Group #28364966 — subgroups

Subgroups — #28364966

Order 13952.

216 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 61 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×#16812645	C₂	1	2	yes	maximal, complemented
2	#20295267	C₂	1	1	yes	maximal, characteristic
3	#20294830	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#13182004	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
7	#16812678	—	2	1	no	—
8	#16812645	—	2	4	no	—
9	C₂×#13182004	C₂²	1	2	yes	—
10	#16812633	C₂²	1	1	yes	characteristic
11	#16812624	—	2	1	no	—
12	C₂²×Q₈×C₁₀₉	—	2	1	no	—
13	C₂×Q₁₆×C₁₀₉	—	2	4	no	—
14	C₂×Q₁₆×C₁₀₉	—	2	1	no	—
15	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₁₈	—	2	1	no	—
18	Dic₄₃₆	—	2	2	no	—
19	C₄⋊Dic₁₀₉	—	2	1	no	—
20	#13182008	—	4	1	no	—
21	C₂²⋊Dic₁₀₉	—	4	1	no	—
22	#13182004	—	2	2	no	—
23	C₂×Q₈×C₁₀₉	—	2	4	no	—
24	C₂×Q₈×C₁₀₉	—	2	2	no	—
25	C₂×Q₈×C₁₀₉	—	4	2	no	—
26	C₂×Q₈×C₁₀₉	—	4	1	no	—
27	C₂²×C₄×C₁₀₉	—	4	1	no	—
28	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
29	Q₁₆×C₁₀₉	—	2	4	no	—
30	Q₁₆×C₁₀₉	—	2	2	no	—
31	Q₁₆×C₁₀₉	D₄	1	4	yes	—
32	C₂×C₈×C₁₀₉	—	2	2	no	—
33	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₂³×C₁₀₉	D₈	1	1	yes	characteristic
36	Q₈×C₁₀₉	—	2	4	no	—
37	Q₈×C₁₀₉	—	4	4	no	—
38	Q₈×C₁₀₉	—	4	2	no	—
39	C₂×C₄×C₁₀₉	—	2	1	no	—
40	C₂×C₄×C₁₀₉	—	2	1	no	—
41	C₂×C₄×C₁₀₉	—	4	4	no	—
42	C₂×C₄×C₁₀₉	—	4	2	no	—
43	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₁₀₉	—	4	1	no	—
46	C₂×Dic₁₀₉	—	4	1	no	—
47	Dic₂₁₈	—	4	1	no	—
48	Dic₂₁₈	—	4	1	no	—
49	C₈×C₁₀₉	—	2	1	no	—
50	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
51	C₂²×C₁₀₉	—	2	1	no	—
52	C₂²×C₁₀₉	—	2	1	no	—
53	C₂²×C₁₀₉	Q₃₂	1	2	yes	complemented
54	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
55	C₄×C₁₀₉	—	2	1	no	—
56	C₄×C₁₀₉	—	4	4	no	—
57	C₄×C₁₀₉	#120	1	1	yes	characteristic
58	C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
59	Dic₁₀₉	—	8	1	no	—
60	Dic₁₀₉	—	8	1	no	—
61	C₂×C₁₀₉	—	2	2	no	—
62	C₂×C₁₀₉	#447	1	1	yes	characteristic
63	C₂×C₁₀₉	C₂×Q₃₂	1	1	yes	characteristic
64	C₂×C₁₀₉	#510	1	1	yes	characteristic
65	#2188	—	109	1	no	maximal, Sylow
66	C₁₀₉	#2188	1	1	yes	characteristic, complemented, Sylow
67	C₂²×Q₁₆	D₁₀₉	1	1	yes	characteristic
68	C₂×Q₃₂	—	109	2	no	—
69	#467	—	109	1	no	—
70	#358	—	109	2	no	—
71	C₂²⋊C₁₆	—	109	1	no	—
72	C₂²×Q₈	—	2	1	no	—
73	C₂×Q₁₆	—	2	4	no	—
74	C₂×Q₁₆	—	2	1	no	—
75	C₂×Q₁₆	—	109	1	no	—
76	C₂×Q₁₆	C₂×D₁₀₉	1	2	yes	—
77	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
78	C₂²⋊Q₈	—	218	1	no	—
79	Q₃₂	—	218	4	no	—
80	C₂×C₁₆	—	109	2	no	—
81	#107	—	109	1	no	—
82	#103	—	218	1	no	—
83	C₂×Q₈	—	2	4	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	4	1	no	—
87	C₂×Q₈	—	218	1	no	—
88	C₂²×C₄	—	4	1	no	—
89	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
90	Q₁₆	—	2	4	no	—
91	Q₁₆	—	2	2	no	—
92	Q₁₆	—	218	2	no	—
93	Q₁₆	C₂²⋊D₁₀₉	1	4	yes	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
96	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
97	C₄⋊C₄	—	436	1	no	—
98	C₄⋊C₄	—	218	1	no	—
99	C₂²⋊C₄	—	436	1	no	—
100	C₁₆	—	218	2	no	—
101	C₂³	#13182014	1	1	yes	characteristic
102	Q₈	—	2	4	no	—
103	Q₈	—	4	4	no	—
104	Q₈	—	4	2	no	—
105	Q₈	—	436	1	no	—
106	Q₈	—	436	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	4	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	436	1	no	—
112	C₂×C₄	—	436	1	no	—
113	C₂×C₄	#13182014	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
115	C₈	—	2	1	no	—
116	C₈	C₂×C₂²⋊D₁₀₉	1	2	yes	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#16812645	1	2	yes	complemented
120	C₂²	C₂×#13182014	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	4	4	no	—
123	C₄	—	872	1	no	—
124	C₄	—	872	1	no	—
125	C₄	#16812668	1	1	yes	characteristic
126	C₄	#16812725	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	#20295160	1	1	yes	characteristic
129	C₂	C₂×#16812645	1	1	yes	characteristic
130	C₂	#20295467	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812645

C₂

yes

maximal, complemented

#20295267

C₂

yes

maximal, characteristic

#20294830

C₂

yes

maximal, complemented

C₂²⋊#13182004

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂×Dic₄₃₆

C₂²

yes

—

—

C₂²

yes

—

#16812633

C₂²

yes

characteristic

#16812624

—

C₂²×Q₈×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

Dic₄₃₆

—

C₄⋊Dic₁₀₉

—

#13182008

—

C₂²⋊Dic₁₀₉

—

#13182004

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

D₈

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

D₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×D₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

Q₃₂

yes

complemented

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#447

yes

characteristic

C₂×C₁₀₉

C₂×Q₃₂

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#2188

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

D₁₀₉

yes

characteristic

C₂×Q₃₂

—

109

—

#467

—

109

—

#358

—

109

—

C₂²⋊C₁₆

—

109

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

109

—

C₂×Q₁₆

C₂×D₁₀₉

yes

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂²⋊Q₈

—

218

—

Q₃₂

—

218

—

C₂×C₁₆

—

109

—

#107

—

109

—

#103

—

218

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

218

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

218

—

Q₁₆

C₂²⋊D₁₀₉

yes

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

436

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

436

—

100

C₁₆

—

218

—

101

C₂³

#13182014

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

105

Q₈

—

436

—

106

Q₈

—

436

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

436

—

112

C₂×C₄

—

436

—

113

C₂×C₄

#13182014

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

115

C₈

—

116

C₈

C₂×C₂²⋊D₁₀₉

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

#16812645

yes

complemented

120

C₂²

C₂×#13182014

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

872

—

124

C₄

—

872

—

125

C₄

#16812668

yes

characteristic

126

C₄

#16812725

yes

characteristic

127

C₂

—

128

C₂

#20295160

yes

characteristic

129

C₂

C₂×#16812645

yes

characteristic

130

C₂

#20295467

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#13182014
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	C₂²×Q₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₈
prime radical	C₂³×C₁₀₉	D₈
hypercenter	C₂²×Q₁₆	D₁₀₉
hypocenter	C₁₀₉	#2188
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

C₂×#13182014

commutator, cosocle

C₂×C₈×C₁₀₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

C₂²×Q₁₆×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₈

prime radical

C₂³×C₁₀₉

D₈

hypercenter

C₂²×Q₁₆

D₁₀₉

hypocenter

C₁₀₉

#2188

LCS

G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
1. C₂×#16812645	C₂	semidirect
3. #20294830	C₂	semidirect
53. C₂²×C₁₀₉	Q₃₂	semidirect
66. C₁₀₉	#2188	semidirect
119. C₂²	#16812645	semidirect

H ≅ G/N

type

1. C₂×#16812645

C₂

semidirect

3. #20294830

C₂

semidirect

53. C₂²×C₁₀₉

Q₃₂

semidirect

66. C₁₀₉

#2188

semidirect

119. C₂²

#16812645

semidirect

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

subgroup	index
C₂×#16812645	2
#20295267	2
#20294830	2
C₂²⋊#13182004	2
C₂²×Q₁₆×C₁₀₉	2
#2188	109

subgroup

index

C₂²×Q₁₆×C₁₀₉

109

Subgroups — #28364966

Conjugacy classes of subgroups

Complements

Maximal subgroups