Finite Group Databasebeta

← #28348035

Subgroups — #28348035

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×#16812141C211yesmaximal, characteristic, complemented
2C22×#13133808C211yesmaximal, characteristic, complemented
3#20293290C211yesmaximal, characteristic, complemented
4#20293288C211yesmaximal, characteristic
5#20293020C211yesmaximal, characteristic, complemented
6#20292884C211yesmaximal, characteristic, complemented
7#20292858C211yesmaximal, characteristic, complemented
8#20292852C211yesmaximal, characteristic, complemented
9#20292835C211yesmaximal, characteristic, complemented
10#20292792C211yesmaximal, characteristic, complemented
11#20292781C211yesmaximal, characteristic, complemented
12#20292768C211yesmaximal, characteristic, complemented
13#20292731C211yesmaximal, characteristic, complemented
14#20292695C211yesmaximal, characteristic, complemented
15#451×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C22×Dic214C2211yescharacteristic
18C2×Q8×D107C2211yescharacteristic
19#16812141—24no—
20C2×#13133808—24no—
21C2×#13133808C2211yescharacteristic
22C2×#13133808C2211yescharacteristic
23C2×#13133808C2211yescharacteristic
24C2×#13133808C2211yescharacteristic
25C2×#13133807C2211yescharacteristic
26C2×#13133807C2211yescharacteristic
27C2×#13133805C2211yescharacteristic
28C2×#13133801C2211yescharacteristic
29C22×C107⋊C8C2211yescharacteristic
30#16812108—21no—
31#16812107C2211yescharacteristic
32#16812098—21no—
33#16812098C2211yescharacteristic
34#16812089C2211yescharacteristic
35#16812087—21no—
36#16812086—21no—
37#16812085—21no—
38#16812085C2211yescharacteristic
39#16812083—21no—
40#16812083C2211yescharacteristic
41Q8×Dic107C2211yescharacteristic
42#16812074C2211yescharacteristic
43C4×C22⋊D107—21no—
44C4×C22⋊D107C2211yescharacteristic
45(C22⋊D107)⋊C4—21no—
46#16812040C2211yescharacteristic
47Dic107⋊C8C2211yescharacteristic
48#16812034C2211yescharacteristic
49#16812034C2211yescharacteristic
50#16812033C2211yescharacteristic
51#16812029C2211yescharacteristic
52(C2×D107)⋊C8C2211yescharacteristic
53#16812019C2211yescharacteristic
54C22×Q8×C107C2211yescharacteristic
55C2×Q16×C107C2211yescharacteristic
56C2×Q16×C107C2211yescharacteristic
57C22⋊Q8×C107C2211yescharacteristic
58#103×C107C2211yescharacteristic
59#103×C107C2211yescharacteristic
60C22⋊C8×C107C2211yescharacteristic
61#13133819—22no—
62#13133819—41no—
63C2×Dic214—22no—
64C2×Dic214—21no—
65C2×Dic214—21no—
66C2×Dic214—41no—
67C2×Dic214C2311yescharacteristic
68C2×Dic214C2311yescharacteristic
69Q8×D107—22no—
70Q8×D107—41no—
71C2×D428—21no—
72C2×D428C2311yescharacteristic
73C2×C22⋊D107—21no—
74C2×C22⋊D107D411yescharacteristic
75C22×Dic107—21no—
76C2×C4×D107—21no—
77C2×C4×D107—21no—
78C2×C4×D107C2311yescharacteristic
79#13133808—22no—
80#13133808—22no—
81#13133808—22no—
82#13133808D414yescomplemented
83#13133807—22no—
84#13133807—41no—
85#13133805—22no—
86#13133801—22no—
87C2×C107⋊C8—22no—
88C2×C107⋊C8C2311yescharacteristic
89C2×C107⋊C8C2311yescharacteristic
90C4⋊Dic107—21no—
91C4⋊Dic107—21no—
92C4⋊Dic107C2311yescharacteristic
93Dic107⋊C4—21no—
94Dic107⋊C4—21no—
95Dic107⋊C4—21no—
96Dic107⋊C4—21no—
97Dic107⋊C4D411yescharacteristic
98C22⋊Dic107D411yescharacteristic
99(C2×D107)⋊C4—21no—
100(C2×D107)⋊C4—21no—
101(C2×D107)⋊C4—21no—
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×Q8×C107C2311yescharacteristic
114C22×C4×C107—21no—
115C22×C4×C107C2311yescharacteristic
116Q16×C107—22no—
117Q16×C107—41no—
118C2×C8×C107C2311yescharacteristic
119C2×C8×C107C2311yescharacteristic
120C4⋊C4×C107—21no—
121C4⋊C4×C107C2311yescharacteristic
122C22⋊C4×C107—21no—
123C23×C107C2×D411yescharacteristic
124C22×D107—21no—
125C22×D107C2×D411yescharacteristic
126Q8×C107—21no—
127Q8×C107—21no—
128Q8×C107—21no—
129Q8×C107—21no—
130Q8×C107—21no—
131Q8×C107—42no—
132Q8×C107C2×D412yes—
133C2×C4×C107—21no—
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—42no—
148C2×Dic107C2×D411yescharacteristic
149C2×Dic107C2×D411yescharacteristic
150C2×Dic107C2×D411yescharacteristic
151D428—21no—
152D428—21no—
153D428—42no—
154C4×D107—21no—
155C4×D107—21no—
156C4×D107—42no—
157C4×D107—42no—
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#13712yescomplemented
171C22×C107C22×D411yescharacteristic
172C2×D107—21no—
173C2×D107—21no—
174C2×D107—21no—
175C2×D107—41no—
176C2×D107—41no—
177C2×D107—41no—
178C4×C107—21no—
179C4×C107—22no—
180C4×C107—41no—
181C4×C107—41no—
182C4×C107—41no—
183C4×C107C22×D411yescharacteristic
184C4×C107H(5,𝔽2)11yescharacteristic
185Dic107—22no—
186Dic107—21no—
187Dic107—21no—
188Dic107—21no—
189Dic107—41no—
190Dic107—41no—
191C2×C107—22no—
192C2×C107D4211yescharacteristic
193C2×C107C2×#13711yescharacteristic
194C2×C107#57811yescharacteristic
195D107—41no—
196D107—81no—
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#451D10711yescharacteristic, complemented
211#448—1071no—
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—2144no—
224C2×Q16C2×D10711yescharacteristic
225C2×Q16C2×D10711yescharacteristic
226C2×SD16—1071no—
227C2×SD16—1071no—
228C2×SD16—1071no—
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—2141no—
238C22⋊Q8—2141no—
239C22⋊Q8—2141no—
240C22⋊Q8C2×D10711yescharacteristic
241#121—1071no—
242#121—2141no—
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—4281no—
267C2×Q8—4281no—
268C2×Q8—2141no—
269C2×Q8—2142no—
270C2×Q8—2142no—
271C2×Q8—2141no—
272C2×Q8C22×D10711yescharacteristic
273C2×Q8C22×D10711yescharacteristic
274C2×Q8C22×D10711yescharacteristic
275C2×D4—1071no—
276C2×D4—1071no—
277C2×D4—2141no—
278C2×D4—2141no—
279C22×C4—21no—
280C22×C4—1071no—
281C22×C4—2141no—
282C22×C4—2141no—
283C22×C4—2141no—
284C22×C4C22×D10711yescharacteristic
285Q16—22no—
286Q16—41no—
287Q16—1074no—
288Q16—2142no—
289Q16—2142no—
290Q16—2142no—
291SD16—4281no—
292SD16—2142no—
293SD16—2142no—
294M16—2142no—
295C2×C8—1071no—
296C2×C8—1071no—
297C2×C8—2142no—
298C2×C8C22×D10711yescharacteristic
299C2×C8C22×D10711yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—1071no—
302C4⋊C4—1071no—
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—1071no—
322C23—2141no—
323C23D4×D10711yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—1072no—
331Q8—4282no—
332Q8—4282no—
333Q8—2141no—
334Q8—2141no—
335Q8—2141no—
336Q8D4×D10712yes—
337D4—4282no—
338D4—4282no—
339D4—2142no—
340D4—2141no—
341D4—2141no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—1071no—
350C2×C4—1071no—
351C2×C4—1071no—
352C2×C4—4282no—
353C2×C4—4282no—
354C2×C4—4282no—
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—4281no—
369C22—4281no—
370C22—4281no—
371C22—2141no—
372C22—2141no—
373C22—2141no—
374C22#1681214112yescomplemented
375C22C2×D4×D10711yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—4281no—
382C4—4281no—
383C4—2141no—
384C4—2141no—
385C4—2141no—
386C4—2142no—
387C4C2×D4×D10711yescharacteristic
388C4#1681215411yescharacteristic
389C2—22no—
390C2—8561no—
391C2—4281no—
392C2#2029325911yescharacteristic
393C2C2×#1681214111yescharacteristic
394C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#451×C107C2
socleC22×C107C22×D4
prime radical#20293288C2
hypercenter#451D107
hypocenterC107#3270
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#16812141C2semidirect
2. C22×#13133808C2semidirect
3. #20293290C2semidirect
5. #20293020C2semidirect
6. #20292884C2semidirect
7. #20292858C2semidirect
8. #20292852C2semidirect
9. #20292835C2semidirect
10. #20292792C2semidirect
11. #20292781C2semidirect
12. #20292768C2semidirect
13. #20292731C2semidirect
14. #20292695C2semidirect
15. #451×C107C2semidirect
82. #13133808D4semidirect
170. C22×C107#137semidirect
198. C107#3270semidirect
210. #451D107semidirect
374. C22#16812141semidirect

Maximal subgroups

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

subgroupindex
C2×#168121412
C22×#131338082
#202932902
#202932882
#202930202
#202928842
#202928582
#202928522
#202928352
#202927922
#202927812
#202927682
#202927312
#202926952
#451×C1072
#3270107