Group #28362888 — subgroups

Subgroups — #28362888

Order 13952.

260 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 121 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×#16812628	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812628	C₂	1	1	yes	maximal, characteristic
3	C₂×#16812609	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#20294872	C₂	1	1	yes	maximal, characteristic
6	#20294781	C₂	1	1	yes	maximal, characteristic
7	#336×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11724784	C₂²	1	1	yes	characteristic
9	C₂²×#11724784	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
12	#16812628	C₄	1	4	yes	complemented
13	#16812628	C₄	1	4	yes	complemented
14	#16812609	—	2	2	no	—
15	C₈×Dic₁₀₉	—	2	2	no	—
16	#16812606	C₄	1	2	yes	—
17	#16812606	C₄	1	2	yes	—
18	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
24	C₂×#11724784	—	2	1	no	—
25	C₂×#11724784	—	2	1	no	—
26	C₂×#11724784	—	2	1	no	—
27	C₂×#11724784	—	2	1	no	—
28	C₂×#11724784	C₂×C₄	1	2	yes	—
29	C₂×#11724784	C₂×C₄	1	2	yes	—
30	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
31	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₀₉	D₄	1	2	yes	—
33	C₄×Dic₁₀₉	Q₈	1	2	yes	—
34	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₁₀₉	—	2	1	no	—
37	C₂×C₈×C₁₀₉	—	2	1	no	—
38	C₂×C₈×C₁₀₉	—	2	1	no	—
39	C₂×C₈×C₁₀₉	—	2	1	no	—
40	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
41	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
42	C₄²×C₁₀₉	—	2	1	no	—
43	C₄²×C₁₀₉	—	2	1	no	—
44	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	—	2	2	no	—
46	C₂×C₄×C₁₀₉	—	2	2	no	—
47	C₂×C₄×C₁₀₉	—	2	2	no	—
48	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
53	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
54	C₂×Dic₁₀₉	—	2	2	no	—
55	C₂×Dic₁₀₉	—	2	2	no	—
56	C₂×Dic₁₀₉	—	2	2	no	—
57	C₂×Dic₁₀₉	—	2	2	no	—
58	C₂×Dic₁₀₉	C₄²	1	2	yes	—
59	C₂×Dic₁₀₉	C₄²	1	2	yes	—
60	C₂×Dic₁₀₉	C₄⋊C₄	1	4	yes	—
61	C₈×C₁₀₉	—	2	2	no	—
62	C₈×C₁₀₉	—	2	2	no	—
63	#11724784	—	2	2	no	—
64	#11724784	—	2	2	no	—
65	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
66	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₁₀₉	#117	1	1	yes	characteristic
68	C₂²×C₁₀₉	#131	1	2	yes	—
69	C₂²×C₁₀₉	#131	1	2	yes	—
70	C₄×C₁₀₉	—	4	2	no	—
71	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
72	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
73	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
74	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
75	Dic₁₀₉	—	2	2	no	—
76	Dic₁₀₉	—	2	2	no	—
77	Dic₁₀₉	—	2	4	no	—
78	C₂×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₁₀₉	#405	1	1	yes	characteristic
80	C₂×C₁₀₉	#405	1	1	yes	characteristic
81	C₂×C₁₀₉	#424	1	2	yes	—
82	C₂×C₁₀₉	#433	1	2	yes	—
83	#1768	—	109	1	no	maximal, Sylow
84	C₁₀₉	#1768	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₄⋊C₈	—	109	1	no	—
86	C₂×C₄⋊C₈	—	109	1	no	—
87	C₂×#97	—	109	1	no	—
88	C₂×C₄×C₈	—	109	1	no	—
89	#376	—	109	1	no	—
90	#336	—	109	1	no	—
91	#336	D₁₀₉	1	1	yes	characteristic
92	C₂²×C₈	—	109	1	no	—
93	C₂²×C₈	—	109	1	no	—
94	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
96	C₂×C₄²	—	109	1	no	—
97	C₂×C₄²	—	109	1	no	—
98	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
99	C₄⋊C₈	—	109	4	no	—
100	C₄⋊C₈	—	109	4	no	—
101	#97	—	218	2	no	—
102	C₄×C₈	—	218	2	no	—
103	#95	—	109	2	no	—
104	#95	—	109	2	no	—
105	C₂²×C₄	—	109	1	no	—
106	C₂²×C₄	—	109	1	no	—
107	C₂²×C₄	—	109	2	no	—
108	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
109	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	109	2	no	—
115	C₂×C₈	—	109	2	no	—
116	C₂×C₈	—	218	1	no	—
117	C₂×C₈	—	218	1	no	—
118	C₂×C₈	—	218	1	no	—
119	C₂×C₈	—	218	1	no	—
120	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
121	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
122	C₄²	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	109	2	no	—
125	C₄²	—	109	2	no	—
126	C₄²	—	109	2	no	—
127	C₄²	—	109	2	no	—
128	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	109	2	no	—
133	C₂×C₄	—	109	2	no	—
134	C₂×C₄	—	109	4	no	—
135	C₂×C₄	—	218	2	no	—
136	C₂×C₄	—	218	2	no	—
137	C₂×C₄	—	218	2	no	—
138	C₂×C₄	—	218	2	no	—
139	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
141	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
142	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
143	C₂×C₄	#13182028	1	1	yes	characteristic
144	C₂×C₄	#13182028	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	—	2	2	no	—
147	C₈	—	218	2	no	—
148	C₈	—	218	2	no	—
149	C₂²	C₄²×D₁₀₉	1	1	yes	characteristic
150	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
151	C₂²	#16812655	1	1	yes	characteristic
152	C₂²	#16812705	1	2	yes	—
153	C₂²	#16812707	1	2	yes	—
154	C₄	—	4	2	no	—
155	C₄	—	218	2	no	—
156	C₄	—	218	2	no	—
157	C₄	—	218	4	no	—
158	C₄	#16812659	1	1	yes	characteristic
159	C₄	#16812659	1	1	yes	characteristic
160	C₄	#16812664	1	1	yes	characteristic
161	C₄	#16812664	1	1	yes	characteristic
162	C₂	#20294877	1	1	yes	characteristic
163	C₂	#20294980	1	1	yes	characteristic
164	C₂	#20294983	1	1	yes	characteristic
165	C₂	#20295051	1	2	yes	—
166	C₂	#20295084	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812628

C₂

yes

maximal, characteristic

C₂×#16812628

C₂

yes

maximal, characteristic

C₂×#16812609

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₀₉

C₂

yes

maximal, characteristic

#20294872

C₂

yes

maximal, characteristic

#20294781

C₂

yes

maximal, characteristic

#336×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

characteristic

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812628

C₄

yes

complemented

#16812628

C₄

yes

complemented

#16812609

—

C₈×Dic₁₀₉

—

#16812606

C₄

yes

—

#16812606

C₄

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄²

yes

—

C₂×Dic₁₀₉

C₄²

yes

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

—

C₂²×C₁₀₉

#131

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

C₄×C₄⋊C₄

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

C₂×C₁₀₉

#433

yes

—

#1768

—

109

maximal, Sylow

C₁₀₉

#1768

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

C₂×C₄⋊C₈

—

109

—

C₂×#97

—

109

—

C₂×C₄×C₈

—

109

—

#376

—

109

—

#336

—

109

—

#336

D₁₀₉

yes

characteristic

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

109

—

100

C₄⋊C₈

—

109

—

101

#97

—

218

—

102

C₄×C₈

—

218

—

103

#95

—

109

—

104

#95

—

109

—

105

C₂²×C₄

—

109

—

106

C₂²×C₄

—

109

—

107

C₂²×C₄

—

109

—

108

C₂²×C₄

C₄×D₁₀₉

yes

—

109

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

109

—

115

C₂×C₈

—

109

—

116

C₂×C₈

—

218

—

117

C₂×C₈

—

218

—

118

C₂×C₈

—

218

—

119

C₂×C₈

—

218

—

120

C₂×C₈

C₄×D₁₀₉

yes

—

121

C₂×C₈

C₄×D₁₀₉

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

109

—

125

C₄²

—

109

—

126

C₄²

—

109

—

127

C₄²

—

109

—

128

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

109

—

133

C₂×C₄

—

109

—

134

C₂×C₄

—

109

—

135

C₂×C₄

—

218

—

136

C₂×C₄

—

218

—

137

C₂×C₄

—

218

—

138

C₂×C₄

—

218

—

139

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

140

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

141

C₂×C₄

D₄×D₁₀₉

yes

characteristic

142

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

143

C₂×C₄

#13182028

yes

characteristic

144

C₂×C₄

#13182028

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

218

—

148

C₈

—

218

—

149

C₂²

C₄²×D₁₀₉

yes

characteristic

150

C₂²

C₄⋊C₄×D₁₀₉

yes

characteristic

151

C₂²

#16812655

yes

characteristic

152

C₂²

#16812705

yes

—

153

C₂²

#16812707

yes

—

154

C₄

—

155

C₄

—

218

—

156

C₄

—

218

—

157

C₄

—

218

—

158

C₄

#16812659

yes

characteristic

159

C₄

#16812659

yes

characteristic

160

C₄

#16812664

yes

characteristic

161

C₄

#16812664

yes

characteristic

162

C₂

#20294877

yes

characteristic

163

C₂

#20294980

yes

characteristic

164

C₂

#20294983

yes

characteristic

165

C₂

#20295051

yes

—

166

C₂

#20295084

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂×C₄²
Fitting	#336×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#336	D₁₀₉
hypocenter	C₁₀₉	#1768
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₉

commutator

C₂²×C₁₀₉

C₂×C₄²

Fitting

#336×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂²×C₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
12. #16812628	C₄	semidirect
13. #16812628	C₄	semidirect
84. C₁₀₉	#1768	semidirect

H ≅ G/N

type

12. #16812628

C₄

semidirect

13. #16812628

C₄

semidirect

84. C₁₀₉

#1768

semidirect

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

subgroup	index
C₂×#16812628	2
C₂×#16812609	2
C₂×C₈×Dic₁₀₉	2
#20294872	2
#20294781	2
#336×C₁₀₉	2
#1768	109

subgroup

index

C₂×C₈×Dic₁₀₉

#336×C₁₀₉

109

Subgroups — #28362888

Conjugacy classes of subgroups

Complements

Maximal subgroups