Finite Group Databasebeta

← #28348422

Subgroups — #28348422

Order 13696.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293281C211yesmaximal, characteristic, complemented
2#20292986C211yesmaximal, characteristic, complemented
3#20292955C211yesmaximal, characteristic, complemented
4#20292943C211yesmaximal, characteristic, complemented
5#20292917C211yesmaximal, characteristic, complemented
6#20292878C211yesmaximal, characteristic, complemented
7#20292858C211yesmaximal, characteristic, complemented
8#20292839C211yesmaximal, characteristic, complemented
9#20292772C211yesmaximal, characteristic, complemented
10#20292761C211yesmaximal, characteristic, complemented
11#20292737C211yesmaximal, characteristic, complemented
12#20292699C211yesmaximal, characteristic, complemented
13#20292628C211yesmaximal, characteristic, complemented
14C2×#16812030C211yesmaximal, characteristic, complemented
15#546×C107C211yesmaximal, characteristic, complemented
16C2×#13133817—21no—
17C2×#13133805C2211yescharacteristic, complemented
18C2×#13133803C2211yescharacteristic
19C2×#13133803C2211yescharacteristic
20C2×#13133802C2211yescharacteristic, complemented
21C22×C107⋊C8C2211yescharacteristic
22#16812109C2211yescharacteristic, complemented
23#16812093—21no—
24#16812093—21no—
25#16812089—21no—
26#16812083C2211yescharacteristic
27#16812083C2211yescharacteristic, complemented
28C4×D428C2211yescharacteristic, complemented
29C2×C4⋊Dic107C2211yescharacteristic
30C2×C4⋊Dic107C2211yescharacteristic, complemented
31#16812040C2211yescharacteristic, complemented
32#16812040C2211yescharacteristic, complemented
33#16812040C2211yescharacteristic, complemented
34C4⋊(C107⋊C8)C2211yescharacteristic, complemented
35#16812033C2211yescharacteristic, complemented
36#16812033C2211yescharacteristic, complemented
37#16812033C2211yescharacteristic, complemented
38#16812030C2211yescharacteristic
39#16812030C2211yescharacteristic
40#16812030C2211yescharacteristic
41#16812030C2211yescharacteristic
42#16812030C2211yescharacteristic
43#16812030C2211yescharacteristic
44#16812029C2211yescharacteristic, complemented
45C22⋊(C107⋊C8)C2211yescharacteristic
46C22⋊(C107⋊C8)C2211yescharacteristic, complemented
47#16812020C2211yescharacteristic, complemented
48C2×#41×C107C2211yescharacteristic
49C22×D4×C107C2211yescharacteristic
50#124×C107C2211yescharacteristic
51#123×C107—21no—
52C22⋊Q8×C107C2211yescharacteristic
53#121×C107—21no—
54#121×C107C2211yescharacteristic
55#120×C107—21no—
56C4×D4×C107C2211yescharacteristic
57C4×D4×C107C2211yescharacteristic
58C2×C22⋊C4×C107—21no—
59#13133817—44no—
60C2×Dic214—21no—
61C2×D428C2311yescharacteristic
62C2×C22⋊D107—21no—
63C2×C22⋊D107—21no—
64C22×Dic107—21no—
65C22×Dic107—21no—
66C2×C4×D107—21no—
67#13133805—41no—
68#13133803—22no—
69#13133803—41no—
70#13133802—41no—
71C2×C107⋊C8—41no—
72C2×C107⋊C8C2311yescharacteristic
73C2×C107⋊C8C2311yescharacteristic
74C2×C107⋊C8C2311yescharacteristic
75C2×C107⋊C8C2311yescharacteristic
76C4⋊Dic107—21no—
77C4⋊Dic107—21no—
78C4⋊Dic107—21no—
79C4⋊Dic107C2311yescharacteristic
80C4⋊Dic107C2311yescharacteristic
81C4⋊Dic107C2311yescharacteristic
82(C2×D107)⋊C4—21no—
83(C2×D107)⋊C4—21no—
84(C2×D107)⋊C4—21no—
85C24×C107—21no—
86#41×C107—22no—
87#41×C107—41no—
88C2×Q8×C107C2311yescharacteristic
89C2×D4×C107—22no—
90C2×D4×C107—21no—
91C2×D4×C107—21no—
92C2×D4×C107—21no—
93C2×D4×C107—41no—
94C2×D4×C107D411yescharacteristic
95C2×D4×C107C2311yescharacteristic
96C2×D4×C107C2311yescharacteristic
97C22×C4×C107—21no—
98C22×C4×C107—21no—
99C22×C4×C107C2311yescharacteristic
100C22×C4×C107C2311yescharacteristic
101C4⋊C4×C107—21no—
102C4⋊C4×C107D411yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C22⋊C4×C107—21no—
105C22⋊C4×C107—21no—
106C22⋊C4×C107—21no—
107C22⋊C4×C107—21no—
108C22⋊C4×C107—21no—
109C22⋊C4×C107D411yescharacteristic
110C22⋊C4×C107D411yescharacteristic
111C42×C107C2311yescharacteristic
112C23×C107—21no—
113C23×C107—21no—
114C23×C107—42no—
115C23×C107—42no—
116C23×C107—81no—
117C23×C107C2×D411yescharacteristic
118C23×C107C2×D411yescharacteristic
119C22×D107—21no—
120Q8×C107—21no—
121Q8×C107—21no—
122D4×C107—22no—
123D4×C107—21no—
124D4×C107—21no—
125D4×C107—21no—
126D4×C107—42no—
127D4×C107—42no—
128D4×C107—81no—
129D4×C107#4112yescomplemented
130C2×C4×C107—21no—
131C2×C4×C107—21no—
132C2×C4×C107—21no—
133C2×C4×C107—21no—
134C2×C4×C107—21no—
135C2×C4×C107—21no—
136C2×C4×C107—42no—
137C2×C4×C107—81no—
138C2×C4×C107C2×D411yescharacteristic
139C2×C4×C107C2×D411yescharacteristic
140C2×C4×C107C2×D411yescharacteristic
141C2×C4×C107C2×D411yescharacteristic
142C2×C4×C107C2411yescharacteristic
143C22⋊D107—42no—
144C22⋊D107—42no—
145C2×Dic107—21no—
146C2×Dic107—21no—
147C2×Dic107—21no—
148C2×Dic107—42no—
149C2×Dic107—42no—
150D428—21no—
151D428—21no—
152C4×D107—42no—
153Dic214—42no—
154C107⋊C8—21no—
155C107⋊C8—21no—
156C107⋊C8—21no—
157C107⋊C8—21no—
158C22×C107—21no—
159C22×C107—21no—
160C22×C107—21no—
161C22×C107—22no—
162C22×C107—21no—
163C22×C107—21no—
164C22×C107—41no—
165C22×C107—41no—
166C22×C107—41no—
167C22×C107—41no—
168C22×C107—41no—
169C22×C107—84no—
170C22×C107AGL(1,ℤ/8)12yes—
171C22×C107C22×D411yescharacteristic
172C2×D107—41no—
173C2×D107—41no—
174C2×D107—41no—
175C4×C107—21no—
176C4×C107—21no—
177C4×C107—21no—
178C4×C107—21no—
179C4×C107—41no—
180C4×C107—41no—
181C4×C107C2×#4111yescharacteristic
182C4×C107H(5,𝔽2)11yescharacteristic
183Dic107—41no—
184Dic107—41no—
185Dic107—41no—
186C2×C107—22no—
187C2×C107—41no—
188C2×C107—42no—
189C2×C107—81no—
190C2×C107#54611yescharacteristic
191C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C107#57711yescharacteristic
193D107—81no—
194#3306—1071nomaximal, Sylow
195C107#330611yescharacteristic, complemented, Sylow
196#547—1071no—
197#546D10711yescharacteristic, complemented
198#489—1071no—
199#483—1071no—
200#481—1071no—
201#476—1071no—
202#469—1071no—
203#465—1071no—
204#463—1071no—
205#449—1071no—
206#447—1071no—
207#442—1071no—
208#435—1071no—
209#417—1071no—
210C2×#102—1071no—
211C2×#41—2141no—
212C2×#41C2×D10711yescharacteristic
213C22×D4C2×D10711yescharacteristic
214C2×SD16—1071no—
215C2×D8—1071no—
216C2×D8—1071no—
217C2×M16—1071no—
218C22×C8—1071no—
219C4⋊Q8—1071no—
220#124C2×D10711yescharacteristic
221#123—21no—
222#123—2141no—
223#123—2141no—
224C22⋊Q8—2141no—
225C22⋊Q8C2×D10711yescharacteristic
226#121—21no—
227#121—1071no—
228#121—1071no—
229#121C2×D10711yescharacteristic
230#120—21no—
231C4×D4—1071no—
232C4×D4C2×D10711yescharacteristic
233C4×D4C2×D10711yescharacteristic
234C2×C4⋊C4—1071no—
235C2×C4⋊C4—1071no—
236C2×C22⋊C4—21no—
237#106—1071no—
238#106—1071no—
239#106—1071no—
240C4⋊C8—1071no—
241#103—1071no—
242#103—1071no—
243#103—1071no—
244#102—1071no—
245#102—1071no—
246#102—1071no—
247#102—1071no—
248#102—1071no—
249#102—1071no—
250#102—1071no—
251C22⋊C8—1071no—
252C22⋊C8—1071no—
253#97—1071no—
254C24—21no—
255#41—22no—
256#41—41no—
257#41—4284no—
258C2×Q8—2141no—
259C2×Q8C22×D10711yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—1071no—
266C2×D4—2141no—
267C2×D4—2141no—
268C2×D4C22⋊D10711yescharacteristic
269C2×D4C22×D10711yescharacteristic
270C2×D4C22×D10711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—2141no—
274C22×C4—2141no—
275C22×C4—2141no—
276C22×C4C22×D10711yescharacteristic
277C22×C4C22×D10711yescharacteristic
278SD16—4281no—
279D8—4281no—
280D8—2142no—
281M16—4281no—
282C2×C8—1071no—
283C2×C8—1071no—
284C2×C8—1071no—
285C2×C8—1071no—
286C2×C8—4281no—
287C4⋊C4—21no—
288C4⋊C4—1071no—
289C4⋊C4—1071no—
290C4⋊C4—1071no—
291C4⋊C4—2141no—
292C4⋊C4—2141no—
293C4⋊C4—2141no—
294C4⋊C4C22⋊D10711yescharacteristic
295C4⋊C4C22×D10711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—2141no—
302C22⋊C4—2141no—
303C22⋊C4—2141no—
304C22⋊C4C22⋊D10711yescharacteristic
305C22⋊C4C22⋊D10711yescharacteristic
306C42C22×D10711yescharacteristic
307C23—21no—
308C23—21no—
309C23—42no—
310C23—42no—
311C23—81no—
312C23—2141no—
313C23C2×C22⋊D10711yescharacteristic
314C23C2×C22⋊D10711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—4282no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—81no—
325D4—4282no—
326D4—4282no—
327D4—2141no—
328D4—2141no—
329D4#1313381712yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—4282no—
339C2×C4—4282no—
340C2×C4—4282no—
341C2×C4—2141no—
342C2×C4—2141no—
343C2×C4—2141no—
344C2×C4C2×C22⋊D10711yescharacteristic
345C2×C4C2×C22⋊D10711yescharacteristic
346C2×C4C2×C22⋊D10711yescharacteristic
347C2×C4C2×C22⋊D10711yescharacteristic
348C2×C4C23×D10711yescharacteristic
349C8—2141no—
350C8—2141no—
351C8—2141no—
352C8—2141no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—4281no—
366C22—4281no—
367C22—4281no—
368C22#1681213412yes—
369C22C22×C22⋊D10711yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—4281no—
377C4—4281no—
378C4—4281no—
379C4C2×#1313381711yescharacteristic
380C4#1681215411yescharacteristic
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—8561no—
386C2#2029327111yescharacteristic
387C2C2×#1681213411yescharacteristic
388C2#2029343811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#546×C107C2
socleC22×C107C22×D4
hypercenter#546D107
hypocenterC107#3306
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #20293281C2semidirect
2. #20292986C2semidirect
3. #20292955C2semidirect
4. #20292943C2semidirect
5. #20292917C2semidirect
6. #20292878C2semidirect
7. #20292858C2semidirect
8. #20292839C2semidirect
9. #20292772C2semidirect
10. #20292761C2semidirect
11. #20292737C2semidirect
12. #20292699C2semidirect
13. #20292628C2semidirect
14. C2×#16812030C2semidirect
15. #546×C107C2semidirect
17. C2×#13133805C22semidirect
20. C2×#13133802C22semidirect
22. #16812109C22semidirect
27. #16812083C22semidirect
28. C4×D428C22semidirect
30. C2×C4⋊Dic107C22semidirect
31. #16812040C22semidirect
32. #16812040C22semidirect
33. #16812040C22semidirect
34. C4⋊(C107⋊C8)C22semidirect
35. #16812033C22semidirect
36. #16812033C22semidirect
37. #16812033C22semidirect
44. #16812029C22semidirect
46. C22⋊(C107⋊C8)C22semidirect
47. #16812020C22semidirect
129. D4×C107#41semidirect
195. C107#3306semidirect
197. #546D107semidirect
329. D4#13133817semidirect

Maximal subgroups

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

subgroupindex
#202932812
#202929862
#202929552
#202929432
#202929172
#202928782
#202928582
#202928392
#202927722
#202927612
#202927372
#202926992
#202926282
C2×#168120302
#546×C1072
#3306107