Finite Group Databasebeta

← #28347973

Subgroups — #28347973

Order 13696.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812138C211yesmaximal, characteristic, complemented
2C2×SD16×D107C211yesmaximal, characteristic, complemented
3Q8×D428C211yesmaximal, characteristic, complemented
4#20293272C211yesmaximal, characteristic
5#20293021C211yesmaximal, characteristic, complemented
6#20292874C211yesmaximal, characteristic, complemented
7#20292869C211yesmaximal, characteristic, complemented
8#20292849C211yesmaximal, characteristic, complemented
9#20292832C211yesmaximal, characteristic, complemented
10#20292790C211yesmaximal, characteristic, complemented
11#20292784C211yesmaximal, characteristic, complemented
12#20292779C211yesmaximal, characteristic, complemented
13#20292767C211yesmaximal, characteristic, complemented
14#20292700C211yesmaximal, characteristic, complemented
15#440×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×Q8×D107C2211yescharacteristic
18C2×Q8×D107C2211yescharacteristic, complemented
19C2×D4×D107C2211yescharacteristic
20#16812138—24no—
21C2×Dic428C2211yescharacteristic, complemented
22C2×Dic428C2211yescharacteristic, complemented
23C2×#13133808C2211yescharacteristic, complemented
24C2×#13133807C2211yescharacteristic
25C2×#13133807C2211yescharacteristic
26C2×#13133806C2211yescharacteristic
27C2×#13133806C2211yescharacteristic
28C2×#13133805C2211yescharacteristic
29SD16×D107—24no—
30C2×#13133801C2211yescharacteristic, complemented
31C2×C8×D107C2211yescharacteristic
32#16812109—21no—
33#16812109C2211yescharacteristic, complemented
34#16812097C2211yescharacteristic
35#16812093—21no—
36#16812089—21no—
37#16812089—21no—
38#16812089C2211yescharacteristic
39#16812089C2211yescharacteristic
40#16812089C2211yescharacteristic, complemented
41#16812084C2211yescharacteristic
42#16812083—21no—
43#16812079—21no—
44C4×D428—21no—
45C4×D428C2211yescharacteristic
46C2×(C2×D107)⋊C4—21no—
47C4⋊(C107⋊C8)C2211yescharacteristic, complemented
48#16812032C2211yescharacteristic, complemented
49#16812032C2211yescharacteristic, complemented
50#16812032C2211yescharacteristic, complemented
51#16812029C2211yescharacteristic
52(C2×D107)⋊C8C2211yescharacteristic
53(C2×D107)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic
55C4×Q8×C107C2211yescharacteristic, complemented
56C4×D4×C107C2211yescharacteristic
57#107×C107C2211yescharacteristic, complemented
58#103×C107C2211yescharacteristic, complemented
59#102×C107C2211yescharacteristic
60#97×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133817—22no—
63#13133817—41no—
64C2×Dic214—21no—
65C2×Dic214—21no—
66C2×Dic214C2311yescharacteristic
67C2×Dic214C2311yescharacteristic
68C2×Dic214C2311yescharacteristic
69Q8×D107—22no—
70Q8×D107—22no—
71Q8×D107—41no—
72Q8×D107—41no—
73C2×D428—21no—
74C2×D428D411yescharacteristic
75C2×D428C2311yescharacteristic
76C2×C22⋊D107—21no—
77C2×C22⋊D107—21no—
78D4×D107—22no—
79D4×D107—41no—
80C22×Dic107—21no—
81C2×C4×D107—21no—
82C2×C4×D107—21no—
83C2×C4×D107C2311yescharacteristic
84C2×C4×D107C2311yescharacteristic
85Dic428—22no—
86Dic428—41no—
87#13133808—22no—
88#13133807—22no—
89#13133807—41no—
90#13133806—22no—
91#13133806—22no—
92#13133805—22no—
93#13133801—22no—
94C2×C107⋊C8C2311yescharacteristic
95C2×C107⋊C8C2311yescharacteristic
96C8×D107—22no—
97C4⋊Dic107—21no—
98C4⋊Dic107—21no—
99C4⋊Dic107—21no—
100C4⋊Dic107—21no—
101C4⋊Dic107—21no—
102C4⋊Dic107D411yescharacteristic
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4—21no—
108(C2×D107)⋊C4—21no—
109(C2×D107)⋊C4D411yescharacteristic
110(C2×D107)⋊C4D411yescharacteristic
111C2×Q8×C107C2311yescharacteristic
112C2×D4×C107C2311yescharacteristic
113C22×C4×C107—21no—
114SD16×C107D414yescomplemented
115C2×C8×C107C2311yescharacteristic
116C2×C8×C107C2311yescharacteristic
117C4⋊C4×C107—21no—
118C4⋊C4×C107C2311yescharacteristic
119C4⋊C4×C107C2311yescharacteristic
120C22⋊C4×C107—21no—
121C42×C107—21no—
122C42×C107C2311yescharacteristic
123C23×C107—21no—
124C22×D107—21no—
125C22×D107—42no—
126C22×D107—42no—
127C22×D107—81no—
128C22×D107C2×D411yescharacteristic
129C22×D107C2×D411yescharacteristic
130Q8×C107—21no—
131Q8×C107C2×D412yes—
132D4×C107—21no—
133D4×C107C2×D412yes—
134C2×C4×C107—21no—
135C2×C4×C107—21no—
136C2×C4×C107—21no—
137C2×C4×C107—81no—
138C2×C4×C107C2×D411yescharacteristic
139C2×C4×C107C2×D411yescharacteristic
140C2×C4×C107C2411yescharacteristic
141C22⋊D107—42no—
142C22⋊D107—42no—
143C2×Dic107—21no—
144C2×Dic107—21no—
145C2×Dic107—21no—
146C2×Dic107—42no—
147C2×Dic107C2×D411yescharacteristic
148C2×Dic107C2×D411yescharacteristic
149D428—22no—
150D428—21no—
151D428—21no—
152D428—81no—
153C4×D107—21no—
154C4×D107—21no—
155C4×D107—21no—
156C4×D107—21no—
157C4×D107—42no—
158C4×D107—42no—
159Dic214—21no—
160Dic214—21no—
161Dic214—21no—
162Dic214—21no—
163Dic214—21no—
164Dic214—21no—
165Dic214—42no—
166Dic214—42no—
167C8×C107—21no—
168C8×C107C2×D412yes—
169C107⋊C8—21no—
170C107⋊C8—21no—
171C22×C107—22no—
172C22×C107—41no—
173C22×C107—41no—
174C22×C107C22×D411yescharacteristic
175C2×D107—21no—
176C2×D107—21no—
177C2×D107—21no—
178C2×D107—21no—
179C2×D107—21no—
180C2×D107—41no—
181C2×D107—41no—
182C2×D107—41no—
183C2×D107—84no—
184C2×D107#13712yescomplemented
185C4×C107—22no—
186C4×C107—21no—
187C4×C107—21no—
188C4×C107—41no—
189C4×C107—41no—
190C4×C107C22×D411yescharacteristic
191C4×C107H(5,𝔽2)11yescharacteristic
192Dic107—21no—
193Dic107—21no—
194Dic107—41no—
195Dic107—41no—
196Dic107—41no—
197C2×C107—42no—
198C2×C107D4211yescharacteristic
199C2×C107C2×#13711yescharacteristic
200C2×C107#57711yescharacteristic
201D107—22no—
202D107—41no—
203D107—81no—
204#3266—1071nomaximal, Sylow
205C107#326611yescharacteristic, complemented, Sylow
206C2×#137—1071no—
207C22×SD16—1071no—
208D4×Q8—1071no—
209#546—1071no—
210#497—1071no—
211#468—1071no—
212#467—1071no—
213#464—1071no—
214#461—1071no—
215#452—1071no—
216#451—1071no—
217#450—1071no—
218#448—1071no—
219#440D10711yescharacteristic, complemented
220#435—1071no—
221C2×#41—1071no—
222C22×Q8—1071no—
223C22×Q8—1071no—
224C22×D4—1071no—
225#137—2144no—
226C2×Q16—1071no—
227C2×Q16—1071no—
228C2×Q16—1071no—
229C2×SD16—1071no—
230C2×SD16—1071no—
231C2×SD16—1071no—
232C2×SD16—1071no—
233C2×SD16—1071no—
234C2×SD16—2144no—
235C2×SD16C2×D10711yescharacteristic
236C2×M16—1071no—
237C22×C8—1071no—
238C4⋊Q8—1071no—
239C4⋊Q8—2141no—
240#124—1071no—
241#123—2141no—
242C22⋊Q8—1071no—
243C22⋊Q8—1071no—
244C22⋊Q8—1071no—
245C22⋊Q8—2141no—
246C22⋊Q8—2141no—
247#121—1071no—
248#121—2141no—
249#120—2141no—
250C4×Q8C2×D10711yescharacteristic, complemented
251C4×D4—1071no—
252C4×D4—2141no—
253C4×D4C2×D10711yescharacteristic
254C2×C22⋊C4—2141no—
255#107C2×D10711yescharacteristic, complemented
256C4⋊C8—1071no—
257#103—1071no—
258#103—1071no—
259#103—1071no—
260#103C2×D10711yescharacteristic, complemented
261#102—1071no—
262#102C2×D10711yescharacteristic
263C22⋊C8—1071no—
264C22⋊C8—1071no—
265#97C2×D10711yescharacteristic, complemented
266C24—2141no—
267#41—4281no—
268#41—2142no—
269C2×Q8—1071no—
270C2×Q8—1071no—
271C2×Q8—1071no—
272C2×Q8—4281no—
273C2×Q8—4281no—
274C2×Q8—2141no—
275C2×Q8—2141no—
276C2×Q8—2142no—
277C2×Q8—2142no—
278C2×Q8C22×D10711yescharacteristic
279C2×D4—1071no—
280C2×D4—1071no—
281C2×D4—4281no—
282C2×D4—2141no—
283C2×D4—2141no—
284C2×D4—2141no—
285C2×D4—2142no—
286C2×D4C22×D10711yescharacteristic
287C22×C4—21no—
288C22×C4—1071no—
289C22×C4—1071no—
290C22×C4—2141no—
291C22×C4—2141no—
292C22×C4—2141no—
293Q16—4281no—
294Q16—2142no—
295Q16—2142no—
296SD16—4281no—
297SD16—2142no—
298SD16—2142no—
299SD16—2142no—
300SD16—2142no—
301SD16D42814yescomplemented
302M16—2142no—
303C2×C8—1071no—
304C2×C8—1071no—
305C2×C8—2142no—
306C2×C8C22×D10711yescharacteristic
307C2×C8C22×D10711yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—1071no—
310C4⋊C4—2141no—
311C4⋊C4—2141no—
312C4⋊C4—2141no—
313C4⋊C4—2141no—
314C4⋊C4—2141no—
315C4⋊C4C22×D10711yescharacteristic
316C4⋊C4C22×D10711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—1071no—
319C22⋊C4—1071no—
320C22⋊C4—2141no—
321C22⋊C4—2141no—
322C22⋊C4—2141no—
323C22⋊C4—2141no—
324C22⋊C4—2141no—
325C22⋊C4—2141no—
326C42—21no—
327C42C22×D10711yescharacteristic
328C23—21no—
329C23—8561no—
330C23—1071no—
331C23—1071no—
332C23—4282no—
333C23—4282no—
334C23—2141no—
335Q8—21no—
336Q8—4282no—
337Q8—4282no—
338Q8—2141no—
339Q8—2141no—
340Q8—2141no—
341Q8—2141no—
342Q8—2141no—
343Q8—2141no—
344Q8C2×D42812yes—
345D4—21no—
346D4—8561no—
347D4—4282no—
348D4—4282no—
349D4—2142no—
350D4—2141no—
351D4—2141no—
352D4C2×D42812yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—81no—
357C2×C4—1071no—
358C2×C4—1071no—
359C2×C4—4282no—
360C2×C4—4282no—
361C2×C4—4282no—
362C2×C4—2141no—
363C2×C4—2141no—
364C2×C4—2141no—
365C2×C4—2141no—
366C2×C4—2141no—
367C2×C4—2141no—
368C2×C4—2141no—
369C2×C4D4×D10711yescharacteristic
370C2×C4D4×D10711yescharacteristic
371C2×C4C23×D10711yescharacteristic
372C8—21no—
373C8—2141no—
374C8—2141no—
375C8C2×D42812yes—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—8564no—
380C22—1072no—
381C22—4281no—
382C22—4281no—
383C22—4281no—
384C22—2141no—
385C22—2141no—
386C22—2141no—
387C22—2141no—
388C22—2141no—
389C22C2×D4×D10711yescharacteristic
390C4—21no—
391C4—21no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—4281no—
396C4—4281no—
397C4—4281no—
398C4—2141no—
399C4—2141no—
400C4C22×D42811yescharacteristic
401C4#1681215511yescharacteristic
402C2—42no—
403C2—8561no—
404C2—4281no—
405C2—2142no—
406C2D4×D42811yescharacteristic
407C2#137×D10711yescharacteristic
408C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#440×C107C2
socleC22×C107C22×D4
prime radical#20293272C2
hypercenter#440D107
hypocenterC107#3266
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. C2×#16812138C2semidirect
2. C2×SD16×D107C2semidirect
3. Q8×D428C2semidirect
5. #20293021C2semidirect
6. #20292874C2semidirect
7. #20292869C2semidirect
8. #20292849C2semidirect
9. #20292832C2semidirect
10. #20292790C2semidirect
11. #20292784C2semidirect
12. #20292779C2semidirect
13. #20292767C2semidirect
14. #20292700C2semidirect
15. #440×C107C2semidirect
18. C2×Q8×D107C22semidirect
21. C2×Dic428C22semidirect
22. C2×Dic428C22semidirect
23. C2×#13133808C22semidirect
30. C2×#13133801C22semidirect
33. #16812109C22semidirect
40. #16812089C22semidirect
47. C4⋊(C107⋊C8)C22semidirect
48. #16812032C22semidirect
49. #16812032C22semidirect
50. #16812032C22semidirect
53. (C2×D107)⋊C8C22semidirect
55. C4×Q8×C107C22semidirect
57. #107×C107C22semidirect
58. #103×C107C22semidirect
60. #97×C107C22semidirect
114. SD16×C107D4semidirect
184. C2×D107#137semidirect
205. C107#3266semidirect
219. #440D107semidirect
250. C4×Q8C2×D107semidirect
255. #107C2×D107semidirect
260. #103C2×D107semidirect
265. #97C2×D107semidirect
301. SD16D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121382
C2×SD16×D1072
Q8×D4282
#202932722
#202930212
#202928742
#202928692
#202928492
#202928322
#202927902
#202927842
#202927792
#202927672
#202927002
#440×C1072
#3266107