Finite Group Databasebeta

← #28346391

Subgroups — #28346391

Order 13696.

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

normal subgroups only

This group has 414 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#20293134C211yesmaximal, characteristic
2C2×#16812067C211yesmaximal, characteristic, complemented
3#20292967C212yesmaximal, complemented
4#20292966C212yesmaximal, complemented
5#20292950C212yesmaximal, complemented
6#20292949C212yesmaximal, complemented
7#20292669C211yesmaximal, characteristic, complemented
8#20292667C211yesmaximal, characteristic, complemented
9#20292628C211yesmaximal, characteristic, complemented
10#20292590C211yesmaximal, characteristic, complemented
11#417×C107C211yesmaximal, characteristic, complemented
12C22×C4×D107C2211yescharacteristic
13C2×#13133802C2211yescharacteristic
14#16812090—21no—
15#16812085C2211yescharacteristic
16#16812085C2211yescharacteristic
17#16812080C2211yescharacteristic
18#16812080C2211yescharacteristic
19#16812077—21no—
20C4×C22⋊D107—21no—
21C4×C22⋊D107—21no—
22#16812068C2211yescharacteristic
23#16812067—21no—
24#16812067—21no—
25#16812067C2212yes—
26#16812067C2212yes—
27C2×C4⋊Dic107C2211yescharacteristic
28C2×(C2×D107)⋊C4—21no—
29C2×C4×Dic107—21no—
30#16812043C2212yescomplemented
31#16812042C2212yescomplemented
32#16812041C2212yescomplemented
33#16812040C2212yescomplemented
34#16812033C2212yescomplemented
35#16812030C2212yescomplemented
36(C2×D107)⋊C8C2212yes—
37(C2×D107)⋊C8C2212yes—
38C2×#41×C107C2211yescharacteristic
39C2×M16×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2211yescharacteristic
41#103×C107C2212yescomplemented
42#102×C107C2212yescomplemented
43C23×D107D411yescharacteristic
44C2×C22⋊D107—21no—
45C2×C22⋊D107—21no—
46C22×Dic107—21no—
47C22×Dic107D411yescharacteristic
48C2×C4×D107—22no—
49C2×C4×D107—22no—
50C2×C4×D107D411yescharacteristic
51C2×C4×D107D411yescharacteristic
52C2×C4×D107C2311yescharacteristic
53C2×C4×D107C2311yescharacteristic
54#13133802—22no—
55C2×C107⋊C8C2312yes—
56C4⋊Dic107—21no—
57C4⋊Dic107C2312yes—
58C4⋊Dic107C2311yescharacteristic
59C4⋊Dic107C2311yescharacteristic
60Dic107⋊C4—21no—
61Dic107⋊C4—21no—
62C22⋊Dic107—21no—
63C22⋊Dic107—21no—
64C22⋊Dic107—21no—
65(C2×D107)⋊C4—22no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—21no—
68(C2×D107)⋊C4—21no—
69C4×Dic107—21no—
70C4×Dic107—22no—
71C4×Dic107—22no—
72#41×C107—22no—
73#41×C107—22no—
74C2×Q8×C107C2311yescharacteristic
75C2×D4×C107—21no—
76C2×D4×C107C2311yescharacteristic
77C22×C4×C107—21no—
78C22×C4×C107—21no—
79C22×C4×C107C2311yescharacteristic
80M16×C107—22no—
81C2×C8×C107C2312yes—
82C4⋊C4×C107—21no—
83C4⋊C4×C107C2312yes—
84C23×C107—21no—
85C23×C107C2×D411yescharacteristic
86C22×D107—22no—
87C22×D107—41no—
88C22×D107—41no—
89C22×D107C2×D411yescharacteristic
90C22×D107C2×D411yescharacteristic
91Q8×C107—21no—
92Q8×C107—21no—
93D4×C107—21no—
94D4×C107—21no—
95D4×C107—42no—
96C2×C4×C107—22no—
97C2×C4×C107—21no—
98C2×C4×C107—42no—
99C2×C4×C107—42no—
100C2×C4×C107C2×D411yescharacteristic
101C2×C4×C107#4112yes—
102C2×C4×C107#4112yes—
103C2×C4×C107C2411yescharacteristic
104C22⋊D107—42no—
105C22⋊D107—42no—
106C2×Dic107—22no—
107C2×Dic107—22no—
108C2×Dic107—21no—
109C2×Dic107—21no—
110C2×Dic107—42no—
111C2×Dic107C2×D411yescharacteristic
112C2×Dic107C2×D411yescharacteristic
113C4×D107—22no—
114C4×D107—22no—
115C4×D107—21no—
116C4×D107—21no—
117C4×D107—21no—
118C4×D107—21no—
119C8×C107—22no—
120C107⋊C8—22no—
121C22×C107—21no—
122C22×C107—21no—
123C22×C107—41no—
124C22×C107—41no—
125C22×C107—41no—
126C22×C107#12312yes—
127C22×C107C22×D411yescharacteristic
128C2×D107—21no—
129C2×D107—21no—
130C2×D107—21no—
131C2×D107—21no—
132C2×D107—21no—
133C2×D107—21no—
134C2×D107—44no—
135C4×C107—41no—
136C4×C107—42no—
137C4×C107#12312yes—
138C4×C107C2×#4111yescharacteristic
139C4×C107C2×#4111yescharacteristic
140Dic107—21no—
141Dic107—21no—
142Dic107—42no—
143Dic107—41no—
144Dic107—41no—
145C2×C107—22no—
146C2×C107—81no—
147C2×C107C2×#12311yescharacteristic
148C2×C107#57511yescharacteristic
149C2×C107#57511yescharacteristic
150D107—41no—
151D107—41no—
152#3093—1071nomaximal, Sylow
153C107#309311yescharacteristic, complemented, Sylow
154#525—1071no—
155C2×#117—1071no—
156#485—1072no—
157#485—1072no—
158#482—1072no—
159#482—1072no—
160#428—1071no—
161#428—1071no—
162#417—1071no—
163#417D10711yescharacteristic, complemented
164#407—1071no—
165C2×#41C2×D10711yescharacteristic
166C23×C4—1071no—
167C2×M16—1071no—
168C2×M16C2×D10711yescharacteristic
169#123—2141no—
170C22⋊Q8—1071no—
171C22⋊Q8—1071no—
172#121—1071no—
173#121—1071no—
174#120—2141no—
175C4×D4—2141no—
176C4×D4—2141no—
177#117—1071no—
178#117—1072no—
179#117—1072no—
180#117—2141no—
181#117—2141no—
182C2×C4⋊C4—1071no—
183C2×C4⋊C4C2×D10711yescharacteristic
184C2×C22⋊C4—2141no—
185C2×C42—2141no—
186#107—1072no—
187#107—1072no—
188#106—1072no—
189#106—1072no—
190#103—1072no—
191#103C2×D10712yescomplemented
192#102—1072no—
193#102C2×D10712yescomplemented
194C22⋊C8—1072no—
195C22⋊C8—1072no—
196C24—1071no—
197#41—22no—
198#41—22no—
199C2×Q8C22×D10711yescharacteristic
200C2×D4—21no—
201C2×D4—2141no—
202C2×D4—2141no—
203C2×D4C22×D10711yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—1071no—
207C22×C4—1071no—
208C22×C4—1071no—
209C22×C4—1071no—
210C22×C4—1071no—
211C22×C4—2141no—
212C22×C4—2142no—
213C22×C4—2142no—
214C22×C4C22×D10711yescharacteristic
215M16—22no—
216M16—2142no—
217C2×C8—1072no—
218C2×C8C22×D10712yes—
219C4⋊C4—21no—
220C4⋊C4—1072no—
221C4⋊C4—1071no—
222C4⋊C4—1071no—
223C4⋊C4—2141no—
224C4⋊C4—2141no—
225C4⋊C4—2141no—
226C4⋊C4C22×D10712yes—
227C22⋊C4—2142no—
228C22⋊C4—2142no—
229C22⋊C4—2141no—
230C22⋊C4—2141no—
231C22⋊C4—2141no—
232C22⋊C4—2141no—
233C22⋊C4—2141no—
234C42—2142no—
235C42—2142no—
236C42—2141no—
237C23—21no—
238C23—1071no—
239C23—1071no—
240C23—4281no—
241C23—4281no—
242C23—2142no—
243C23D4×D10711yescharacteristic
244Q8—21no—
245Q8—21no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—4282no—
250D4—4282no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—1071no—
256C2×C4—1071no—
257C2×C4—4282no—
258C2×C4—2142no—
259C2×C4—2141no—
260C2×C4—2141no—
261C2×C4—2142no—
262C2×C4—2142no—
263C2×C4—2142no—
264C2×C4—2141no—
265C2×C4—2141no—
266C2×C4—2141no—
267C2×C4—2141no—
268C2×C4D4×D10711yescharacteristic
269C2×C4#1313381712yes—
270C2×C4#1313381812yes—
271C2×C4C23×D10711yescharacteristic
272C8—22no—
273C8—2142no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—4284no—
280C22—2141no—
281C22—2141no—
282C22—2141no—
283C22—2141no—
284C22—2141no—
285C22—2141no—
286C22#1681209112yes—
287C22C2×D4×D10711yescharacteristic
288C4—41no—
289C4—42no—
290C4—4281no—
291C4—4282no—
292C4—4281no—
293C4—2141no—
294C4—2141no—
295C4#1681209112yes—
296C4C2×#1313381711yescharacteristic
297C4C2×#1313381811yescharacteristic
298C2—22no—
299C2—81no—
300C2—4281no—
301C2—4281no—
302C2C2×#1681209111yescharacteristic
303C2#2029342311yescharacteristic
304C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#417×C107C2
socleC22×C107C22×D4
prime radical#20293134C2
hypercenter#417D107
hypocenterC107#3093
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#16812067C2semidirect
3. #20292967C2semidirect
4. #20292966C2semidirect
5. #20292950C2semidirect
6. #20292949C2semidirect
7. #20292669C2semidirect
8. #20292667C2semidirect
9. #20292628C2semidirect
10. #20292590C2semidirect
11. #417×C107C2semidirect
30. #16812043C22semidirect
31. #16812042C22semidirect
32. #16812041C22semidirect
33. #16812040C22semidirect
34. #16812033C22semidirect
35. #16812030C22semidirect
41. #103×C107C22semidirect
42. #102×C107C22semidirect
153. C107#3093semidirect
163. #417D107semidirect
191. #103C2×D107semidirect
193. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931342
C2×#168120672
#202929672
#202929662
#202929502
#202929492
#202926692
#202926672
#202926282
#202925902
#417×C1072
#3093107