Group #28362749 — subgroups

Subgroups — #28362749

Order 13952.

424 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 205 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	2	yes	maximal
2	#20294789	C₂	1	4	yes	maximal, complemented
3	C₂³×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812658	—	2	4	no	—
5	#16812658	C₄	1	8	yes	complemented
6	C₂×C₄⋊Dic₁₀₉	C₂²	1	4	yes	—
7	C₂×C₂²⋊Dic₁₀₉	—	2	2	no	—
8	C₂×C₄×Dic₁₀₉	—	2	2	no	—
9	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₀₉	—	2	4	no	—
11	C₂²×C₈×C₁₀₉	C₄	1	4	yes	complemented
12	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
13	C₂²×Dic₁₀₉	—	2	4	no	—
14	C₄⋊Dic₁₀₉	—	2	4	no	—
15	C₄⋊Dic₁₀₉	C₂×C₄	1	8	yes	—
16	C₂²⋊Dic₁₀₉	—	2	8	no	—
17	C₄×Dic₁₀₉	—	2	8	no	—
18	C₂⁴×C₁₀₉	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₉	D₄	1	4	yes	—
23	C₂²×C₄×C₁₀₉	Q₈	1	2	yes	—
24	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₀₉	—	2	8	no	—
26	C₂×C₈×C₁₀₉	—	2	4	no	—
27	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
28	C₂³×C₁₀₉	—	2	4	no	—
29	C₂³×C₁₀₉	C₂²⋊C₄	1	4	yes	—
30	C₂³×C₁₀₉	C₄⋊C₄	1	2	yes	—
31	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₄²	1	4	yes	—
33	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
34	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₁₀₉	C₄⋊C₄	1	8	yes	—
36	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₁₀₉	—	2	8	no	—
42	C₂×Dic₁₀₉	—	4	8	no	—
43	C₈×C₁₀₉	—	2	4	no	—
44	C₂²×C₁₀₉	—	2	8	no	—
45	C₂²×C₁₀₉	—	2	4	no	—
46	C₂²×C₁₀₉	#95	1	4	yes	—
47	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	2	yes	—
48	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₀₉	#135	1	4	yes	—
50	C₄×C₁₀₉	#95	1	4	yes	—
51	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
52	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
54	Dic₁₀₉	—	4	8	no	—
55	C₂×C₁₀₉	—	2	4	no	—
56	C₂×C₁₀₉	C₂×#95	1	1	yes	characteristic
57	C₂×C₁₀₉	#416	1	4	yes	—
58	C₂×C₁₀₉	#427	1	2	yes	—
59	#1735	—	109	1	no	maximal, Sylow
60	C₁₀₉	#1735	1	1	yes	characteristic, complemented, Sylow
61	C₂³×C₈	D₁₀₉	1	1	yes	characteristic
62	C₂×#117	—	109	2	no	—
63	#340	—	109	4	no	—
64	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
65	C₂²×C₈	—	2	4	no	—
66	C₂²×C₈	Dic₁₀₉	1	4	yes	complemented
67	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
68	#117	—	109	8	no	—
69	#117	—	218	4	no	—
70	C₂×C₄⋊C₄	—	109	4	no	—
71	C₂×C₂²⋊C₄	—	218	2	no	—
72	C₂×C₄²	—	218	2	no	—
73	C₂⁴	D₄₃₆	1	1	yes	characteristic
74	C₂²×C₄	—	218	4	no	—
75	C₂²×C₄	Dic₂₁₈	1	2	yes	—
76	C₂²×C₄	C₄×D₁₀₉	1	4	yes	—
77	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
79	C₂²×C₄	C₂²⋊D₁₀₉	1	4	yes	—
80	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
81	C₂×C₈	—	2	8	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	C₂×Dic₁₀₉	1	4	yes	—
84	C₄⋊C₄	—	109	8	no	—
85	C₄⋊C₄	—	218	4	no	—
86	C₂²⋊C₄	—	218	8	no	—
87	C₄²	—	218	8	no	—
88	C₂³	—	2	4	no	—
89	C₂³	#13182006	1	4	yes	—
90	C₂³	C₄⋊Dic₁₀₉	1	2	yes	—
91	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
92	C₂×C₄	—	436	8	no	—
93	C₂×C₄	—	218	8	no	—
94	C₂×C₄	C₄×Dic₁₀₉	1	4	yes	—
95	C₂×C₄	#13182006	1	4	yes	—
96	C₂×C₄	C₂²⋊Dic₁₀₉	1	4	yes	—
97	C₂×C₄	#13182008	1	8	yes	—
98	C₂×C₄	C₄⋊Dic₁₀₉	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
100	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
102	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₂²	—	2	8	no	—
105	C₂²	—	2	4	no	—
106	C₂²	#16812606	1	4	yes	—
107	C₂²	C₂×#13182006	1	2	yes	—
108	C₂²	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
109	C₂²	#16812721	1	4	yes	—
110	C₄	—	436	8	no	—
111	C₄	#16812606	1	4	yes	—
112	C₄	C₂×C₄×Dic₁₀₉	1	1	yes	characteristic
113	C₄	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
114	C₄	C₂×#13182008	1	2	yes	—
115	C₂	—	2	4	no	—
116	C₂	C₂×#16812606	1	1	yes	characteristic
117	C₂	#20295023	1	4	yes	—
118	C₂	#20295064	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812658

C₂

yes

maximal

#20294789

C₂

yes

maximal, complemented

C₂³×C₈×C₁₀₉

C₂

yes

maximal, characteristic

#16812658

—

#16812658

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₉

—

C₂×C₄×Dic₁₀₉

—

C₂³×C₄×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

—

C₂²×C₈×C₁₀₉

C₄

yes

complemented

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂⁴×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂²⋊C₄

yes

—

C₂³×C₁₀₉

C₄⋊C₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄²

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#95

yes

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#135

yes

—

C₄×C₁₀₉

#95

yes

—

C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#95

yes

characteristic

C₂×C₁₀₉

#416

yes

—

C₂×C₁₀₉

#427

yes

—

#1735

—

109

maximal, Sylow

C₁₀₉

#1735

yes

characteristic, complemented, Sylow

C₂³×C₈

D₁₀₉

yes

characteristic

C₂×#117

—

109

—

#340

—

109

—

C₂³×C₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

Dic₁₀₉

yes

complemented

C₂²×C₈

C₂×D₁₀₉

yes

—

#117

—

109

—

#117

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

218

—

C₂×C₄²

—

218

—

C₂⁴

D₄₃₆

yes

characteristic

C₂²×C₄

—

218

—

C₂²×C₄

Dic₂₁₈

yes

—

C₂²×C₄

C₄×D₁₀₉

yes

—

C₂²×C₄

D₄₃₆

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₀₉

yes

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

218

—

C₂³

—

C₂³

#13182006

yes

—

C₂³

C₄⋊Dic₁₀₉

yes

—

C₂³

C₂×D₄₃₆

yes

characteristic

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

C₄×Dic₁₀₉

yes

—

C₂×C₄

#13182006

yes

—

C₂×C₄

C₂²⋊Dic₁₀₉

yes

—

C₂×C₄

#13182008

yes

—

C₂×C₄

C₄⋊Dic₁₀₉

yes

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

100

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

102

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

103

C₈

—

104

C₂²

—

105

C₂²

—

106

C₂²

#16812606

yes

—

107

C₂²

C₂×#13182006

yes

—

108

C₂²

C₂×C₄⋊Dic₁₀₉

yes

characteristic

109

C₂²

#16812721

yes

—

110

C₄

—

436

—

111

C₄

#16812606

yes

—

112

C₄

C₂×C₄×Dic₁₀₉

yes

characteristic

113

C₄

C₂×C₂²⋊Dic₁₀₉

yes

characteristic

114

C₄

C₂×#13182008

yes

—

115

C₂

—

116

C₂

C₂×#16812606

yes

characteristic

117

C₂

#20295023

yes

—

118

C₂

#20295064

yes

—

	subgroup	G/H
center	C₂²×C₄	D₄₃₆
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂³×C₈×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂×D₄
prime radical	C₂⁴×C₁₀₉	D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂³×C₈	D₁₀₉
hypocenter	C₁₀₉	#1735
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂²×C₄

D₄₃₆

commutator

C₄×C₁₀₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂³×C₈×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂×D₄

prime radical

C₂⁴×C₁₀₉

D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₂³×C₈

D₁₀₉

hypocenter

C₁₀₉

#1735

LCS

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
2. #20294789	C₂	semidirect
5. #16812658	C₄	semidirect
11. C₂²×C₈×C₁₀₉	C₄	semidirect
60. C₁₀₉	#1735	semidirect
66. C₂²×C₈	Dic₁₀₉	semidirect

H ≅ G/N

type

2. #20294789

C₂

semidirect

5. #16812658

C₄

semidirect

11. C₂²×C₈×C₁₀₉

C₄

semidirect

60. C₁₀₉

#1735

semidirect

66. C₂²×C₈

Dic₁₀₉

semidirect

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

subgroup	index
C₂×#16812658	2
#20294789	2
C₂³×C₈×C₁₀₉	2
#1735	109

subgroup

index

C₂×#16812658

#20294789

C₂³×C₈×C₁₀₉

#1735

109

Subgroups — #28362749

Conjugacy classes of subgroups

Complements

Maximal subgroups