Finite Group Databasebeta

← #28348292

Subgroups — #28348292

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#20293312C211yesmaximal, characteristic, complemented
2#20293271C211yesmaximal, characteristic
3#20292980C211yesmaximal, characteristic, complemented
4#20292957C211yesmaximal, characteristic, complemented
5#20292949C211yesmaximal, characteristic, complemented
6#20292925C211yesmaximal, characteristic, complemented
7#20292868C211yesmaximal, characteristic, complemented
8#20292855C211yesmaximal, characteristic, complemented
9#20292848C211yesmaximal, characteristic, complemented
10#20292790C211yesmaximal, characteristic, complemented
11#20292778C211yesmaximal, characteristic, complemented
12#20292718C211yesmaximal, characteristic, complemented
13C8×C22⋊D107C211yesmaximal, characteristic, complemented
14#20292620C211yesmaximal, characteristic, complemented
15C2×#102×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×Dic428C2211yescharacteristic, complemented
18C2×#13133806C2211yescharacteristic
19C2×#13133806C2211yescharacteristic
20C2×C8×D107C2211yescharacteristic, complemented
21#16812104—21no—
22#16812100C2211yescharacteristic, complemented
23#16812099—21no—
24#16812096C2211yescharacteristic
25#16812092—21no—
26#16812090—21no—
27#16812089C2211yescharacteristic, complemented
28#16812088C2211yescharacteristic
29#16812088C2211yescharacteristic
30#16812081—21no—
31#16812080C2211yescharacteristic
32#16812078—21no—
33#16812074C2211yescharacteristic, complemented
34D4×Dic107C2211yescharacteristic
35C4×C22⋊D107C2211yescharacteristic
36#16812068—21no—
37#16812067C2211yescharacteristic, complemented
38C2×C22⋊Dic107—21no—
39#16812043C2211yescharacteristic, complemented
40#16812040C2211yescharacteristic, complemented
41#16812040C2211yescharacteristic, complemented
42Dic107⋊C8C2211yescharacteristic, complemented
43#16812034C2211yescharacteristic, complemented
44#16812034C2211yescharacteristic, complemented
45#16812032C2211yescharacteristic, complemented
46#16812032C2211yescharacteristic, complemented
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic, complemented
51C8×Dic107C2211yescharacteristic, complemented
52C22×D4×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic
54C2×C4⋊C4×C107C2211yescharacteristic
55#102×C107C2211yescharacteristic
56#102×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59#13133817—22no—
60#13133817—41no—
61C2×Dic214C2311yescharacteristic
62C2×Dic214C2311yescharacteristic
63C2×C22⋊D107—21no—
64C2×C22⋊D107—21no—
65C2×C22⋊D107D411yescharacteristic
66C22×Dic107—21no—
67C22×Dic107—21no—
68C2×C4×D107C2311yescharacteristic
69Dic428—41no—
70#13133806—22no—
71#13133806—41no—
72C2×C107⋊C8C2311yescharacteristic
73C2×C107⋊C8C2311yescharacteristic
74C8×D107—41no—
75C4⋊Dic107—21no—
76C4⋊Dic107—21no—
77C4⋊Dic107C2311yescharacteristic
78C4⋊Dic107C2311yescharacteristic
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4D411yescharacteristic
84C22⋊Dic107—21no—
85C22⋊Dic107—21no—
86C22⋊Dic107—21no—
87C22⋊Dic107—21no—
88C22⋊Dic107—21no—
89C22⋊Dic107—21no—
90C22⋊Dic107D411yescharacteristic
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4D411yescharacteristic
94C4×Dic107—21no—
95C4×Dic107—21no—
96C4×Dic107C2311yescharacteristic
97C24×C107—21no—
98C2×D4×C107—22no—
99C2×D4×C107—21no—
100C2×D4×C107—41no—
101C2×D4×C107C2311yescharacteristic
102C2×D4×C107C2311yescharacteristic
103C22×C4×C107—21no—
104C22×C4×C107C2311yescharacteristic
105C2×C8×C107—41no—
106C2×C8×C107C2311yescharacteristic
107C2×C8×C107C2311yescharacteristic
108C4⋊C4×C107—21no—
109C4⋊C4×C107C2311yescharacteristic
110C4⋊C4×C107C2311yescharacteristic
111C23×C107—21no—
112C23×C107—21no—
113C23×C107—42no—
114C23×C107—42no—
115C23×C107—81no—
116C23×C107C2×D411yescharacteristic
117C22×D107C2×D411yescharacteristic
118D4×C107—21no—
119D4×C107—21no—
120D4×C107—21no—
121D4×C107—42no—
122D4×C107#4112yes—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—42no—
128C2×C4×C107C2×D411yescharacteristic
129C2×C4×C107C2411yescharacteristic
130C22⋊D107—22no—
131C22⋊D107—42no—
132C22⋊D107—81no—
133C2×Dic107—21no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107—21no—
137C2×Dic107—42no—
138C2×Dic107—81no—
139C2×Dic107C2×D411yescharacteristic
140C2×Dic107C2×D411yescharacteristic
141C2×Dic107C2×D411yescharacteristic
142C4×D107—21no—
143C4×D107—21no—
144Dic214—21no—
145Dic214—21no—
146Dic214—21no—
147Dic214—21no—
148C8×C107—21no—
149C8×C107—21no—
150C107⋊C8—21no—
151C107⋊C8—21no—
152C22×C107—21no—
153C22×C107—21no—
154C22×C107—22no—
155C22×C107—41no—
156C22×C107—41no—
157C22×C107—41no—
158C22×C107—41no—
159C22×C107—41no—
160C22×C107—84no—
161C22×C107#13512yes—
162C22×C107C22×D411yescharacteristic
163C2×D107—21no—
164C2×D107—21no—
165C2×D107—21no—
166C4×C107—21no—
167C4×C107—41no—
168C4×C107—41no—
169C4×C107C2×#4111yescharacteristic
170C4×C107H(5,𝔽2)11yescharacteristic
171Dic107—21no—
172Dic107—21no—
173Dic107—21no—
174Dic107—41no—
175Dic107—41no—
176Dic107—41no—
177Dic107—41no—
178C2×C107—22no—
179C2×C107—42no—
180C2×C107—81no—
181C2×C107#54611yescharacteristic
182C2×C107C2×#13511yescharacteristic
183C2×C107#57711yescharacteristic
184D107—41no—
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×D4—1071no—
200#416—1071no—
201C2×#102D10711yescharacteristic, complemented
202C2×#41—1071no—
203C22×D4C2×D10711yescharacteristic
204C2×Q16—1071no—
205C2×SD16—1071no—
206C2×SD16—1071no—
207C22×C8—1071no—
208C22×C8C2×D10711yescharacteristic
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×D4—1071no—
224#117—1071no—
225#117—2141no—
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#102C2×D10711yescharacteristic
239#102C2×D10711yescharacteristic
240#102C2×D10711yescharacteristic
241#102C2×D10711yescharacteristic
242C22⋊C8—1071no—
243C22⋊C8—1071no—
244C4×C8—1071no—
245C24—21no—
246#41—4281no—
247#41—2142no—
248C2×Q8—1071no—
249C2×Q8—1071no—
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—1071no—
254C2×D4—2141no—
255C2×D4—2141no—
256C2×D4C22×D10711yescharacteristic
257C2×D4C22×D10711yescharacteristic
258C22×C4—21no—
259C22×C4—1071no—
260C22×C4—2141no—
261C22×C4—2141no—
262C22×C4C22×D10711yescharacteristic
263Q16—4281no—
264SD16—4281no—
265SD16—2142no—
266C2×C8—41no—
267C2×C8—1071no—
268C2×C8—1071no—
269C2×C8—4281no—
270C2×C8C22×D10711yescharacteristic
271C2×C8C22×D10711yescharacteristic
272C4⋊C4—21no—
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⋊C4C22×D10711yescharacteristic
283C4⋊C4C22×D10711yescharacteristic
284C22⋊C4—1071no—
285C22⋊C4—1071no—
286C22⋊C4—2141no—
287C22⋊C4—2141no—
288C22⋊C4—2141no—
289C22⋊C4—2141no—
290C22⋊C4—2141no—
291C22⋊C4—2141no—
292C22⋊C4—2141no—
293C22⋊C4—2141no—
294C42—1071no—
295C42—2141no—
296C42—2141no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—1071no—
303C23D4×D10711yescharacteristic
304Q8—2141no—
305Q8—2141no—
306Q8—2141no—
307Q8—2141no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—8561no—
313D4—4282no—
314D4—2142no—
315D4#1313381812yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—8561no—
322C2×C4—1071no—
323C2×C4—1071no—
324C2×C4—1071no—
325C2×C4—4282no—
326C2×C4—2141no—
327C2×C4—2141no—
328C2×C4—2141no—
329C2×C4—2141no—
330C2×C4—2141no—
331C2×C4—2141no—
332C2×C4D4×D10711yescharacteristic
333C2×C4C23×D10711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2141no—
337C8—2141no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—84no—
347C22—2141no—
348C22—2141no—
349C22—2141no—
350C22#1681212812yes—
351C22C2×D4×D10711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—4281no—
356C4—4281no—
357C4—4281no—
358C4—4281no—
359C4—2141no—
360C4—2141no—
361C4—2141no—
362C4C2×#1313381811yescharacteristic
363C4#1681215411yescharacteristic
364C2—22no—
365C2—42no—
366C2—81no—
367C2—4281no—
368C2#2029326711yescharacteristic
369C2C2×#1681212811yescharacteristic
370C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#102×C107C2
socleC22×C107C22×D4
prime radical#20293271C2
hypercenterC2×#102D107
hypocenterC107#3296
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #20293312C2semidirect
3. #20292980C2semidirect
4. #20292957C2semidirect
5. #20292949C2semidirect
6. #20292925C2semidirect
7. #20292868C2semidirect
8. #20292855C2semidirect
9. #20292848C2semidirect
10. #20292790C2semidirect
11. #20292778C2semidirect
12. #20292718C2semidirect
13. C8×C22⋊D107C2semidirect
14. #20292620C2semidirect
15. C2×#102×C107C2semidirect
17. C2×Dic428C22semidirect
20. C2×C8×D107C22semidirect
22. #16812100C22semidirect
27. #16812089C22semidirect
33. #16812074C22semidirect
37. #16812067C22semidirect
39. #16812043C22semidirect
40. #16812040C22semidirect
41. #16812040C22semidirect
42. Dic107⋊C8C22semidirect
43. #16812034C22semidirect
44. #16812034C22semidirect
45. #16812032C22semidirect
46. #16812032C22semidirect
50. (C2×D107)⋊C8C22semidirect
51. C8×Dic107C22semidirect
186. C107#3296semidirect
201. C2×#102D107semidirect

Maximal subgroups

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

subgroupindex
#202933122
#202932712
#202929802
#202929572
#202929492
#202929252
#202928682
#202928552
#202928482
#202927902
#202927782
#202927182
C8×C22⋊D1072
#202926202
C2×#102×C1072
#3296107