Finite Group Databasebeta

← #28348290

Subgroups — #28348290

Order 13696.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293273C211yesmaximal, characteristic
2#20292984C211yesmaximal, characteristic, complemented
3#20292957C211yesmaximal, characteristic, complemented
4#20292951C211yesmaximal, characteristic, complemented
5#20292924C211yesmaximal, characteristic, complemented
6#20292866C211yesmaximal, characteristic, complemented
7#20292858C211yesmaximal, characteristic, complemented
8#20292849C211yesmaximal, characteristic, complemented
9#20292787C211yesmaximal, characteristic, complemented
10#20292777C211yesmaximal, characteristic, complemented
11C4×#13133805C211yesmaximal, characteristic, complemented
12D4×C107⋊C8C211yesmaximal, characteristic, complemented
13#20292621C211yesmaximal, characteristic, complemented
14C2×#16812029C211yesmaximal, characteristic, complemented
15#552×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C22×D428C2211yescharacteristic
18C2×#13133808C2211yescharacteristic, complemented
19C2×#13133805C2211yescharacteristic
20C2×#13133805C2211yescharacteristic
21C22×C107⋊C8C2211yescharacteristic
22C22×C107⋊C8C2211yescharacteristic, complemented
23#16812097C2211yescharacteristic
24#16812094—21no—
25#16812088C2211yescharacteristic
26#16812083C2211yescharacteristic
27#16812082—21no—
28#16812077—21no—
29C4×D428C2211yescharacteristic
30C22⋊C4×D107—21no—
31#16812042C2211yescharacteristic, complemented
32#16812040C2211yescharacteristic, complemented
33#16812040C2211yescharacteristic, complemented
34C4⋊(C107⋊C8)C2211yescharacteristic, complemented
35#16812033C2211yescharacteristic, complemented
36#16812032C2211yescharacteristic, complemented
37#16812032C2211yescharacteristic, complemented
38#16812032C2211yescharacteristic, complemented
39#16812029C2211yescharacteristic
40#16812029C2211yescharacteristic
41#16812029C2211yescharacteristic
42#16812029C2211yescharacteristic
43#16812029C2211yescharacteristic
44#16812029C2211yescharacteristic
45C22⋊(C107⋊C8)C2211yescharacteristic
46C22⋊(C107⋊C8)C2211yescharacteristic, complemented
47C4×C107⋊C8C2211yescharacteristic, complemented
48#126×C107—21no—
49#125×C107—21no—
50#125×C107C2211yescharacteristic, complemented
51#123×C107—21no—
52C22⋊Q8×C107C2211yescharacteristic
53C22⋊Q8×C107C2211yescharacteristic, complemented
54C4×Q8×C107C2211yescharacteristic, complemented
55C4×D4×C107C2211yescharacteristic
56#117×C107—21no—
57#117×C107C2211yescharacteristic, complemented
58C2×C4⋊C4×C107C2211yescharacteristic
59C23×D107—21no—
60#13133818—22no—
61#13133818—41no—
62C2×Dic214C2311yescharacteristic
63C2×D428—22no—
64C2×D428—21no—
65C2×D428—41no—
66C2×D428C2311yescharacteristic
67C2×D428C2311yescharacteristic
68C2×C22⋊D107—21no—
69C2×C22⋊D107—21no—
70C2×C4×D107—21no—
71C2×C4×D107—21no—
72#13133808—41no—
73#13133805—22no—
74#13133805—41no—
75C2×C107⋊C8—41no—
76C2×C107⋊C8—41no—
77C2×C107⋊C8C2311yescharacteristic
78C2×C107⋊C8C2311yescharacteristic
79C2×C107⋊C8C2311yescharacteristic
80C2×C107⋊C8C2311yescharacteristic
81C4⋊Dic107C2311yescharacteristic
82Dic107⋊C4—21no—
83C22⋊Dic107—21no—
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88C2×Q8×C107C2311yescharacteristic
89C2×D4×C107D411yescharacteristic
90C22×C4×C107—21no—
91C22×C4×C107C2311yescharacteristic
92C22×C4×C107C2311yescharacteristic
93C4⋊C4×C107—21no—
94C4⋊C4×C107—21no—
95C4⋊C4×C107—21no—
96C4⋊C4×C107—21no—
97C4⋊C4×C107—21no—
98C4⋊C4×C107—21no—
99C4⋊C4×C107D411yescharacteristic
100C4⋊C4×C107C2311yescharacteristic
101C4⋊C4×C107C2311yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C22⋊C4×C107—21no—
104C22⋊C4×C107—21no—
105C22⋊C4×C107—21no—
106C22⋊C4×C107D411yescharacteristic
107C22⋊C4×C107D411yescharacteristic
108C42×C107—21no—
109C42×C107—21no—
110C42×C107C2311yescharacteristic
111C23×C107C2×D411yescharacteristic
112C23×C107C2×D411yescharacteristic
113C22×D107—21no—
114C22×D107—21no—
115C22×D107—42no—
116C22×D107—42no—
117C22×D107—81no—
118Q8×C107—21no—
119Q8×C107—21no—
120D4×C107—22no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107—42no—
130C2×C4×C107C2×D411yescharacteristic
131C2×C4×C107C2×D411yescharacteristic
132C2×C4×C107C2×D411yescharacteristic
133C2×C4×C107C2×D411yescharacteristic
134C2×C4×C107C2411yescharacteristic
135C22⋊D107—42no—
136C22⋊D107—81no—
137C2×Dic107—21no—
138C2×Dic107—21no—
139D428—21no—
140D428—21no—
141D428—21no—
142D428—42no—
143D428#4112yes—
144C4×D107—42no—
145C4×D107—81no—
146Dic214—21no—
147Dic214—21no—
148C107⋊C8—21no—
149C107⋊C8—21no—
150C107⋊C8—21no—
151C107⋊C8—21no—
152C22×C107—21no—
153C22×C107—21no—
154C22×C107—21no—
155C22×C107—21no—
156C22×C107—21no—
157C22×C107#13512yes—
158C22×C107C22×D411yescharacteristic
159C2×D107—22no—
160C2×D107—41no—
161C2×D107—41no—
162C2×D107—41no—
163C2×D107—41no—
164C2×D107—41no—
165C2×D107—84no—
166C4×C107—21no—
167C4×C107—21no—
168C4×C107—21no—
169C4×C107—21no—
170C4×C107—41no—
171C4×C107—41no—
172C4×C107—41no—
173C4×C107—41no—
174C4×C107C2×#4111yescharacteristic
175C4×C107H(5,𝔽2)11yescharacteristic
176Dic107—41no—
177Dic107—41no—
178C2×C107—22no—
179C2×C107—41no—
180C2×C107#54611yescharacteristic
181C2×C107C2×#13511yescharacteristic
182C2×C107#57711yescharacteristic
183D107—42no—
184D107—81no—
185#3296—1071nomaximal, Sylow
186C107#329611yescharacteristic, complemented, Sylow
187#552D10711yescharacteristic, complemented
188#546—1071no—
189#488—1071no—
190#483—1071no—
191#482—1071no—
192#478—1071no—
193#467—1071no—
194#465—1071no—
195#464—1071no—
196#452—1071no—
197#450—1071no—
198C4×SD16—1071no—
199C8×D4—1071no—
200#416—1071no—
201C2×#102—1071no—
202C2×#41—1071no—
203C22×D4—1071no—
204C2×Q16—1071no—
205C2×SD16—1071no—
206C2×SD16—1071no—
207C22×C8—1071no—
208C22×C8—1071no—
209#126—21no—
210#125—21no—
211#125C2×D10711yescharacteristic, complemented
212#124—1071no—
213#123—21no—
214#123—2141no—
215C22⋊Q8—1071no—
216C22⋊Q8C2×D10711yescharacteristic
217C22⋊Q8C2×D10711yescharacteristic, complemented
218#121—1071no—
219#121—2141no—
220#120—2141no—
221C4×Q8C2×D10711yescharacteristic, complemented
222C4×D4—1071no—
223C4×D4C2×D10711yescharacteristic
224#117—21no—
225#117C2×D10711yescharacteristic, complemented
226C2×C4⋊C4C2×D10711yescharacteristic
227C2×C22⋊C4—2141no—
228#107—1071no—
229#106—1071no—
230#106—1071no—
231C4⋊C8—1071no—
232#103—1071no—
233#103—1071no—
234#103—1071no—
235#103—1071no—
236#102—1071no—
237#102—1071no—
238#102—1071no—
239#102—1071no—
240#102—1071no—
241#102—1071no—
242C22⋊C8—1071no—
243C22⋊C8—1071no—
244C4×C8—1071no—
245C24—2141no—
246#41—4281no—
247#41—2142no—
248C2×Q8—1071no—
249C2×Q8C22×D10711yescharacteristic
250C2×D4—1071no—
251C2×D4—1071no—
252C2×D4—4281no—
253C2×D4—2142no—
254C2×D4—2141no—
255C2×D4—2141no—
256C2×D4—2141no—
257C2×D4C22⋊D10711yescharacteristic
258C22×C4—21no—
259C22×C4—2141no—
260C22×C4—2141no—
261C22×C4C22×D10711yescharacteristic
262C22×C4C22×D10711yescharacteristic
263Q16—4281no—
264SD16—4281no—
265SD16—2142no—
266C2×C8—1071no—
267C2×C8—1071no—
268C2×C8—1071no—
269C2×C8—1071no—
270C2×C8—4281no—
271C2×C8—4281no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—1071no—
279C4⋊C4—2141no—
280C4⋊C4C22⋊D10711yescharacteristic
281C4⋊C4C22×D10711yescharacteristic
282C4⋊C4C22×D10711yescharacteristic
283C4⋊C4C22×D10711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—2141no—
288C22⋊C4—2141no—
289C22⋊C4—2141no—
290C22⋊C4—2141no—
291C22⋊C4—2141no—
292C22⋊C4C22⋊D10711yescharacteristic
293C22⋊C4C22⋊D10711yescharacteristic
294C42—21no—
295C42—21no—
296C42C22×D10711yescharacteristic
297C23—8561no—
298C23—4282no—
299C23—4282no—
300C23—2141no—
301C23—2141no—
302C23C2×C22⋊D10711yescharacteristic
303C23C2×C22⋊D10711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2141no—
307Q8—2141no—
308D4—22no—
309D4—8561no—
310D4—1072no—
311D4—4282no—
312D4—4282no—
313D4—2141no—
314D4—2141no—
315D4—2141no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—8561no—
326C2×C4—4282no—
327C2×C4—2141no—
328C2×C4—2141no—
329C2×C4C2×C22⋊D10711yescharacteristic
330C2×C4C2×C22⋊D10711yescharacteristic
331C2×C4C2×C22⋊D10711yescharacteristic
332C2×C4C2×C22⋊D10711yescharacteristic
333C2×C4C23×D10711yescharacteristic
334C8—2141no—
335C8—2141no—
336C8—2141no—
337C8—2141no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—8564no—
344C22—4281no—
345C22—4281no—
346C22—4281no—
347C22—4281no—
348C22—4281no—
349C22—2142no—
350C22#1681212912yes—
351C22C22×C22⋊D10711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—4281no—
361C4—4281no—
362C4#41×D10711yescharacteristic
363C4#1681215511yescharacteristic
364C2—22no—
365C2—41no—
366C2—8561no—
367C2—4282no—
368C2#2029326511yescharacteristic
369C2C2×#1681212911yescharacteristic
370C2#2029343511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#552×C107C2
socleC22×C107C22×D4
prime radical#20293273C2
hypercenter#552D107
hypocenterC107#3296
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #20292984C2semidirect
3. #20292957C2semidirect
4. #20292951C2semidirect
5. #20292924C2semidirect
6. #20292866C2semidirect
7. #20292858C2semidirect
8. #20292849C2semidirect
9. #20292787C2semidirect
10. #20292777C2semidirect
11. C4×#13133805C2semidirect
12. D4×C107⋊C8C2semidirect
13. #20292621C2semidirect
14. C2×#16812029C2semidirect
15. #552×C107C2semidirect
18. C2×#13133808C22semidirect
22. C22×C107⋊C8C22semidirect
31. #16812042C22semidirect
32. #16812040C22semidirect
33. #16812040C22semidirect
34. C4⋊(C107⋊C8)C22semidirect
35. #16812033C22semidirect
36. #16812032C22semidirect
37. #16812032C22semidirect
38. #16812032C22semidirect
46. C22⋊(C107⋊C8)C22semidirect
47. C4×C107⋊C8C22semidirect
50. #125×C107C22semidirect
53. C22⋊Q8×C107C22semidirect
54. C4×Q8×C107C22semidirect
57. #117×C107C22semidirect
186. C107#3296semidirect
187. #552D107semidirect
211. #125C2×D107semidirect
217. C22⋊Q8C2×D107semidirect
221. C4×Q8C2×D107semidirect
225. #117C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202932732
#202929842
#202929572
#202929512
#202929242
#202928662
#202928582
#202928492
#202927872
#202927772
C4×#131338052
D4×C107⋊C82
#202926212
C2×#168120292
#552×C1072
#3296107