Finite Group Databasebeta

← #28347975

Subgroups — #28347975

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
2C22×#13133806C211yesmaximal, characteristic, complemented
3#20293290C211yesmaximal, characteristic, complemented
4#20293271C211yesmaximal, characteristic
5#20293018C211yesmaximal, characteristic, complemented
6#20292872C211yesmaximal, characteristic, complemented
7#20292866C211yesmaximal, characteristic, complemented
8#20292848C211yesmaximal, characteristic, complemented
9#20292830C211yesmaximal, characteristic, complemented
10#20292790C211yesmaximal, characteristic, complemented
11#20292785C211yesmaximal, characteristic, complemented
12#20292764C211yesmaximal, characteristic, complemented
13#20292724C211yesmaximal, characteristic, complemented
14#20292695C211yesmaximal, characteristic, complemented
15#450×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C22×Dic214C2211yescharacteristic
18C2×Q8×D107C2211yescharacteristic, complemented
19#16812138—24no—
20C2×Dic428C2211yescharacteristic, complemented
21C2×Dic428C2211yescharacteristic, complemented
22C2×#13133808C2211yescharacteristic, complemented
23C2×#13133806—24no—
24C2×#13133806C2211yescharacteristic
25C2×#13133806C2211yescharacteristic
26C2×#13133806C2211yescharacteristic
27C2×#13133806C2211yescharacteristic
28C2×#13133801C2211yescharacteristic, complemented
29C22×C107⋊C8C2211yescharacteristic
30#16812108—21no—
31#16812107C2211yescharacteristic, complemented
32#16812096C2211yescharacteristic
33#16812090—21no—
34#16812089C2211yescharacteristic, complemented
35#16812088C2211yescharacteristic
36#16812087—21no—
37#16812086—21no—
38#16812081—21no—
39#16812080C2211yescharacteristic
40#16812078—21no—
41#16812074C2211yescharacteristic, complemented
42D4×Dic107C2211yescharacteristic
43C4×C22⋊D107C2211yescharacteristic
44(C22⋊D107)⋊C4—21no—
45C2×C22⋊Dic107—21no—
46#16812042C2211yescharacteristic, complemented
47Dic107⋊C8C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812034C2211yescharacteristic, complemented
50#16812033C2211yescharacteristic, complemented
51#16812032C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53#16812019C2211yescharacteristic, complemented
54C22×D4×C107C2211yescharacteristic
55C2×SD16×C107C2211yescharacteristic
56C2×SD16×C107C2211yescharacteristic
57C22⋊Q8×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59#102×C107C2211yescharacteristic
60C22⋊C8×C107C2211yescharacteristic
61#13133817—22no—
62#13133817—41no—
63C2×Dic214—22no—
64C2×Dic214—21no—
65C2×Dic214—21no—
66C2×Dic214—41no—
67C2×Dic214C2311yescharacteristic
68C2×Dic214C2311yescharacteristic
69C2×Dic214C2311yescharacteristic
70Q8×D107—22no—
71Q8×D107—41no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74C2×C22⋊D107D411yescharacteristic
75C22×Dic107—21no—
76C22×Dic107—21no—
77C22×Dic107—21no—
78C2×C4×D107—21no—
79C2×C4×D107C2311yescharacteristic
80Dic428—22no—
81Dic428—41no—
82#13133808—22no—
83#13133806—22no—
84#13133806—22no—
85#13133806—22no—
86#13133806D414yescomplemented
87#13133801—22no—
88C2×C107⋊C8—22no—
89C2×C107⋊C8C2311yescharacteristic
90C2×C107⋊C8C2311yescharacteristic
91C4⋊Dic107—21no—
92C4⋊Dic107C2311yescharacteristic
93Dic107⋊C4—21no—
94Dic107⋊C4—21no—
95Dic107⋊C4—21no—
96Dic107⋊C4—21no—
97Dic107⋊C4D411yescharacteristic
98C22⋊Dic107—21no—
99C22⋊Dic107—21no—
100C22⋊Dic107—21no—
101C22⋊Dic107—21no—
102C22⋊Dic107—21no—
103C22⋊Dic107D411yescharacteristic
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4D411yescharacteristic
106C4×Dic107—21no—
107C4×Dic107C2311yescharacteristic
108C24×C107—21no—
109C2×Q8×C107C2311yescharacteristic
110C2×D4×C107—22no—
111C2×D4×C107—21no—
112C2×D4×C107—41no—
113C2×D4×C107C2311yescharacteristic
114C2×D4×C107C2311yescharacteristic
115C22×C4×C107C2311yescharacteristic
116SD16×C107—22no—
117SD16×C107—41no—
118C2×C8×C107C2311yescharacteristic
119C2×C8×C107C2311yescharacteristic
120C4⋊C4×C107—21no—
121C4⋊C4×C107C2311yescharacteristic
122C22⋊C4×C107—21no—
123C23×C107—21no—
124C23×C107—21no—
125C23×C107—42no—
126C23×C107—42no—
127C23×C107—81no—
128C23×C107C2×D411yescharacteristic
129C22×D107C2×D411yescharacteristic
130Q8×C107—21no—
131Q8×C107—21no—
132D4×C107—21no—
133D4×C107—21no—
134D4×C107—21no—
135D4×C107—42no—
136D4×C107C2×D412yes—
137C2×C4×C107—21no—
138C2×C4×C107—21no—
139C2×C4×C107—21no—
140C2×C4×C107—21no—
141C2×C4×C107C2×D411yescharacteristic
142C2×C4×C107C2411yescharacteristic
143C22⋊D107—22no—
144C22⋊D107—42no—
145C22⋊D107—81no—
146C2×Dic107—21no—
147C2×Dic107—21no—
148C2×Dic107—21no—
149C2×Dic107—21no—
150C2×Dic107—42no—
151C2×Dic107—42no—
152C2×Dic107—81no—
153C2×Dic107C2×D411yescharacteristic
154C2×Dic107C2×D411yescharacteristic
155C2×Dic107C2×D411yescharacteristic
156C4×D107—21no—
157C4×D107—21no—
158C4×D107—42no—
159Dic214—21no—
160Dic214—21no—
161Dic214—21no—
162Dic214—21no—
163Dic214—21no—
164Dic214—42no—
165Dic214—42no—
166Dic214C2×D412yes—
167C8×C107—21no—
168C8×C107—21no—
169C107⋊C8—21no—
170C107⋊C8C2×D412yes—
171C22×C107—21no—
172C22×C107—21no—
173C22×C107—22no—
174C22×C107—41no—
175C22×C107—41no—
176C22×C107—41no—
177C22×C107—41no—
178C22×C107—41no—
179C22×C107—84no—
180C22×C107#13712yescomplemented
181C22×C107C22×D411yescharacteristic
182C2×D107—21no—
183C2×D107—21no—
184C2×D107—21no—
185C4×C107—21no—
186C4×C107—41no—
187C4×C107—41no—
188C4×C107C22×D411yescharacteristic
189C4×C107H(5,𝔽2)11yescharacteristic
190Dic107—22no—
191Dic107—21no—
192Dic107—21no—
193Dic107—21no—
194Dic107—41no—
195Dic107—41no—
196Dic107—41no—
197C2×C107—22no—
198C2×C107—42no—
199C2×C107—81no—
200C2×C107D4211yescharacteristic
201C2×C107C2×#13711yescharacteristic
202C2×C107#57711yescharacteristic
203D107—41no—
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#450D10711yescharacteristic, complemented
218#448—1071no—
219#440—1071no—
220#435—1071no—
221C2×#41—1071no—
222C22×Q8—1071no—
223C22×Q8—1071no—
224C22×D4C2×D10711yescharacteristic
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—2144no—
234C2×SD16C2×D10711yescharacteristic
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—2141no—
245C22⋊Q8—2141no—
246C22⋊Q8C2×D10711yescharacteristic
247#121—1071no—
248#121—2141no—
249#120—2141no—
250C4×Q8—1071no—
251C4×D4—1071no—
252C4×D4—1071no—
253C4×D4—2141no—
254C2×C22⋊C4—2141no—
255#107—1071no—
256C4⋊C8—1071no—
257#103—1071no—
258#103—1071no—
259#103—1071no—
260#103—1071no—
261#102C2×D10711yescharacteristic
262#102C2×D10711yescharacteristic
263C22⋊C8—1071no—
264C22⋊C8C2×D10711yescharacteristic
265#97—1071no—
266C24—21no—
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—2142no—
276C2×Q8—2142no—
277C2×Q8—2141no—
278C2×Q8C22×D10711yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—1071no—
283C2×D4—2141no—
284C2×D4—2141no—
285C2×D4C22×D10711yescharacteristic
286C2×D4C22×D10711yescharacteristic
287C22×C4—1071no—
288C22×C4—2141no—
289C22×C4—2141no—
290C22×C4—2141no—
291C22×C4—2141no—
292C22×C4C22×D10711yescharacteristic
293Q16—4281no—
294Q16—2142no—
295Q16—2142no—
296SD16—22no—
297SD16—41no—
298SD16—1074no—
299SD16—2142no—
300SD16—2142no—
301SD16—2142no—
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—1071no—
311C4⋊C4—2141no—
312C4⋊C4—2141no—
313C4⋊C4—2141no—
314C4⋊C4—2141no—
315C4⋊C4—2141no—
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—1071no—
327C42—2141no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1071no—
334C23D4×D10711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—1072no—
338Q8—4282no—
339Q8—4282no—
340Q8—2141no—
341Q8—2141no—
342Q8—2141no—
343Q8—2141no—
344Q8—2141no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—8561no—
350D4—4282no—
351D4—2142no—
352D4D4×D10712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—8561no—
358C2×C4—1071no—
359C2×C4—1071no—
360C2×C4—1071no—
361C2×C4—4282no—
362C2×C4—4282no—
363C2×C4—4282no—
364C2×C4—2141no—
365C2×C4—2141no—
366C2×C4—2141no—
367C2×C4—2141no—
368C2×C4—2141no—
369C2×C4—2141no—
370C2×C4D4×D10711yescharacteristic
371C2×C4C23×D10711yescharacteristic
372C8—21no—
373C8—21no—
374C8—1072no—
375C8—2141no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—2141no—
386C22—2141no—
387C22—2141no—
388C22#1681213812yescomplemented
389C22C2×D4×D10711yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—4281no—
394C4—4281no—
395C4—4281no—
396C4—2141no—
397C4—2141no—
398C4—2141no—
399C4—2142no—
400C4C2×D4×D10711yescharacteristic
401C4#1681215411yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—4281no—
406C2#2029325911yescharacteristic
407C2C2×#1681213811yescharacteristic
408C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#450×C107C2
socleC22×C107C22×D4
prime radical#20293271C2
hypercenter#450D107
hypocenterC107#3266
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#16812138C2semidirect
2. C22×#13133806C2semidirect
3. #20293290C2semidirect
5. #20293018C2semidirect
6. #20292872C2semidirect
7. #20292866C2semidirect
8. #20292848C2semidirect
9. #20292830C2semidirect
10. #20292790C2semidirect
11. #20292785C2semidirect
12. #20292764C2semidirect
13. #20292724C2semidirect
14. #20292695C2semidirect
15. #450×C107C2semidirect
18. C2×Q8×D107C22semidirect
20. C2×Dic428C22semidirect
21. C2×Dic428C22semidirect
22. C2×#13133808C22semidirect
28. C2×#13133801C22semidirect
31. #16812107C22semidirect
34. #16812089C22semidirect
41. #16812074C22semidirect
46. #16812042C22semidirect
47. Dic107⋊C8C22semidirect
48. #16812034C22semidirect
49. #16812034C22semidirect
50. #16812033C22semidirect
51. #16812032C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. #16812019C22semidirect
86. #13133806D4semidirect
180. C22×C107#137semidirect
205. C107#3266semidirect
217. #450D107semidirect
388. C22#16812138semidirect

Maximal subgroups

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

subgroupindex
C2×#168121382
C22×#131338062
#202932902
#202932712
#202930182
#202928722
#202928662
#202928482
#202928302
#202927902
#202927852
#202927642
#202927242
#202926952
#450×C1072
#3266107