Group #28378322 — subgroups

Subgroups — C₂×#20295876 #28378322

Order 13952.

280 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 106 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	1	yes	maximal, characteristic
2	C₂×C₄×#11724786	C₂	1	1	yes	maximal, characteristic
3	#20295876	C₂	1	4	yes	maximal, direct product factor
4	C₂×#16812664	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724786	—	2	1	no	—
6	C₂²×#11724786	—	2	1	no	—
7	C₄⋊#11724786	—	2	1	no	—
8	C₄⋊#11724786	C₂²	1	2	yes	—
9	C₄×#11724786	—	2	1	no	—
10	C₄×#11724786	C₂²	1	2	yes	—
11	C₂²×Dic₂₁₈	C₄	1	1	yes	characteristic
12	#16812664	—	2	1	no	—
13	#16812664	—	2	1	no	—
14	#16812664	C₄	1	2	yes	—
15	#16812664	C₂²	1	2	yes	—
16	C₂×#13182008	—	2	1	no	—
17	C₂×C₄×Dic₁₀₉	—	2	1	no	—
18	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
20	C₂×#11724786	—	2	4	no	—
21	C₂×#11724786	—	2	2	no	—
22	C₂×#11724786	—	2	2	no	—
23	C₂×#11724786	—	4	2	no	—
24	C₂×Dic₂₁₈	—	2	1	no	—
25	C₂×Dic₂₁₈	—	2	2	no	—
26	C₂×Dic₂₁₈	C₈	1	4	yes	complemented
27	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
28	C₂²×Dic₁₀₉	—	2	1	no	—
29	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
30	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
31	#13182008	—	2	2	no	—
32	#13182008	—	2	2	no	—
33	C₄×Dic₁₀₉	—	2	2	no	—
34	C₄×Dic₁₀₉	—	2	2	no	—
35	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
37	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
38	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₉	—	2	1	no	—
40	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₄×C₁₀₉	—	2	1	no	—
42	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
43	C₂³×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	—	2	2	no	—
45	C₂×C₄×C₁₀₉	—	4	2	no	—
46	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
48	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
49	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×Dic₁₀₉	—	2	4	no	—
51	C₂×Dic₁₀₉	—	2	2	no	—
52	C₂×Dic₁₀₉	—	2	2	no	—
53	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₁₀₉	SD₁₆	1	2	yes	—
56	C₂×Dic₁₀₉	Q₁₆	1	2	yes	—
57	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
59	Dic₂₁₈	—	2	4	no	—
60	Dic₂₁₈	—	2	2	no	—
61	Dic₂₁₈	C₂×C₈	1	4	yes	complemented
62	#11724786	—	2	4	no	—
63	#11724786	—	4	2	no	—
64	C₂²×C₁₀₉	C₂²⋊C₈	1	2	yes	—
65	C₂²×C₁₀₉	#103	1	2	yes	—
66	C₂²×C₁₀₉	#104	1	2	yes	—
67	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₁₀₉	—	4	2	no	—
69	C₄×C₁₀₉	C₂²⋊C₈	1	2	yes	—
70	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
71	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
72	Dic₁₀₉	—	2	4	no	—
73	Dic₁₀₉	—	2	1	no	—
74	Dic₁₀₉	—	2	1	no	—
75	Dic₁₀₉	#103	1	2	yes	—
76	Dic₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
77	Dic₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₁₀₉	Q₈⋊C₈	1	4	yes	complemented
79	C₂×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₁₀₉	C₂×#103	1	1	yes	characteristic
81	C₂×C₁₀₉	C₂×#104	1	1	yes	characteristic
82	C₂×Q₈⋊C₈	—	109	1	no	maximal, Sylow
83	C₁₀₉	C₂×Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
84	C₂×C₄×Q₈	—	109	1	no	—
85	C₂×C₄⋊C₈	—	109	1	no	—
86	C₂×C₄×C₈	—	109	1	no	—
87	Q₈⋊C₈	—	109	4	no	—
88	C₂²×Q₈	—	109	1	no	—
89	C₂²×C₈	—	218	1	no	—
90	C₂²×C₈	—	218	1	no	—
91	C₄×Q₈	—	109	2	no	—
92	C₄×Q₈	—	109	2	no	—
93	C₄×Q₈	—	218	1	no	—
94	C₄×Q₈	—	218	1	no	—
95	C₂×C₄⋊C₄	—	218	1	no	—
96	C₂×C₄⋊C₄	#89137	1	1	yes	characteristic
97	C₂×C₄²	—	109	1	no	—
98	C₂×C₄²	—	218	1	no	—
99	C₄⋊C₈	—	109	2	no	—
100	C₄⋊C₈	—	218	1	no	—
101	C₄×C₈	—	109	2	no	—
102	C₄×C₈	—	218	1	no	—
103	C₂×Q₈	—	109	4	no	—
104	C₂×Q₈	—	109	2	no	—
105	C₂×Q₈	—	218	1	no	—
106	C₂×Q₈	—	218	2	no	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	—	109	1	no	—
109	C₂²×C₄	—	109	1	no	—
110	C₂²×C₄	—	218	1	no	—
111	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
112	C₂×C₈	—	436	2	no	—
113	C₂×C₈	—	218	2	no	—
114	C₂×C₈	—	218	2	no	—
115	C₂×C₈	—	218	4	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	218	2	no	—
118	C₄⋊C₄	—	218	2	no	—
119	C₄⋊C₄	C₂×#89137	1	2	yes	—
120	C₄²	—	109	1	no	—
121	C₄²	—	109	1	no	—
122	C₄²	—	109	1	no	—
123	C₄²	—	109	1	no	—
124	C₄²	—	218	2	no	—
125	C₄²	—	218	2	no	—
126	C₂³	C₂²⋊#89137	1	1	yes	characteristic
127	Q₈	—	109	4	no	—
128	Q₈	—	218	4	no	—
129	Q₈	—	218	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	109	2	no	—
133	C₂×C₄	—	109	2	no	—
134	C₂×C₄	—	109	1	no	—
135	C₂×C₄	—	109	1	no	—
136	C₂×C₄	—	109	1	no	—
137	C₂×C₄	—	109	1	no	—
138	C₂×C₄	—	218	4	no	—
139	C₂×C₄	—	218	2	no	—
140	C₂×C₄	—	218	2	no	—
141	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
142	C₂×C₄	D₁₀₉⋊C₈	1	2	yes	—
143	C₂×C₄	#13182039	1	2	yes	—
144	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
145	C₈	—	436	2	no	—
146	C₈	—	218	4	no	—
147	C₂²	#16812756	1	2	yes	—
148	C₂²	#16812762	1	2	yes	—
149	C₂²	#16812763	1	2	yes	—
150	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
151	C₄	—	4	2	no	—
152	C₄	—	109	1	no	—
153	C₄	—	109	2	no	—
154	C₄	—	109	1	no	—
155	C₄	—	218	4	no	—
156	C₄	—	218	1	no	—
157	C₄	—	218	1	no	—
158	C₄	#16812756	1	2	yes	—
159	C₄	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
160	C₄	C₂×#13182039	1	1	yes	characteristic
161	C₂	#20295876	1	4	yes	direct product factor
162	C₂	C₂×#16812756	1	1	yes	characteristic
163	C₂	C₂×#16812762	1	1	yes	characteristic
164	C₂	C₂×#16812763	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11724786

C₂

yes

maximal, characteristic

C₂×C₄×#11724786

C₂

yes

maximal, characteristic

#20295876

C₂

yes

maximal, direct product factor

C₂×#16812664

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₄×#11724786

—

C₄×#11724786

C₂²

yes

—

C₂²×Dic₂₁₈

C₄

yes

—

—

C₄

yes

—

#16812664

C₂²

yes

—

C₂×#13182008

—

C₂×C₄×Dic₁₀₉

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₄

yes

—

—

—

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₈

yes

complemented

C₂×Dic₂₁₈

C₂×C₄

yes

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

D₄

yes

characteristic

C₂²×Dic₁₀₉

D₄

yes

characteristic

#13182008

—

#13182008

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

M₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

SD₁₆

yes

—

C₂×Dic₁₀₉

Q₁₆

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

C₂×C₈

yes

complemented

#11724786

—

#11724786

—

C₂²×C₁₀₉

C₂²⋊C₈

yes

—

C₂²×C₁₀₉

#103

yes

—

C₂²×C₁₀₉

#104

yes

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊C₈

yes

—

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#103

yes

—

Dic₁₀₉

C₂×SD₁₆

yes

characteristic

Dic₁₀₉

C₂×Q₁₆

yes

characteristic

C₂×C₁₀₉

Q₈⋊C₈

yes

complemented

C₂×C₁₀₉

C₂×C₂²⋊C₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×Q₈⋊C₈

—

109

maximal, Sylow

C₁₀₉

C₂×Q₈⋊C₈

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

109

—

C₂×C₄⋊C₈

—

109

—

C₂×C₄×C₈

—

109

—

Q₈⋊C₈

—

109

—

C₂²×Q₈

—

109

—

C₂²×C₈

—

218

—

C₂²×C₈

—

218

—

C₄×Q₈

—

109

—

C₄×Q₈

—

109

—

C₄×Q₈

—

218

—

C₄×Q₈

—

218

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

#89137

yes

characteristic

C₂×C₄²

—

109

—

C₂×C₄²

—

218

—

C₄⋊C₈

—

109

—

100

C₄⋊C₈

—

218

—

101

C₄×C₈

—

109

—

102

C₄×C₈

—

218

—

103

C₂×Q₈

—

109

—

104

C₂×Q₈

—

109

—

105

C₂×Q₈

—

218

—

106

C₂×Q₈

—

218

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

—

109

C₂²×C₄

—

109

—

110

C₂²×C₄

—

218

—

111

C₂²×C₄

C₂×#89137

yes

characteristic

112

C₂×C₈

—

436

—

113

C₂×C₈

—

218

—

114

C₂×C₈

—

218

—

115

C₂×C₈

—

218

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

218

—

118

C₄⋊C₄

—

218

—

119

C₄⋊C₄

C₂×#89137

yes

—

120

C₄²

—

109

—

121

C₄²

—

109

—

122

C₄²

—

109

—

123

C₄²

—

109

—

124

C₄²

—

218

—

125

C₄²

—

218

—

126

C₂³

C₂²⋊#89137

yes

characteristic

127

Q₈

—

109

—

128

Q₈

—

218

—

129

Q₈

—

218

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

109

—

133

C₂×C₄

—

109

—

134

C₂×C₄

—

109

—

135

C₂×C₄

—

109

—

136

C₂×C₄

—

109

—

137

C₂×C₄

—

109

—

138

C₂×C₄

—

218

—

139

C₂×C₄

—

218

—

140

C₂×C₄

—

218

—

141

C₂×C₄

C₂²⋊#89137

yes

characteristic

142

C₂×C₄

D₁₀₉⋊C₈

yes

—

143

C₂×C₄

#13182039

yes

—

144

C₂×C₄

C₂²×#89137

yes

characteristic

145

C₈

—

436

—

146

C₈

—

218

—

147

C₂²

#16812756

yes

—

148

C₂²

#16812762

yes

—

149

C₂²

#16812763

yes

—

150

C₂²

C₂×C₂²⋊#89137

yes

characteristic

151

C₄

—

152

C₄

—

109

—

153

C₄

—

109

—

154

C₄

—

109

—

155

C₄

—

218

—

156

C₄

—

218

—

157

C₄

—

218

—

158

C₄

#16812756

yes

—

159

C₄

C₂×D₁₀₉⋊C₈

yes

characteristic

160

C₄

C₂×#13182039

yes

characteristic

161

C₂

#20295876

yes

direct product factor

162

C₂

C₂×#16812756

yes

characteristic

163

C₂

C₂×#16812762

yes

characteristic

164

C₂

C₂×#16812763

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊#89137
commutator	C₄×C₁₀₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₂×C₄⋊C₄×C₁₀₉	C₄
socle, prime radical	C₂³×C₁₀₉	C₂²⋊C₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₂×C₄⋊C₄	#89137
hypocenter	C₁₀₉	C₂×Q₈⋊C₈
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²⋊#89137

commutator

C₄×C₁₀₉

C₂²×C₈

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₂×C₄⋊C₄×C₁₀₉

C₄

socle, prime radical

C₂³×C₁₀₉

C₂²⋊C₄

cosocle

C₄×Dic₁₀₉

C₂³

hypercenter

C₂×C₄⋊C₄

#89137

hypocenter

C₁₀₉

C₂×Q₈⋊C₈

LCS

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
3. #20295876	C₂	direct (N × H)
26. C₂×Dic₂₁₈	C₈	semidirect
61. Dic₂₁₈	C₂×C₈	semidirect
78. C₂×C₁₀₉	Q₈⋊C₈	semidirect
83. C₁₀₉	C₂×Q₈⋊C₈	semidirect
161. C₂	#20295876	direct (N × H)

H ≅ G/N

type

3. #20295876

C₂

direct (N × H)

26. C₂×Dic₂₁₈

C₈

semidirect

61. Dic₂₁₈

C₂×C₈

semidirect

78. C₂×C₁₀₉

Q₈⋊C₈

semidirect

83. C₁₀₉

C₂×Q₈⋊C₈

semidirect

161. C₂

#20295876

direct (N × H)

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

subgroup	index
C₂×C₄⋊#11724786	2
C₂×C₄×#11724786	2
#20295876	2
C₂×#16812664	2
C₂×Q₈⋊C₈	109

subgroup

index

C₂×Q₈⋊C₈

109

Subgroups — C₂×#20295876 #28378322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295876 #28378322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295876 #28378322