Finite Group Databasebeta

← #28348024

Subgroups — #28348024

Order 13696.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812138C211yesmaximal, characteristic, complemented
2C22×#13133808C211yesmaximal, characteristic, complemented
3Q8×C22⋊D107C211yesmaximal, characteristic, complemented
4#20293283C211yesmaximal, characteristic
5#20293018C211yesmaximal, characteristic, complemented
6#20292884C211yesmaximal, characteristic, complemented
7#20292854C211yesmaximal, characteristic, complemented
8#20292847C211yesmaximal, characteristic, complemented
9#20292836C211yesmaximal, characteristic, complemented
10#20292786C211yesmaximal, characteristic, complemented
11#20292785C211yesmaximal, characteristic, complemented
12#20292784C211yesmaximal, characteristic, complemented
13#20292731C211yesmaximal, characteristic, complemented
14#20292695C211yesmaximal, characteristic, complemented
15#448×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C22×Dic214C2211yescharacteristic
18C2×Q8×D107C2211yescharacteristic
19#16812138—24no—
20C2×Dic428C2211yescharacteristic
21C2×Dic428C2211yescharacteristic
22C2×#13133808—24no—
23C2×#13133808C2211yescharacteristic
24C2×#13133808C2211yescharacteristic
25C2×#13133808C2211yescharacteristic
26C2×#13133808C2211yescharacteristic
27C2×#13133806C2211yescharacteristic
28C2×#13133801C2211yescharacteristic
29C22×C107⋊C8C2211yescharacteristic
30#16812107—21no—
31#16812107C2211yescharacteristic
32#16812096C2211yescharacteristic
33#16812095—21no—
34#16812089C2211yescharacteristic
35#16812088—21no—
36#16812088C2211yescharacteristic
37#16812086—21no—
38#16812085—21no—
39#16812085C2211yescharacteristic
40#16812081—21no—
41Q8×Dic107C2211yescharacteristic
42#16812074C2211yescharacteristic
43D4×Dic107—21no—
44C4×C22⋊D107—21no—
45C4×C22⋊D107C2211yescharacteristic
46#16812040C2211yescharacteristic
47Dic107⋊C8C2211yescharacteristic
48#16812034C2211yescharacteristic
49#16812034C2211yescharacteristic
50#16812032C2211yescharacteristic
51#16812030C2211yescharacteristic
52(C2×D107)⋊C8C2211yescharacteristic
53#16812019C2211yescharacteristic
54C22×Q8×C107C2211yescharacteristic
55C2×SD16×C107C2211yescharacteristic
56C2×SD16×C107C2211yescharacteristic
57#121×C107C2211yescharacteristic
58#103×C107C2211yescharacteristic
59#103×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⋊D107D411yescharacteristic
74C22×Dic107—21no—
75C22×Dic107—21no—
76C2×C4×D107—21no—
77C2×C4×D107C2311yescharacteristic
78Dic428—22no—
79Dic428—41no—
80#13133808—22no—
81#13133808—22no—
82#13133808—22no—
83#13133808D414yescomplemented
84#13133806—22no—
85#13133801—22no—
86C2×C107⋊C8—22no—
87C2×C107⋊C8C2311yescharacteristic
88C2×C107⋊C8C2311yescharacteristic
89C4⋊Dic107—21no—
90C4⋊Dic107—21no—
91C4⋊Dic107C2311yescharacteristic
92Dic107⋊C4—21no—
93Dic107⋊C4—21no—
94Dic107⋊C4—21no—
95Dic107⋊C4—21no—
96Dic107⋊C4—21no—
97Dic107⋊C4D411yescharacteristic
98C22⋊Dic107—21no—
99C22⋊Dic107—21no—
100C22⋊Dic107—21no—
101C22⋊Dic107D411yescharacteristic
102(C2×D107)⋊C4—21no—
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4D411yescharacteristic
105C4×Dic107—21no—
106C4×Dic107—21no—
107C4×Dic107C2311yescharacteristic
108C2×Q8×C107—21no—
109C2×Q8×C107—22no—
110C2×Q8×C107—41no—
111C2×Q8×C107C2311yescharacteristic
112C2×Q8×C107C2311yescharacteristic
113C2×D4×C107—21no—
114C2×D4×C107C2311yescharacteristic
115C22×C4×C107—21no—
116C22×C4×C107C2311yescharacteristic
117SD16×C107—22no—
118SD16×C107—41no—
119C2×C8×C107C2311yescharacteristic
120C2×C8×C107C2311yescharacteristic
121C4⋊C4×C107C2311yescharacteristic
122C22⋊C4×C107—21no—
123C23×C107—21no—
124C23×C107C2×D411yescharacteristic
125C22×D107C2×D411yescharacteristic
126Q8×C107—21no—
127Q8×C107—21no—
128Q8×C107—21no—
129Q8×C107—42no—
130Q8×C107C2×D412yes—
131D4×C107—21no—
132D4×C107—21no—
133D4×C107—42no—
134C2×C4×C107—21no—
135C2×C4×C107—21no—
136C2×C4×C107—21no—
137C2×C4×C107—21no—
138C2×C4×C107—21no—
139C2×C4×C107—42no—
140C2×C4×C107C2×D411yescharacteristic
141C2×C4×C107C2411yescharacteristic
142C22⋊D107—22no—
143C22⋊D107—42no—
144C2×Dic107—21no—
145C2×Dic107—21no—
146C2×Dic107—21no—
147C2×Dic107—21no—
148C2×Dic107—42no—
149C2×Dic107—42no—
150C2×Dic107C2×D411yescharacteristic
151C2×Dic107C2×D411yescharacteristic
152C2×Dic107C2×D411yescharacteristic
153C4×D107—21no—
154C4×D107—21no—
155C4×D107—42no—
156Dic214—21no—
157Dic214—21no—
158Dic214—21no—
159Dic214—21no—
160Dic214—21no—
161Dic214—42no—
162Dic214—42no—
163Dic214C2×D412yes—
164C8×C107—21no—
165C8×C107—21no—
166C107⋊C8—21no—
167C107⋊C8C2×D412yes—
168C22×C107—21no—
169C22×C107—21no—
170C22×C107—41no—
171C22×C107—41no—
172C22×C107—41no—
173C22×C107#13712yescomplemented
174C22×C107C22×D411yescharacteristic
175C2×D107—21no—
176C2×D107—21no—
177C2×D107—21no—
178C4×C107—21no—
179C4×C107—22no—
180C4×C107—41no—
181C4×C107—41no—
182C4×C107C22×D411yescharacteristic
183C4×C107H(5,𝔽2)11yescharacteristic
184Dic107—22no—
185Dic107—21no—
186Dic107—21no—
187Dic107—21no—
188Dic107—41no—
189Dic107—41no—
190Dic107—41no—
191C2×C107—22no—
192C2×C107—81no—
193C2×C107D4211yescharacteristic
194C2×C107C2×#13711yescharacteristic
195C2×C107#57811yescharacteristic
196D107—41no—
197#3270—1071nomaximal, Sylow
198C107#327011yescharacteristic, complemented, Sylow
199C2×#137—1071no—
200C22×Q16—1071no—
201D4×Q8—1071no—
202#548—1071no—
203#497—1071no—
204#470—1071no—
205#465—1071no—
206#464—1071no—
207#462—1071no—
208#452—1071no—
209#451—1071no—
210#451—1071no—
211#448D10711yescharacteristic, complemented
212#441—1071no—
213#435—1071no—
214C2×#41—1071no—
215C22×Q8—1071no—
216C22×Q8—1071no—
217C22×Q8C2×D10711yescharacteristic
218#137—2144no—
219C2×Q16—1071no—
220C2×Q16—1071no—
221C2×Q16—1071no—
222C2×Q16—1071no—
223C2×Q16—1071no—
224C2×Q16—1071no—
225C2×Q16—2144no—
226C2×SD16—1071no—
227C2×SD16C2×D10711yescharacteristic
228C2×SD16C2×D10711yescharacteristic
229C2×M16—1071no—
230C22×C8—1071no—
231C4⋊Q8—1071no—
232C4⋊Q8—2141no—
233#124—1071no—
234#124—2141no—
235C22⋊Q8—1071no—
236C22⋊Q8—1071no—
237C22⋊Q8—1071no—
238C22⋊Q8—2141no—
239C22⋊Q8—2141no—
240C22⋊Q8—2141no—
241#121—2141no—
242#121C2×D10711yescharacteristic
243C4×Q8—1071no—
244C4×Q8—1071no—
245C4×D4—1071no—
246C4×D4—2141no—
247C4×D4—2141no—
248#106—1071no—
249C4⋊C8—1071no—
250#103—1071no—
251#103—1071no—
252#103—1071no—
253#103C2×D10711yescharacteristic
254#103C2×D10711yescharacteristic
255#102—1071no—
256C22⋊C8—1071no—
257C22⋊C8C2×D10711yescharacteristic
258#97—1071no—
259#41—4281no—
260#41—2142no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—1071no—
265C2×Q8—1071no—
266C2×Q8—1071no—
267C2×Q8—4281no—
268C2×Q8—4281no—
269C2×Q8—2141no—
270C2×Q8—2142no—
271C2×Q8—2142no—
272C2×Q8—2141no—
273C2×Q8C22×D10711yescharacteristic
274C2×Q8C22×D10711yescharacteristic
275C2×D4—21no—
276C2×D4—1071no—
277C2×D4—2141no—
278C2×D4C22×D10711yescharacteristic
279C22×C4—21no—
280C22×C4—1071no—
281C22×C4—2141no—
282C22×C4—2141no—
283C22×C4—2141no—
284C22×C4C22×D10711yescharacteristic
285Q16—1074no—
286Q16—4281no—
287Q16—2142no—
288Q16—2142no—
289Q16—2142no—
290Q16—2142no—
291SD16—22no—
292SD16—41no—
293SD16—2142no—
294M16—2142no—
295C2×C8—1071no—
296C2×C8—1071no—
297C2×C8—2142no—
298C2×C8C22×D10711yescharacteristic
299C2×C8C22×D10711yescharacteristic
300C4⋊C4—1071no—
301C4⋊C4—1071no—
302C4⋊C4—2141no—
303C4⋊C4—2141no—
304C4⋊C4—2141no—
305C4⋊C4—2141no—
306C4⋊C4—2141no—
307C4⋊C4—2141no—
308C4⋊C4—2141no—
309C4⋊C4C22×D10711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—1071no—
312C22⋊C4—1071no—
313C22⋊C4—2141no—
314C22⋊C4—2141no—
315C22⋊C4—2141no—
316C22⋊C4—2141no—
317C22⋊C4—2141no—
318C42—1071no—
319C42—2141no—
320C42—2141no—
321C23—21no—
322C23—1071no—
323C23D4×D10711yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—42no—
328Q8—1072no—
329Q8—4282no—
330Q8—4282no—
331Q8—2141no—
332Q8—2141no—
333Q8—2141no—
334Q8—2141no—
335Q8—2141no—
336Q8D4×D10712yes—
337D4—21no—
338D4—21no—
339D4—42no—
340D4—4282no—
341D4—2142no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—1071no—
349C2×C4—1071no—
350C2×C4—1071no—
351C2×C4—4282no—
352C2×C4—4282no—
353C2×C4—4282no—
354C2×C4—2141no—
355C2×C4—2141no—
356C2×C4—2141no—
357C2×C4—2141no—
358C2×C4—2141no—
359C2×C4—2141no—
360C2×C4D4×D10711yescharacteristic
361C2×C4C23×D10711yescharacteristic
362C8—21no—
363C8—21no—
364C8—1072no—
365C8—2141no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—2141no—
372C22—2141no—
373C22—2141no—
374C22#1681213812yescomplemented
375C22C2×D4×D10711yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—4281no—
381C4—4281no—
382C4—4281no—
383C4—2141no—
384C4—2141no—
385C4—2141no—
386C4—2142no—
387C4C2×D4×D10711yescharacteristic
388C4#1681215411yescharacteristic
389C2—22no—
390C2—81no—
391C2—4281no—
392C2#2029325911yescharacteristic
393C2C2×#1681213811yescharacteristic
394C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#448×C107C2
socleC22×C107C22×D4
prime radical#20293283C2
hypercenter#448D107
hypocenterC107#3270
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#16812138C2semidirect
2. C22×#13133808C2semidirect
3. Q8×C22⋊D107C2semidirect
5. #20293018C2semidirect
6. #20292884C2semidirect
7. #20292854C2semidirect
8. #20292847C2semidirect
9. #20292836C2semidirect
10. #20292786C2semidirect
11. #20292785C2semidirect
12. #20292784C2semidirect
13. #20292731C2semidirect
14. #20292695C2semidirect
15. #448×C107C2semidirect
83. #13133808D4semidirect
173. C22×C107#137semidirect
198. C107#3270semidirect
211. #448D107semidirect
374. 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×#131338082
Q8×C22⋊D1072
#202932832
#202930182
#202928842
#202928542
#202928472
#202928362
#202927862
#202927852
#202927842
#202927312
#202926952
#448×C1072
#3270107