Group #28378818 — subgroups

Subgroups — #28378818

Order 13952.

292 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 130 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×C₄⋊#11724786	C₂	1	2	yes	maximal
2	C₂×#16812754	C₂	1	2	yes	maximal
3	#20295892	C₂	1	2	yes	maximal
4	C₄²×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724786	C₂²	1	4	yes	—
6	C₄⋊#11724786	C₄	1	8	yes	complemented
7	#16812754	—	2	4	no	—
8	C₂×C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₀₉	C₄	1	1	yes	characteristic
15	C₂×#11724786	—	2	4	no	—
16	C₂×#11724786	—	2	4	no	—
17	C₂×#11724786	C₂×C₄	1	8	yes	—
18	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
22	C₄×Dic₁₀₉	—	2	2	no	—
23	C₄×Dic₁₀₉	—	2	2	no	—
24	C₄×Dic₁₀₉	—	2	2	no	—
25	C₄×Dic₁₀₉	—	2	2	no	—
26	C₄×Dic₁₀₉	C₂×C₄	1	4	yes	—
27	C₄×Dic₁₀₉	D₄	1	2	yes	—
28	C₄×Dic₁₀₉	Q₈	1	2	yes	—
29	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₁₀₉	—	2	2	no	—
33	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	—	2	1	no	—
35	C₂×C₄×C₁₀₉	—	2	1	no	—
36	C₂×C₄×C₁₀₉	—	2	1	no	—
37	C₂×C₄×C₁₀₉	—	2	1	no	—
38	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
39	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
41	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
42	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
43	C₂×Dic₁₀₉	—	2	1	no	—
44	C₂×Dic₁₀₉	—	2	1	no	—
45	C₂×Dic₁₀₉	—	2	1	no	—
46	C₂×Dic₁₀₉	—	2	1	no	—
47	C₂×Dic₁₀₉	—	2	1	no	—
48	C₂×Dic₁₀₉	—	2	1	no	—
49	C₂×Dic₁₀₉	C₄²	1	2	yes	—
50	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
51	C₂×Dic₁₀₉	M₁₆	1	2	yes	—
52	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
53	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
55	C₂×Dic₁₀₉	#41	1	2	yes	—
56	#11724786	—	2	8	no	—
57	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
58	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₀₉	#117	1	1	yes	characteristic
60	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
61	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
62	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
63	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
64	C₄×C₁₀₉	—	2	2	no	—
65	C₄×C₁₀₉	—	2	2	no	—
66	C₄×C₁₀₉	#97	1	4	yes	complemented
67	Dic₁₀₉	—	2	2	no	—
68	Dic₁₀₉	—	2	2	no	—
69	Dic₁₀₉	—	2	2	no	—
70	Dic₁₀₉	C₄×D₄	1	2	yes	—
71	Dic₁₀₉	C₄×Q₈	1	2	yes	—
72	C₂×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₁₀₉	C₂×#97	1	1	yes	characteristic
74	C₂×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
75	C₂×C₁₀₉	#423	1	1	yes	characteristic
76	C₂×C₁₀₉	#423	1	1	yes	characteristic
77	C₂×C₁₀₉	#432	1	1	yes	characteristic
78	C₂×C₁₀₉	#432	1	1	yes	characteristic
79	#1767	—	109	1	no	maximal, Sylow
80	C₁₀₉	#1767	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₄⋊C₈	—	109	2	no	—
82	C₂×#97	—	109	2	no	—
83	C₄³	—	109	1	no	—
84	#336	—	109	2	no	—
85	C₂²×C₈	—	109	4	no	—
86	C₂×C₄²	—	109	1	no	—
87	C₂×C₄²	—	109	1	no	—
88	C₂×C₄²	—	109	1	no	—
89	C₂×C₄²	—	109	1	no	—
90	C₂×C₄²	—	109	1	no	—
91	C₂×C₄²	—	109	1	no	—
92	C₂×C₄²	#89137	1	1	yes	characteristic
93	C₄⋊C₈	—	109	8	no	—
94	#97	—	218	4	no	—
95	C₂²×C₄	—	109	1	no	—
96	C₂²×C₄	—	109	1	no	—
97	C₂²×C₄	—	109	1	no	—
98	C₂²×C₄	—	109	1	no	—
99	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
100	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
101	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
102	C₂×C₈	—	109	8	no	—
103	C₂×C₈	—	218	4	no	—
104	C₂×C₈	—	218	4	no	—
105	C₄²	—	2	2	no	—
106	C₄²	—	109	4	no	—
107	C₄²	—	109	2	no	—
108	C₄²	—	109	2	no	—
109	C₄²	—	218	2	no	—
110	C₄²	—	218	2	no	—
111	C₄²	—	218	2	no	—
112	C₄²	—	218	2	no	—
113	C₂³	C₂²×#89137	1	1	yes	characteristic
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	109	2	no	—
119	C₂×C₄	—	109	1	no	—
120	C₂×C₄	—	109	1	no	—
121	C₂×C₄	—	109	2	no	—
122	C₂×C₄	—	109	2	no	—
123	C₂×C₄	—	109	2	no	—
124	C₂×C₄	—	109	2	no	—
125	C₂×C₄	—	218	1	no	—
126	C₂×C₄	—	218	1	no	—
127	C₂×C₄	—	218	1	no	—
128	C₂×C₄	—	218	1	no	—
129	C₂×C₄	—	218	1	no	—
130	C₂×C₄	—	218	1	no	—
131	C₂×C₄	C₄×#89137	1	2	yes	—
132	C₂×C₄	C₄⋊#89137	1	2	yes	—
133	C₂×C₄	#13182038	1	2	yes	—
134	C₂×C₄	#13182039	1	2	yes	—
135	C₂×C₄	#13182039	1	2	yes	—
136	C₈	—	218	8	no	—
137	C₂²	C₂×C₄×#89137	1	1	yes	characteristic
138	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
139	C₂²	#16812778	1	1	yes	characteristic
140	C₂²	C₂×#13182038	1	1	yes	characteristic
141	C₂²	D₁₀₉⋊M₁₆	1	1	yes	characteristic
142	C₂²	C₂×#13182039	1	1	yes	characteristic
143	C₂²	C₂×#13182039	1	1	yes	characteristic
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	109	2	no	—
147	C₄	—	109	2	no	—
148	C₄	—	218	2	no	—
149	C₄	—	218	2	no	—
150	C₄	—	218	2	no	—
151	C₄	#16812754	1	4	yes	complemented
152	C₂	C₄×C₄⋊#89137	1	1	yes	characteristic
153	C₂	C₂×#16812754	1	1	yes	characteristic
154	C₂	C₄×#13182039	1	1	yes	characteristic
155	C₂	#20296029	1	1	yes	characteristic
156	C₂	#20296030	1	1	yes	characteristic
157	C₂	#20296047	1	1	yes	characteristic
158	C₂	#20296048	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11724786

C₂

yes

maximal

C₂×#16812754

C₂

yes

maximal

#20295892

C₂

yes

maximal

C₄²×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724786

C₂²

yes

—

C₄⋊#11724786

C₄

yes

complemented

#16812754

—

C₂×C₄×Dic₁₀₉

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₄

yes

—

—

C₂×C₄

yes

—

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂³

yes

characteristic

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

M₁₆

yes

—

C₂×C₄×C₁₀₉

M₁₆

yes

—

C₂×C₄×C₁₀₉

M₁₆

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄²

yes

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

M₁₆

yes

—

C₂×Dic₁₀₉

C₂²×C₄

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₉

#41

yes

—

#11724786

—

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#117

yes

characteristic

C₂²×C₁₀₉

C₂×M₁₆

yes

characteristic

C₂²×C₁₀₉

C₂×M₁₆

yes

characteristic

C₂²×C₁₀₉

C₂×M₁₆

yes

characteristic

C₂²×C₁₀₉

C₂×M₁₆

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#97

yes

complemented

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₄×D₄

yes

—

Dic₁₀₉

C₄×Q₈

yes

—

C₂×C₁₀₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

C₂×#97

yes

characteristic

C₂×C₁₀₉

C₄×M₁₆

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1767

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

C₂×#97

—

109

—

C₄³

—

109

—

#336

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

#89137

yes

characteristic

C₄⋊C₈

—

109

—

#97

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×#89137

yes

characteristic

100

C₂²×C₄

C₂×#89137

yes

characteristic

101

C₂²×C₄

C₂×#89137

yes

characteristic

102

C₂×C₈

—

109

—

103

C₂×C₈

—

218

—

104

C₂×C₈

—

218

—

105

C₄²

—

106

C₄²

—

109

—

107

C₄²

—

109

—

108

C₄²

—

109

—

109

C₄²

—

218

—

110

C₄²

—

218

—

111

C₄²

—

218

—

112

C₄²

—

218

—

113

C₂³

C₂²×#89137

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

109

—

119

C₂×C₄

—

109

—

120

C₂×C₄

—

109

—

121

C₂×C₄

—

109

—

122

C₂×C₄

—

109

—

123

C₂×C₄

—

109

—

124

C₂×C₄

—

109

—

125

C₂×C₄

—

218

—

126

C₂×C₄

—

218

—

127

C₂×C₄

—

218

—

128

C₂×C₄

—

218

—

129

C₂×C₄

—

218

—

130

C₂×C₄

—

218

—

131

C₂×C₄

C₄×#89137

yes

—

132

C₂×C₄

C₄⋊#89137

yes

—

133

C₂×C₄

#13182038

yes

—

134

C₂×C₄

#13182039

yes

—

135

C₂×C₄

#13182039

yes

—

136

C₈

—

218

—

137

C₂²

C₂×C₄×#89137

yes

characteristic

138

C₂²

C₂×C₄⋊#89137

yes

characteristic

139

C₂²

#16812778

yes

characteristic

140

C₂²

C₂×#13182038

yes

characteristic

141

C₂²

D₁₀₉⋊M₁₆

yes

characteristic

142

C₂²

C₂×#13182039

yes

characteristic

143

C₂²

C₂×#13182039

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

109

—

147

C₄

—

109

—

148

C₄

—

218

—

149

C₄

—

218

—

150

C₄

—

218

—

151

C₄

#16812754

yes

complemented

152

C₂

C₄×C₄⋊#89137

yes

characteristic

153

C₂

C₂×#16812754

yes

characteristic

154

C₂

C₄×#13182039

yes

characteristic

155

C₂

#20296029

yes

characteristic

156

C₂

#20296030

yes

characteristic

157

C₂

#20296047

yes

characteristic

158

C₂

#20296048

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#89137
commutator	C₂²×C₁₀₉	C₂×C₄²
Fitting	C₂×C₄²×C₁₀₉	C₄
socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂²×Dic₁₀₉	C₂³
hypercenter	C₂×C₄²	#89137
hypocenter	C₁₀₉	#1767
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂²×#89137

commutator

C₂²×C₁₀₉

C₂×C₄²

Fitting

C₂×C₄²×C₁₀₉

C₄

socle, prime radical

C₂³×C₁₀₉

C₂²×C₄

cosocle

C₂²×Dic₁₀₉

C₂³

C₂×C₄²

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
6. C₄⋊#11724786	C₄	semidirect
66. C₄×C₁₀₉	#97	semidirect
80. C₁₀₉	#1767	semidirect
151. C₄	#16812754	semidirect

H ≅ G/N

type

6. C₄⋊#11724786

C₄

semidirect

66. C₄×C₁₀₉

#97

semidirect

80. C₁₀₉

#1767

semidirect

151. C₄

#16812754

semidirect

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

subgroup	index
C₂×C₄⋊#11724786	2
C₂×#16812754	2
#20295892	2
C₄²×Dic₁₀₉	2
#1767	109

subgroup

index

C₄²×Dic₁₀₉

109

Subgroups — #28378818

Conjugacy classes of subgroups

Complements

Maximal subgroups