Finite Group Databasebeta

← #28347025

Subgroups — #28347025

Order 13696.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293161C211yesmaximal, characteristic, complemented
2#20293026C211yesmaximal, characteristic, complemented
3#20293009C211yesmaximal, characteristic, complemented
4#20292984C211yesmaximal, characteristic, complemented
5#20292981C211yesmaximal, characteristic, complemented
6#20292866C211yesmaximal, characteristic, complemented
7#20292861C211yesmaximal, characteristic
8#20292858C211yesmaximal, characteristic, complemented
9#20292854C211yesmaximal, characteristic, complemented
10C4×#13133808C211yesmaximal, characteristic, complemented
11C4×#13133805C211yesmaximal, characteristic, complemented
12C4×#13133806C211yesmaximal, characteristic, complemented
13C4×#13133803C211yesmaximal, characteristic, complemented
14C2×C4×C107⋊C8C211yesmaximal, characteristic, complemented
15#529×C107C211yesmaximal, characteristic, complemented
16C2×#13133808C2211yescharacteristic
17C2×#13133806C2211yescharacteristic
18C2×#13133805C2211yescharacteristic
19C2×#13133803C2211yescharacteristic
20C22×C107⋊C8C2211yescharacteristic
21C22×C107⋊C8C2211yescharacteristic
22#16812103—21no—
23#16812101C2211yescharacteristic
24#16812097C2211yescharacteristic
25#16812092—21no—
26#16812088C2211yescharacteristic
27#16812083C2211yescharacteristic
28C4×Dic214C2211yescharacteristic
29C4×D428C2211yescharacteristic
30C4×C22⋊D107—21no—
31#16812066—21no—
32#16812042C2211yescharacteristic
33#16812042C2211yescharacteristic
34#16812040C2211yescharacteristic
35#16812040C2211yescharacteristic
36#16812033C2211yescharacteristic
37#16812033C2211yescharacteristic
38#16812032C2211yescharacteristic
39#16812032C2211yescharacteristic
40#16812030C2211yescharacteristic
41#16812030C2211yescharacteristic
42#16812029C2211yescharacteristic
43#16812029C2211yescharacteristic
44C4×C107⋊C8C2211yescharacteristic
45C4×C107⋊C8C2211yescharacteristic
46C4×C107⋊C8C2211yescharacteristic
47C4×C107⋊C8C2211yescharacteristic
48#126×C107—21no—
49#125×C107C2211yescharacteristic
50#124×C107C2211yescharacteristic
51#123×C107—21no—
52C22⋊Q8×C107C2211yescharacteristic
53#121×C107C2211yescharacteristic
54C4×Q8×C107C2211yescharacteristic
55C4×D4×C107—21no—
56C4×D4×C107C2211yescharacteristic
57#117×C107—21no—
58C2×C42×C107C2211yescharacteristic
59C2×Dic214C2311yescharacteristic
60C2×D428C2311yescharacteristic
61C2×C22⋊D107—21no—
62C2×C4×D107—21no—
63#13133808—22no—
64#13133806—22no—
65#13133805—22no—
66#13133803—22no—
67C2×C107⋊C8—22no—
68C2×C107⋊C8—22no—
69C2×C107⋊C8C2311yescharacteristic
70C2×C107⋊C8C2311yescharacteristic
71C2×C107⋊C8C2311yescharacteristic
72C2×C107⋊C8C2311yescharacteristic
73C4⋊Dic107C2311yescharacteristic
74C4⋊Dic107C2311yescharacteristic
75Dic107⋊C4—21no—
76Dic107⋊C4—21no—
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79C22⋊Dic107—21no—
80(C2×D107)⋊C4—21no—
81(C2×D107)⋊C4—21no—
82(C2×D107)⋊C4—21no—
83(C2×D107)⋊C4—21no—
84C4×Dic107—21no—
85C2×Q8×C107C2311yescharacteristic
86C2×D4×C107—21no—
87C2×D4×C107C2311yescharacteristic
88C22×C4×C107—21no—
89C22×C4×C107D411yescharacteristic
90C22×C4×C107D411yescharacteristic
91C22×C4×C107C2311yescharacteristic
92C4⋊C4×C107—21no—
93C4⋊C4×C107—21no—
94C4⋊C4×C107—21no—
95C4⋊C4×C107—21no—
96C4⋊C4×C107C2311yescharacteristic
97C4⋊C4×C107C2311yescharacteristic
98C22⋊C4×C107—21no—
99C22⋊C4×C107—21no—
100C22⋊C4×C107—21no—
101C22⋊C4×C107—21no—
102C22⋊C4×C107—21no—
103C42×C107—21no—
104C42×C107D411yescharacteristic
105C42×C107D411yescharacteristic
106C42×C107C2311yescharacteristic
107C42×C107C2311yescharacteristic
108C23×C107—21no—
109C23×C107C2×D411yescharacteristic
110C22×D107—21no—
111Q8×C107—21no—
112Q8×C107—21no—
113D4×C107—21no—
114D4×C107—21no—
115D4×C107—42no—
116C2×C4×C107—22no—
117C2×C4×C107—22no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—42no—
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2×D411yescharacteristic
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2×D411yescharacteristic
129C2×C4×C107C2411yescharacteristic
130C22⋊D107—42no—
131C2×Dic107—21no—
132C2×Dic107—21no—
133C2×Dic107—21no—
134D428—21no—
135D428—21no—
136C4×D107—42no—
137Dic214—21no—
138Dic214—21no—
139C107⋊C8—21no—
140C107⋊C8—21no—
141C107⋊C8#4112yes—
142C107⋊C8#4112yes—
143C22×C107—21no—
144C22×C107—21no—
145C22×C107—41no—
146C22×C107—41no—
147C22×C107—41no—
148C22×C107#13512yes—
149C22×C107C22×D411yescharacteristic
150C2×D107—41no—
151C2×D107—41no—
152C2×D107—41no—
153C4×C107—21no—
154C4×C107—21no—
155C4×C107—21no—
156C4×C107—21no—
157C4×C107—41no—
158C4×C107—41no—
159C4×C107—41no—
160C4×C107#13512yes—
161C4×C107C2×#4111yescharacteristic
162C4×C107C2×#4111yescharacteristic
163Dic107—41no—
164Dic107—41no—
165Dic107—41no—
166C2×C107—22no—
167C2×C107—81no—
168C2×C107#53211yescharacteristic
169C2×C107C2×#13511yescharacteristic
170C2×C107C2×#13511yescharacteristic
171D107—81no—
172#3168—1071nomaximal, Sylow
173C107#316811yescharacteristic, complemented, Sylow
174#529—1071no—
175#529D10711yescharacteristic, complemented
176#499—1071no—
177#495—1071no—
178#488—1071no—
179#488—1071no—
180#467—1071no—
181#466—1071no—
182#465—1071no—
183#465—1071no—
184C4×Q16—1071no—
185C4×SD16—1071no—
186C4×SD16—1071no—
187C4×D8—1071no—
188C2×C4×C8—1071no—
189C2×Q16—1071no—
190C2×SD16—1071no—
191C2×SD16—1071no—
192C2×D8—1071no—
193C22×C8—1071no—
194C22×C8—1071no—
195#126—21no—
196#126—2141no—
197#125—1071no—
198#125C2×D10711yescharacteristic
199#124—1071no—
200#124C2×D10711yescharacteristic
201#123—21no—
202#123—2141no—
203C22⋊Q8—1071no—
204C22⋊Q8C2×D10711yescharacteristic
205#121—1071no—
206#121C2×D10711yescharacteristic
207C4×Q8—1071no—
208C4×Q8C2×D10711yescharacteristic
209C4×D4—21no—
210C4×D4—1071no—
211C4×D4—2141no—
212C4×D4C2×D10711yescharacteristic
213#117—21no—
214#117—2141no—
215C2×C42C2×D10711yescharacteristic
216#107—1071no—
217#107—1071no—
218#106—1071no—
219#106—1071no—
220#103—1071no—
221#103—1071no—
222#103—1071no—
223#103—1071no—
224#102—1071no—
225#102—1071no—
226#102—1071no—
227#102—1071no—
228C4×C8—1071no—
229C4×C8—1071no—
230C4×C8—1071no—
231C4×C8—1071no—
232C2×Q8—1071no—
233C2×Q8C22×D10711yescharacteristic
234C2×D4—21no—
235C2×D4—1071no—
236C2×D4—2141no—
237C2×D4C22×D10711yescharacteristic
238C22×C4—21no—
239C22×C4—2141no—
240C22×C4C22⋊D10711yescharacteristic
241C22×C4C22⋊D10711yescharacteristic
242C22×C4C22×D10711yescharacteristic
243Q16—2142no—
244SD16—2142no—
245SD16—2142no—
246D8—2142no—
247C2×C8—1071no—
248C2×C8—1071no—
249C2×C8—1071no—
250C2×C8—1071no—
251C2×C8—2142no—
252C2×C8—2142no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—1071no—
258C4⋊C4—1071no—
259C4⋊C4—2141no—
260C4⋊C4—2141no—
261C4⋊C4—2141no—
262C4⋊C4—2141no—
263C4⋊C4C22×D10711yescharacteristic
264C4⋊C4C22×D10711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—2141no—
271C22⋊C4—2141no—
272C22⋊C4—2141no—
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C42—21no—
276C42—2141no—
277C42C22⋊D10711yescharacteristic
278C42C22⋊D10711yescharacteristic
279C42C22×D10711yescharacteristic
280C42C22×D10711yescharacteristic
281C23—21no—
282C23—2141no—
283C23C2×C22⋊D10711yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—2141no—
287Q8—2141no—
288D4—21no—
289D4—21no—
290D4—42no—
291D4—4282no—
292D4—2141no—
293D4—2141no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—4282no—
303C2×C4—2141no—
304C2×C4—2141no—
305C2×C4—2141no—
306C2×C4C2×C22⋊D10711yescharacteristic
307C2×C4C2×C22⋊D10711yescharacteristic
308C2×C4C2×C22⋊D10711yescharacteristic
309C2×C4C2×C22⋊D10711yescharacteristic
310C2×C4C2×C22⋊D10711yescharacteristic
311C2×C4C23×D10711yescharacteristic
312C8—1072no—
313C8—1072no—
314C8—2141no—
315C8—2141no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—4281no—
322C22—4281no—
323C22—4281no—
324C22#1681212912yes—
325C22C22×C22⋊D10711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—4281no—
334C4—4281no—
335C4—4281no—
336C4#1681212912yes—
337C4#41×D10711yescharacteristic
338C4#41×D10711yescharacteristic
339C2—22no—
340C2—81no—
341C2—8561no—
342C2#2029317311yescharacteristic
343C2C2×#1681212911yescharacteristic
344C2C2×#1681212911yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#529×C107C2
socleC22×C107C22×D4
prime radical#20292861C2
hypercenter#529D107
hypocenterC107#3168
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #20293161C2semidirect
2. #20293026C2semidirect
3. #20293009C2semidirect
4. #20292984C2semidirect
5. #20292981C2semidirect
6. #20292866C2semidirect
8. #20292858C2semidirect
9. #20292854C2semidirect
10. C4×#13133808C2semidirect
11. C4×#13133805C2semidirect
12. C4×#13133806C2semidirect
13. C4×#13133803C2semidirect
14. C2×C4×C107⋊C8C2semidirect
15. #529×C107C2semidirect
173. C107#3168semidirect
175. #529D107semidirect

Maximal subgroups

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

subgroupindex
#202931612
#202930262
#202930092
#202929842
#202929812
#202928662
#202928612
#202928582
#202928542
C4×#131338082
C4×#131338052
C4×#131338062
C4×#131338032
C2×C4×C107⋊C82
#529×C1072
#3168107