Group #28379269 — subgroups

Subgroups — #28379269

Order 13952.

220 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 72 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	D₁₀₉⋊M₃₂	C₂	1	2	yes	maximal, complemented
2	#20295923	C₂	1	2	yes	maximal, complemented
3	#20295922	C₂	1	2	yes	maximal, complemented
4	C₂×#16812705	C₂	1	1	yes	maximal, characteristic
5	#16812774	—	2	2	no	—
6	#16812773	C₂²	1	4	yes	—
7	D₁₀₉⋊C₁₆	—	2	2	no	—
8	C₂×#13182029	C₄	1	1	yes	characteristic
9	#16812705	—	2	2	no	—
10	#16812705	—	2	2	no	—
11	C₂×#13182013	C₂²	1	1	yes	characteristic
12	C₂×#13182012	C₄	1	1	yes	characteristic
13	C₂×#13182012	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
15	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₀₉	C₄	1	1	yes	characteristic
17	#13182032	—	2	4	no	—
18	#13182029	—	2	2	no	—
19	#13182029	—	2	2	no	—
20	C₂×Dic₂₁₈	C₈	1	1	yes	characteristic
21	C₂×D₄₃₆	C₈	1	1	yes	characteristic
22	C₂×C₂²⋊D₁₀₉	C₈	1	1	yes	characteristic
23	C₂×C₂²⋊D₁₀₉	C₈	1	1	yes	characteristic
24	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
26	#13182013	—	2	2	no	—
27	#13182012	—	2	2	no	—
28	#13182012	—	2	2	no	—
29	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
30	C₂×#11724784	C₂³	1	1	yes	characteristic
31	C₈×D₁₀₉	—	2	1	no	—
32	C₈×D₁₀₉	—	2	1	no	—
33	C₈×D₁₀₉	D₄	1	2	yes	—
34	C₈×D₁₀₉	D₄	1	2	yes	—
35	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₀₉	—	2	1	no	—
37	C₂×C₈×C₁₀₉	—	2	1	no	—
38	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
42	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	—	2	1	no	—
44	C₂×C₄×C₁₀₉	—	2	1	no	—
45	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	C₂²⋊D₁₀₉	—	2	2	no	—
48	C₂²⋊D₁₀₉	—	2	2	no	—
49	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
50	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
51	D₄₃₆	—	2	2	no	—
52	C₄×D₁₀₉	—	2	1	no	—
53	C₄×D₁₀₉	—	2	1	no	—
54	C₄×D₁₀₉	C₂²⋊C₄	1	2	yes	—
55	C₄×D₁₀₉	M₁₆	1	2	yes	—
56	Dic₂₁₈	—	2	2	no	—
57	C₈×C₁₀₉	—	2	1	no	—
58	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
60	#11724784	—	2	1	no	—
61	#11724784	C₂×D₄	1	2	yes	—
62	C₂²×C₁₀₉	—	2	1	no	—
63	C₂²×C₁₀₉	—	4	1	no	—
64	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
65	C₂×D₁₀₉	—	2	1	no	—
66	C₂×D₁₀₉	—	4	1	no	—
67	C₂×D₁₀₉	—	4	1	no	—
68	C₂×D₁₀₉	C₂²⋊C₈	1	2	yes	—
69	C₄×C₁₀₉	—	2	1	no	—
70	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
72	Dic₁₀₉	—	2	1	no	—
73	Dic₁₀₉	C₂²⋊C₈	1	2	yes	—
74	C₂×C₁₀₉	—	2	1	no	—
75	C₂×C₁₀₉	—	4	1	no	—
76	C₂×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
77	D₁₀₉	—	2	2	no	—
78	D₁₀₉	—	4	1	no	—
79	#2115	—	109	1	no	maximal, Sylow
80	C₁₀₉	#2115	1	1	yes	characteristic, complemented, Sylow
81	C₂×#131	—	109	1	no	—
82	C₂×M₃₂	—	109	2	no	—
83	#349	—	109	2	no	—
84	#349	—	109	2	no	—
85	C₂×#41	—	109	1	no	—
86	#131	—	218	2	no	—
87	#131	—	218	2	no	—
88	C₂×M₁₆	—	109	1	no	—
89	C₂×M₁₆	—	109	1	no	—
90	C₂×M₁₆	—	109	1	no	—
91	C₂²×C₈	—	109	1	no	—
92	C₂²×C₈	—	109	1	no	—
93	C₂²×C₈	#89137	1	1	yes	characteristic
94	M₃₂	—	109	4	no	—
95	M₃₂	—	218	2	no	—
96	C₂×C₁₆	—	218	2	no	—
97	#41	—	218	2	no	—
98	#41	—	218	2	no	—
99	C₂×Q₈	—	109	1	no	—
100	C₂×D₄	—	109	1	no	—
101	C₂×D₄	—	109	1	no	—
102	C₂×D₄	—	109	1	no	—
103	C₂²×C₄	—	109	1	no	—
104	C₂²×C₄	—	109	1	no	—
105	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
106	M₁₆	—	218	2	no	—
107	M₁₆	—	218	2	no	—
108	M₁₆	—	218	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	109	1	no	—
112	C₂×C₈	—	109	2	no	—
113	C₂×C₈	—	109	2	no	—
114	C₂×C₈	—	109	1	no	—
115	C₂×C₈	—	218	1	no	—
116	C₂×C₈	—	218	1	no	—
117	C₂×C₈	C₂×#89137	1	1	yes	characteristic
118	C₂×C₈	C₂×#89137	1	1	yes	characteristic
119	C₁₆	—	218	4	no	—
120	C₂³	—	109	1	no	—
121	C₂³	—	109	1	no	—
122	C₂³	D₁₀₉⋊C₈	1	1	yes	characteristic
123	Q₈	—	218	2	no	—
124	D₄	—	218	2	no	—
125	D₄	—	218	2	no	—
126	D₄	—	218	2	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	109	1	no	—
130	C₂×C₄	—	109	2	no	—
131	C₂×C₄	—	109	2	no	—
132	C₂×C₄	—	109	1	no	—
133	C₂×C₄	—	218	1	no	—
134	C₂×C₄	—	218	1	no	—
135	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
136	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	109	2	no	—
139	C₈	—	218	1	no	—
140	C₈	C₂²⋊#89137	1	1	yes	characteristic
141	C₈	C₂²⋊#89137	1	1	yes	characteristic
142	C₂²	—	2	1	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	109	2	no	—
145	C₂²	—	436	1	no	—
146	C₂²	—	436	1	no	—
147	C₂²	—	218	1	no	—
148	C₂²	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	109	2	no	—
151	C₄	—	218	1	no	—
152	C₄	C₂×C₂²⋊#89137	1	1	yes	characteristic
153	C₄	D₁₀₉⋊M₁₆	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	4	1	no	—
156	C₂	—	436	1	no	—
157	C₂	—	218	2	no	—
158	C₂	D₁₀₉⋊(C₂²⋊C₈)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₀₉⋊M₃₂

C₂

yes

maximal, complemented

#20295923

C₂

yes

maximal, complemented

#20295922

C₂

yes

maximal, complemented

C₂×#16812705

C₂

yes

maximal, characteristic

#16812774

—

#16812773

C₂²

yes

—

D₁₀₉⋊C₁₆

—

C₂×#13182029

C₄

yes

—

—

C₂²

yes

characteristic

C₂×#13182012

C₄

yes

characteristic

C₂×#13182012

C₂²

yes

characteristic

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₄

yes

—

—

—

C₂×Dic₂₁₈

C₈

yes

characteristic

C₂×D₄₃₆

C₈

yes

characteristic

C₂×C₂²⋊D₁₀₉

C₈

yes

characteristic

C₂×C₂²⋊D₁₀₉

C₈

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

—

—

C₂×C₄

yes

characteristic

C₂×#11724784

C₂³

yes

characteristic

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₈×D₁₀₉

D₄

yes

—

C₈×D₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂³×C₁₀₉

C₂×C₈

yes

characteristic

C₂²×D₁₀₉

C₂×C₈

yes

characteristic

C₂²×D₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂²⋊D₁₀₉

—

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂²⋊C₄

yes

—

C₄×D₁₀₉

M₁₆

yes

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₈×C₁₀₉

C₂²⋊C₄

yes

characteristic

#11724784

—

#11724784

C₂×D₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²⋊C₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₂²⋊C₈

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₂²⋊C₈

yes

characteristic

D₁₀₉

—

D₁₀₉

—

#2115

—

109

maximal, Sylow

C₁₀₉

#2115

yes

characteristic, complemented, Sylow

C₂×#131

—

109

—

C₂×M₃₂

—

109

—

#349

—

109

—

#349

—

109

—

C₂×#41

—

109

—

#131

—

218

—

#131

—

218

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

#89137

yes

characteristic

M₃₂

—

109

—

M₃₂

—

218

—

C₂×C₁₆

—

218

—

#41

—

218

—

#41

—

218

—

C₂×Q₈

—

109

—

100

C₂×D₄

—

109

—

101

C₂×D₄

—

109

—

102

C₂×D₄

—

109

—

103

C₂²×C₄

—

109

—

104

C₂²×C₄

—

109

—

105

C₂²×C₄

C₂×#89137

yes

characteristic

106

M₁₆

—

218

—

107

M₁₆

—

218

—

108

M₁₆

—

218

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

109

—

112

C₂×C₈

—

109

—

113

C₂×C₈

—

109

—

114

C₂×C₈

—

109

—

115

C₂×C₈

—

218

—

116

C₂×C₈

—

218

—

117

C₂×C₈

C₂×#89137

yes

characteristic

118

C₂×C₈

C₂×#89137

yes

characteristic

119

C₁₆

—

218

—

120

C₂³

—

109

—

121

C₂³

—

109

—

122

C₂³

D₁₀₉⋊C₈

yes

characteristic

123

Q₈

—

218

—

124

D₄

—

218

—

125

D₄

—

218

—

126

D₄

—

218

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

109

—

130

C₂×C₄

—

109

—

131

C₂×C₄

—

109

—

132

C₂×C₄

—

109

—

133

C₂×C₄

—

218

—

134

C₂×C₄

—

218

—

135

C₂×C₄

D₁₀₉⋊C₈

yes

characteristic

136

C₂×C₄

C₂²×#89137

yes

characteristic

137

C₈

—

138

C₈

—

109

—

139

C₈

—

218

—

140

C₈

C₂²⋊#89137

yes

characteristic

141

C₈

C₂²⋊#89137

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

109

—

145

C₂²

—

436

—

146

C₂²

—

436

—

147

C₂²

—

218

—

148

C₂²

C₂×D₁₀₉⋊C₈

yes

characteristic

149

C₄

—

150

C₄

—

109

—

151

C₄

—

218

—

152

C₄

C₂×C₂²⋊#89137

yes

characteristic

153

C₄

D₁₀₉⋊M₁₆

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

—

436

—

157

C₂

—

218

—

158

C₂

D₁₀₉⋊(C₂²⋊C₈)

yes

characteristic

	subgroup	G/H
center	C₈	C₂²⋊#89137
commutator	C₂²×C₁₀₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₂²×C₈×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×C₂²⋊C₈
prime radical	C₂×#13182029	C₄
cosocle	C₂×#11724784	C₂³
hypercenter	C₂²×C₈	#89137
hypocenter	C₁₀₉	#2115
LCS	G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈

subgroup

G/H

center

C₈

C₂²⋊#89137

commutator

C₂²×C₁₀₉

C₂²×C₈

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₂²×C₈×C₁₀₉

C₄

socle

C₂×C₁₀₉

C₂×C₂²⋊C₈

C₄

C₂³

C₂²×C₈

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈

N	H ≅ G/N	type
1. D₁₀₉⋊M₃₂	C₂	semidirect
2. #20295923	C₂	semidirect
3. #20295922	C₂	semidirect
80. C₁₀₉	#2115	semidirect

H ≅ G/N

type

1. D₁₀₉⋊M₃₂

C₂

semidirect

2. #20295923

C₂

semidirect

3. #20295922

C₂

semidirect

80. C₁₀₉

#2115

semidirect

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

subgroup	index
D₁₀₉⋊M₃₂	2
#20295923	2
#20295922	2
C₂×#16812705	2
#2115	109

subgroup

index

D₁₀₉⋊M₃₂

109

Subgroups — #28379269

Conjugacy classes of subgroups

Complements

Maximal subgroups