Group #28379130 — subgroups

Subgroups — #28379130

Order 13952.

260 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 110 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×#16812766	C₂	1	2	yes	maximal
2	C₂×C₄⋊#11724786	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11724786	C₂	1	1	yes	maximal, characteristic
4	#20295892	C₂	1	2	yes	maximal
5	C₄⋊C₄×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11724786	C₂²	1	2	yes	—
7	C₂²×#11724786	C₂²	1	1	yes	characteristic
8	C₂²×#11724786	C₂²	1	1	yes	characteristic
9	#16812766	—	2	4	no	—
10	C₄⋊#11724786	—	2	2	no	—
11	C₄×#11724786	—	2	2	no	—
12	C₂×C₄⋊Dic₁₀₉	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	#16812606	C₄	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
17	C₂×#11724786	—	2	1	no	—
18	C₂×#11724786	—	2	1	no	—
19	C₂×#11724786	—	2	1	no	—
20	C₂×#11724786	—	2	1	no	—
21	C₂×#11724786	—	2	2	no	—
22	C₂×#11724786	—	2	2	no	—
23	C₂×#11724786	—	2	2	no	—
24	C₂×#11724786	D₄	1	2	yes	—
25	C₂×#11724786	Q₈	1	2	yes	—
26	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
28	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₁₀₉	C₈	1	4	yes	complemented
30	C₄×Dic₁₀₉	—	2	2	no	—
31	C₄×Dic₁₀₉	—	2	2	no	—
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	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₄×C₁₀₉	—	2	2	no	—
37	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄×C₁₀₉	—	2	2	no	—
41	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
44	C₂×Dic₁₀₉	—	2	2	no	—
45	C₂×Dic₁₀₉	—	2	1	no	—
46	C₂×Dic₁₀₉	—	2	1	no	—
47	C₂×Dic₁₀₉	—	2	2	no	—
48	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
49	C₂×Dic₁₀₉	C₂×C₈	1	4	yes	—
50	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
55	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
56	#11724786	—	2	2	no	—
57	#11724786	—	2	2	no	—
58	#11724786	—	4	2	no	—
59	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
63	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
64	C₂²×C₁₀₉	#131	1	1	yes	characteristic
65	C₂²×C₁₀₉	#131	1	1	yes	characteristic
66	C₄×C₁₀₉	—	4	2	no	—
67	C₄×C₁₀₉	C₄⋊C₈	1	4	yes	complemented
68	Dic₁₀₉	—	2	2	no	—
69	Dic₁₀₉	—	2	4	no	—
70	Dic₁₀₉	#121	1	1	yes	characteristic
71	Dic₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
72	Dic₁₀₉	#125	1	1	yes	characteristic
73	Dic₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₁₀₉	#389	1	1	yes	characteristic
75	C₂×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₁₀₉	#424	1	1	yes	characteristic
77	C₂×C₁₀₉	C₈×D₄	1	1	yes	characteristic
78	C₂×C₁₀₉	#436	1	1	yes	characteristic
79	C₂×C₁₀₉	C₈×Q₈	1	1	yes	characteristic
80	C₂×C₁₀₉	#446	1	1	yes	characteristic
81	#1939	—	109	1	no	maximal, Sylow
82	C₁₀₉	#1939	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄⋊C₈	—	109	1	no	—
84	C₂×C₄⋊C₈	—	109	2	no	—
85	C₂×C₄×C₈	—	109	1	no	—
86	C₄×C₄⋊C₄	—	109	1	no	—
87	#336	—	109	2	no	—
88	C₂²×C₈	—	109	1	no	—
89	C₂²×C₈	—	109	1	no	—
90	C₂²×C₈	—	109	2	no	—
91	C₂×C₄⋊C₄	—	109	1	no	—
92	C₂×C₄⋊C₄	#89137	1	1	yes	characteristic
93	C₂×C₄²	—	109	1	no	—
94	C₂×C₄²	—	109	2	no	—
95	C₄⋊C₈	—	218	2	no	—
96	C₄⋊C₈	—	218	4	no	—
97	C₄×C₈	—	218	2	no	—
98	#95	—	109	2	no	—
99	C₂²×C₄	—	109	1	no	—
100	C₂²×C₄	—	109	1	no	—
101	C₂²×C₄	—	109	2	no	—
102	C₂²×C₄	C₂×#89137	1	2	yes	—
103	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
104	C₂×C₈	—	109	2	no	—
105	C₂×C₈	—	109	2	no	—
106	C₂×C₈	—	218	2	no	—
107	C₂×C₈	—	218	2	no	—
108	C₂×C₈	—	218	2	no	—
109	C₂×C₈	—	218	1	no	—
110	C₂×C₈	—	218	1	no	—
111	C₂×C₈	—	218	1	no	—
112	C₂×C₈	—	218	1	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	109	4	no	—
115	C₄²	—	109	2	no	—
116	C₄²	—	109	2	no	—
117	C₄²	—	218	2	no	—
118	C₄²	—	218	2	no	—
119	C₂³	C₂²×#89137	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	109	1	no	—
124	C₂×C₄	—	109	1	no	—
125	C₂×C₄	—	109	1	no	—
126	C₂×C₄	—	109	1	no	—
127	C₂×C₄	—	109	1	no	—
128	C₂×C₄	—	109	1	no	—
129	C₂×C₄	—	109	2	no	—
130	C₂×C₄	—	109	4	no	—
131	C₂×C₄	—	218	2	no	—
132	C₂×C₄	—	218	1	no	—
133	C₂×C₄	—	218	1	no	—
134	C₂×C₄	—	218	2	no	—
135	C₂×C₄	C₄⋊#89137	1	2	yes	—
136	C₂×C₄	D₁₀₉⋊C₈	1	2	yes	—
137	C₂×C₄	#13182038	1	2	yes	—
138	C₈	—	436	2	no	—
139	C₈	—	218	2	no	—
140	C₈	—	218	2	no	—
141	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
142	C₂²	D₄×#89137	1	1	yes	characteristic
143	C₂²	Q₈×#89137	1	1	yes	characteristic
144	C₂²	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
145	C₂²	C₂×#13182038	1	1	yes	characteristic
146	C₂²	#16812786	1	1	yes	characteristic
147	C₂²	#16812787	1	1	yes	characteristic
148	C₄	—	4	2	no	—
149	C₄	—	109	1	no	—
150	C₄	—	109	1	no	—
151	C₄	—	109	1	no	—
152	C₄	—	109	1	no	—
153	C₄	—	218	2	no	—
154	C₄	—	218	4	no	—
155	C₄	#16812766	1	4	yes	complemented
156	C₂	#20295977	1	1	yes	characteristic
157	C₂	C₂×#16812766	1	1	yes	characteristic
158	C₂	#20296033	1	1	yes	characteristic
159	C₂	#20296054	1	1	yes	characteristic
160	C₂	#20296061	1	1	yes	characteristic
161	C₂	#20296075	1	1	yes	characteristic
162	C₂	#20296077	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812766

C₂

yes

maximal

C₂×C₄⋊#11724786

C₂

yes

maximal, characteristic

C₂×C₄×#11724786

C₂

yes

maximal, characteristic

#20295892

C₂

yes

maximal

C₄⋊C₄×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724786

C₂²

yes

—

C₂²×#11724786

C₂²

yes

characteristic

C₂²×#11724786

C₂²

yes

—

—

—

C₂×C₄⋊Dic₁₀₉

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812606

C₄

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₄

yes

—

—

—

—

—

—

—

D₄

yes

—

C₂×#11724786

Q₈

yes

—

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

—

C₂²×Dic₁₀₉

C₂³

yes

characteristic

C₄⋊Dic₁₀₉

C₈

yes

complemented

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

M₁₆

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₉

C₂×Q₈

yes

C₂×Dic₁₀₉

yes

C₂×Dic₁₀₉

yes

—

—

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₂²×C₁₀₉

C₄×Q₈

yes

characteristic

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₂²×C₁₀₉

C₂×M₁₆

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄⋊C₈

yes

complemented

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#121

yes

characteristic

Dic₁₀₉

C₂²⋊Q₈

yes

characteristic

Dic₁₀₉

#125

yes

characteristic

Dic₁₀₉

C₄⋊Q₈

yes

characteristic

C₂×C₁₀₉

#389

yes

characteristic

C₂×C₁₀₉

C₂×C₄⋊C₈

yes

characteristic

C₂×C₁₀₉

#424

yes

characteristic

C₂×C₁₀₉

C₈×D₄

yes

characteristic

C₂×C₁₀₉

#436

yes

characteristic

C₂×C₁₀₉

C₈×Q₈

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1939

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

C₂×C₄⋊C₈

—

109

—

C₂×C₄×C₈

—

109

—

C₄×C₄⋊C₄

—

109

—

#336

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

#89137

yes

characteristic

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

C₄×C₈

—

218

—

#95

—

109

—

C₂²×C₄

—

109

—

100

C₂²×C₄

—

109

—

101

C₂²×C₄

—

109

—

102

C₂²×C₄

C₂×#89137

yes

—

103

C₂²×C₄

C₂×#89137

yes

characteristic

104

C₂×C₈

—

109

—

105

C₂×C₈

—

109

—

106

C₂×C₈

—

218

—

107

C₂×C₈

—

218

—

108

C₂×C₈

—

218

—

109

C₂×C₈

—

218

—

110

C₂×C₈

—

218

—

111

C₂×C₈

—

218

—

112

C₂×C₈

—

218

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

109

—

115

C₄²

—

109

—

116

C₄²

—

109

—

117

C₄²

—

218

—

118

C₄²

—

218

—

119

C₂³

C₂²×#89137

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

109

—

124

C₂×C₄

—

109

—

125

C₂×C₄

—

109

—

126

C₂×C₄

—

109

—

127

C₂×C₄

—

109

—

128

C₂×C₄

—

109

—

129

C₂×C₄

—

109

—

130

C₂×C₄

—

109

—

131

C₂×C₄

—

218

—

132

C₂×C₄

—

218

—

133

C₂×C₄

—

218

—

134

C₂×C₄

—

218

—

135

C₂×C₄

C₄⋊#89137

yes

—

136

C₂×C₄

D₁₀₉⋊C₈

yes

—

137

C₂×C₄

#13182038

yes

—

138

C₈

—

436

—

139

C₈

—

218

—

140

C₈

—

218

—

141

C₂²

C₂×C₄⋊#89137

yes

characteristic

142

C₂²

D₄×#89137

yes

characteristic

143

C₂²

Q₈×#89137

yes

characteristic

144

C₂²

C₂×D₁₀₉⋊C₈

yes

characteristic

145

C₂²

C₂×#13182038

yes

characteristic

146

C₂²

#16812786

yes

characteristic

147

C₂²

#16812787

yes

characteristic

148

C₄

—

149

C₄

—

109

—

150

C₄

—

109

—

151

C₄

—

109

—

152

C₄

—

109

—

153

C₄

—

218

—

154

C₄

—

218

—

155

C₄

#16812766

yes

complemented

156

C₂

#20295977

yes

characteristic

157

C₂

C₂×#16812766

yes

characteristic

158

C₂

#20296033

yes

characteristic

159

C₂

#20296054

yes

characteristic

160

C₂

#20296061

yes

characteristic

161

C₂

#20296075

yes

characteristic

162

C₂

#20296077

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#89137
commutator	C₂²×C₁₀₉	C₂²×C₈
Fitting	C₂×C₄⋊C₄×C₁₀₉	C₄
socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂²×Dic₁₀₉	C₂³
hypercenter	C₂×C₄⋊C₄	#89137
hypocenter	C₁₀₉	#1939
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×#89137

commutator

C₂²×C₁₀₉

C₂²×C₈

Fitting

C₂×C₄⋊C₄×C₁₀₉

C₄

socle, prime radical

C₂³×C₁₀₉

C₂²×C₄

cosocle

C₂²×Dic₁₀₉

C₂³

C₂×C₄⋊C₄

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
29. C₄⋊Dic₁₀₉	C₈	semidirect
67. C₄×C₁₀₉	C₄⋊C₈	semidirect
82. C₁₀₉	#1939	semidirect
155. C₄	#16812766	semidirect

H ≅ G/N

type

29. C₄⋊Dic₁₀₉

C₈

semidirect

67. C₄×C₁₀₉

C₄⋊C₈

semidirect

82. C₁₀₉

#1939

semidirect

155. C₄

#16812766

semidirect

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

subgroup	index
C₂×#16812766	2
C₂×C₄⋊#11724786	2
C₂×C₄×#11724786	2
#20295892	2
C₄⋊C₄×Dic₁₀₉	2
#1939	109

subgroup

index

C₄⋊C₄×Dic₁₀₉

109

Subgroups — #28379130

Conjugacy classes of subgroups

Complements

Maximal subgroups