Finite Group Databasebeta

← #28348297

Subgroups — #28348297

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#20293316C211yesmaximal, characteristic, complemented
2#20293265C211yesmaximal, characteristic
3#20292983C211yesmaximal, characteristic, complemented
4#20292956C211yesmaximal, characteristic, complemented
5#20292951C211yesmaximal, characteristic, complemented
6#20292929C211yesmaximal, characteristic, complemented
7#20292867C211yesmaximal, characteristic, complemented
8#20292856C211yesmaximal, characteristic, complemented
9#20292852C211yesmaximal, characteristic, complemented
10#20292789C211yesmaximal, characteristic, complemented
11#20292779C211yesmaximal, characteristic, complemented
12#20292718C211yesmaximal, characteristic, complemented
13C8×C22⋊D107C211yesmaximal, characteristic, complemented
14#102×D107C211yesmaximal, characteristic, complemented
15#416×C107C211yesmaximal, characteristic, complemented
16C2×D4×D107C2211yescharacteristic
17C2×#13133808C2211yescharacteristic, complemented
18C2×#13133807C2211yescharacteristic
19C2×#13133806C2211yescharacteristic
20C2×C8×D107C2211yescharacteristic
21#16812102—21no—
22#16812101—21no—
23#16812100C2211yescharacteristic, complemented
24#16812098C2211yescharacteristic
25#16812092—21no—
26#16812091—21no—
27#16812089C2211yescharacteristic
28#16812088C2211yescharacteristic, complemented
29#16812085C2211yescharacteristic
30#16812083C2211yescharacteristic
31#16812081—21no—
32#16812077—21no—
33#16812074C2211yescharacteristic, complemented
34D4×Dic107C2211yescharacteristic
35C4×C22⋊D107C2211yescharacteristic
36#16812066—21no—
37C4⋊C4×D107C2211yescharacteristic
38C2×(C2×D107)⋊C4—21no—
39#16812042C2211yescharacteristic, complemented
40#16812041C2211yescharacteristic, complemented
41#16812040C2211yescharacteristic, complemented
42Dic107⋊C8C2211yescharacteristic, complemented
43#16812034C2211yescharacteristic, complemented
44#16812033C2211yescharacteristic, complemented
45#16812031C2211yescharacteristic
46#16812030C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic, complemented
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic, complemented
52C2×#41×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic, complemented
54#117×C107C2211yescharacteristic, complemented
55#103×C107C2211yescharacteristic, complemented
56#103×C107C2211yescharacteristic, complemented
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C23×D107—21no—
60C2×Dic214C2311yescharacteristic
61C2×D428C2311yescharacteristic
62C2×C22⋊D107—21no—
63C2×C22⋊D107—21no—
64C2×C22⋊D107D411yescharacteristic
65D4×D107—22no—
66D4×D107—41no—
67C22×Dic107—21no—
68C2×C4×D107—21no—
69C2×C4×D107C2311yescharacteristic
70#13133808—41no—
71#13133807—41no—
72#13133806—22no—
73C2×C107⋊C8C2311yescharacteristic
74C2×C107⋊C8C2311yescharacteristic
75C8×D107—41no—
76C4⋊Dic107—21no—
77C4⋊Dic107—21no—
78C4⋊Dic107C2311yescharacteristic
79C4⋊Dic107C2311yescharacteristic
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4D411yescharacteristic
86C22⋊Dic107—21no—
87C22⋊Dic107—21no—
88C22⋊Dic107D411yescharacteristic
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4D411yescharacteristic
95C4×Dic107—21no—
96C4×Dic107C2311yescharacteristic
97#41×C107—22no—
98#41×C107—41no—
99C2×Q8×C107C2311yescharacteristic
100C2×D4×C107—21no—
101C2×D4×C107C2311yescharacteristic
102C22×C4×C107—21no—
103C22×C4×C107C2311yescharacteristic
104C2×C8×C107—41no—
105C2×C8×C107C2311yescharacteristic
106C2×C8×C107C2311yescharacteristic
107C4⋊C4×C107C2311yescharacteristic
108C4⋊C4×C107C2311yescharacteristic
109C22⋊C4×C107—21no—
110C42×C107—21no—
111C23×C107—21no—
112C23×C107C2×D411yescharacteristic
113C22×D107—21no—
114C22×D107—42no—
115C22×D107—42no—
116C22×D107—81no—
117C22×D107C2×D411yescharacteristic
118Q8×C107—21no—
119Q8×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—21no—
128C2×C4×C107—42no—
129C2×C4×C107C2×D411yescharacteristic
130C2×C4×C107C2411yescharacteristic
131C22⋊D107—22no—
132C22⋊D107—42no—
133C22⋊D107—81no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107—21no—
137C2×Dic107—81no—
138C2×Dic107C2×D411yescharacteristic
139C2×Dic107C2×D411yescharacteristic
140C2×Dic107C2×D411yescharacteristic
141D428—21no—
142D428—21no—
143C4×D107—21no—
144C4×D107—21no—
145C4×D107—42no—
146Dic214—21no—
147Dic214—21no—
148C8×C107—21no—
149C8×C107—21no—
150C107⋊C8—21no—
151C107⋊C8—21no—
152C22×C107—22no—
153C22×C107—21no—
154C22×C107—21no—
155C22×C107—21no—
156C22×C107—41no—
157C22×C107—41no—
158C22×C107C22×D411yescharacteristic
159C2×D107—21no—
160C2×D107—21no—
161C2×D107—41no—
162C2×D107—41no—
163C2×D107—41no—
164C2×D107—84no—
165C2×D107#13512yes—
166C4×C107—21no—
167C4×C107—41no—
168C4×C107—41no—
169C4×C107—41no—
170C4×C107C2×#4111yescharacteristic
171C4×C107H(5,𝔽2)11yescharacteristic
172Dic107—21no—
173Dic107—21no—
174Dic107—21no—
175Dic107—41no—
176Dic107—41no—
177Dic107—41no—
178C2×C107—42no—
179C2×C107—41no—
180C2×C107#54611yescharacteristic
181C2×C107C2×#13511yescharacteristic
182C2×C107#57711yescharacteristic
183D107—22no—
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×D4—1071no—
200#416D10711yescharacteristic, complemented
201C2×#102—1071no—
202C2×#41C2×D10711yescharacteristic
203C22×D4—1071no—
204C2×Q16—1071no—
205C2×SD16—1071no—
206C2×SD16—1071no—
207C22×C8—1071no—
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×D4—1071no—
224#117—2141no—
225#117C2×D10711yescharacteristic, complemented
226C2×C4⋊C4—1071no—
227C2×C22⋊C4—2141no—
228#107—1071no—
229#106—1071no—
230#106—1071no—
231C4⋊C8—1071no—
232#103—1071no—
233#103—1071no—
234#103C2×D10711yescharacteristic, complemented
235#103C2×D10711yescharacteristic, complemented
236#102—1071no—
237#102—1071no—
238#102—1071no—
239#102—1071no—
240#102C2×D10711yescharacteristic
241#102C2×D10711yescharacteristic
242C22⋊C8—1071no—
243C22⋊C8—1071no—
244C4×C8—1071no—
245C24—2141no—
246#41—22no—
247#41—41no—
248C2×Q8—1071no—
249C2×Q8C22×D10711yescharacteristic
250C2×D4—21no—
251C2×D4—1071no—
252C2×D4—1071no—
253C2×D4—4281no—
254C2×D4—2141no—
255C2×D4—2141no—
256C2×D4—2142no—
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—1071no—
273C4⋊C4—1071no—
274C4⋊C4—1071no—
275C4⋊C4—2141no—
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—21no—
285C22⋊C4—1071no—
286C22⋊C4—1071no—
287C22⋊C4—2141no—
288C22⋊C4—2141no—
289C22⋊C4—2141no—
290C22⋊C4—2141no—
291C22⋊C4—2141no—
292C22⋊C4—2141no—
293C22⋊C4—2141no—
294C42—21no—
295C42—1071no—
296C42—2141no—
297C23—21no—
298C23—8561no—
299C23—1071no—
300C23—4282no—
301C23—4282no—
302C23—2141no—
303C23D4×D10711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2141no—
307Q8—2141no—
308D4—21no—
309D4—42no—
310D4—8561no—
311D4—4282no—
312D4—2142no—
313D4—2141no—
314D4—2141no—
315D4#1313381812yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—8561no—
323C2×C4—1071no—
324C2×C4—1071no—
325C2×C4—1071no—
326C2×C4—4282no—
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—21no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—8564no—
345C22—1072no—
346C22—4281no—
347C22—4281no—
348C22—4281no—
349C22—2141no—
350C22—2141no—
351C22C2×D4×D10711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—4281no—
357C4—4281no—
358C4—4281no—
359C4—2141no—
360C4—2141no—
361C4—2141no—
362C4C2×#1313381811yescharacteristic
363C4#1681215411yescharacteristic
364C2—41no—
365C2—42no—
366C2—8561no—
367C2—2142no—
368C2#2029326711yescharacteristic
369C2#135×D10711yescharacteristic
370C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#416×C107C2
socleC22×C107C22×D4
prime radical#20293265C2
hypercenter#416D107
hypocenterC107#3296
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #20293316C2semidirect
3. #20292983C2semidirect
4. #20292956C2semidirect
5. #20292951C2semidirect
6. #20292929C2semidirect
7. #20292867C2semidirect
8. #20292856C2semidirect
9. #20292852C2semidirect
10. #20292789C2semidirect
11. #20292779C2semidirect
12. #20292718C2semidirect
13. C8×C22⋊D107C2semidirect
14. #102×D107C2semidirect
15. #416×C107C2semidirect
17. C2×#13133808C22semidirect
23. #16812100C22semidirect
28. #16812088C22semidirect
33. #16812074C22semidirect
39. #16812042C22semidirect
40. #16812041C22semidirect
41. #16812040C22semidirect
42. Dic107⋊C8C22semidirect
43. #16812034C22semidirect
44. #16812033C22semidirect
49. C22⋊(C107⋊C8)C22semidirect
51. C8×Dic107C22semidirect
53. C22×C8×C107C22semidirect
54. #117×C107C22semidirect
55. #103×C107C22semidirect
56. #103×C107C22semidirect
186. C107#3296semidirect
200. #416D107semidirect
208. C22×C8C2×D107semidirect
225. #117C2×D107semidirect
234. #103C2×D107semidirect
235. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202933162
#202932652
#202929832
#202929562
#202929512
#202929292
#202928672
#202928562
#202928522
#202927892
#202927792
#202927182
C8×C22⋊D1072
#102×D1072
#416×C1072
#3296107