Finite Group Databasebeta

← #28348025

Subgroups — #28348025

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×Q16×D107C211yesmaximal, characteristic, complemented
2Q8×C22⋊D107C211yesmaximal, characteristic, complemented
3#20293284C211yesmaximal, characteristic
4#20293018C211yesmaximal, characteristic, complemented
5#20292885C211yesmaximal, characteristic, complemented
6#20292853C211yesmaximal, characteristic, complemented
7#20292847C211yesmaximal, characteristic, complemented
8#20292835C211yesmaximal, characteristic, complemented
9#20292789C211yesmaximal, characteristic, complemented
10#20292783C211yesmaximal, characteristic, complemented
11#20292781C211yesmaximal, characteristic, complemented
12#20292764C211yesmaximal, characteristic, complemented
13#20292734C211yesmaximal, characteristic, complemented
14#20292696C211yesmaximal, characteristic, complemented
15C2×#137×C107C211yesmaximal, characteristic, complemented
16C2×Q8×D107C2211yescharacteristic
17C2×Q8×D107C2211yescharacteristic
18C2×Dic428C2211yescharacteristic
19C2×#13133808C2211yescharacteristic
20C2×#13133808C2211yescharacteristic
21C2×#13133808C2211yescharacteristic
22Q16×D107—24no—
23C2×#13133806C2211yescharacteristic
24C2×C8×D107C2211yescharacteristic
25#16812107—21no—
26#16812107C2211yescharacteristic
27#16812096—21no—
28#16812096C2211yescharacteristic
29#16812088—21no—
30#16812088—21no—
31#16812088C2211yescharacteristic
32#16812088C2211yescharacteristic
33#16812085—21no—
34#16812085C2211yescharacteristic
35#16812080—21no—
36#16812080C2211yescharacteristic
37Q8×Dic107C2211yescharacteristic
38Q8×Dic107C2211yescharacteristic
39C4×C22⋊D107—21no—
40C4×C22⋊D107—21no—
41C4×C22⋊D107C2211yescharacteristic
42#16812041C2211yescharacteristic
43Dic107⋊C8C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812034C2211yescharacteristic
46#16812033C2211yescharacteristic
47#16812033C2211yescharacteristic
48#16812033C2211yescharacteristic
49#16812030C2211yescharacteristic
50C22⋊(C107⋊C8)C2211yescharacteristic
51(C2×D107)⋊C8C2211yescharacteristic
52#16812019C2211yescharacteristic
53C2×#41×C107C2211yescharacteristic
54C22×Q8×C107C2211yescharacteristic
55#137×C107—24no—
56C2×Q16×C107C2211yescharacteristic
57C2×Q16×C107C2211yescharacteristic
58C2×SD16×C107C2211yescharacteristic
59C2×SD16×C107C2211yescharacteristic
60C2×M16×C107C2211yescharacteristic
61C2×Dic214—21no—
62C2×Dic214—21no—
63C2×Dic214C2311yescharacteristic
64C2×Dic214C2311yescharacteristic
65Q8×D107—22no—
66Q8×D107—22no—
67Q8×D107—41no—
68Q8×D107—41no—
69C2×C22⋊D107—21no—
70C2×C22⋊D107D411yescharacteristic
71C2×C4×D107—21no—
72C2×C4×D107—21no—
73C2×C4×D107C2311yescharacteristic
74Dic428—22no—
75#13133808—22no—
76#13133808—22no—
77#13133808—41no—
78#13133806—41no—
79C2×C107⋊C8C2311yescharacteristic
80C2×C107⋊C8C2311yescharacteristic
81C8×D107—22no—
82C4⋊Dic107—21no—
83C4⋊Dic107—21no—
84C4⋊Dic107C2311yescharacteristic
85C4⋊Dic107C2311yescharacteristic
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88Dic107⋊C4—21no—
89Dic107⋊C4—21no—
90Dic107⋊C4—21no—
91Dic107⋊C4D411yescharacteristic
92C22⋊Dic107—21no—
93C22⋊Dic107—21no—
94C22⋊Dic107—21no—
95C22⋊Dic107—21no—
96C22⋊Dic107—21no—
97C22⋊Dic107D411yescharacteristic
98(C2×D107)⋊C4—21no—
99(C2×D107)⋊C4D411yescharacteristic
100C4×Dic107—21no—
101C4×Dic107—21no—
102C4×Dic107C2311yescharacteristic
103#41×C107—22no—
104#41×C107—41no—
105C2×Q8×C107—21no—
106C2×Q8×C107—22no—
107C2×Q8×C107—41no—
108C2×Q8×C107C2311yescharacteristic
109C2×Q8×C107C2311yescharacteristic
110C2×Q8×C107C2311yescharacteristic
111C2×D4×C107—21no—
112C2×D4×C107C2311yescharacteristic
113C22×C4×C107—21no—
114C22×C4×C107—21no—
115C22×C4×C107C2311yescharacteristic
116Q16×C107—22no—
117Q16×C107D414yescomplemented
118SD16×C107—22no—
119SD16×C107—22no—
120M16×C107—22no—
121C2×C8×C107C2311yescharacteristic
122C2×C8×C107C2311yescharacteristic
123C23×C107—21no—
124C23×C107C2×D411yescharacteristic
125C22×D107C2×D411yescharacteristic
126Q8×C107—21no—
127Q8×C107—21no—
128Q8×C107—21no—
129Q8×C107—21no—
130Q8×C107—42no—
131Q8×C107C2×D412yes—
132Q8×C107C2×D412yes—
133D4×C107—21no—
134D4×C107—21no—
135D4×C107—42no—
136C2×C4×C107—21no—
137C2×C4×C107—21no—
138C2×C4×C107—21no—
139C2×C4×C107—21no—
140C2×C4×C107—21no—
141C2×C4×C107—42no—
142C2×C4×C107—42no—
143C2×C4×C107C2×D411yescharacteristic
144C2×C4×C107C2411yescharacteristic
145C22⋊D107—22no—
146C22⋊D107—42no—
147C2×Dic107—21no—
148C2×Dic107—21no—
149C2×Dic107—21no—
150C2×Dic107—21no—
151C2×Dic107C2×D411yescharacteristic
152C2×Dic107C2×D411yescharacteristic
153C2×Dic107C2×D411yescharacteristic
154C4×D107—21no—
155C4×D107—21no—
156C4×D107—42no—
157C4×D107—42no—
158Dic214—21no—
159Dic214—21no—
160Dic214—21no—
161Dic214—21no—
162Dic214—42no—
163Dic214—42no—
164C8×C107—21no—
165C8×C107C2×D412yes—
166C107⋊C8—21no—
167C107⋊C8—21no—
168C22×C107—21no—
169C22×C107—21no—
170C22×C107—21no—
171C22×C107—41no—
172C22×C107—41no—
173C22×C107—41no—
174C22×C107C22×D411yescharacteristic
175C2×D107—21no—
176C2×D107—21no—
177C2×D107#13712yescomplemented
178C4×C107—22no—
179C4×C107—22no—
180C4×C107—21no—
181C4×C107—41no—
182C4×C107C22×D411yescharacteristic
183C4×C107H(5,𝔽2)11yescharacteristic
184Dic107—21no—
185Dic107—21no—
186Dic107—21no—
187Dic107—41no—
188Dic107—41no—
189Dic107—41no—
190Dic107—41no—
191C2×C107—41no—
192C2×C107—81no—
193C2×C107D4211yescharacteristic
194C2×C107C2×#13711yescharacteristic
195C2×C107#57811yescharacteristic
196D107—22no—
197#3270—1071nomaximal, Sylow
198C107#327011yescharacteristic, complemented, Sylow
199C2×#137D10711yescharacteristic, complemented
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#448—1071no—
212#441—1071no—
213#435—1071no—
214C2×#41C2×D10711yescharacteristic
215C22×Q8—1071no—
216C22×Q8—1071no—
217C22×Q8C2×D10711yescharacteristic
218#137—24no—
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×SD16C2×D10711yescharacteristic
228C2×SD16C2×D10711yescharacteristic
229C2×M16C2×D10711yescharacteristic
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—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#103—1071no—
254#103—1071no—
255#102—1071no—
256C22⋊C8—1071no—
257C22⋊C8—1071no—
258#97—1071no—
259#41—22no—
260#41—41no—
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—2141no—
270C2×Q8—2142no—
271C2×Q8—2142no—
272C2×Q8C22×D10711yescharacteristic
273C2×Q8C22×D10711yescharacteristic
274C2×Q8C22×D10711yescharacteristic
275C2×D4—21no—
276C2×D4—1071no—
277C2×D4—2141no—
278C2×D4C22×D10711yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—1071no—
282C22×C4—2141no—
283C22×C4—2141no—
284C22×C4C22×D10711yescharacteristic
285Q16—22no—
286Q16—4281no—
287Q16—2142no—
288Q16—2142no—
289Q16—2142no—
290Q16C22⋊D10714yescomplemented
291SD16—22no—
292SD16—22no—
293SD16—4281no—
294M16—22no—
295C2×C8—1071no—
296C2×C8—1071no—
297C2×C8—2142no—
298C2×C8C22×D10711yescharacteristic
299C2×C8C22×D10711yescharacteristic
300C4⋊C4—1071no—
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⋊C4—2141no—
310C22⋊C4—1071no—
311C22⋊C4—1071no—
312C22⋊C4—2141no—
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—21no—
328Q8—42no—
329Q8—4282no—
330Q8—4282no—
331Q8—2141no—
332Q8—2141no—
333Q8—2141no—
334Q8—2141no—
335Q8C2×C22⋊D10712yes—
336Q8C2×C22⋊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—42no—
349C2×C4—1071no—
350C2×C4—1071no—
351C2×C4—1071no—
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—2141no—
364C8—2141no—
365C8C2×C22⋊D10712yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—1072no—
373C22—2141no—
374C22—2141no—
375C22C2×D4×D10711yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—4281no—
381C4—4281no—
382C4—4281no—
383C4—4281no—
384C4—2141no—
385C4—2141no—
386C4—2141no—
387C4C22×C22⋊D10711yescharacteristic
388C4#1681215411yescharacteristic
389C2—41no—
390C2—81no—
391C2—2142no—
392C2D4×C22⋊D10711yescharacteristic
393C2#137×D10711yescharacteristic
394C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#137×C107C2
socleC22×C107C22×D4
prime radical#20293284C2
hypercenterC2×#137D107
hypocenterC107#3270
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D107C2semidirect
2. Q8×C22⋊D107C2semidirect
4. #20293018C2semidirect
5. #20292885C2semidirect
6. #20292853C2semidirect
7. #20292847C2semidirect
8. #20292835C2semidirect
9. #20292789C2semidirect
10. #20292783C2semidirect
11. #20292781C2semidirect
12. #20292764C2semidirect
13. #20292734C2semidirect
14. #20292696C2semidirect
15. C2×#137×C107C2semidirect
117. Q16×C107D4semidirect
177. C2×D107#137semidirect
198. C107#3270semidirect
199. C2×#137D107semidirect
290. Q16C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1072
Q8×C22⋊D1072
#202932842
#202930182
#202928852
#202928532
#202928472
#202928352
#202927892
#202927832
#202927812
#202927642
#202927342
#202926962
C2×#137×C1072
#3270107