Group #28362171 — subgroups

Subgroups — #28362171

Order 13952.

206 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic
2	#20295405	C₂	1	1	yes	maximal, characteristic
3	#20295318	C₂	1	1	yes	maximal, characteristic
4	#20294774	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#11724784	C₂	1	1	yes	maximal, characteristic
6	#20294756	C₂	1	1	yes	maximal, characteristic
7	#477×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182019	—	2	2	no	—
9	#16812699	—	2	1	no	—
10	#16812699	C₂²	1	1	yes	characteristic
11	#16812698	—	2	2	no	—
12	#16812698	—	2	1	no	—
13	Q₈×Dic₁₀₉	—	2	1	no	—
14	C₄×Dic₂₁₈	—	2	1	no	—
15	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
16	#16812632	—	2	1	no	—
17	C₄⋊#11724784	C₂²	1	1	yes	characteristic
18	#16812622	—	2	1	no	—
19	C₄×#11724784	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
22	#107×C₁₀₉	—	2	1	no	—
23	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	#103×C₁₀₉	—	2	1	no	—
25	C₂×Dic₂₁₈	—	2	2	no	—
26	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
27	C₂×Dic₂₁₈	D₄	1	2	yes	—
28	#13182019	—	4	4	no	—
29	C₂×#11724784	—	2	1	no	—
30	C₂×#11724784	—	2	1	no	—
31	C₄⋊Dic₁₀₉	—	2	2	no	—
32	C₄⋊Dic₁₀₉	—	2	1	no	—
33	C₄⋊Dic₁₀₉	—	2	1	no	—
34	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
35	#13182008	—	2	2	no	—
36	#13182008	—	2	1	no	—
37	C₄×Dic₁₀₉	—	2	2	no	—
38	C₄×Dic₁₀₉	—	2	1	no	—
39	C₂×Q₈×C₁₀₉	—	2	1	no	—
40	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
41	C₂×C₈×C₁₀₉	—	2	1	no	—
42	C₄⋊C₄×C₁₀₉	—	2	1	no	—
43	C₄⋊C₄×C₁₀₉	—	2	1	no	—
44	C₄⋊C₄×C₁₀₉	—	2	1	no	—
45	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
46	C₄²×C₁₀₉	—	2	1	no	—
47	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
48	Q₈×C₁₀₉	—	2	1	no	—
49	Q₈×C₁₀₉	—	4	2	no	—
50	Q₈×C₁₀₉	Q₁₆	1	2	yes	complemented
51	C₂×C₄×C₁₀₉	—	2	1	no	—
52	C₂×C₄×C₁₀₉	—	2	1	no	—
53	C₂×C₄×C₁₀₉	—	4	1	no	—
54	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₁₀₉	—	2	1	no	—
58	C₂×Dic₁₀₉	—	2	2	no	—
59	C₂×Dic₁₀₉	—	2	1	no	—
60	Dic₂₁₈	—	2	1	no	—
61	Dic₂₁₈	—	4	2	no	—
62	Dic₂₁₈	—	8	2	no	—
63	Dic₂₁₈	Q₁₆	1	2	yes	complemented
64	C₈×C₁₀₉	—	4	1	no	—
65	#11724784	—	2	2	no	—
66	#11724784	—	4	1	no	—
67	C₂²×C₁₀₉	#120	1	1	yes	characteristic
68	C₄×C₁₀₉	—	2	2	no	—
69	C₄×C₁₀₉	—	2	1	no	—
70	C₄×C₁₀₉	—	4	1	no	—
71	C₄×C₁₀₉	—	8	1	no	—
72	C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
73	C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₁₀₉	#137	1	1	yes	characteristic
75	C₄×C₁₀₉	#137	1	1	yes	characteristic
76	Dic₁₀₉	—	2	2	no	—
77	Dic₁₀₉	—	4	2	no	—
78	Dic₁₀₉	—	4	1	no	—
79	C₂×C₁₀₉	#451	1	1	yes	characteristic
80	C₂×C₁₀₉	#451	1	1	yes	characteristic
81	C₂×C₁₀₉	#453	1	1	yes	characteristic
82	#1633	—	109	1	no	maximal, Sylow
83	C₁₀₉	#1633	1	1	yes	characteristic, complemented, Sylow
84	Q₈²	—	109	1	no	—
85	#494	—	109	1	no	—
86	#477	—	109	1	no	—
87	#477	D₁₀₉	1	1	yes	characteristic
88	#333	—	109	1	no	—
89	Q₈⋊C₈	—	109	1	no	—
90	Q₈⋊C₈	—	109	1	no	—
91	C₂×Q₁₆	—	218	2	no	—
92	C₄⋊Q₈	—	109	1	no	—
93	C₄⋊Q₈	—	218	1	no	—
94	C₄⋊Q₈	—	218	2	no	—
95	C₄⋊Q₈	—	218	1	no	—
96	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
97	C₄×Q₈	—	109	1	no	—
98	C₄×Q₈	—	218	1	no	—
99	C₄×Q₈	—	218	1	no	—
100	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
101	#107	—	2	1	no	—
102	#107	—	218	1	no	—
103	C₄⋊C₈	—	109	1	no	—
104	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
105	#103	—	2	1	no	—
106	#103	—	218	1	no	—
107	C₄×C₈	—	109	1	no	—
108	C₂×Q₈	—	2	1	no	—
109	C₂×Q₈	—	109	1	no	—
110	C₂×Q₈	—	109	2	no	—
111	C₂×Q₈	—	218	2	no	—
112	C₂×Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
113	Q₁₆	—	436	4	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	218	1	no	—
116	C₂×C₈	—	218	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	109	1	no	—
121	C₄⋊C₄	—	218	2	no	—
122	C₄⋊C₄	—	218	2	no	—
123	C₄⋊C₄	—	218	1	no	—
124	C₄⋊C₄	—	218	1	no	—
125	C₄⋊C₄	—	218	1	no	—
126	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
127	C₄²	—	2	1	no	—
128	C₄²	—	218	2	no	—
129	C₄²	—	218	1	no	—
130	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
131	Q₈	—	2	1	no	—
132	Q₈	—	4	2	no	—
133	Q₈	—	872	2	no	—
134	Q₈	—	109	2	no	—
135	Q₈	—	436	2	no	—
136	Q₈	—	218	1	no	—
137	Q₈	#13182019	1	2	yes	complemented
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	218	1	no	—
142	C₂×C₄	—	218	2	no	—
143	C₂×C₄	—	218	1	no	—
144	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
145	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
147	C₈	—	4	1	no	—
148	C₈	—	436	1	no	—
149	C₈	—	218	2	no	—
150	C₂²	#16812667	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	1	no	—
153	C₄	—	4	1	no	—
154	C₄	—	8	1	no	—
155	C₄	—	436	1	no	—
156	C₄	—	436	2	no	—
157	C₄	—	218	2	no	—
158	C₄	Q₁₆×D₁₀₉	1	1	yes	characteristic
159	C₄	C₂×#13182019	1	1	yes	characteristic
160	C₄	#16812729	1	1	yes	characteristic
161	C₄	#16812731	1	1	yes	characteristic
162	C₂	#20295180	1	1	yes	characteristic
163	C₂	#20295181	1	1	yes	characteristic
164	C₂	#20295192	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₂₁₈

C₂

yes

maximal, characteristic

#20295405

C₂

yes

maximal, characteristic

#20295318

C₂

yes

maximal, characteristic

#20294774

C₂

yes

maximal, characteristic

Q₈⋊#11724784

C₂

yes

maximal, characteristic

#20294756

C₂

yes

maximal, characteristic

#477×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182019

—

#16812699

—

#16812699

C₂²

yes

characteristic

#16812698

—

#16812698

—

Q₈×Dic₁₀₉

—

C₄×Dic₂₁₈

—

C₄×Dic₂₁₈

C₂²

yes

characteristic

#16812632

—

C₄⋊#11724784

C₂²

yes

characteristic

#16812622

—

C₄×#11724784

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

C₄×Q₈×C₁₀₉

C₂²

yes

characteristic

#107×C₁₀₉

—

C₄⋊C₈×C₁₀₉

C₂²

yes

characteristic

#103×C₁₀₉

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

D₄

yes

characteristic

C₂×Dic₂₁₈

D₄

yes

—

#13182019

—

C₂×#11724784

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

D₄

yes

characteristic

#13182008

—

#13182008

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

Q₁₆

yes

complemented

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

Q₁₆

yes

complemented

C₈×C₁₀₉

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

#120

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×Q₁₆

yes

characteristic

C₄×C₁₀₉

C₂×Q₁₆

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1633

yes

characteristic, complemented, Sylow

Q₈²

—

109

—

#494

—

109

—

#477

—

109

—

#477

D₁₀₉

yes

characteristic

#333

—

109

—

Q₈⋊C₈

—

109

—

Q₈⋊C₈

—

109

—

C₂×Q₁₆

—

218

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

C₄×Q₈

—

109

—

C₄×Q₈

—

218

—

C₄×Q₈

—

218

—

100

C₄×Q₈

C₂×D₁₀₉

yes

characteristic

101

#107

—

102

#107

—

218

—

103

C₄⋊C₈

—

109

—

104

C₄⋊C₈

C₂×D₁₀₉

yes

characteristic

105

#103

—

106

#103

—

218

—

107

C₄×C₈

—

109

—

108

C₂×Q₈

—

109

C₂×Q₈

—

109

—

110

C₂×Q₈

—

109

—

111

C₂×Q₈

—

218

—

112

C₂×Q₈

C₂²⋊D₁₀₉

yes

characteristic

113

Q₁₆

—

436

—

114

C₂×C₈

—

115

C₂×C₈

—

218

—

116

C₂×C₈

—

218

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

109

—

121

C₄⋊C₄

—

218

—

122

C₄⋊C₄

—

218

—

123

C₄⋊C₄

—

218

—

124

C₄⋊C₄

—

218

—

125

C₄⋊C₄

—

218

—

126

C₄⋊C₄

C₂²⋊D₁₀₉

yes

characteristic

127

C₄²

—

128

C₄²

—

218

—

129

C₄²

—

218

—

130

C₄²

C₂²×D₁₀₉

yes

characteristic

131

Q₈

—

132

Q₈

—

133

Q₈

—

872

—

134

Q₈

—

109

—

135

Q₈

—

436

—

136

Q₈

—

218

—

137

Q₈

#13182019

yes

complemented

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

218

—

142

C₂×C₄

—

218

—

143

C₂×C₄

—

218

—

144

C₂×C₄

D₄×D₁₀₉

yes

characteristic

145

C₂×C₄

D₄×D₁₀₉

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

147

C₈

—

148

C₈

—

436

—

149

C₈

—

218

—

150

C₂²

#16812667

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

436

—

156

C₄

—

436

—

157

C₄

—

218

—

158

C₄

Q₁₆×D₁₀₉

yes

characteristic

159

C₄

C₂×#13182019

yes

characteristic

160

C₄

#16812729

yes

characteristic

161

C₄

#16812731

yes

characteristic

162

C₂

#20295180

yes

characteristic

163

C₂

#20295181

yes

characteristic

164

C₂

#20295192

yes

characteristic

	subgroup	G/H
center	C₂²	#16812667
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#477×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	#120
hypercenter	#477	D₁₀₉
hypocenter	C₁₀₉	#1633
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#16812667

commutator, cosocle

C₄²×C₁₀₉

C₂³

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#477×C₁₀₉

C₂

C₂²×C₁₀₉

D₁₀₉

C₁₀₉

G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
50. Q₈×C₁₀₉	Q₁₆	semidirect
63. Dic₂₁₈	Q₁₆	semidirect
83. C₁₀₉	#1633	semidirect
137. Q₈	#13182019	semidirect

H ≅ G/N

type

50. Q₈×C₁₀₉

Q₁₆

semidirect

63. Dic₂₁₈

Q₁₆

semidirect

83. C₁₀₉

#1633

semidirect

137. Q₈

#13182019

semidirect

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

subgroup	index
Q₈×Dic₂₁₈	2
#20295405	2
#20295318	2
#20294774	2
Q₈⋊#11724784	2
#20294756	2
#477×C₁₀₉	2
#1633	109

subgroup

index

Q₈×Dic₂₁₈

#477×C₁₀₉

109

Subgroups — #28362171

Conjugacy classes of subgroups

Complements

Maximal subgroups