Finite Group Databasebeta

← #28346401

Subgroups — #28346401

Order 13696.

462 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812083C211yesmaximal, characteristic
2#20292951C214yesmaximal, complemented
3#20292945C212yesmaximal, complemented
4#20292940C212yesmaximal, complemented
5#20292670C212yesmaximal, complemented
6#20292631C212yesmaximal, complemented
7#20292591C211yesmaximal, characteristic, complemented
8C2×#117×C107C211yesmaximal, characteristic, complemented
9C22×D428C2211yescharacteristic
10C22×C22⋊D107—21no—
11C2×#13133802C2212yes—
12#16812083—21no—
13#16812083—21no—
14#16812083C2212yes—
15#16812083C2212yes—
16C2×C4⋊Dic107C2211yescharacteristic
17C2×(C2×D107)⋊C4—21no—
18#16812042C2214yescomplemented
19#16812040C2214yescomplemented
20#16812029C2214yescomplemented
21#16812029C2214yescomplemented
22C22⋊(C107⋊C8)C2212yescomplemented
23C22⋊(C107⋊C8)C2212yescomplemented
24C23×C4×C107C2211yescharacteristic
25#117×C107—21no—
26#117×C107—21no—
27#117×C107C2212yes—
28#117×C107C2212yescomplemented
29C2×C4⋊C4×C107C2211yescharacteristic, complemented
30C2×C4⋊C4×C107C2211yescharacteristic, complemented
31C2×C22⋊C4×C107—21no—
32C2×C42×C107—21no—
33C23×D107—21no—
34C2×D428—24no—
35C2×D428—21no—
36C2×D428C2312yes—
37C2×C22⋊D107—22no—
38C2×C22⋊D107—22no—
39C2×C22⋊D107—44no—
40C22×Dic107—21no—
41#13133802—24no—
42C2×C107⋊C8C2314yes—
43C4⋊Dic107—21no—
44C4⋊Dic107C2312yes—
45(C2×D107)⋊C4—22no—
46(C2×D107)⋊C4—22no—
47C24×C107D411yescharacteristic
48C22×C4×C107—24no—
49C22×C4×C107—21no—
50C22×C4×C107—21no—
51C22×C4×C107D411yescharacteristic
52C22×C4×C107D411yescharacteristic
53C22×C4×C107D411yescharacteristic
54C22×C4×C107C2311yescharacteristic
55C22×C4×C107C2311yescharacteristic
56C22×C4×C107C2311yescharacteristic
57C4⋊C4×C107—21no—
58C4⋊C4×C107—21no—
59C4⋊C4×C107C2312yes—
60C4⋊C4×C107C2312yes—
61C22⋊C4×C107—22no—
62C22⋊C4×C107—22no—
63C42×C107—22no—
64C42×C107—22no—
65C23×C107—22no—
66C23×C107—22no—
67C23×C107—41no—
68C23×C107C2×D411yescharacteristic
69C23×C107C2×D411yescharacteristic
70C23×C107C2×D411yescharacteristic
71C22×D107—22no—
72C22×D107—42no—
73C22×D107—42no—
74C22×D107—42no—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—42no—
83C2×C4×C107—42no—
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107#4114yes—
88C2×C4×C107C2411yescharacteristic
89C22⋊D107—44no—
90C22⋊D107—44no—
91C2×Dic107—22no—
92C2×Dic107—42no—
93D428—24no—
94D428—42no—
95C107⋊C8—24no—
96C22×C107—21no—
97C22×C107—21no—
98C22×C107—21no—
99C22×C107—21no—
100C22×C107—21no—
101C22×C107—21no—
102C22×C107—21no—
103C22×C107—44no—
104C22×C107#12312yes—
105C22×C107AGL(1,ℤ/8)12yes—
106C22×C107C22×D411yescharacteristic
107C2×D107—42no—
108C2×D107—42no—
109C2×D107—42no—
110C2×D107—84no—
111C4×C107—21no—
112C4×C107—21no—
113C4×C107—42no—
114C4×C107—42no—
115C4×C107#12312yes—
116C4×C107C2×#4112yes—
117Dic107—42no—
118C2×C107—22no—
119C2×C107—22no—
120C2×C107—41no—
121C2×C107C2×#12311yescharacteristic
122C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
123C2×C107#57511yescharacteristic
124D107—82no—
125#3094—1071nomaximal, Sylow
126C107#309411yescharacteristic, complemented, Sylow
127C2×#121—1071no—
128C2×#117D10711yescharacteristic, complemented
129#482—1074no—
130#481—1072no—
131#480—1072no—
132#428—1072no—
133#418—1072no—
134#407—1071no—
135C22×D4—1071no—
136C22×D4—2141no—
137C23×C4C2×D10711yescharacteristic
138C2×M16—1072no—
139#121—1072no—
140#121—1072no—
141#121—2141no—
142#121—2141no—
143#117—21no—
144#117—21no—
145#117C2×D10712yes—
146#117C2×D10712yescomplemented
147C2×C4⋊C4—1071no—
148C2×C4⋊C4C2×D10711yescharacteristic, complemented
149C2×C4⋊C4C2×D10711yescharacteristic, complemented
150C2×C22⋊C4—21no—
151C2×C22⋊C4—2141no—
152C2×C42—21no—
153#107—1074no—
154#106—1074no—
155#102—1074no—
156#102—1074no—
157C22⋊C8—1072no—
158C22⋊C8—1072no—
159C24—2141no—
160C24C22⋊D10711yescharacteristic
161C2×D4—1072no—
162C2×D4—4284no—
163C2×D4—2142no—
164C2×D4—2142no—
165C2×D4—2141no—
166C2×D4—2144no—
167C22×C4—24no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—2141no—
171C22×C4C22⋊D10711yescharacteristic
172C22×C4C22⋊D10711yescharacteristic
173C22×C4C22⋊D10711yescharacteristic
174C22×C4C22×D10711yescharacteristic
175C22×C4C22×D10711yescharacteristic
176C22×C4C22×D10711yescharacteristic
177M16—2144no—
178C2×C8—1074no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—1072no—
182C4⋊C4—2141no—
183C4⋊C4C22×D10712yes—
184C4⋊C4C22×D10712yes—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—2142no—
188C22⋊C4—2142no—
189C42—22no—
190C42—22no—
191C23—22no—
192C23—22no—
193C23—41no—
194C23—4282no—
195C23—4282no—
196C23—4282no—
197C23—2142no—
198C23C2×C22⋊D10711yescharacteristic
199C23C2×C22⋊D10711yescharacteristic
200C23C2×C22⋊D10711yescharacteristic
201D4—4284no—
202D4—4284no—
203D4—4282no—
204D4—2144no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—4282no—
215C2×C4—2142no—
216C2×C4C2×C22⋊D10711yescharacteristic
217C2×C4C2×C22⋊D10711yescharacteristic
218C2×C4C2×C22⋊D10711yescharacteristic
219C2×C4#1313381814yes—
220C2×C4C23×D10711yescharacteristic
221C8—2144no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—8564no—
231C22—4282no—
232C22—4282no—
233C22—4282no—
234C22#1681209212yes—
235C22#1681213412yes—
236C22C22×C22⋊D10711yescharacteristic
237C4—21no—
238C4—21no—
239C4—42no—
240C4—42no—
241C4—4282no—
242C4#1681209212yes—
243C4C2×#1313381812yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—8562no—
248C2C2×#1681209211yescharacteristic
249C2C2×#1681213411yescharacteristic
250C2#2029342511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#117×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812083C2
hypercenterC2×#117D107
hypocenterC107#3094
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20292951C2semidirect
3. #20292945C2semidirect
4. #20292940C2semidirect
5. #20292670C2semidirect
6. #20292631C2semidirect
7. #20292591C2semidirect
8. C2×#117×C107C2semidirect
18. #16812042C22semidirect
19. #16812040C22semidirect
20. #16812029C22semidirect
21. #16812029C22semidirect
22. C22⋊(C107⋊C8)C22semidirect
23. C22⋊(C107⋊C8)C22semidirect
28. #117×C107C22semidirect
29. C2×C4⋊C4×C107C22semidirect
30. C2×C4⋊C4×C107C22semidirect
126. C107#3094semidirect
128. C2×#117D107semidirect
146. #117C2×D107semidirect
148. C2×C4⋊C4C2×D107semidirect
149. C2×C4⋊C4C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120832
#202929512
#202929452
#202929402
#202926702
#202926312
#202925912
C2×#117×C1072
#3094107