Group #28363078 — subgroups

Subgroups — #28363078

Order 13952.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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	2	yes	maximal, complemented
2	#20295000	C₂	1	2	yes	maximal
3	#20294783	C₂	1	2	yes	maximal, complemented
4	C₂×#131×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#16812658	C₂²	1	4	yes	—
6	#16812633	—	2	2	no	—
7	#16812631	—	2	2	no	—
8	#16812614	—	2	4	no	—
9	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
10	#131×C₁₀₉	—	2	2	no	—
11	#131×C₁₀₉	—	2	2	no	—
12	C₂×M₁₆×C₁₀₉	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₀₉	C₄	1	2	yes	complemented
15	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₀₉	—	2	4	no	—
17	C₂²⋊Dic₁₀₉	—	2	4	no	—
18	C₄×Dic₁₀₉	—	2	4	no	—
19	#41×C₁₀₉	—	2	2	no	—
20	#41×C₁₀₉	—	2	2	no	—
21	C₂×Q₈×C₁₀₉	Q₈	1	1	yes	characteristic
22	C₂×D₄×C₁₀₉	D₄	1	2	yes	—
23	C₂×D₄×C₁₀₉	Q₈	1	1	yes	characteristic
24	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₀₉	—	2	2	no	—
27	M₁₆×C₁₀₉	—	2	4	no	—
28	C₂×C₈×C₁₀₉	—	2	2	no	—
29	C₂×C₈×C₁₀₉	—	2	2	no	—
30	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
34	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
35	C₂³×C₁₀₉	C₄⋊C₄	1	2	yes	—
36	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₁₀₉	—	2	2	no	—
38	D₄×C₁₀₉	—	2	4	no	—
39	D₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄×C₁₀₉	—	2	2	no	—
41	C₂×C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	#41	1	2	yes	—
47	C₂×Dic₁₀₉	—	4	4	no	—
48	C₈×C₁₀₉	—	2	2	no	—
49	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₁₀₉	—	2	2	no	—
53	C₂²×C₁₀₉	—	4	1	no	—
54	C₂²×C₁₀₉	—	4	2	no	—
55	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₀₉	#121	1	2	yes	—
57	C₄×C₁₀₉	—	2	2	no	—
58	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₁₀₉	#117	1	1	yes	characteristic
60	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
61	Dic₁₀₉	—	8	4	no	—
62	C₂×C₁₀₉	—	2	2	no	—
63	C₂×C₁₀₉	—	2	1	no	—
64	C₂×C₁₀₉	—	4	2	no	—
65	C₂×C₁₀₉	#380	1	1	yes	characteristic
66	#1812	—	109	1	no	maximal, Sylow
67	C₁₀₉	#1812	1	1	yes	characteristic, complemented, Sylow
68	C₂×#131	D₁₀₉	1	1	yes	characteristic
69	#427	—	109	2	no	—
70	#410	—	109	2	no	—
71	H(3,ℤ/4)	—	109	2	no	—
72	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
73	#131	—	2	2	no	—
74	#131	—	2	2	no	—
75	C₂×M₁₆	Dic₁₀₉	1	2	yes	complemented
76	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
77	C₂²×C₈	Dic₁₀₉	1	2	yes	complemented
78	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
79	#117	—	109	4	no	—
80	#107	—	218	2	no	—
81	#106	—	218	2	no	—
82	#99	—	218	4	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	Dic₂₁₈	1	1	yes	characteristic
86	C₂×D₄	Dic₂₁₈	1	1	yes	characteristic
87	C₂×D₄	D₄₃₆	1	2	yes	—
88	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	C₂×Dic₁₀₉	1	2	yes	—
95	C₂×C₈	C₂×Dic₁₀₉	1	1	yes	characteristic
96	C₂×C₈	C₂×Dic₁₀₉	1	1	yes	characteristic
97	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
98	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
99	C₄⋊C₄	—	218	4	no	—
100	C₂²⋊C₄	—	218	4	no	—
101	C₄²	—	218	4	no	—
102	C₂³	C₄⋊Dic₁₀₉	1	2	yes	—
103	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	436	4	no	—
110	C₂×C₄	C₄⋊Dic₁₀₉	1	2	yes	—
111	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
113	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
114	C₂×C₄	#13182029	1	2	yes	—
115	C₈	—	2	2	no	—
116	C₈	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
117	C₈	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
118	C₈	C₂²⋊Dic₁₀₉	1	2	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	—	4	1	no	—
121	C₂²	—	4	2	no	—
122	C₂²	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
123	C₂²	#16812673	1	2	yes	—
124	C₄	—	2	2	no	—
125	C₄	—	872	4	no	—
126	C₄	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
127	C₄	#16812658	1	1	yes	characteristic
128	C₄	#16812678	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	—	2	1	no	—
131	C₂	—	4	2	no	—
132	C₂	#20294886	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295064

C₂

yes

maximal, complemented

#20295000

C₂

yes

maximal

#20294783

C₂

yes

maximal, complemented

C₂×#131×C₁₀₉

C₂

yes

maximal, characteristic

#16812658

C₂²

yes

—

#16812633

—

#16812631

—

#16812614

—

C₂×#41×C₁₀₉

C₂²

yes

characteristic

#131×C₁₀₉

—

#131×C₁₀₉

—

C₂×M₁₆×C₁₀₉

C₄

yes

complemented

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₄

yes

complemented

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

Q₈

yes

characteristic

C₂×D₄×C₁₀₉

D₄

yes

—

C₂×D₄×C₁₀₉

Q₈

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₄⋊C₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₈×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₈×C₁₀₉

C₂²⋊C₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#121

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

#117

yes

characteristic

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#380

yes

characteristic

#1812

—

109

maximal, Sylow

C₁₀₉

#1812

yes

characteristic, complemented, Sylow

C₂×#131

D₁₀₉

yes

characteristic

#427

—

109

—

#410

—

109

—

H(3,ℤ/4)

—

109

—

C₂×#41

C₂×D₁₀₉

yes

characteristic

#131

—

#131

—

C₂×M₁₆

Dic₁₀₉

yes

complemented

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

Dic₁₀₉

yes

complemented

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

—

109

—

#107

—

218

—

#106

—

218

—

#99

—

218

—

#41

—

#41

—

C₂×Q₈

Dic₂₁₈

yes

characteristic

C₂×D₄

Dic₂₁₈

yes

characteristic

C₂×D₄

D₄₃₆

yes

—

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₀₉

yes

—

C₂×C₈

C₂×Dic₁₀₉

yes

characteristic

C₂×C₈

C₂×Dic₁₀₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₀₉

yes

—

C₂×C₈

C₂²⋊D₁₀₉

yes

—

C₄⋊C₄

—

218

—

100

C₂²⋊C₄

—

218

—

101

C₄²

—

218

—

102

C₂³

C₄⋊Dic₁₀₉

yes

—

103

C₂³

C₂×D₄₃₆

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

436

—

110

C₂×C₄

C₄⋊Dic₁₀₉

yes

—

111

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

113

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

114

C₂×C₄

#13182029

yes

—

115

C₈

—

116

C₈

C₂²⋊Dic₁₀₉

yes

characteristic

117

C₈

C₂²⋊Dic₁₀₉

yes

characteristic

118

C₈

C₂²⋊Dic₁₀₉

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×C₄⋊Dic₁₀₉

yes

characteristic

123

C₂²

#16812673

yes

—

124

C₄

—

125

C₄

—

872

—

126

C₄

C₂×C₂²⋊Dic₁₀₉

yes

characteristic

127

C₄

#16812658

yes

characteristic

128

C₄

#16812678

yes

—

129

C₂

—

130

C₂

—

131

C₂

—

132

C₂

#20294886

yes

characteristic

	subgroup	G/H
center	C₄	#16812658
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×#131×C₁₀₉	C₂
socle	C₂×C₁₀₉	#380
prime radical	C₂×#41×C₁₀₉	C₂²
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×#131	D₁₀₉
hypocenter	C₁₀₉	#1812
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

subgroup

G/H

center

C₄

#16812658

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×#131×C₁₀₉

C₂

socle

C₂×C₁₀₉

#380

prime radical

C₂×#41×C₁₀₉

C₂²

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

N	H ≅ G/N	type
1. #20295064	C₂	semidirect
3. #20294783	C₂	semidirect
12. C₂×M₁₆×C₁₀₉	C₄	semidirect
14. C₂²×C₈×C₁₀₉	C₄	semidirect
67. C₁₀₉	#1812	semidirect
75. C₂×M₁₆	Dic₁₀₉	semidirect
77. C₂²×C₈	Dic₁₀₉	semidirect

H ≅ G/N

type

1. #20295064

C₂

semidirect

3. #20294783

C₂

semidirect

12. C₂×M₁₆×C₁₀₉

C₄

semidirect

14. C₂²×C₈×C₁₀₉

C₄

semidirect

67. C₁₀₉

#1812

semidirect

75. C₂×M₁₆

Dic₁₀₉

semidirect

77. C₂²×C₈

Dic₁₀₉

semidirect

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

subgroup	index
#20295064	2
#20295000	2
#20294783	2
C₂×#131×C₁₀₉	2
#1812	109

subgroup

index

C₂×#131×C₁₀₉

109

Subgroups — #28363078

Conjugacy classes of subgroups

Complements

Maximal subgroups