Finite Group Databasebeta

← #28346086

Subgroups — #28346086

Order 13696.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812138C211yesmaximal, characteristic, complemented
2C2×M16×D107C211yesmaximal, characteristic, complemented
3#20293137C211yesmaximal, characteristic
4C22⋊Q8×D107C211yesmaximal, characteristic, complemented
5#20292885C211yesmaximal, characteristic, complemented
6#20292883C211yesmaximal, characteristic, complemented
7#20292871C211yesmaximal, characteristic, complemented
8#20292869C211yesmaximal, characteristic, complemented
9#20292867C211yesmaximal, characteristic, complemented
10#20292855C211yesmaximal, characteristic, complemented
11#20292853C211yesmaximal, characteristic, complemented
12#20292668C211yesmaximal, characteristic, complemented
13#20292634C211yesmaximal, characteristic, complemented
14#20292625C211yesmaximal, characteristic, complemented
15#468×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×Q8×D107C2211yescharacteristic
18C22×C4×D107C2211yescharacteristic
19#16812138—24no—
20C2×Dic428C2211yescharacteristic, complemented
21C2×#13133808C2211yescharacteristic, complemented
22C2×#13133806C2211yescharacteristic, complemented
23C2×#13133802C2211yescharacteristic
24C2×#13133801C2211yescharacteristic
25C2×#13133801C2211yescharacteristic
26M16×D107—24no—
27C2×C8×D107C2211yescharacteristic
28C2×C8×D107C2211yescharacteristic
29#16812091—21no—
30#16812089C2211yescharacteristic
31#16812089C2211yescharacteristic
32#16812088C2211yescharacteristic
33#16812088C2211yescharacteristic
34#16812087—21no—
35#16812086—21no—
36#16812085C2211yescharacteristic
37#16812080C2211yescharacteristic
38#16812077—21no—
39D4×Dic107—21no—
40(C22⋊D107)⋊C4—21no—
41#16812067C2211yescharacteristic
42C4⋊C4×D107—21no—
43C4⋊C4×D107C2211yescharacteristic
44C22⋊C4×D107—21no—
45#16812042C2211yescharacteristic, complemented
46#16812041C2211yescharacteristic, complemented
47#16812040C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812034C2211yescharacteristic, complemented
50#16812033C2211yescharacteristic, complemented
51#16812032C2211yescharacteristic, complemented
52#16812032C2211yescharacteristic, complemented
53#16812030C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic, complemented
55C2×M16×C107C2211yescharacteristic
56C22⋊Q8×C107C2211yescharacteristic
57#121×C107C2211yescharacteristic
58#107×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107D411yescharacteristic
62#13133817—22no—
63#13133817—22no—
64C2×Dic214—21no—
65C2×Dic214C2311yescharacteristic
66C2×Dic214C2311yescharacteristic
67Q8×D107—22no—
68Q8×D107—22no—
69C2×C22⋊D107—21no—
70C2×C22⋊D107—21no—
71C22×Dic107—21no—
72C22×Dic107D411yescharacteristic
73C2×C4×D107—22no—
74C2×C4×D107—22no—
75C2×C4×D107—21no—
76C2×C4×D107—21no—
77C2×C4×D107D411yescharacteristic
78C2×C4×D107D411yescharacteristic
79C2×C4×D107C2311yescharacteristic
80C2×C4×D107C2311yescharacteristic
81Dic428—22no—
82#13133808—22no—
83#13133806—22no—
84#13133802—22no—
85#13133801—22no—
86#13133801D414yescomplemented
87C2×C107⋊C8C2311yescharacteristic
88C2×C107⋊C8C2311yescharacteristic
89C8×D107—22no—
90C8×D107—22no—
91C4⋊Dic107—21no—
92C4⋊Dic107—21no—
93C4⋊Dic107C2311yescharacteristic
94C4⋊Dic107C2311yescharacteristic
95Dic107⋊C4—21no—
96Dic107⋊C4—21no—
97Dic107⋊C4—21no—
98Dic107⋊C4—21no—
99C22⋊Dic107—21no—
100C22⋊Dic107—21no—
101C22⋊Dic107—21no—
102(C2×D107)⋊C4—21no—
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106C4×Dic107—21no—
107C2×Q8×C107C2311yescharacteristic
108C2×D4×C107—21no—
109C2×D4×C107C2311yescharacteristic
110C22×C4×C107C2311yescharacteristic
111SD16×C107—22no—
112M16×C107—22no—
113C2×C8×C107C2311yescharacteristic
114C2×C8×C107C2311yescharacteristic
115C4⋊C4×C107—21no—
116C4⋊C4×C107C2311yescharacteristic
117C4⋊C4×C107C2311yescharacteristic
118C22⋊C4×C107—21no—
119C22⋊C4×C107—21no—
120C23×C107—21no—
121C23×C107C2×D411yescharacteristic
122C22×D107—22no—
123C22×D107—22no—
124C22×D107—41no—
125C22×D107C2×D411yescharacteristic
126C22×D107C2×D411yescharacteristic
127Q8×C107—21no—
128Q8×C107—21no—
129D4×C107—21no—
130D4×C107—21no—
131D4×C107—42no—
132C2×C4×C107—21no—
133C2×C4×C107—21no—
134C2×C4×C107—21no—
135C2×C4×C107—21no—
136C2×C4×C107—21no—
137C2×C4×C107C2×D411yescharacteristic
138C2×C4×C107C2411yescharacteristic
139C22⋊D107—42no—
140C22⋊D107—42no—
141C2×Dic107—22no—
142C2×Dic107—21no—
143C2×Dic107—21no—
144C2×Dic107—21no—
145C2×Dic107—21no—
146C2×Dic107—42no—
147C2×Dic107C2×D411yescharacteristic
148C2×Dic107C2×D411yescharacteristic
149C4×D107—22no—
150C4×D107—21no—
151C4×D107—21no—
152C4×D107—22no—
153C4×D107—42no—
154C4×D107—42no—
155C4×D107C2×D412yes—
156C4×D107#4112yes—
157Dic214—21no—
158Dic214—21no—
159Dic214—21no—
160Dic214—21no—
161Dic214—42no—
162C8×C107—21no—
163C8×C107C2×D412yes—
164C107⋊C8—21no—
165C107⋊C8C2×D412yes—
166C22×C107—21no—
167C22×C107—21no—
168C22×C107—21no—
169C22×C107—41no—
170C22×C107—41no—
171C22×C107—41no—
172C22×C107C22×D411yescharacteristic
173C2×D107—21no—
174C2×D107—21no—
175C2×D107—21no—
176C2×D107—21no—
177C2×D107—44no—
178C2×D107#12112yes—
179C2×D107#13712yescomplemented
180C4×C107—21no—
181C4×C107—41no—
182C4×C107—41no—
183C4×C107—41no—
184C4×C107C22×D411yescharacteristic
185C4×C107C2×#4111yescharacteristic
186Dic107—21no—
187Dic107—41no—
188Dic107—41no—
189Dic107—41no—
190Dic107—41no—
191Dic107#12112yes—
192C2×C107—41no—
193C2×C107—81no—
194C2×C107C2×#12111yescharacteristic
195C2×C107C2×#13711yescharacteristic
196C2×C107#57511yescharacteristic
197D107—22no—
198D107—22no—
199#3056—1071nomaximal, Sylow
200C107#305611yescharacteristic, complemented, Sylow
201C2×#137—1071no—
202C22×M16—1071no—
203#525—1071no—
204C2×C22⋊Q8—1071no—
205#470—1071no—
206#470—1071no—
207#468—1071no—
208#468D10711yescharacteristic, complemented
209#467—1071no—
210#467—1071no—
211#465—1071no—
212#465—1071no—
213#428—1071no—
214#419—1071no—
215#417—1071no—
216C2×#41—1071no—
217C22×Q8—1071no—
218C23×C4—1071no—
219#137—2144no—
220C2×Q16—1071no—
221C2×Q16—1071no—
222C2×SD16—1071no—
223C2×SD16C2×D10711yescharacteristic, complemented
224C2×M16—1071no—
225C2×M16—1071no—
226C2×M16—1071no—
227C2×M16—2144no—
228C2×M16C2×D10711yescharacteristic
229C22×C8—1071no—
230C22×C8—1071no—
231#123—2141no—
232C22⋊Q8—1071no—
233C22⋊Q8—1071no—
234C22⋊Q8—1071no—
235C22⋊Q8—1071no—
236C22⋊Q8—1071no—
237C22⋊Q8—2141no—
238C22⋊Q8—2141no—
239C22⋊Q8C2×D10711yescharacteristic
240#121—1071no—
241#121C2×D10711yescharacteristic
242#120—2141no—
243C4×D4—2141no—
244C4×D4—2141no—
245#117—1071no—
246C2×C4⋊C4—1071no—
247C2×C4⋊C4—2141no—
248C2×C22⋊C4—2141no—
249#107—1071no—
250#107C2×D10711yescharacteristic, complemented
251#106—1071no—
252#106—1071no—
253#103—1071no—
254#103—1071no—
255#103—1071no—
256#103—1071no—
257#103—1071no—
258#103C2×D10711yescharacteristic, complemented
259#102—1071no—
260#102C2×D10711yescharacteristic, complemented
261C24—1071no—
262#41—2142no—
263#41—2142no—
264C2×Q8—1071no—
265C2×Q8—1071no—
266C2×Q8—2141no—
267C2×Q8—2142no—
268C2×Q8—2142no—
269C2×Q8C22×D10711yescharacteristic
270C2×D4—21no—
271C2×D4—2141no—
272C2×D4—2141no—
273C2×D4C22×D10711yescharacteristic
274C22×C4—1071no—
275C22×C4—1071no—
276C22×C4—1071no—
277C22×C4—1071no—
278C22×C4—1071no—
279C22×C4—2142no—
280C22×C4—2142no—
281C22×C4—2141no—
282C22×C4—2141no—
283C22×C4—2141no—
284C22×C4C22×D10711yescharacteristic
285Q16—2142no—
286Q16—2142no—
287SD16—22no—
288SD16—2142no—
289M16—22no—
290M16—1074no—
291M16—2142no—
292M16—2142no—
293C2×C8—1071no—
294C2×C8—1071no—
295C2×C8—2142no—
296C2×C8—2142no—
297C2×C8C22×D10711yescharacteristic
298C2×C8C22×D10711yescharacteristic
299C4⋊C4—21no—
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⋊C4C22×D10711yescharacteristic
309C4⋊C4C22×D10711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—2141no—
313C22⋊C4—2141no—
314C22⋊C4—2141no—
315C22⋊C4—2141no—
316C22⋊C4—2141no—
317C22⋊C4—2141no—
318C22⋊C4—2141no—
319C42—2141no—
320C23—21no—
321C23—1071no—
322C23—1071no—
323C23—4281no—
324C23—2142no—
325C23—2142no—
326C23D4×D10711yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—4282no—
330Q8—2141no—
331Q8—2141no—
332Q8—2141no—
333Q8—2141no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—4282no—
338D4—4282no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—1071no—
345C2×C4—1072no—
346C2×C4—1072no—
347C2×C4—1071no—
348C2×C4—4282no—
349C2×C4—4282no—
350C2×C4—4282no—
351C2×C4—2142no—
352C2×C4—2142no—
353C2×C4—2142no—
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—1072no—
364C8—2141no—
365C8D4×D10712yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—1072no—
373C22—1072no—
374C22—4284no—
375C22—2141no—
376C22—2141no—
377C22—2141no—
378C22—2141no—
379C22C2×D4×D10711yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—1072no—
385C4—4281no—
386C4—4281no—
387C4—4281no—
388C4—4281no—
389C4—2141no—
390C4C2×D4×D10711yescharacteristic
391C4#41×D10711yescharacteristic
392C2—41no—
393C2—81no—
394C2—2142no—
395C2—2142no—
396C2#121×D10711yescharacteristic
397C2#137×D10711yescharacteristic
398C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#468×C107C2
socleC22×C107C22×D4
prime radical#20293137C2
hypercenter#468D107
hypocenterC107#3056
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×#16812138C2semidirect
2. C2×M16×D107C2semidirect
4. C22⋊Q8×D107C2semidirect
5. #20292885C2semidirect
6. #20292883C2semidirect
7. #20292871C2semidirect
8. #20292869C2semidirect
9. #20292867C2semidirect
10. #20292855C2semidirect
11. #20292853C2semidirect
12. #20292668C2semidirect
13. #20292634C2semidirect
14. #20292625C2semidirect
15. #468×C107C2semidirect
20. C2×Dic428C22semidirect
21. C2×#13133808C22semidirect
22. C2×#13133806C22semidirect
45. #16812042C22semidirect
46. #16812041C22semidirect
47. #16812040C22semidirect
48. #16812034C22semidirect
49. #16812034C22semidirect
50. #16812033C22semidirect
51. #16812032C22semidirect
52. #16812032C22semidirect
53. #16812030C22semidirect
54. C2×SD16×C107C22semidirect
58. #107×C107C22semidirect
59. #103×C107C22semidirect
60. #102×C107C22semidirect
86. #13133801D4semidirect
179. C2×D107#137semidirect
200. C107#3056semidirect
208. #468D107semidirect
223. C2×SD16C2×D107semidirect
250. #107C2×D107semidirect
258. #103C2×D107semidirect
260. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121382
C2×M16×D1072
#202931372
C22⋊Q8×D1072
#202928852
#202928832
#202928712
#202928692
#202928672
#202928552
#202928532
#202926682
#202926342
#202926252
#468×C1072
#3056107