Group #28373572 — subgroups

Subgroups — #28373572

Order 13952.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#20295654	C₂	1	2	yes	maximal
2	#20295433	C₂	1	4	yes	maximal, complemented
3	#20295430	C₂	1	2	yes	maximal, complemented
4	#20295427	C₂	1	2	yes	maximal, complemented
5	#20295069	C₂	1	2	yes	maximal
6	C₂×#16812633	C₂	1	1	yes	maximal, characteristic
7	#20294981	C₂	1	1	yes	maximal, characteristic
8	#427×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#13182013	C₂²	1	1	yes	characteristic
10	#16812699	—	2	2	no	—
11	#16812698	C₂²	1	4	yes	—
12	#16812691	—	2	2	no	—
13	#16812690	C₂²	1	4	yes	—
14	#16812676	—	2	4	no	—
15	#16812658	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
17	#16812633	C₂²	1	2	yes	—
18	#16812633	C₂²	1	2	yes	—
19	#16812632	C₂²	1	4	yes	—
20	#16812630	C₂²	1	4	yes	—
21	#16812609	C₂²	1	2	yes	—
22	C₈×Dic₁₀₉	C₂²	1	2	yes	—
23	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	#117×C₁₀₉	C₂²	1	2	yes	—
25	#107×C₁₀₉	C₂²	1	1	yes	characteristic
26	#107×C₁₀₉	C₂²	1	1	yes	characteristic
27	#106×C₁₀₉	C₂²	1	2	yes	—
28	C₂×Dic₂₁₈	—	2	4	no	—
29	C₂²×Dic₁₀₉	—	2	2	no	—
30	#13182013	Q₈	1	4	yes	complemented
31	C₂×#11724784	C₂³	1	2	yes	—
32	C₄⋊Dic₁₀₉	—	2	2	no	—
33	C₄⋊Dic₁₀₉	—	2	4	no	—
34	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
35	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
36	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
37	#13182008	—	2	4	no	—
38	#13182008	—	2	4	no	—
39	C₂²⋊Dic₁₀₉	D₄	1	2	yes	—
40	C₄×Dic₁₀₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₁₀₉	Q₈	1	4	yes	complemented
43	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
46	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
47	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
48	C₄²×C₁₀₉	—	2	2	no	—
49	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₀₉	—	2	2	no	—
51	C₂×C₄×C₁₀₉	—	2	2	no	—
52	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₁₀₉	—	2	4	no	—
57	C₂×Dic₁₀₉	—	8	2	no	—
58	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
59	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
60	Dic₂₁₈	—	8	4	no	—
61	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
62	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
63	#11724784	C₂×Q₈	1	4	yes	—
64	C₂²×C₁₀₉	—	2	1	no	—
65	C₂²×C₁₀₉	—	2	1	no	—
66	C₂²×C₁₀₉	C₄⋊Q₈	1	2	yes	—
67	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₁₀₉	—	4	2	no	—
69	C₄×C₁₀₉	—	4	2	no	—
70	C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
71	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
72	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
73	Dic₁₀₉	—	2	2	no	—
74	Dic₁₀₉	—	2	2	no	—
75	Dic₁₀₉	—	4	4	no	—
76	C₂×C₁₀₉	—	2	2	no	—
77	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₁₀₉	#576	1	1	yes	characteristic
79	C₂×C₁₀₉	#578	1	1	yes	characteristic
80	#3164	—	109	1	no	maximal, Sylow
81	C₁₀₉	#3164	1	1	yes	characteristic, complemented, Sylow
82	#544	—	109	2	no	—
83	#501	—	109	4	no	—
84	#500	—	109	2	no	—
85	#499	—	109	2	no	—
86	#428	—	109	2	no	—
87	#427	D₁₀₉	1	1	yes	characteristic
88	C₂×#107	—	109	1	no	—
89	#405	—	109	1	no	—
90	C₂×M₁₆	—	109	1	no	—
91	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
92	C₄⋊Q₈	—	109	4	no	—
93	C₄⋊Q₈	—	218	2	no	—
94	#125	—	109	4	no	—
95	#125	—	218	2	no	—
96	C₂²⋊Q₈	—	218	4	no	—
97	#117	—	109	1	no	—
98	#117	C₂×D₁₀₉	1	2	yes	—
99	C₂×C₄⋊C₄	—	109	2	no	—
100	#107	—	109	2	no	—
101	#107	—	109	2	no	—
102	#107	—	109	4	no	—
103	#107	C₂×D₁₀₉	1	1	yes	characteristic
104	#107	C₂×D₁₀₉	1	1	yes	characteristic
105	#106	—	109	4	no	—
106	#106	C₂×D₁₀₉	1	2	yes	—
107	#97	—	109	2	no	—
108	C₄×C₈	—	109	2	no	—
109	C₂×Q₈	—	218	4	no	—
110	C₂²×C₄	—	218	2	no	—
111	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
112	M₁₆	—	109	4	no	—
113	C₂×C₈	—	109	2	no	—
114	C₂×C₈	Dic₂₁₈	1	4	yes	complemented
115	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
117	C₄⋊C₄	—	109	1	no	—
118	C₄⋊C₄	—	109	1	no	—
119	C₄⋊C₄	—	109	4	no	—
120	C₄⋊C₄	—	218	4	no	—
121	C₄⋊C₄	—	218	4	no	—
122	C₄⋊C₄	—	218	4	no	—
123	C₄⋊C₄	—	218	2	no	—
124	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
125	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	—	109	2	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	109	2	no	—
130	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
131	Q₈	—	872	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	872	2	no	—
135	C₂×C₄	—	109	2	no	—
136	C₂×C₄	—	109	2	no	—
137	C₂×C₄	—	218	4	no	—
138	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
139	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
140	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
141	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
142	C₈	—	109	4	no	—
143	C₈	C₂×Dic₂₁₈	1	2	yes	—
144	C₈	C₂×Dic₂₁₈	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#16812698	1	2	yes	—
148	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
149	C₄	—	4	2	no	—
150	C₄	—	4	2	no	—
151	C₄	—	436	4	no	—
152	C₄	—	218	2	no	—
153	C₄	—	218	2	no	—
154	C₄	#16812698	1	2	yes	—
155	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
156	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	C₂×#16812698	1	1	yes	characteristic
159	C₂	#20295828	1	1	yes	characteristic
160	C₂	#20295840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295654

C₂

yes

maximal

#20295433

C₂

yes

maximal, complemented

#20295430

C₂

yes

maximal, complemented

#20295427

C₂

yes

maximal, complemented

#20295069

C₂

yes

maximal

C₂×#16812633

C₂

yes

maximal, characteristic

#20294981

C₂

yes

maximal, characteristic

#427×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

characteristic

#16812699

—

#16812698

C₂²

yes

—

#16812691

—

#16812690

C₂²

yes

—

#16812676

—

#16812658

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

#16812633

C₂²

yes

—

#16812633

C₂²

yes

—

#16812632

C₂²

yes

—

#16812630

C₂²

yes

—

#16812609

C₂²

yes

—

C₈×Dic₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₂²

yes

—

#107×C₁₀₉

C₂²

yes

characteristic

#107×C₁₀₉

C₂²

yes

characteristic

#106×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

—

#13182013

Q₈

yes

complemented

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

D₄

yes

characteristic

C₄⋊Dic₁₀₉

D₄

yes

characteristic

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

—

#13182008

—

C₂²⋊Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

Q₈

yes

complemented

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

—

Dic₂₁₈

—

C₈×C₁₀₉

C₂×Q₈

yes

—

C₈×C₁₀₉

C₂×Q₈

yes

—

#11724784

C₂×Q₈

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄⋊Q₈

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄⋊Q₈

yes

—

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄⋊Q₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#3164

yes

characteristic, complemented, Sylow

#544

—

109

—

#501

—

109

—

#500

—

109

—

#499

—

109

—

#428

—

109

—

#427

D₁₀₉

yes

characteristic

C₂×#107

—

109

—

#405

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

218

—

#125

—

109

—

#125

—

218

—

C₂²⋊Q₈

—

218

—

#117

—

109

—

#117

C₂×D₁₀₉

yes

—

C₂×C₄⋊C₄

—

109

—

100

#107

—

109

—

101

#107

—

109

—

102

#107

—

109

—

103

#107

C₂×D₁₀₉

yes

characteristic

104

#107

C₂×D₁₀₉

yes

characteristic

105

#106

—

109

—

106

#106

C₂×D₁₀₉

yes

—

107

#97

—

109

—

108

C₄×C₈

—

109

—

109

C₂×Q₈

—

218

—

110

C₂²×C₄

—

218

—

111

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

112

M₁₆

—

109

—

113

C₂×C₈

—

109

—

114

C₂×C₈

Dic₂₁₈

yes

complemented

115

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

116

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

117

C₄⋊C₄

—

109

—

118

C₄⋊C₄

—

109

—

119

C₄⋊C₄

—

109

—

120

C₄⋊C₄

—

218

—

121

C₄⋊C₄

—

218

—

122

C₄⋊C₄

—

218

—

123

C₄⋊C₄

—

218

—

124

C₄⋊C₄

C₂²×D₁₀₉

yes

—

125

C₄⋊C₄

C₂²×D₁₀₉

yes

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

109

—

128

C₄²

—

129

C₄²

—

109

—

130

C₂³

D₄×D₁₀₉

yes

characteristic

131

Q₈

—

872

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

872

—

135

C₂×C₄

—

109

—

136

C₂×C₄

—

109

—

137

C₂×C₄

—

218

—

138

C₂×C₄

D₄×D₁₀₉

yes

characteristic

139

C₂×C₄

Q₈×D₁₀₉

yes

—

140

C₂×C₄

C₂×Dic₂₁₈

yes

—

141

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

142

C₈

—

109

—

143

C₈

C₂×Dic₂₁₈

yes

—

144

C₈

C₂×Dic₂₁₈

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

#16812698

yes

—

148

C₂²

C₂×D₄×D₁₀₉

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

436

—

152

C₄

—

218

—

153

C₄

—

218

—

154

C₄

#16812698

yes

—

155

C₄

C₂×Q₈×D₁₀₉

yes

characteristic

156

C₄

C₂²×Dic₂₁₈

yes

characteristic

157

C₂

—

158

C₂

C₂×#16812698

yes

characteristic

159

C₂

#20295828

yes

characteristic

160

C₂

#20295840

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#427×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	#427	D₁₀₉
hypocenter	C₁₀₉	#3164
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#427×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

prime radical

C₂³×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #20295433	C₂	semidirect
3. #20295430	C₂	semidirect
4. #20295427	C₂	semidirect
30. #13182013	Q₈	semidirect
42. C₂×C₈×C₁₀₉	Q₈	semidirect
81. C₁₀₉	#3164	semidirect
114. C₂×C₈	Dic₂₁₈	semidirect

H ≅ G/N

type

2. #20295433

C₂

semidirect

3. #20295430

C₂

semidirect

4. #20295427

C₂

semidirect

30. #13182013

Q₈

semidirect

42. C₂×C₈×C₁₀₉

Q₈

semidirect

81. C₁₀₉

#3164

semidirect

114. C₂×C₈

Dic₂₁₈

semidirect

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

subgroup	index
#20295654	2
#20295433	2
#20295430	2
#20295427	2
#20295069	2
C₂×#16812633	2
#20294981	2
#427×C₁₀₉	2
#3164	109

subgroup

index

#427×C₁₀₉

109

Subgroups — #28373572

Conjugacy classes of subgroups

Complements

Maximal subgroups