Finite Group Databasebeta

← #28348298

Subgroups — #28348298

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#20293317C211yesmaximal, characteristic, complemented
2#20293273C211yesmaximal, characteristic
3#20292982C211yesmaximal, characteristic, complemented
4#20292958C211yesmaximal, characteristic, complemented
5#20292952C211yesmaximal, characteristic, complemented
6#20292928C211yesmaximal, characteristic, complemented
7#20292868C211yesmaximal, characteristic, complemented
8#20292856C211yesmaximal, characteristic, complemented
9#20292850C211yesmaximal, characteristic, complemented
10#20292791C211yesmaximal, characteristic, complemented
11#20292780C211yesmaximal, characteristic, complemented
12C4×#13133807C211yesmaximal, characteristic, complemented
13#20292624C211yesmaximal, characteristic, complemented
14C2×#16812031C211yesmaximal, characteristic, complemented
15C8×D4×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C22×D428C2211yescharacteristic
18C2×Dic428C2211yescharacteristic, complemented
19C2×#13133807C2211yescharacteristic
20C2×#13133807C2211yescharacteristic
21#16812102—21no—
22#16812101C2211yescharacteristic, complemented
23#16812099—21no—
24#16812097C2211yescharacteristic
25#16812094—21no—
26#16812090—21no—
27#16812088C2211yescharacteristic
28#16812088C2211yescharacteristic
29#16812088C2211yescharacteristic, complemented
30#16812083C2211yescharacteristic
31#16812082—21no—
32#16812077—21no—
33C4×Dic214C2211yescharacteristic, complemented
34C4×D428C2211yescharacteristic
35#16812068C2211yescharacteristic, complemented
36#16812065—21no—
37C2×C4⋊Dic107C2211yescharacteristic
38C22⋊C4×D107—21no—
39#16812043C2211yescharacteristic, complemented
40#16812041C2211yescharacteristic, complemented
41#16812041C2211yescharacteristic, complemented
42#16812034C2211yescharacteristic, complemented
43#16812034C2211yescharacteristic, complemented
44#16812034C2211yescharacteristic, complemented
45#16812034C2211yescharacteristic, complemented
46#16812031C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812031C2211yescharacteristic
49#16812031C2211yescharacteristic
50#16812031C2211yescharacteristic
51#16812031C2211yescharacteristic
52C22×C8×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic, complemented
54C4×D4×C107C2211yescharacteristic
55C4⋊C8×C107C2211yescharacteristic, complemented
56C22⋊C8×C107C2211yescharacteristic
57C22⋊C8×C107C2211yescharacteristic, complemented
58C4×C8×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60#13133818—22no—
61#13133818—41no—
62C2×Dic214C2311yescharacteristic
63C2×Dic214C2311yescharacteristic
64C2×D428—22no—
65C2×D428—21no—
66C2×D428—41no—
67C2×D428C2311yescharacteristic
68C2×D428C2311yescharacteristic
69C2×C22⋊D107—21no—
70C2×C22⋊D107—21no—
71C22×Dic107—21no—
72C2×C4×D107—21no—
73C2×C4×D107—21no—
74Dic428—41no—
75#13133807—22no—
76#13133807—41no—
77C4⋊Dic107—21no—
78C4⋊Dic107C2311yescharacteristic
79C4⋊Dic107C2311yescharacteristic
80C4⋊Dic107C2311yescharacteristic
81C4⋊Dic107C2311yescharacteristic
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88C22⋊Dic107—21no—
89C22⋊Dic107—21no—
90C22⋊Dic107—21no—
91C22⋊Dic107—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96C4×Dic107—21no—
97C4×Dic107—21no—
98C2×D4×C107D411yescharacteristic
99C22×C4×C107C2311yescharacteristic
100C22×C4×C107C2311yescharacteristic
101C2×C8×C107—41no—
102C2×C8×C107—41no—
103C2×C8×C107C2311yescharacteristic
104C2×C8×C107C2311yescharacteristic
105C2×C8×C107C2311yescharacteristic
106C2×C8×C107C2311yescharacteristic
107C4⋊C4×C107D411yescharacteristic
108C22⋊C4×C107D411yescharacteristic
109C22⋊C4×C107D411yescharacteristic
110C42×C107C2311yescharacteristic
111C23×C107C2×D411yescharacteristic
112C23×C107C2×D411yescharacteristic
113C22×D107—21no—
114C22×D107—21no—
115C22×D107—42no—
116C22×D107—42no—
117C22×D107—81no—
118D4×C107—22no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107C2×D411yescharacteristic
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2×D411yescharacteristic
127C2×C4×C107C2411yescharacteristic
128C22⋊D107—42no—
129C22⋊D107—81no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—21no—
133C2×Dic107—21no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107—42no—
137D428—21no—
138D428—21no—
139D428—21no—
140D428—42no—
141D428#4112yes—
142C4×D107—42no—
143C4×D107—81no—
144Dic214—21no—
145Dic214—21no—
146Dic214—21no—
147Dic214—21no—
148C8×C107—21no—
149C8×C107—21no—
150C8×C107—21no—
151C8×C107—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×C107C2×#4111yescharacteristic
171C4×C107H(5,𝔽2)11yescharacteristic
172Dic107—41no—
173Dic107—41no—
174Dic107—41no—
175Dic107—41no—
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#552—1071no—
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×D4D10711yescharacteristic, complemented
200#416—1071no—
201C2×#102—1071no—
202C2×#41—1071no—
203C22×D4—1071no—
204C2×Q16—1071no—
205C2×SD16—1071no—
206C2×SD16—1071no—
207C22×C8C2×D10711yescharacteristic
208C22×C8C2×D10711yescharacteristic, complemented
209#126—2141no—
210#125—1071no—
211#125—2141no—
212#124—1071no—
213#123—2141no—
214#123—2141no—
215C22⋊Q8—1071no—
216C22⋊Q8—1071no—
217C22⋊Q8—1071no—
218#121—1071no—
219#121—2141no—
220#120—2141no—
221C4×Q8—1071no—
222C4×D4—1071no—
223C4×D4C2×D10711yescharacteristic
224#117—1071no—
225#117—2141no—
226C2×C4⋊C4—1071no—
227C2×C22⋊C4—2141no—
228#107—1071no—
229#106—1071no—
230#106—1071no—
231C4⋊C8C2×D10711yescharacteristic, complemented
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⋊C8C2×D10711yescharacteristic
243C22⋊C8C2×D10711yescharacteristic, complemented
244C4×C8C2×D10711yescharacteristic, complemented
245C24—2141no—
246#41—4281no—
247#41—2142no—
248C2×Q8—1071no—
249C2×Q8—1071no—
250C2×D4—1071no—
251C2×D4—1071no—
252C2×D4—4281no—
253C2×D4—2142no—
254C2×D4—2141no—
255C2×D4—2141no—
256C2×D4—2141no—
257C2×D4D42811yescharacteristic
258C22×C4—2141no—
259C22×C4—2141no—
260C22×C4—2141no—
261C22×C4C22×D10711yescharacteristic
262C22×C4C22×D10711yescharacteristic
263Q16—4281no—
264SD16—4281no—
265SD16—2142no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C22×D10711yescharacteristic
269C2×C8C22×D10711yescharacteristic
270C2×C8C22×D10711yescharacteristic
271C2×C8C22×D10711yescharacteristic
272C4⋊C4—1071no—
273C4⋊C4—1071no—
274C4⋊C4—1071no—
275C4⋊C4—1071no—
276C4⋊C4—2141no—
277C4⋊C4—2141no—
278C4⋊C4—2141no—
279C4⋊C4—2141no—
280C4⋊C4—2141no—
281C4⋊C4—2141no—
282C4⋊C4—2141no—
283C4⋊C4D42811yescharacteristic
284C22⋊C4—2141no—
285C22⋊C4—2141no—
286C22⋊C4—2141no—
287C22⋊C4—2141no—
288C22⋊C4—2141no—
289C22⋊C4—2141no—
290C22⋊C4—2141no—
291C22⋊C4—2141no—
292C22⋊C4D42811yescharacteristic
293C22⋊C4D42811yescharacteristic
294C42—2141no—
295C42—2141no—
296C42C22×D10711yescharacteristic
297C23—8561no—
298C23—4282no—
299C23—4282no—
300C23—2141no—
301C23—2141no—
302C23C2×D42811yescharacteristic
303C23C2×D42811yescharacteristic
304Q8—2141no—
305Q8—2141no—
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—8561no—
321C2×C4—4282no—
322C2×C4—4282no—
323C2×C4—2141no—
324C2×C4—2141no—
325C2×C4—2141no—
326C2×C4—2141no—
327C2×C4—2141no—
328C2×C4—2141no—
329C2×C4C2×D42811yescharacteristic
330C2×C4C2×D42811yescharacteristic
331C2×C4C2×D42811yescharacteristic
332C2×C4C2×D42811yescharacteristic
333C2×C4C23×D10711yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
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#1681213112yes—
351C22C22×D42811yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—4281no—
357C4—4281no—
358C4—4281no—
359C4—4281no—
360C4—4281no—
361C4—4281no—
362C4#41×D10711yescharacteristic
363C4#1681215411yescharacteristic
364C2—22no—
365C2—41no—
366C2—8561no—
367C2—4282no—
368C2#2029327211yescharacteristic
369C2C2×#1681213111yescharacteristic
370C2#2029343911yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC8×D4×C107C2
socleC22×C107C22×D4
prime radical#20293273C2
hypercenterC8×D4D107
hypocenterC107#3296
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20293317C2semidirect
3. #20292982C2semidirect
4. #20292958C2semidirect
5. #20292952C2semidirect
6. #20292928C2semidirect
7. #20292868C2semidirect
8. #20292856C2semidirect
9. #20292850C2semidirect
10. #20292791C2semidirect
11. #20292780C2semidirect
12. C4×#13133807C2semidirect
13. #20292624C2semidirect
14. C2×#16812031C2semidirect
15. C8×D4×C107C2semidirect
18. C2×Dic428C22semidirect
22. #16812101C22semidirect
29. #16812088C22semidirect
33. C4×Dic214C22semidirect
35. #16812068C22semidirect
39. #16812043C22semidirect
40. #16812041C22semidirect
41. #16812041C22semidirect
42. #16812034C22semidirect
43. #16812034C22semidirect
44. #16812034C22semidirect
45. #16812034C22semidirect
53. C22×C8×C107C22semidirect
55. C4⋊C8×C107C22semidirect
57. C22⋊C8×C107C22semidirect
58. C4×C8×C107C22semidirect
186. C107#3296semidirect
199. C8×D4D107semidirect
208. C22×C8C2×D107semidirect
231. C4⋊C8C2×D107semidirect
243. C22⋊C8C2×D107semidirect
244. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202933172
#202932732
#202929822
#202929582
#202929522
#202929282
#202928682
#202928562
#202928502
#202927912
#202927802
C4×#131338072
#202926242
C2×#168120312
C8×D4×C1072
#3296107