Finite Group Databasebeta

← #28346709

Subgroups — #28346709

Order 13696.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293237C211yesmaximal, characteristic, complemented
2#20293101C211yesmaximal, characteristic
3#20292968C211yesmaximal, characteristic, complemented
4#20292930C211yesmaximal, characteristic, complemented
5#20292926C211yesmaximal, characteristic, complemented
6#20292831C211yesmaximal, characteristic, complemented
7#20292831C211yesmaximal, characteristic, complemented
8#20292780C211yesmaximal, characteristic
9#20292777C211yesmaximal, characteristic
10#20292730C211yesmaximal, characteristic, complemented
11#20292726C211yesmaximal, characteristic, complemented
12#20292714C211yesmaximal, characteristic, complemented
13#20292712C211yesmaximal, characteristic, complemented
14#20292685C211yesmaximal, characteristic, complemented
15#485×C107C211yesmaximal, characteristic, complemented
16C22×D428C2211yescharacteristic
17C2×#13133807C2211yescharacteristic
18C2×#13133807C2211yescharacteristic
19C2×#13133805C2211yescharacteristic
20C2×#13133805C2211yescharacteristic
21#16812107C2211yescharacteristic, complemented
22#16812102—21no—
23#16812102—21no—
24#16812100—21no—
25#16812100C2211yescharacteristic, complemented
26#16812099—21no—
27#16812088C2211yescharacteristic
28Q8×Dic107C2211yescharacteristic, complemented
29#16812074C2211yescharacteristic, complemented
30C4×D428—21no—
31C4×D428—21no—
32#16812071C2211yescharacteristic
33#16812071C2211yescharacteristic
34#16812071C2211yescharacteristic
35#16812071C2211yescharacteristic
36#16812068C2211yescharacteristic
37C22⋊C4×D107—21no—
38C22⋊C4×D107—21no—
39#16812042C2211yescharacteristic, complemented
40#16812040C2211yescharacteristic, complemented
41Dic107⋊C8C2211yescharacteristic, complemented
42Dic107⋊C8C2211yescharacteristic, complemented
43#16812032C2211yescharacteristic, complemented
44#16812032C2211yescharacteristic, complemented
45#16812031C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50#16812019C2211yescharacteristic, complemented
51C8×Dic107C2211yescharacteristic, complemented
52C22⋊Q8×C107C2211yescharacteristic
53#117×C107C2211yescharacteristic
54#107×C107C2211yescharacteristic, complemented
55#106×C107C2211yescharacteristic, complemented
56#103×C107C2211yescharacteristic, complemented
57#103×C107C2211yescharacteristic, complemented
58C22⋊C8×C107C2211yescharacteristic
59C23×D107—21no—
60C2×Dic214C2311yescharacteristic
61C2×D428—21no—
62C2×D428—22no—
63C2×D428—22no—
64C2×D428C2311yescharacteristic
65C2×D428C2311yescharacteristic
66C2×C4×D107—21no—
67C2×C4×D107—21no—
68C2×C4×D107—21no—
69C2×C4×D107—21no—
70#13133807—22no—
71#13133807—22no—
72#13133805—22no—
73#13133805—22no—
74C2×C107⋊C8C2311yescharacteristic
75C2×C107⋊C8C2311yescharacteristic
76C4⋊Dic107—21no—
77C4⋊Dic107—21no—
78C4⋊Dic107D411yescharacteristic
79C4⋊Dic107D411yescharacteristic
80C4⋊Dic107C2311yescharacteristic
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86C22⋊Dic107—21no—
87C22⋊Dic107D411yescharacteristic
88C22⋊Dic107D411yescharacteristic
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93C4×Dic107—21no—
94C4×Dic107—21no—
95C4×Dic107C2311yescharacteristic
96C4×Dic107C2311yescharacteristic
97C2×Q8×C107C2311yescharacteristic
98C22×C4×C107C2311yescharacteristic
99C2×C8×C107C2311yescharacteristic
100C2×C8×C107C2311yescharacteristic
101C4⋊C4×C107—21no—
102C4⋊C4×C107C2311yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C22⋊C4×C107—21no—
106C22⋊C4×C107—21no—
107C42×C107—21no—
108C23×C107C2×D411yescharacteristic
109C22×D107—21no—
110C22×D107—21no—
111C22×D107—24no—
112C22×D107—81no—
113C22×D107—81no—
114Q8×C107—21no—
115Q8×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107C2×D411yescharacteristic
123C2×C4×C107C2411yescharacteristic
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107C2×D411yescharacteristic
127C2×Dic107C2×D411yescharacteristic
128C2×Dic107C2×D411yescharacteristic
129C2×Dic107C2×D411yescharacteristic
130D428—24no—
131D428—21no—
132D428—21no—
133D428#4112yes—
134D428#4112yes—
135C4×D107—81no—
136C4×D107—81no—
137C4×D107—81no—
138C4×D107—81no—
139Dic214—21no—
140Dic214—21no—
141C8×C107—21no—
142C8×C107—21no—
143C107⋊C8—21no—
144C107⋊C8—21no—
145C22×C107—21no—
146C22×C107—21no—
147C22×C107—21no—
148C22×C107C22×D411yescharacteristic
149C2×D107—22no—
150C2×D107—22no—
151C2×D107—41no—
152C2×D107—41no—
153C2×D107—41no—
154C2×D107—41no—
155C2×D107—84no—
156C4×C107—21no—
157C4×C107—41no—
158C4×C107—41no—
159C4×C107—41no—
160C4×C107—41no—
161C4×C107C2×#4111yescharacteristic
162C4×C107C2×#4111yescharacteristic
163Dic107—21no—
164Dic107—21no—
165Dic107—21no—
166Dic107—21no—
167Dic107—41no—
168Dic107—41no—
169C2×C107—41no—
170C2×C107#52511yescharacteristic
171C2×C107#57711yescharacteristic
172C2×C107#57711yescharacteristic
173D107—42no—
174D107—42no—
175#3123—1071nomaximal, Sylow
176C107#312311yescharacteristic, complemented, Sylow
177#541—1071no—
178#518—1071no—
179#485—1071no—
180#485D10711yescharacteristic, complemented
181#478—1071no—
182#478—1071no—
183#461—1071no—
184#461—1071no—
185#450—1071no—
186#450—1071no—
187#440—1071no—
188#440—1071no—
189C4×SD16—1071no—
190C4×SD16—1071no—
191#433—1071no—
192C22×D4—1071no—
193C2×SD16—1071no—
194C2×SD16—1071no—
195C2×SD16—1071no—
196C2×SD16—1071no—
197C4⋊Q8—1071no—
198#126—2141no—
199#126—2141no—
200#125—1071no—
201#125—2141no—
202#125—2141no—
203C22⋊Q8—1071no—
204C22⋊Q8C2×D10711yescharacteristic
205C4×Q8—1071no—
206C4×Q8—1071no—
207C4×D4—1071no—
208C4×D4—1071no—
209C4×D4—1071no—
210C4×D4—1071no—
211C4×D4—2141no—
212C4×D4—2141no—
213#117—1071no—
214#117C2×D10711yescharacteristic
215C2×C22⋊C4—2141no—
216C2×C22⋊C4—2141no—
217#107—1071no—
218#107C2×D10711yescharacteristic, complemented
219#106—1071no—
220#106C2×D10711yescharacteristic, complemented
221C4⋊C8—1071no—
222C4⋊C8—1071no—
223#103—1071no—
224#103—1071no—
225#103C2×D10711yescharacteristic, complemented
226#103C2×D10711yescharacteristic, complemented
227#102—1071no—
228#102—1071no—
229#102—1071no—
230#102—1071no—
231C22⋊C8—1071no—
232C22⋊C8C2×D10711yescharacteristic
233#97—1071no—
234C4×C8—1071no—
235C24—2141no—
236C2×Q8—1071no—
237C2×Q8C22×D10711yescharacteristic
238C2×D4—1071no—
239C2×D4—1071no—
240C2×D4—2142no—
241C2×D4—2142no—
242C2×D4—2141no—
243C22×C4—2141no—
244C22×C4—2141no—
245C22×C4—2141no—
246C22×C4—2141no—
247C22×C4C22×D10711yescharacteristic
248SD16—2142no—
249SD16—2142no—
250SD16—2142no—
251SD16—2142no—
252C2×C8—1071no—
253C2×C8—1071no—
254C2×C8C22×D10711yescharacteristic
255C2×C8C22×D10711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—1071no—
258C4⋊C4—1071no—
259C4⋊C4—1071no—
260C4⋊C4—2141no—
261C4⋊C4—2141no—
262C4⋊C4—2141no—
263C4⋊C4—2141no—
264C4⋊C4—2141no—
265C4⋊C4—2141no—
266C4⋊C4—2141no—
267C4⋊C4C22×D10711yescharacteristic
268C4⋊C4C22×D10711yescharacteristic
269C4⋊C4C22×D10711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—1071no—
273C22⋊C4—1071no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C22⋊C4—2141no—
277C22⋊C4—2141no—
278C22⋊C4—2141no—
279C42—21no—
280C42—1071no—
281C42—1071no—
282C42—2141no—
283C42—2141no—
284C23—8561no—
285C23—8561no—
286C23—2144no—
287C23—2141no—
288C23—2141no—
289C23D4×D10711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—2141no—
293Q8—2141no—
294D4—1072no—
295D4—1072no—
296D4—2144no—
297D4—2141no—
298D4—2141no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—8561no—
306C2×C4—8561no—
307C2×C4—8561no—
308C2×C4—8561no—
309C2×C4—1071no—
310C2×C4—1071no—
311C2×C4—1071no—
312C2×C4—1071no—
313C2×C4—2141no—
314C2×C4—2141no—
315C2×C4D4×D10711yescharacteristic
316C2×C4C23×D10711yescharacteristic
317C8—21no—
318C8—21no—
319C8—2141no—
320C8—2141no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—8564no—
325C22—4281no—
326C22—4281no—
327C22—4281no—
328C22—4281no—
329C22—2142no—
330C22—2142no—
331C22C2×D4×D10711yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—4281no—
338C4—4281no—
339C4—2141no—
340C4—2141no—
341C4—2141no—
342C4—2141no—
343C4#41×D10711yescharacteristic
344C4#41×D10711yescharacteristic
345C2—41no—
346C2—4282no—
347C2—4282no—
348C2#2029313511yescharacteristic
349C2#2029343411yescharacteristic
350C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#485×C107C2
socleC22×C107C22×D4
prime radicalC22×D428C22
hypercenter#485D107
hypocenterC107#3123
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #20293237C2semidirect
3. #20292968C2semidirect
4. #20292930C2semidirect
5. #20292926C2semidirect
6. #20292831C2semidirect
7. #20292831C2semidirect
10. #20292730C2semidirect
11. #20292726C2semidirect
12. #20292714C2semidirect
13. #20292712C2semidirect
14. #20292685C2semidirect
15. #485×C107C2semidirect
21. #16812107C22semidirect
25. #16812100C22semidirect
28. Q8×Dic107C22semidirect
29. #16812074C22semidirect
39. #16812042C22semidirect
40. #16812040C22semidirect
41. Dic107⋊C8C22semidirect
42. Dic107⋊C8C22semidirect
43. #16812032C22semidirect
44. #16812032C22semidirect
50. #16812019C22semidirect
51. C8×Dic107C22semidirect
54. #107×C107C22semidirect
55. #106×C107C22semidirect
56. #103×C107C22semidirect
57. #103×C107C22semidirect
176. C107#3123semidirect
180. #485D107semidirect
218. #107C2×D107semidirect
220. #106C2×D107semidirect
225. #103C2×D107semidirect
226. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202932372
#202931012
#202929682
#202929302
#202929262
#202928312
#202927802
#202927772
#202927302
#202927262
#202927142
#202927122
#202926852
#485×C1072
#3123107