Group #28372797 — subgroups

Subgroups — C₂×#20295311 #28372797

Order 13952.

416 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 181 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×#16812699	C₂	1	1	yes	maximal, characteristic
2	#20295311	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16812630	C₂	1	2	yes	maximal
4	C₂×C₄⋊#11724784	C₂	1	1	yes	maximal, characteristic
5	C₂×#16812623	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₈	—	2	1	no	—
8	C₂²×#11724784	C₂²	1	2	yes	—
9	#16812699	—	2	2	no	—
10	#16812699	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
12	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
13	#16812630	C₂²	1	8	yes	—
14	C₄⋊#11724784	C₂²	1	4	yes	—
15	#16812623	C₂²	1	8	yes	—
16	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₀₉	—	2	2	no	—
18	C₄×Q₈×C₁₀₉	C₂²	1	4	yes	—
19	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
20	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₁₀₉	—	2	1	no	—
22	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₁₈	—	2	4	no	—
24	C₂×Dic₂₁₈	—	4	4	no	—
25	C₂²×Dic₁₀₉	—	2	2	no	—
26	C₂×#11724784	—	2	4	no	—
27	C₂×#11724784	C₂³	1	4	yes	—
28	C₄⋊Dic₁₀₉	—	2	4	no	—
29	C₄⋊Dic₁₀₉	—	2	2	no	—
30	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
31	C₂×Q₈×C₁₀₉	—	2	1	no	—
32	C₂×Q₈×C₁₀₉	—	2	2	no	—
33	C₂×Q₈×C₁₀₉	Q₈	1	4	yes	complemented
34	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₀₉	—	2	1	no	—
36	C₂²×C₄×C₁₀₉	—	2	1	no	—
37	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₁₀₉	—	2	4	no	—
41	C₄⋊C₄×C₁₀₉	—	2	1	no	—
42	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
43	C₄²×C₁₀₉	—	2	4	no	—
44	C₄²×C₁₀₉	D₄	1	2	yes	—
45	C₄²×C₁₀₉	C₂³	1	2	yes	—
46	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₁₀₉	—	2	4	no	—
48	Q₈×C₁₀₉	—	2	2	no	—
49	Q₈×C₁₀₉	C₂×Q₈	1	4	yes	complemented
50	C₂×C₄×C₁₀₉	—	2	4	no	—
51	C₂×C₄×C₁₀₉	—	2	2	no	—
52	C₂×C₄×C₁₀₉	—	2	2	no	—
53	C₂×C₄×C₁₀₉	—	2	2	no	—
54	C₂×C₄×C₁₀₉	—	4	2	no	—
55	C₂×C₄×C₁₀₉	SD₁₆	1	4	yes	—
56	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₁₀₉	#41	1	2	yes	—
61	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₁₀₉	—	2	4	no	—
63	C₂×Dic₁₀₉	—	4	4	no	—
64	Dic₂₁₈	—	4	8	no	—
65	#11724784	—	2	4	no	—
66	C₂²×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
67	C₂²×C₁₀₉	C₂×SD₁₆	1	2	yes	—
68	C₂²×C₁₀₉	#137	1	2	yes	—
69	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
70	C₄×C₁₀₉	—	2	4	no	—
71	C₄×C₁₀₉	—	2	2	no	—
72	C₄×C₁₀₉	—	4	2	no	—
73	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
74	C₄×C₁₀₉	C₂×SD₁₆	1	4	yes	—
75	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
76	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
77	Dic₁₀₉	—	4	4	no	—
78	C₂×C₁₀₉	#475	1	4	yes	complemented
79	C₂×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₁₀₉	C₂²×SD₁₆	1	1	yes	characteristic
81	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
82	C₂×#475	—	109	1	no	maximal, Sylow
83	C₁₀₉	C₂×#475	1	1	yes	characteristic, complemented, Sylow
84	C₂×C₄⋊Q₈	—	109	1	no	—
85	C₂×C₄×Q₈	D₁₀₉	1	1	yes	characteristic
86	#475	—	109	8	no	—
87	C₂×#106	—	109	2	no	—
88	C₂×C₄⋊C₈	—	109	1	no	—
89	C₂×#103	—	109	2	no	—
90	C₂²×Q₈	—	218	1	no	—
91	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
92	C₂²×C₈	—	109	2	no	—
93	C₄⋊Q₈	—	109	4	no	—
94	C₄⋊Q₈	—	218	2	no	—
95	C₄×Q₈	—	2	2	no	—
96	C₄×Q₈	C₂×D₁₀₉	1	4	yes	—
97	C₂×C₄⋊C₄	—	2	1	no	—
98	C₂×C₄⋊C₄	—	109	2	no	—
99	C₂×C₄⋊C₄	—	218	1	no	—
100	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
101	C₂×C₄²	—	2	1	no	—
102	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
103	#106	—	109	8	no	—
104	C₄⋊C₈	—	109	4	no	—
105	#103	—	109	8	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	2	2	no	—
108	C₂×Q₈	—	436	4	no	—
109	C₂×Q₈	—	218	4	no	—
110	C₂×Q₈	Dic₂₁₈	1	4	yes	complemented
111	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	218	2	no	—
115	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
116	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
118	C₂×C₈	—	109	4	no	—
119	C₂×C₈	—	218	4	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	109	4	no	—
123	C₄⋊C₄	—	218	2	no	—
124	C₄⋊C₄	—	218	4	no	—
125	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
126	C₄²	—	2	4	no	—
127	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
128	C₄²	C₂²×D₁₀₉	1	2	yes	—
129	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
130	Q₈	—	2	4	no	—
131	Q₈	—	2	2	no	—
132	Q₈	—	436	8	no	—
133	Q₈	C₂×Dic₂₁₈	1	4	yes	complemented
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	436	4	no	—
140	C₂×C₄	—	218	4	no	—
141	C₂×C₄	#13182016	1	4	yes	—
142	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
145	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
146	C₂×C₄	#13182029	1	2	yes	—
147	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
148	C₈	—	218	4	no	—
149	C₂²	#16812678	1	2	yes	—
150	C₂²	C₂×#13182016	1	2	yes	—
151	C₂²	#16812729	1	2	yes	—
152	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
153	C₄	—	2	4	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	—	436	4	no	—
157	C₄	#16812678	1	2	yes	—
158	C₄	C₂×#13182016	1	4	yes	—
159	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
160	C₄	C₂×#13182029	1	1	yes	characteristic
161	C₂	#20295311	1	4	yes	direct product factor
162	C₂	C₂×#16812678	1	1	yes	characteristic
163	C₂	C₂²×#13182016	1	1	yes	characteristic
164	C₂	C₂×#16812729	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812699

C₂

yes

maximal, characteristic

#20295311

C₂

yes

maximal, direct product factor

C₂×#16812630

C₂

yes

maximal

C₂×C₄⋊#11724784

C₂

yes

maximal, characteristic

C₂×#16812623

C₂

yes

maximal

C₂×C₄×Q₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

—

C₂²×#11724784

C₂²

yes

—

#16812699

—

#16812699

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

#16812630

C₂²

yes

—

C₄⋊#11724784

C₂²

yes

—

#16812623

C₂²

yes

—

C₂²×Q₈×C₁₀₉

C₂²

yes

characteristic

C₄×Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

—

C₂×#11724784

—

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

Q₈

yes

complemented

C₂×Q₈×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

C₂×Q₈

yes

complemented

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

SD₁₆

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

#11724784

—

C₂²×C₁₀₉

C₂²⋊Q₈

yes

—

C₂²×C₁₀₉

C₂×SD₁₆

yes

—

C₂²×C₁₀₉

#137

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

C₂×SD₁₆

yes

—

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

C₄×C₁₀₉

C₂×#41

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

#475

yes

complemented

C₂×C₁₀₉

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₁₀₉

C₂²×SD₁₆

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

C₂×#475

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

C₂×C₄×Q₈

D₁₀₉

yes

characteristic

#475

—

109

—

C₂×#106

—

109

—

C₂×C₄⋊C₈

—

109

—

C₂×#103

—

109

—

C₂²×Q₈

—

218

—

C₂²×Q₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

218

—

C₄×Q₈

—

C₄×Q₈

C₂×D₁₀₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

100

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

101

C₂×C₄²

—

102

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

103

#106

—

109

—

104

C₄⋊C₈

—

109

—

105

#103

—

109

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

436

—

109

C₂×Q₈

—

218

—

110

C₂×Q₈

Dic₂₁₈

yes

complemented

111

C₂×Q₈

C₂²×D₁₀₉

yes

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

218

—

115

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

116

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

117

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

118

C₂×C₈

—

109

—

119

C₂×C₈

—

218

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

109

—

123

C₄⋊C₄

—

218

—

124

C₄⋊C₄

—

218

—

125

C₄⋊C₄

C₂²×D₁₀₉

yes

—

126

C₄²

—

127

C₄²

C₂²⋊D₁₀₉

yes

—

128

C₄²

C₂²×D₁₀₉

yes

—

129

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

130

Q₈

—

131

Q₈

—

132

Q₈

—

436

—

133

Q₈

C₂×Dic₂₁₈

yes

complemented

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

436

—

140

C₂×C₄

—

218

—

141

C₂×C₄

#13182016

yes

—

142

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

143

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

145

C₂×C₄

C₂×Dic₂₁₈

yes

—

146

C₂×C₄

#13182029

yes

—

147

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

148

C₈

—

218

—

149

C₂²

#16812678

yes

—

150

C₂²

C₂×#13182016

yes

—

151

C₂²

#16812729

yes

—

152

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

436

—

157

C₄

#16812678

yes

—

158

C₄

C₂×#13182016

yes

—

159

C₄

C₂²×Dic₂₁₈

yes

characteristic

160

C₄

C₂×#13182029

yes

characteristic

161

C₂

#20295311

yes

direct product factor

162

C₂

C₂×#16812678

yes

characteristic

163

C₂

C₂²×#13182016

yes

characteristic

164

C₂

C₂×#16812729

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×C₄×Q₈×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	C₂×C₄×Q₈	D₁₀₉
hypocenter	C₁₀₉	C₂×#475
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂×C₄×Q₈×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

hypercenter

C₂×C₄×Q₈

D₁₀₉

hypocenter

C₁₀₉

C₂×#475

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
2. #20295311	C₂	direct (N × H)
33. C₂×Q₈×C₁₀₉	Q₈	semidirect
49. Q₈×C₁₀₉	C₂×Q₈	semidirect
78. C₂×C₁₀₉	#475	semidirect
83. C₁₀₉	C₂×#475	semidirect
110. C₂×Q₈	Dic₂₁₈	semidirect
133. Q₈	C₂×Dic₂₁₈	semidirect
161. C₂	#20295311	direct (N × H)

H ≅ G/N

type

2. #20295311

C₂

direct (N × H)

33. C₂×Q₈×C₁₀₉

Q₈

semidirect

49. Q₈×C₁₀₉

C₂×Q₈

semidirect

78. C₂×C₁₀₉

#475

semidirect

83. C₁₀₉

C₂×#475

semidirect

110. C₂×Q₈

Dic₂₁₈

semidirect

133. Q₈

C₂×Dic₂₁₈

semidirect

161. C₂

#20295311

direct (N × H)

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

subgroup	index
C₂×#16812699	2
#20295311	2
C₂×#16812630	2
C₂×C₄⋊#11724784	2
C₂×#16812623	2
C₂×C₄×Q₈×C₁₀₉	2
C₂×#475	109

subgroup

index

C₂×C₄×Q₈×C₁₀₉

109

Subgroups — C₂×#20295311 #28372797

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295311 #28372797

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295311 #28372797