Finite Group Databasebeta

← #28346422

Subgroups — #28346422

Order 13696.

418 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#20293196C211yesmaximal, characteristic
2#20293107C211yesmaximal, characteristic, complemented
3#20292954C212yesmaximal, complemented
4#20292953C212yesmaximal, complemented
5#20292947C212yesmaximal, complemented
6#20292942C212yesmaximal, complemented
7#20292670C211yesmaximal, characteristic, complemented
8#20292631C211yesmaximal, characteristic, complemented
9C2×#16812031C211yesmaximal, characteristic, complemented
10#20292593C211yesmaximal, characteristic, complemented
11#427×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×D428C2211yescharacteristic
14C2×#13133802C2211yescharacteristic
15#16812106—21no—
16#16812097—21no—
17#16812084C2212yes—
18#16812084C2212yes—
19#16812079—22no—
20C4×Dic214—21no—
21C4×D428—21no—
22(C22⋊D107)⋊C4—22no—
23#16812067C2212yescomplemented
24C2×C4⋊Dic107C2211yescharacteristic
25C4⋊C4×D107C2212yescomplemented
26#16812042C2212yescomplemented
27#16812040C2212yescomplemented
28#16812031C2212yes—
29#16812031C2212yes—
30#16812029C2212yes—
31#16812029C2212yes—
32(C2×D107)⋊C8C2212yescomplemented
33(C2×D107)⋊C8C2212yescomplemented
34C22×C8×C107C2211yescharacteristic
35#117×C107C2211yescharacteristic
36#117×C107C2211yescharacteristic
37#107×C107C2211yescharacteristic, complemented
38#107×C107C2211yescharacteristic, complemented
39#106×C107C2211yescharacteristic, complemented
40#106×C107C2211yescharacteristic, complemented
41C23×D107—21no—
42#13133818—22no—
43#13133818—22no—
44C2×Dic214D411yescharacteristic
45C2×D428—22no—
46C2×D428—22no—
47C2×D428—21no—
48C2×D428—22no—
49C2×D428—22no—
50C2×D428D411yescharacteristic
51C2×D428C2312yes—
52C2×C22⋊D107D412yes—
53C22×Dic107—21no—
54C2×C4×D107—22no—
55C2×C4×D107C2312yes—
56#13133802—22no—
57C2×C107⋊C8C2312yes—
58C4⋊Dic107—21no—
59C4⋊Dic107C2312yes—
60Dic107⋊C4—22no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—22no—
64C4×Dic107—22no—
65C22×C4×C107C2311yescharacteristic
66C2×C8×C107—22no—
67C2×C8×C107C2311yescharacteristic
68C2×C8×C107C2311yescharacteristic
69C4⋊C4×C107C2311yescharacteristic
70C4⋊C4×C107C2311yescharacteristic
71C4⋊C4×C107C2311yescharacteristic
72C4⋊C4×C107C2311yescharacteristic
73C22⋊C4×C107—21no—
74C22⋊C4×C107—21no—
75C42×C107—21no—
76C42×C107—21no—
77C23×C107C2×D411yescharacteristic
78C22×D107—22no—
79C22×D107—42no—
80C22×D107—81no—
81C22×D107—81no—
82C22×D107C2×D412yes—
83C2×C4×C107—21no—
84C2×C4×C107—21no—
85C2×C4×C107—21no—
86C2×C4×C107—21no—
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107#4112yes—
89C2×C4×C107#4112yes—
90C2×C4×C107C2411yescharacteristic
91C22⋊D107—24no—
92C2×Dic107—22no—
93C2×Dic107—81no—
94C2×Dic107C2×D412yes—
95D428—22no—
96D428—22no—
97D428—22no—
98D428—42no—
99D428—82no—
100D428—82no—
101C4×D107—22no—
102C4×D107—22no—
103C4×D107—82no—
104Dic214—22no—
105C8×C107—21no—
106C8×C107—21no—
107C107⋊C8—22no—
108C22×C107—21no—
109C22×C107—21no—
110C22×C107#12312yes—
111C22×C107C22×D411yescharacteristic
112C2×D107—22no—
113C2×D107—22no—
114C2×D107—22no—
115C2×D107—42no—
116C2×D107—42no—
117C2×D107—42no—
118C2×D107—84no—
119C4×C107—41no—
120C4×C107—41no—
121C4×C107—41no—
122C4×C107—41no—
123C4×C107#12312yes—
124C4×C107C2×#4111yescharacteristic
125C4×C107C2×#4111yescharacteristic
126Dic107—22no—
127Dic107—42no—
128C2×C107—22no—
129C2×C107C2×#12311yescharacteristic
130C2×C107#57611yescharacteristic
131C2×C107#57711yescharacteristic
132D107—42no—
133D107—82no—
134#3096—1071nomaximal, Sylow
135C107#309611yescharacteristic, complemented, Sylow
136#535—1071no—
137#520—1071no—
138#482—1072no—
139#482—1072no—
140#481—1072no—
141#480—1072no—
142#428—1071no—
143#427D10711yescharacteristic, complemented
144#418—1071no—
145C2×#102—1071no—
146#408—1071no—
147C2×#41—1071no—
148C22×D4—1071no—
149C2×M16—1071no—
150C22×C8C2×D10711yescharacteristic
151#127—2141no—
152#124—2141no—
153#121—1072no—
154#121—1072no—
155#120—2142no—
156C4×Q8—2141no—
157C4×D4—2141no—
158C4×D4—2142no—
159#117—1072no—
160#117C2×D10711yescharacteristic
161#117C2×D10711yescharacteristic
162C2×C4⋊C4—1071no—
163C2×C4⋊C4—1072no—
164#107—1072no—
165#107C2×D10711yescharacteristic, complemented
166#107C2×D10711yescharacteristic, complemented
167#106—1072no—
168#106C2×D10711yescharacteristic, complemented
169#106C2×D10711yescharacteristic, complemented
170#102—1072no—
171#102—1072no—
172#102—1072no—
173#102—1072no—
174C22⋊C8—1072no—
175C22⋊C8—1072no—
176C24—2141no—
177#41—2142no—
178#41—2142no—
179C2×Q8—1071no—
180C2×D4—1072no—
181C2×D4—1071no—
182C2×D4—1072no—
183C2×D4—2142no—
184C2×D4—2142no—
185C2×D4—2141no—
186C2×D4—2142no—
187C2×D4—2142no—
188C22×C4—1072no—
189C22×C4—2142no—
190C22×C4—2141no—
191C22×C4C22×D10711yescharacteristic
192M16—2142no—
193C2×C8—22no—
194C2×C8—1072no—
195C2×C8C22×D10711yescharacteristic
196C2×C8C22×D10711yescharacteristic
197C4⋊C4—1072no—
198C4⋊C4—2142no—
199C4⋊C4—2141no—
200C4⋊C4C22×D10711yescharacteristic
201C4⋊C4C22×D10711yescharacteristic
202C4⋊C4C22×D10711yescharacteristic
203C4⋊C4C22×D10711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—2142no—
207C22⋊C4—2142no—
208C22⋊C4—2142no—
209C42—21no—
210C42—21no—
211C42—2142no—
212C23—8561no—
213C23—8561no—
214C23—1072no—
215C23—4282no—
216C23—2142no—
217C23D4×D10711yescharacteristic
218Q8—2142no—
219D4—8562no—
220D4—8562no—
221D4—4282no—
222D4—2144no—
223D4—2142no—
224D4—2142no—
225D4—2142no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—8562no—
231C2×C4—8561no—
232C2×C4—1072no—
233C2×C4—2142no—
234C2×C4—2142no—
235C2×C4—2142no—
236C2×C4D4×D10711yescharacteristic
237C2×C4#1313381812yes—
238C2×C4#1313381912yes—
239C2×C4C23×D10711yescharacteristic
240C8—21no—
241C8—21no—
242C8—2142no—
243C22—21no—
244C22—21no—
245C22—8564no—
246C22—4282no—
247C22—4282no—
248C22—4282no—
249C22—2142no—
250C22—2142no—
251C22—2142no—
252C22#1681209412yes—
253C22C2×D4×D10711yescharacteristic
254C4—41no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—4282no—
259C4—2142no—
260C4#1681209412yes—
261C4C2×#1313381811yescharacteristic
262C4C2×#1313381911yescharacteristic
263C2—22no—
264C2—8562no—
265C2—4282no—
266C2C2×#1681209411yescharacteristic
267C2#2029342911yescharacteristic
268C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#427×C107C2
socleC22×C107C22×D4
prime radical#20293196C2
hypercenter#427D107
hypocenterC107#3096
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
2. #20293107C2semidirect
3. #20292954C2semidirect
4. #20292953C2semidirect
5. #20292947C2semidirect
6. #20292942C2semidirect
7. #20292670C2semidirect
8. #20292631C2semidirect
9. C2×#16812031C2semidirect
10. #20292593C2semidirect
11. #427×C107C2semidirect
23. #16812067C22semidirect
25. C4⋊C4×D107C22semidirect
26. #16812042C22semidirect
27. #16812040C22semidirect
32. (C2×D107)⋊C8C22semidirect
33. (C2×D107)⋊C8C22semidirect
37. #107×C107C22semidirect
38. #107×C107C22semidirect
39. #106×C107C22semidirect
40. #106×C107C22semidirect
135. C107#3096semidirect
143. #427D107semidirect
165. #107C2×D107semidirect
166. #107C2×D107semidirect
168. #106C2×D107semidirect
169. #106C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931962
#202931072
#202929542
#202929532
#202929472
#202929422
#202926702
#202926312
C2×#168120312
#202925932
#427×C1072
#3096107