C₄

semidirect

61. C₁₀₉

#1953

semidirect

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

subgroup	index
#20295074	2
C₂×#16812635	2
#20294981	2
#20294797	2
C₂×#108×C₁₀₉	2
#1953	109

subgroup

index

C₂×#108×C₁₀₉

109

Subgroups — #28363820

Conjugacy classes of subgroups

Complements

Maximal subgroups