Group #28363783 — subgroups

Subgroups — #28363783

Order 13952.

200 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 89 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	#20295064	C₂	1	1	yes	maximal, characteristic
2	#20295054	C₂	1	2	yes	maximal
3	C₂×#16812609	C₂	1	1	yes	maximal, characteristic
4	#20294787	C₂	1	2	yes	maximal
5	C₂×#108×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11724784	C₂²	1	1	yes	characteristic
7	#16812658	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	#16812633	C₄	1	2	yes	complemented
10	#16812631	C₄	1	2	yes	complemented
11	#16812628	—	2	4	no	—
12	#16812609	—	2	2	no	—
13	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	#108×C₁₀₉	—	2	2	no	—
16	C₂²×Dic₁₀₉	—	2	1	no	—
17	C₂×#11724784	—	2	1	no	—
18	C₂×#11724784	—	2	1	no	—
19	C₂×#11724784	D₄	1	2	yes	—
20	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
21	C₂²⋊Dic₁₀₉	—	2	2	no	—
22	C₄×Dic₁₀₉	—	2	1	no	—
23	C₄×Dic₁₀₉	—	2	2	no	—
24	C₄×Dic₁₀₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₀₉	—	2	2	no	—
27	M₁₆×C₁₀₉	—	2	2	no	—
28	C₂×C₈×C₁₀₉	—	2	2	no	—
29	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
32	C₂×C₈×C₁₀₉	Q₈	1	2	yes	—
33	C₂³×C₁₀₉	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₂×Q₈	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×Dic₁₀₉	—	2	4	no	—
41	C₂×Dic₁₀₉	—	2	2	no	—
42	C₂×Dic₁₀₉	—	4	2	no	—
43	C₈×C₁₀₉	—	4	2	no	—
44	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
45	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
46	#11724784	—	2	2	no	—
47	C₂²×C₁₀₉	—	2	2	no	—
48	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₀₉	#125	1	1	yes	characteristic
51	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
53	C₄×C₁₀₉	#121	1	1	yes	characteristic
54	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
55	Dic₁₀₉	—	4	2	no	—
56	Dic₁₀₉	—	4	4	no	—
57	C₂×C₁₀₉	—	2	1	no	—
58	C₂×C₁₀₉	—	2	1	no	—
59	C₂×C₁₀₉	#389	1	1	yes	characteristic
60	C₂×C₁₀₉	#444	1	2	yes	—
61	#1946	—	109	1	no	maximal, Sylow
62	C₁₀₉	#1946	1	1	yes	characteristic, complemented, Sylow
63	C₂×#108	D₁₀₉	1	1	yes	characteristic
64	#427	—	109	1	no	—
65	#424	—	109	2	no	—
66	C₂×#97	—	109	1	no	—
67	#339	—	109	2	no	—
68	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
69	C₂²×C₈	—	109	1	no	—
70	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
71	#117	—	109	2	no	—
72	C₂×C₄²	—	109	1	no	—
73	#108	—	2	2	no	—
74	#107	—	109	2	no	—
75	#106	—	109	2	no	—
76	C₄⋊C₈	—	218	4	no	—
77	#97	—	218	2	no	—
78	C₂²×C₄	—	218	1	no	—
79	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
80	M₁₆	—	2	2	no	—
81	M₁₆	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	109	2	no	—
84	C₂×C₈	—	218	1	no	—
85	C₂×C₈	—	218	1	no	—
86	C₂×C₈	Dic₂₁₈	1	2	yes	—
87	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
88	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
89	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
90	C₄⋊C₄	—	109	4	no	—
91	C₂²⋊C₄	—	218	2	no	—
92	C₄²	—	109	2	no	—
93	C₄²	—	218	2	no	—
94	C₄²	—	218	1	no	—
95	C₂³	#13182030	1	1	yes	characteristic
96	C₂×C₄	—	436	2	no	—
97	C₂×C₄	—	218	4	no	—
98	C₂×C₄	—	218	2	no	—
99	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
104	C₂×C₄	#13182028	1	1	yes	characteristic
105	C₈	—	4	2	no	—
106	C₈	—	218	2	no	—
107	C₈	#13182008	1	2	yes	—
108	C₈	#13182008	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	#16812661	1	1	yes	characteristic
111	C₂²	#16812675	1	1	yes	characteristic
112	C₂²	#16812690	1	1	yes	characteristic
113	C₄	—	436	2	no	—
114	C₄	—	436	4	no	—
115	C₄	C₂×#13182008	1	1	yes	characteristic
116	C₄	#16812664	1	1	yes	characteristic
117	C₄	#16812670	1	1	yes	characteristic
118	C₄	#16812698	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#20294922	1	1	yes	characteristic
122	C₂	#20295147	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295064

C₂

yes

maximal, characteristic

#20295054

C₂

yes

maximal

C₂×#16812609

C₂

yes

maximal, characteristic

#20294787

C₂

yes

maximal

C₂×#108×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

characteristic

#16812658

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812633

C₄

yes

complemented

#16812631

C₄

yes

complemented

#16812628

—

#16812609

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#108×C₁₀₉

—

C₂²×Dic₁₀₉

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

D₄

yes

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

Q₈

yes

—

C₂³×C₁₀₉

#41

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

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₀₉

#125

yes

characteristic

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

C₄×C₁₀₉

#121

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#389

yes

characteristic

C₂×C₁₀₉

#444

yes

—

#1946

—

109

maximal, Sylow

C₁₀₉

#1946

yes

characteristic, complemented, Sylow

C₂×#108

D₁₀₉

yes

characteristic

#427

—

109

—

#424

—

109

—

C₂×#97

—

109

—

#339

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

—

109

—

C₂×C₄²

—

109

—

#108

—

#107

—

109

—

#106

—

109

—

C₄⋊C₈

—

218

—

#97

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

Dic₂₁₈

yes

—

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₀₉

yes

—

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

218

—

C₄²

—

218

—

C₂³

#13182030

yes

characteristic

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

100

C₂×C₄

D₄×D₁₀₉

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

102

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

103

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

104

C₂×C₄

#13182028

yes

characteristic

105

C₈

—

106

C₈

—

218

—

107

C₈

#13182008

yes

—

108

C₈

#13182008

yes

—

109

C₂²

—

110

C₂²

#16812661

yes

characteristic

111

C₂²

#16812675

yes

characteristic

112

C₂²

#16812690

yes

characteristic

113

C₄

—

436

—

114

C₄

—

436

—

115

C₄

C₂×#13182008

yes

characteristic

116

C₄

#16812664

yes

characteristic

117

C₄

#16812670

yes

characteristic

118

C₄

#16812698

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#20294922

yes

characteristic

122

C₂

#20295147

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×#108×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₄×D₄
prime radical	C₂³×C₁₀₉	#41
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×#108	D₁₀₉
hypocenter	C₁₀₉	#1946
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×#108×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₄×D₄

prime radical

C₂³×C₁₀₉

#41

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #16812633	C₄	semidirect
10. #16812631	C₄	semidirect
62. C₁₀₉	#1946	semidirect

H ≅ G/N

type

9. #16812633

C₄

semidirect

10. #16812631

C₄

semidirect

62. C₁₀₉

#1946

semidirect

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

subgroup	index
#20295064	2
#20295054	2
C₂×#16812609	2
#20294787	2
C₂×#108×C₁₀₉	2
#1946	109

subgroup

index

C₂×#108×C₁₀₉

109

Subgroups — #28363783

Conjugacy classes of subgroups

Complements

Maximal subgroups