Group #28364539 — subgroups

Subgroups — #28364539

Order 13952.

176 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20295068	C₂	1	2	yes	maximal
2	C₂×#16812633	C₂	1	1	yes	maximal, characteristic
3	#20295052	C₂	1	1	yes	maximal, characteristic
4	#20294794	C₂	1	2	yes	maximal
5	#341×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	1	yes	characteristic
7	#16812658	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
9	#16812633	—	2	2	no	—
10	#16812633	—	2	2	no	—
11	#16812631	—	2	2	no	—
12	#16812628	—	2	2	no	—
13	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₉	—	2	2	no	—
16	#13182013	—	2	2	no	—
17	C₂×#11724784	—	2	1	no	—
18	C₄⋊Dic₁₀₉	—	2	2	no	—
19	C₄⋊Dic₁₀₉	—	2	2	no	—
20	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
21	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
22	C₂²⋊Dic₁₀₉	—	2	1	no	—
23	C₄×Dic₁₀₉	—	2	1	no	—
24	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₀₉	—	2	2	no	—
26	M₁₆×C₁₀₉	Q₈	1	4	yes	—
27	C₂×C₈×C₁₀₉	—	2	2	no	—
28	C₂×C₈×C₁₀₉	—	2	2	no	—
29	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
30	C₂³×C₁₀₉	#41	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₁₀₉	#41	1	2	yes	—
35	C₂×Dic₁₀₉	—	2	1	no	—
36	C₂×Dic₁₀₉	—	2	1	no	—
37	C₂×Dic₁₀₉	—	4	2	no	—
38	C₂×Dic₁₀₉	—	8	2	no	—
39	C₈×C₁₀₉	—	2	4	no	—
40	C₈×C₁₀₉	—	2	2	no	—
41	#11724784	—	4	1	no	—
42	C₂²×C₁₀₉	—	2	1	no	—
43	C₂²×C₁₀₉	—	2	1	no	—
44	C₂²×C₁₀₉	#121	1	1	yes	characteristic
45	C₂²×C₁₀₉	#125	1	2	yes	—
46	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₁₀₉	#123	1	1	yes	characteristic
48	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
49	Dic₁₀₉	—	4	1	no	—
50	Dic₁₀₉	—	4	1	no	—
51	Dic₁₀₉	—	8	2	no	—
52	C₂×C₁₀₉	—	2	2	no	—
53	C₂×C₁₀₉	#398	1	1	yes	characteristic
54	C₂×C₁₀₉	#472	1	1	yes	characteristic
55	C₂×C₁₀₉	#473	1	1	yes	characteristic
56	#2089	—	109	1	no	maximal, Sylow
57	C₁₀₉	#2089	1	1	yes	characteristic, complemented, Sylow
58	#428	—	109	2	no	—
59	C₂×#107	—	109	1	no	—
60	#424	—	109	1	no	—
61	#343	—	109	2	no	—
62	#341	D₁₀₉	1	1	yes	characteristic
63	C₂×M₁₆	—	109	1	no	—
64	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
65	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
66	#117	—	109	1	no	—
67	C₂×C₄⋊C₄	—	109	2	no	—
68	#107	—	218	2	no	—
69	#107	—	218	2	no	—
70	#106	—	218	2	no	—
71	C₄⋊C₈	—	218	2	no	—
72	C₂²×C₄	—	218	2	no	—
73	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
74	M₁₆	—	2	2	no	—
75	M₁₆	—	218	2	no	—
76	M₁₆	Dic₂₁₈	1	4	yes	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	218	1	no	—
80	C₂×C₈	D₄₃₆	1	2	yes	—
81	C₄⋊C₄	—	109	1	no	—
82	C₄⋊C₄	—	109	1	no	—
83	C₄⋊C₄	—	218	2	no	—
84	C₄⋊C₄	—	218	2	no	—
85	C₂²⋊C₄	—	218	1	no	—
86	C₄²	—	218	1	no	—
87	C₂³	#13182030	1	1	yes	characteristic
88	C₂×C₄	—	872	2	no	—
89	C₂×C₄	—	436	2	no	—
90	C₂×C₄	—	218	1	no	—
91	C₂×C₄	—	218	1	no	—
92	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
93	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
95	C₂×C₄	#13182028	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	2	2	no	—
98	C₈	—	436	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	#16812674	1	1	yes	characteristic
102	C₂²	#16812690	1	2	yes	—
103	C₄	—	872	2	no	—
104	C₄	—	436	1	no	—
105	C₄	—	436	1	no	—
106	C₄	#16812676	1	2	yes	—
107	C₄	#16812683	1	1	yes	characteristic
108	C₄	#16812699	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#20294958	1	1	yes	characteristic
111	C₂	#20295297	1	1	yes	characteristic
112	C₂	#20295302	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295068

C₂

yes

maximal

C₂×#16812633

C₂

yes

maximal, characteristic

#20295052

C₂

yes

maximal, characteristic

#20294794

C₂

yes

maximal

#341×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

characteristic

#16812658

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

—

—

—

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

—

#13182013

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

D₄

yes

characteristic

C₄⋊Dic₁₀₉

D₄

yes

characteristic

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

Q₈

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

#41

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

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#121

yes

characteristic

C₂²×C₁₀₉

#125

yes

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

#123

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#2089

yes

characteristic, complemented, Sylow

#428

—

109

—

C₂×#107

—

109

—

#424

—

109

—

#343

—

109

—

#341

D₁₀₉

yes

characteristic

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

—

109

—

C₂×C₄⋊C₄

—

109

—

#107

—

218

—

#107

—

218

—

#106

—

218

—

C₄⋊C₈

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

218

—

M₁₆

Dic₂₁₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

218

—

C₂×C₈

D₄₃₆

yes

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

218

—

C₂³

#13182030

yes

characteristic

C₂×C₄

—

872

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

D₄×D₁₀₉

yes

characteristic

C₂×C₄

C₂×D₄₃₆

yes

characteristic

C₂×C₄

C₂×Dic₂₁₈

yes

—

C₂×C₄

#13182028

yes

—

C₈

—

C₈

—

C₈

—

436

—

C₂²

—

100

C₂²

—

101

C₂²

#16812674

yes

characteristic

102

C₂²

#16812690

yes

—

103

C₄

—

872

—

104

C₄

—

436

—

105

C₄

—

436

—

106

C₄

#16812676

yes

—

107

C₄

#16812683

yes

characteristic

108

C₄

#16812699

yes

characteristic

109

C₂

—

110

C₂

#20294958

yes

characteristic

111

C₂

#20295297

yes

characteristic

112

C₂

#20295302

yes

characteristic

	subgroup	G/H
center	C₂²	#16812674
commutator, cosocle	C₂²×C₄×C₁₀₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#341×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#121
prime radical	C₂³×C₁₀₉	#41
hypercenter	#341	D₁₀₉
hypocenter	C₁₀₉	#2089
LCS	G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#16812674

commutator, cosocle

C₂²×C₄×C₁₀₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#341×C₁₀₉

C₂

C₂²×C₁₀₉

C₂³×C₁₀₉

D₁₀₉

C₁₀₉

G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
57. C₁₀₉	#2089	semidirect

H ≅ G/N

type

57. C₁₀₉

#2089

semidirect

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

subgroup	index
#20295068	2
C₂×#16812633	2
#20295052	2
#20294794	2
#341×C₁₀₉	2
#2089	109

subgroup

index

#341×C₁₀₉

109

Subgroups — #28364539

Conjugacy classes of subgroups

Complements

Maximal subgroups