Group #28371900 — subgroups

Subgroups — #28371900

Order 13952.

456 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 221 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	#20295504	C₂	1	2	yes	maximal
2	#20295133	C₂	1	4	yes	maximal, complemented
3	Q₁₆×Dic₁₀₉	C₂	1	4	yes	maximal, complemented
4	#20295064	C₂	1	1	yes	maximal, characteristic
5	#20295035	C₂	1	2	yes	maximal
6	#20294981	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182013	C₂²	1	1	yes	characteristic
9	Q₈×Dic₁₀₉	—	2	4	no	—
10	Q₈×Dic₁₀₉	C₂²	1	8	yes	—
11	#16812658	—	2	4	no	—
12	#16812658	C₂²	1	2	yes	—
13	#16812658	C₂²	1	1	yes	characteristic
14	#16812633	C₂²	1	2	yes	—
15	#16812631	C₂²	1	2	yes	—
16	#16812623	C₂²	1	8	yes	—
17	#16812609	C₂²	1	2	yes	—
18	C₈×Dic₁₀₉	C₂²	1	2	yes	—
19	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
20	C₂×Q₁₆×C₁₀₉	C₄	1	8	yes	complemented
21	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
22	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
23	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	#13182013	—	2	2	no	—
25	C₂×#11724784	C₂³	1	2	yes	—
26	C₄⋊Dic₁₀₉	—	2	8	no	—
27	C₄⋊Dic₁₀₉	D₄	1	2	yes	—
28	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
29	C₂²⋊Dic₁₀₉	—	2	2	no	—
30	C₂²⋊Dic₁₀₉	D₄	1	2	yes	—
31	C₄×Dic₁₀₉	—	2	2	no	—
32	C₄×Dic₁₀₉	—	2	8	no	—
33	C₄×Dic₁₀₉	C₂³	1	2	yes	—
34	C₂×Q₈×C₁₀₉	—	2	2	no	—
35	C₂×Q₈×C₁₀₉	—	2	4	no	—
36	C₂×Q₈×C₁₀₉	C₂×C₄	1	8	yes	—
37	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
38	C₂²×C₄×C₁₀₉	—	2	2	no	—
39	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
40	Q₁₆×C₁₀₉	C₂×C₄	1	8	yes	—
41	Q₁₆×C₁₀₉	C₂×C₄	1	8	yes	—
42	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
43	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₁₀₉	—	2	8	no	—
47	Q₈×C₁₀₉	—	2	4	no	—
48	Q₈×C₁₀₉	C₂²×C₄	1	8	yes	—
49	C₂×C₄×C₁₀₉	—	2	8	no	—
50	C₂×C₄×C₁₀₉	—	2	4	no	—
51	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₉	#41	1	2	yes	—
54	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₁₀₉	—	2	4	no	—
56	C₂×Dic₁₀₉	C₂×D₄	1	4	yes	—
57	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
58	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
59	#11724784	—	2	2	no	—
60	C₂²×C₁₀₉	—	2	2	no	—
61	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
62	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₁₀₉	—	2	8	no	—
64	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
65	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
67	Dic₁₀₉	—	2	4	no	—
68	Dic₁₀₉	—	4	4	no	—
69	C₂×C₁₀₉	—	2	2	no	—
70	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
71	C₂×C₁₀₉	#578	1	2	yes	—
72	#2950	—	109	1	no	maximal, Sylow
73	C₁₀₉	#2950	1	1	yes	characteristic, complemented, Sylow
74	C₂²×Q₁₆	D₁₀₉	1	1	yes	characteristic
75	#519	—	109	2	no	—
76	#441	—	109	4	no	—
77	C₄×Q₁₆	—	109	4	no	—
78	#427	—	109	1	no	—
79	#419	—	109	2	no	—
80	#405	—	109	1	no	—
81	C₂²×Q₈	C₂×D₁₀₉	1	2	yes	—
82	C₂×Q₁₆	Dic₁₀₉	1	8	yes	complemented
83	C₂×Q₁₆	C₂×D₁₀₉	1	2	yes	—
84	C₂×Q₁₆	C₂×D₁₀₉	1	2	yes	—
85	C₂×M₁₆	—	109	1	no	—
86	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
87	C₄×Q₈	—	109	8	no	—
88	C₄×Q₈	—	218	4	no	—
89	#117	—	109	1	no	—
90	#117	—	109	2	no	—
91	#117	—	218	4	no	—
92	#107	—	109	2	no	—
93	#106	—	109	2	no	—
94	#103	—	109	8	no	—
95	#97	—	109	2	no	—
96	C₄×C₈	—	109	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	C₂×Dic₁₀₉	1	8	yes	—
100	C₂×Q₈	C₂²×D₁₀₉	1	4	yes	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
103	Q₁₆	C₂×Dic₁₀₉	1	8	yes	—
104	Q₁₆	C₂×Dic₁₀₉	1	8	yes	—
105	M₁₆	—	218	2	no	—
106	C₂×C₈	—	109	2	no	—
107	C₂×C₈	C₂×Dic₁₀₉	1	4	yes	—
108	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
110	C₄⋊C₄	—	109	2	no	—
111	C₄⋊C₄	—	109	4	no	—
112	C₄⋊C₄	—	218	8	no	—
113	C₂²⋊C₄	—	109	2	no	—
114	C₂²⋊C₄	—	218	2	no	—
115	C₄²	—	109	2	no	—
116	C₄²	—	218	8	no	—
117	C₄²	—	218	2	no	—
118	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
119	Q₈	—	2	8	no	—
120	Q₈	—	2	4	no	—
121	Q₈	C₂²×Dic₁₀₉	1	8	yes	—
122	C₂×C₄	—	2	8	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	109	4	no	—
125	C₂×C₄	—	218	4	no	—
126	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
127	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
128	C₂×C₄	#13182028	1	2	yes	—
129	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
130	C₈	—	218	2	no	—
131	C₈	C₂²×Dic₁₀₉	1	2	yes	—
132	C₈	C₂²×Dic₁₀₉	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	D₄×Dic₁₀₉	1	2	yes	—
135	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
136	C₄	—	2	8	no	—
137	C₄	—	436	4	no	—
138	C₄	—	218	4	no	—
139	C₄	D₄×Dic₁₀₉	1	2	yes	—
140	C₄	C₂³×Dic₁₀₉	1	1	yes	characteristic
141	C₄	C₂×#13182028	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×D₄×Dic₁₀₉	1	1	yes	characteristic
144	C₂	#20295839	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295504

C₂

yes

maximal

#20295133

C₂

yes

maximal, complemented

Q₁₆×Dic₁₀₉

C₂

yes

maximal, complemented

#20295064

C₂

yes

maximal, characteristic

#20295035

C₂

yes

maximal

#20294981

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

characteristic

Q₈×Dic₁₀₉

—

Q₈×Dic₁₀₉

C₂²

yes

—

#16812658

—

#16812658

C₂²

yes

—

#16812658

C₂²

yes

characteristic

#16812633

C₂²

yes

—

#16812631

C₂²

yes

—

#16812623

C₂²

yes

—

#16812609

C₂²

yes

—

C₈×Dic₁₀₉

C₂²

yes

—

C₂²×Q₈×C₁₀₉

C₂²

yes

—

C₂×Q₁₆×C₁₀₉

C₄

yes

complemented

C₂×Q₁₆×C₁₀₉

C₂²

yes

—

C₂×Q₁₆×C₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#13182013

—

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

D₄

yes

—

C₄⋊Dic₁₀₉

C₂³

yes

—

C₂²⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂³

yes

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

C₂×Q₈×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

Q₁₆×C₁₀₉

C₂×C₄

yes

—

Q₁₆×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

—

C₈×C₁₀₉

C₂²×C₄

yes

—

C₈×C₁₀₉

C₂²×C₄

yes

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×D₄

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

—

C₄×C₁₀₉

C₂³×C₄

yes

characteristic

C₄×C₁₀₉

C₂×#41

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄×D₄

yes

characteristic

C₂×C₁₀₉

#578

yes

—

#2950

—

109

maximal, Sylow

C₁₀₉

#2950

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

D₁₀₉

yes

characteristic

#519

—

109

—

#441

—

109

—

C₄×Q₁₆

—

109

—

#427

—

109

—

#419

—

109

—

#405

—

109

—

C₂²×Q₈

C₂×D₁₀₉

yes

—

C₂×Q₁₆

Dic₁₀₉

yes

complemented

C₂×Q₁₆

C₂×D₁₀₉

yes

—

C₂×Q₁₆

C₂×D₁₀₉

yes

—

C₂×M₁₆

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₄×Q₈

—

109

—

C₄×Q₈

—

218

—

#117

—

109

—

#117

—

109

—

#117

—

218

—

#107

—

109

—

#106

—

109

—

#103

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₀₉

yes

—

100

C₂×Q₈

C₂²×D₁₀₉

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

103

Q₁₆

C₂×Dic₁₀₉

yes

—

104

Q₁₆

C₂×Dic₁₀₉

yes

—

105

M₁₆

—

218

—

106

C₂×C₈

—

109

—

107

C₂×C₈

C₂×Dic₁₀₉

yes

—

108

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

109

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

110

C₄⋊C₄

—

109

—

111

C₄⋊C₄

—

109

—

112

C₄⋊C₄

—

218

—

113

C₂²⋊C₄

—

109

—

114

C₂²⋊C₄

—

218

—

115

C₄²

—

109

—

116

C₄²

—

218

—

117

C₄²

—

218

—

118

C₂³

D₄×D₁₀₉

yes

characteristic

119

Q₈

—

120

Q₈

—

121

Q₈

C₂²×Dic₁₀₉

yes

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

109

—

125

C₂×C₄

—

218

—

126

C₂×C₄

C₂²×Dic₁₀₉

yes

—

127

C₂×C₄

D₄×D₁₀₉

yes

characteristic

128

C₂×C₄

#13182028

yes

—

129

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

130

C₈

—

218

—

131

C₈

C₂²×Dic₁₀₉

yes

—

132

C₈

C₂²×Dic₁₀₉

yes

—

133

C₂²

—

134

C₂²

D₄×Dic₁₀₉

yes

—

135

C₂²

C₂×D₄×D₁₀₉

yes

characteristic

136

C₄

—

137

C₄

—

436

—

138

C₄

—

218

—

139

C₄

D₄×Dic₁₀₉

yes

—

140

C₄

C₂³×Dic₁₀₉

yes

characteristic

141

C₄

C₂×#13182028

yes

characteristic

142

C₂

—

143

C₂

C₂×D₄×Dic₁₀₉

yes

characteristic

144

C₂

#20295839

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂²×Q₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂²×Q₁₆	D₁₀₉
hypocenter	C₁₀₉	#2950
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₁₀₉

commutator

C₄×C₁₀₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂²×Q₁₆×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂×C₄×C₁₀₉

C₂⁴

hypercenter

C₂²×Q₁₆

D₁₀₉

hypocenter

C₁₀₉

#2950

LCS

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. #20295133	C₂	semidirect
3. Q₁₆×Dic₁₀₉	C₂	semidirect
20. C₂×Q₁₆×C₁₀₉	C₄	semidirect
73. C₁₀₉	#2950	semidirect
82. C₂×Q₁₆	Dic₁₀₉	semidirect

H ≅ G/N

type

2. #20295133

C₂

semidirect

3. Q₁₆×Dic₁₀₉

C₂

semidirect

20. C₂×Q₁₆×C₁₀₉

C₄

semidirect

73. C₁₀₉

#2950

semidirect

82. C₂×Q₁₆

Dic₁₀₉

semidirect

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

subgroup	index
#20295504	2
#20295133	2
Q₁₆×Dic₁₀₉	2
#20295064	2
#20295035	2
#20294981	2
C₂²×Q₁₆×C₁₀₉	2
#2950	109

subgroup

index

Q₁₆×Dic₁₀₉

C₂²×Q₁₆×C₁₀₉

109

Subgroups — #28371900

Conjugacy classes of subgroups

Complements

Maximal subgroups