Finite Group Databasebeta

← #28347978

Subgroups — #28347978

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×#16812142C211yesmaximal, characteristic, complemented
2C2×SD16×D107C211yesmaximal, characteristic, complemented
3#20293290C211yesmaximal, characteristic, complemented
4#20293274C211yesmaximal, characteristic
5#20293018C211yesmaximal, characteristic, complemented
6#20292869C211yesmaximal, characteristic, complemented
7#20292848C211yesmaximal, characteristic, complemented
8#20292831C211yesmaximal, characteristic, complemented
9#20292787C211yesmaximal, characteristic, complemented
10#20292784C211yesmaximal, characteristic, complemented
11#20292779C211yesmaximal, characteristic, complemented
12#20292765C211yesmaximal, characteristic, complemented
13#20292726C211yesmaximal, characteristic, complemented
14#20292696C211yesmaximal, characteristic, complemented
15#468×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C22×Dic214C2211yescharacteristic, complemented
18C2×Q8×D107C2211yescharacteristic
19C2×D4×D107C2211yescharacteristic
20#16812142—24no—
21C2×Dic428C2211yescharacteristic, complemented
22C2×Dic428C2211yescharacteristic, complemented
23C2×#13133808C2211yescharacteristic, complemented
24C2×#13133807C2211yescharacteristic
25C2×#13133807C2211yescharacteristic
26C2×#13133806C2211yescharacteristic
27C2×#13133806C2211yescharacteristic
28C2×#13133805C2211yescharacteristic
29SD16×D107—24no—
30C2×C8×D107C2211yescharacteristic
31#16812108—21no—
32#16812107C2211yescharacteristic, complemented
33#16812096C2211yescharacteristic
34#16812091—21no—
35#16812089C2211yescharacteristic
36#16812089C2211yescharacteristic
37#16812087—21no—
38#16812086—21no—
39#16812080—21no—
40#16812077—21no—
41#16812074C2211yescharacteristic, complemented
42#16812071C2211yescharacteristic
43C4×C22⋊D107C2211yescharacteristic
44(C22⋊D107)⋊C4—21no—
45C22⋊C4×D107—21no—
46Dic107⋊C8C2211yescharacteristic, complemented
47#16812032C2211yescharacteristic, complemented
48#16812032C2211yescharacteristic, complemented
49#16812032C2211yescharacteristic, complemented
50#16812029C2211yescharacteristic
51C22⋊(C107⋊C8)C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic
53#16812019C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic
55C2×M16×C107C2211yescharacteristic, complemented
56C22⋊Q8×C107C2211yescharacteristic, complemented
57#121×C107C2211yescharacteristic
58#107×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic
61C23×D107—21no—
62#13133818—22no—
63#13133818—41no—
64C2×Dic214—22no—
65C2×Dic214—21no—
66C2×Dic214—21no—
67C2×Dic214—41no—
68C2×Dic214C2311yescharacteristic
69C2×Dic214C2311yescharacteristic
70C2×Dic214C2311yescharacteristic
71Q8×D107—22no—
72Q8×D107—41no—
73C2×D428C2311yescharacteristic
74C2×C22⋊D107—21no—
75C2×C22⋊D107—21no—
76C2×C22⋊D107D411yescharacteristic
77D4×D107—22no—
78D4×D107—41no—
79C22×Dic107—21no—
80C2×C4×D107—21no—
81C2×C4×D107—21no—
82C2×C4×D107—21no—
83C2×C4×D107C2311yescharacteristic
84Dic428—22no—
85Dic428—22no—
86#13133808—41no—
87#13133807—22no—
88#13133807D414yescomplemented
89#13133806—22no—
90#13133806—41no—
91#13133805—22no—
92C2×C107⋊C8C2311yescharacteristic
93C2×C107⋊C8C2311yescharacteristic
94C8×D107—22no—
95C4⋊Dic107—21no—
96C4⋊Dic107—21no—
97Dic107⋊C4—21no—
98Dic107⋊C4—21no—
99Dic107⋊C4—21no—
100Dic107⋊C4D411yescharacteristic
101C22⋊Dic107—21no—
102C22⋊Dic107—21no—
103C22⋊Dic107D411yescharacteristic
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4D411yescharacteristic
108C4×Dic107—21no—
109C4×Dic107C2311yescharacteristic
110C2×Q8×C107C2311yescharacteristic
111C2×D4×C107—21no—
112C2×D4×C107C2311yescharacteristic
113C22×C4×C107C2311yescharacteristic
114SD16×C107—22no—
115M16×C107—22no—
116C2×C8×C107C2311yescharacteristic
117C2×C8×C107C2311yescharacteristic
118C4⋊C4×C107—21no—
119C4⋊C4×C107C2311yescharacteristic
120C4⋊C4×C107C2311yescharacteristic
121C22⋊C4×C107—21no—
122C22⋊C4×C107—21no—
123C23×C107—21no—
124C23×C107C2×D411yescharacteristic
125C22×D107—21no—
126C22×D107—42no—
127C22×D107—42no—
128C22×D107—81no—
129C22×D107C2×D411yescharacteristic
130Q8×C107—21no—
131Q8×C107—21no—
132D4×C107—21no—
133D4×C107—21no—
134D4×C107—81no—
135C2×C4×C107—21no—
136C2×C4×C107—21no—
137C2×C4×C107—21no—
138C2×C4×C107—21no—
139C2×C4×C107—21no—
140C2×C4×C107C2×D411yescharacteristic
141C2×C4×C107C2411yescharacteristic
142C22⋊D107—22no—
143C22⋊D107—42no—
144C22⋊D107—42no—
145C2×Dic107—21no—
146C2×Dic107—21no—
147C2×Dic107—21no—
148C2×Dic107—42no—
149C2×Dic107C2×D411yescharacteristic
150C2×Dic107C2×D411yescharacteristic
151C2×Dic107C2×D411yescharacteristic
152D428—21no—
153D428C2×D412yes—
154C4×D107—21no—
155C4×D107—21no—
156C4×D107—42no—
157C4×D107—42no—
158C4×D107—81no—
159Dic214—21no—
160Dic214—21no—
161Dic214—21no—
162Dic214—21no—
163Dic214—21no—
164Dic214—42no—
165Dic214—42no—
166Dic214C2×D412yes—
167C8×C107—21no—
168C8×C107C2×D412yes—
169C107⋊C8—21no—
170C107⋊C8—21no—
171C22×C107—21no—
172C22×C107—21no—
173C22×C107—21no—
174C22×C107—41no—
175C22×C107—41no—
176C22×C107—41no—
177C22×C107C22×D411yescharacteristic
178C2×D107—21no—
179C2×D107—21no—
180C2×D107—22no—
181C2×D107—41no—
182C2×D107—41no—
183C2×D107—84no—
184C2×D107#13712yescomplemented
185C4×C107—21no—
186C4×C107—41no—
187C4×C107—41no—
188C4×C107—41no—
189C4×C107C22×D411yescharacteristic
190C4×C107H(5,𝔽2)11yescharacteristic
191Dic107—22no—
192Dic107—21no—
193Dic107—21no—
194Dic107—21no—
195Dic107—41no—
196Dic107—41no—
197C2×C107—41no—
198C2×C107—81no—
199C2×C107D4211yescharacteristic
200C2×C107C2×#13711yescharacteristic
201C2×C107#57711yescharacteristic
202D107—22no—
203D107—42no—
204#3266—1071nomaximal, Sylow
205C107#326611yescharacteristic, complemented, Sylow
206C2×#137—1071no—
207C22×SD16—1071no—
208D4×Q8—1071no—
209#546—1071no—
210#497—1071no—
211#468D10711yescharacteristic, complemented
212#467—1071no—
213#464—1071no—
214#461—1071no—
215#452—1071no—
216#451—1071no—
217#450—1071no—
218#448—1071no—
219#440—1071no—
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×M16C2×D10711yescharacteristic, complemented
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, complemented
247#121—2141no—
248#121C2×D10711yescharacteristic
249#120—2141no—
250C4×Q8—1071no—
251C4×D4—1071no—
252C4×D4—1071no—
253C4×D4—2141no—
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#97—1071no—
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—2142no—
276C2×Q8—2142no—
277C2×Q8—2141no—
278C2×Q8C22×D10711yescharacteristic
279C2×D4—21no—
280C2×D4—1071no—
281C2×D4—1071no—
282C2×D4—4281no—
283C2×D4—2141no—
284C2×D4—2142no—
285C2×D4—2141no—
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—1074no—
298SD16—4281no—
299SD16—2142no—
300SD16—2142no—
301SD16—2142no—
302M16—22no—
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—21no—
319C22⋊C4—1071no—
320C22⋊C4—1071no—
321C22⋊C4—2141no—
322C22⋊C4—2141no—
323C22⋊C4—2141no—
324C22⋊C4—2141no—
325C22⋊C4—2141no—
326C42—1071no—
327C42—2141no—
328C23—21no—
329C23—8561no—
330C23—1071no—
331C23—4282no—
332C23—4282no—
333C23—2141no—
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—81no—
348D4—1072no—
349D4—4282no—
350D4—4282no—
351D4—2142no—
352D4—2141no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—8561no—
359C2×C4—1071no—
360C2×C4—1071no—
361C2×C4—1071no—
362C2×C4—4282no—
363C2×C4—4282no—
364C2×C4—4282no—
365C2×C4—2141no—
366C2×C4—2141no—
367C2×C4—2141no—
368C2×C4—2141no—
369C2×C4—2141no—
370C2×C4D4×D10711yescharacteristic
371C2×C4C23×D10711yescharacteristic
372C8—21no—
373C8—2141no—
374C8—2141no—
375C8D4×D10712yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—8564no—
383C22—1072no—
384C22—4281no—
385C22—4281no—
386C22—2142no—
387C22—2141no—
388C22—2141no—
389C22C2×D4×D10711yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—4281no—
395C4—4281no—
396C4—2141no—
397C4—2141no—
398C4—2141no—
399C4—2142no—
400C4C2×D4×D10711yescharacteristic
401C4#1681215511yescharacteristic
402C2—41no—
403C2—81no—
404C2—4282no—
405C2—2142no—
406C2#2029326111yescharacteristic
407C2#137×D10711yescharacteristic
408C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#468×C107C2
socleC22×C107C22×D4
prime radical#20293274C2
hypercenter#468D107
hypocenterC107#3266
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#16812142C2semidirect
2. C2×SD16×D107C2semidirect
3. #20293290C2semidirect
5. #20293018C2semidirect
6. #20292869C2semidirect
7. #20292848C2semidirect
8. #20292831C2semidirect
9. #20292787C2semidirect
10. #20292784C2semidirect
11. #20292779C2semidirect
12. #20292765C2semidirect
13. #20292726C2semidirect
14. #20292696C2semidirect
15. #468×C107C2semidirect
17. C22×Dic214C22semidirect
21. C2×Dic428C22semidirect
22. C2×Dic428C22semidirect
23. C2×#13133808C22semidirect
32. #16812107C22semidirect
41. #16812074C22semidirect
46. Dic107⋊C8C22semidirect
47. #16812032C22semidirect
48. #16812032C22semidirect
49. #16812032C22semidirect
51. C22⋊(C107⋊C8)C22semidirect
53. #16812019C22semidirect
55. C2×M16×C107C22semidirect
56. C22⋊Q8×C107C22semidirect
58. #107×C107C22semidirect
59. #103×C107C22semidirect
88. #13133807D4semidirect
184. C2×D107#137semidirect
205. C107#3266semidirect
211. #468D107semidirect
236. C2×M16C2×D107semidirect
246. C22⋊Q8C2×D107semidirect
255. #107C2×D107semidirect
260. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
C2×SD16×D1072
#202932902
#202932742
#202930182
#202928692
#202928482
#202928312
#202927872
#202927842
#202927792
#202927652
#202927262
#202926962
#468×C1072
#3266107