Group #28363113 — subgroups

Subgroups — #28363113

Order 13952.

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

normal subgroups only

This group has 272 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	C₂×#16812658	C₂	1	1	yes	maximal, characteristic
2	#20294990	C₂	1	1	yes	maximal, characteristic
3	#20294794	C₂	1	4	yes	maximal, complemented
4	#407×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182013	C₂²	1	2	yes	—
6	#16812658	—	2	2	no	—
7	#16812658	—	2	2	no	—
8	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
9	C₂×C₂²⋊Dic₁₀₉	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₀₉	C₄	1	2	yes	—
11	C₂²⋊#11724784	—	2	2	no	—
12	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
15	C₂²×Dic₁₀₉	C₂×C₄	1	4	yes	—
16	#13182013	—	2	4	no	—
17	C₂×#11724784	—	2	2	no	—
18	C₄⋊Dic₁₀₉	—	2	2	no	—
19	C₄⋊Dic₁₀₉	—	2	2	no	—
20	C₂²⋊Dic₁₀₉	—	2	4	no	—
21	C₄×Dic₁₀₉	—	2	4	no	—
22	C₂⁴×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₀₉	—	2	4	no	—
24	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
26	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
27	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₀₉	—	2	4	no	—
29	C₂×C₈×C₁₀₉	—	2	2	no	—
30	C₂³×C₁₀₉	—	2	2	no	—
31	C₂³×C₁₀₉	—	4	2	no	—
32	C₂³×C₁₀₉	C₂²⋊C₄	1	2	yes	—
33	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	—	2	2	no	—
35	C₂×C₄×C₁₀₉	—	2	4	no	—
36	C₂×C₄×C₁₀₉	—	2	4	no	—
37	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	#41	1	4	yes	—
41	C₂×Dic₁₀₉	—	2	4	no	—
42	C₂×Dic₁₀₉	—	4	4	no	—
43	C₈×C₁₀₉	—	4	2	no	—
44	#11724784	—	4	2	no	—
45	C₂²×C₁₀₉	—	2	2	no	—
46	C₂²×C₁₀₉	—	2	2	no	—
47	C₂²×C₁₀₉	—	2	4	no	—
48	C₂²×C₁₀₉	—	4	4	no	—
49	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₁₀₉	#117	1	2	yes	—
51	C₄×C₁₀₉	—	2	2	no	—
52	C₄×C₁₀₉	#123	1	2	yes	—
53	C₄×C₁₀₉	#123	1	2	yes	—
54	Dic₁₀₉	—	4	4	no	—
55	C₂×C₁₀₉	—	2	2	no	—
56	C₂×C₁₀₉	—	4	2	no	—
57	C₂×C₁₀₉	#381	1	1	yes	characteristic
58	C₂×C₁₀₉	#413	1	2	yes	—
59	#1822	—	109	1	no	maximal, Sylow
60	C₁₀₉	#1822	1	1	yes	characteristic, complemented, Sylow
61	C₂×#117	—	109	1	no	—
62	#407	—	109	1	no	—
63	#407	D₁₀₉	1	1	yes	characteristic
64	#343	—	109	4	no	—
65	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
66	C₂×M₁₆	—	109	2	no	—
67	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
68	#117	—	218	2	no	—
69	#117	—	218	2	no	—
70	C₂×C₄⋊C₄	—	109	2	no	—
71	C₂×C₂²⋊C₄	—	109	2	no	—
72	C₂×C₄²	—	109	2	no	—
73	C₂²⋊C₈	—	2	2	no	—
74	C₂²⋊C₈	—	218	2	no	—
75	C₂⁴	C₄×D₁₀₉	1	1	yes	characteristic
76	C₂²×C₄	—	2	4	no	—
77	C₂²×C₄	—	109	4	no	—
78	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
79	C₂²×C₄	D₄₃₆	1	2	yes	—
80	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
82	M₁₆	—	2	4	no	—
83	M₁₆	—	218	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	218	2	no	—
86	C₄⋊C₄	—	218	2	no	—
87	C₄⋊C₄	—	218	2	no	—
88	C₂²⋊C₄	—	218	4	no	—
89	C₄²	—	218	4	no	—
90	C₂³	—	2	2	no	—
91	C₂³	—	4	2	no	—
92	C₂³	#13182006	1	2	yes	—
93	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	436	4	no	—
98	C₂×C₄	—	218	4	no	—
99	C₂×C₄	#13182006	1	2	yes	—
100	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
101	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
102	C₂×C₄	#13182028	1	4	yes	—
103	C₈	—	4	2	no	—
104	C₈	—	436	2	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	4	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	4	4	no	—
109	C₂²	C₂×#13182006	1	1	yes	characteristic
110	C₂²	#16812655	1	2	yes	—
111	C₄	—	2	2	no	—
112	C₄	—	436	4	no	—
113	C₄	#16812680	1	2	yes	—
114	C₄	#16812683	1	2	yes	—
115	C₂	—	2	2	no	—
116	C₂	—	4	2	no	—
117	C₂	#20294887	1	1	yes	characteristic
118	C₂	#20295008	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812658

C₂

yes

maximal, characteristic

#20294990

C₂

yes

maximal, characteristic

#20294794

C₂

yes

maximal, complemented

#407×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

#16812658

—

#16812658

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₉

C₄

yes

—

C₂×C₄×Dic₁₀₉

C₄

yes

—

C₂²⋊#11724784

—

C₂³×C₄×C₁₀₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂²⋊C₈×C₁₀₉

—

C₂²×Dic₁₀₉

C₂×C₄

yes

—

#13182013

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂⁴×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂²⋊C₄

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

#117

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#123

yes

—

C₄×C₁₀₉

#123

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#381

yes

characteristic

C₂×C₁₀₉

#413

yes

—

#1822

—

109

maximal, Sylow

C₁₀₉

#1822

yes

characteristic, complemented, Sylow

C₂×#117

—

109

—

#407

—

109

—

#407

D₁₀₉

yes

characteristic

#343

—

109

—

C₂³×C₄

C₂×D₁₀₉

yes

characteristic

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

#117

—

218

—

#117

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₄²

—

109

—

C₂²⋊C₈

—

C₂²⋊C₈

—

218

—

C₂⁴

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

109

—

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

D₄₃₆

yes

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

218

—

C₂³

—

C₂³

—

C₂³

#13182006

yes

—

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

#13182006

yes

—

100

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

101

C₂×C₄

C₂×D₄₃₆

yes

characteristic

102

C₂×C₄

#13182028

yes

—

103

C₈

—

104

C₈

—

436

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×#13182006

yes

characteristic

110

C₂²

#16812655

yes

—

111

C₄

—

112

C₄

—

436

—

113

C₄

#16812680

yes

—

114

C₄

#16812683

yes

—

115

C₂

—

116

C₂

—

117

C₂

#20294887

yes

characteristic

118

C₂

#20295008

yes

—

	subgroup	G/H
center	C₂²	C₂×#13182006
commutator	C₂³×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#407×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₀₉	C₂×C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#407	D₁₀₉
hypocenter	C₁₀₉	#1822
LCS	G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂×#13182006

commutator

C₂³×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#407×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₁₀₉

C₂×C₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
3. #20294794	C₂	semidirect
60. C₁₀₉	#1822	semidirect

H ≅ G/N

type

3. #20294794

C₂

semidirect

60. C₁₀₉

#1822

semidirect

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

subgroup	index
C₂×#16812658	2
#20294990	2
#20294794	2
#407×C₁₀₉	2
#1822	109

subgroup

index

#407×C₁₀₉

109

Subgroups — #28363113

Conjugacy classes of subgroups

Complements

Maximal subgroups