Finite Group Databasebeta

← #28348278

Subgroups — #28348278

Order 13696.

462 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
1#20293335C211yesmaximal, characteristic, complemented
2D4×C22⋊D107C211yesmaximal, characteristic
3#20292972C211yesmaximal, characteristic, complemented
4#20292946C211yesmaximal, characteristic, complemented
5#20292944C211yesmaximal, characteristic, complemented
6#20292915C211yesmaximal, characteristic, complemented
7#20292858C211yesmaximal, characteristic, complemented
8#20292853C211yesmaximal, characteristic, complemented
9#20292825C211yesmaximal, characteristic, complemented
10#20292776C211yesmaximal, characteristic, complemented
11SD16×Dic107C211yesmaximal, characteristic, complemented
12(C22⋊D107)⋊C8C211yesmaximal, characteristic, complemented
13C2×#16812030C211yesmaximal, characteristic, complemented
14#102×D107C211yesmaximal, characteristic, complemented
15#450×C107C211yesmaximal, characteristic, complemented
16C22×C22⋊D107—21no—
17C2×D4×D107C2211yescharacteristic
18C2×#13133805C2211yescharacteristic, complemented
19C22×C107⋊C8C2211yescharacteristic, complemented
20C2×C8×D107C2211yescharacteristic, complemented
21#16812108C2211yescharacteristic, complemented
22#16812105C2211yescharacteristic
23#16812099—21no—
24#16812089—21no—
25#16812086—21no—
26#16812085C2211yescharacteristic, complemented
27#16812083C2211yescharacteristic
28#16812081—21no—
29#16812080C2211yescharacteristic
30#16812078—21no—
31#16812077—21no—
32Q8×Dic107C2211yescharacteristic, complemented
33D4×Dic107C2211yescharacteristic
34C4×C22⋊D107C2211yescharacteristic
35(C22⋊D107)⋊C4—21no—
36C2×C4⋊Dic107C2211yescharacteristic, complemented
37C4⋊C4×D107C2211yescharacteristic, complemented
38#16812041C2211yescharacteristic, complemented
39#16812041C2211yescharacteristic, complemented
40#16812040C2211yescharacteristic, complemented
41Dic107⋊C8C2211yescharacteristic, complemented
42#16812033C2211yescharacteristic, complemented
43#16812031C2211yescharacteristic, complemented
44#16812031C2211yescharacteristic, complemented
45#16812030C2211yescharacteristic
46#16812030C2211yescharacteristic
47#16812030C2211yescharacteristic, complemented
48#16812030C2211yescharacteristic, complemented
49#16812029C2211yescharacteristic, complemented
50(C2×D107)⋊C8C2211yescharacteristic, complemented
51C8×Dic107C2211yescharacteristic, complemented
52C22×D4×C107C2211yescharacteristic
53C2×SD16×C107C2211yescharacteristic
54C2×SD16×C107C2211yescharacteristic, complemented
55C22⋊Q8×C107C2211yescharacteristic, complemented
56#102×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic, complemented
58C22⋊C8×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60C2×Dic214—21no—
61C2×D428C2311yescharacteristic
62C2×C22⋊D107—21no—
63C2×C22⋊D107—21no—
64C2×C22⋊D107—21no—
65C2×C22⋊D107—21no—
66C2×C22⋊D107—21no—
67C2×C22⋊D107—44no—
68C2×C22⋊D107D411yescharacteristic
69D4×D107—22no—
70D4×D107—41no—
71C22×Dic107—21no—
72C22×Dic107—21no—
73C2×C4×D107—21no—
74C2×C4×D107C2311yescharacteristic
75#13133805—41no—
76C2×C107⋊C8—41no—
77C2×C107⋊C8C2311yescharacteristic
78C2×C107⋊C8C2311yescharacteristic
79C8×D107—41no—
80C4⋊Dic107—21no—
81C4⋊Dic107—21no—
82C4⋊Dic107C2311yescharacteristic
83C4⋊Dic107C2311yescharacteristic
84C4⋊Dic107C2311yescharacteristic
85Dic107⋊C4—21no—
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88Dic107⋊C4D411yescharacteristic
89C22⋊Dic107—21no—
90C22⋊Dic107—21no—
91C22⋊Dic107D411yescharacteristic
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4D411yescharacteristic
95C4×Dic107—21no—
96C4×Dic107C2311yescharacteristic
97C24×C107—21no—
98C2×Q8×C107C2311yescharacteristic
99C2×D4×C107—22no—
100C2×D4×C107—21no—
101C2×D4×C107—41no—
102C2×D4×C107C2311yescharacteristic
103C2×D4×C107C2311yescharacteristic
104C22×C4×C107C2311yescharacteristic
105SD16×C107—22no—
106SD16×C107—41no—
107C2×C8×C107C2311yescharacteristic
108C2×C8×C107C2311yescharacteristic
109C4⋊C4×C107—21no—
110C4⋊C4×C107C2311yescharacteristic
111C22⋊C4×C107—21no—
112C23×C107—21no—
113C23×C107—21no—
114C23×C107—42no—
115C23×C107—42no—
116C23×C107—81no—
117C23×C107C2×D411yescharacteristic
118C22×D107—21no—
119C22×D107—42no—
120C22×D107—42no—
121C22×D107—81no—
122C22×D107C2×D411yescharacteristic
123Q8×C107—21no—
124Q8×C107—21no—
125D4×C107—21no—
126D4×C107—21no—
127D4×C107—21no—
128D4×C107—42no—
129D4×C107#4112yes—
130C2×C4×C107—21no—
131C2×C4×C107—21no—
132C2×C4×C107—21no—
133C2×C4×C107—21no—
134C2×C4×C107C2×D411yescharacteristic
135C2×C4×C107C2411yescharacteristic
136C22⋊D107—42no—
137C22⋊D107—42no—
138C22⋊D107—42no—
139C22⋊D107—42no—
140C22⋊D107—81no—
141C22⋊D107SD1614yescomplemented
142C2×Dic107—21no—
143C2×Dic107—21no—
144C2×Dic107—21no—
145C2×Dic107—42no—
146C2×Dic107—81no—
147C2×Dic107C2×D411yescharacteristic
148C2×Dic107C2×D411yescharacteristic
149C2×Dic107C2×D411yescharacteristic
150D428—21no—
151D428—21no—
152C4×D107—21no—
153C4×D107—21no—
154C4×D107—81no—
155Dic214—81no—
156C8×C107—21no—
157C8×C107—21no—
158C107⋊C8—21no—
159C107⋊C8—21no—
160C22×C107—21no—
161C22×C107—21no—
162C22×C107—22no—
163C22×C107—41no—
164C22×C107—41no—
165C22×C107—41no—
166C22×C107—41no—
167C22×C107—41no—
168C22×C107—84no—
169C22×C107C2×SD1612yes—
170C22×C107C22×D411yescharacteristic
171C2×D107—21no—
172C2×D107—21no—
173C2×D107—41no—
174C2×D107—41no—
175C2×D107—41no—
176C2×D107—84no—
177C2×D107C2×SD1612yes—
178C4×C107—21no—
179C4×C107—41no—
180C4×C107—41no—
181C4×C107C2×#4111yescharacteristic
182C4×C107H(5,𝔽2)11yescharacteristic
183Dic107—21no—
184Dic107—21no—
185Dic107—41no—
186Dic107—41no—
187Dic107—41no—
188Dic107C2×SD1612yes—
189C2×C107—22no—
190C2×C107—42no—
191C2×C107—81no—
192C2×C107#54611yescharacteristic
193C2×C107C22×SD1611yescharacteristic
194C2×C107#57611yescharacteristic
195D107—22no—
196D107—81no—
197#3295—1071nomaximal, Sylow
198C107#329511yescharacteristic, complemented, Sylow
199#554—1071no—
200D42—1071no—
201#486—1071no—
202#481—1071no—
203#481—1071no—
204#476—1071no—
205#465—1071no—
206#465—1071no—
207#460—1071no—
208#450—1071no—
209#450D10711yescharacteristic, complemented
210C4×SD16—1071no—
211C8×D4—1071no—
212C2×#102—1071no—
213C2×#102—1071no—
214C22×D4—1071no—
215C22×D4—2141no—
216C22×D4C2×D10711yescharacteristic
217C2×SD16—1071no—
218C2×SD16C2×D10711yescharacteristic
219C2×SD16C2×D10711yescharacteristic, complemented
220C22×C8—1071no—
221C22×C8—1071no—
222C4⋊Q8—1071no—
223#127—1071no—
224#125—2141no—
225C22⋊Q8—1071no—
226C22⋊Q8—2141no—
227C22⋊Q8—2141no—
228C22⋊Q8C2×D10711yescharacteristic, complemented
229#121—1071no—
230#121—1071no—
231#121—2141no—
232#120—2141no—
233#120—2141no—
234C4×Q8—1071no—
235C4×D4—1071no—
236C4×D4—1071no—
237C4×D4—2141no—
238C2×C4⋊C4—1071no—
239C2×C4⋊C4—1071no—
240#106—1071no—
241#106—1071no—
242#106—1071no—
243C4⋊C8—1071no—
244#103—1071no—
245#102—1071no—
246#102—1071no—
247#102—1071no—
248#102—1071no—
249#102—1071no—
250#102—1071no—
251#102—1071no—
252#102C2×D10711yescharacteristic
253#102C2×D10711yescharacteristic, complemented
254C22⋊C8—1071no—
255C22⋊C8C2×D10711yescharacteristic, complemented
256C4×C8—1071no—
257C24—21no—
258C24—2141no—
259C2×Q8—2141no—
260C2×Q8C22×D10711yescharacteristic
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—1071no—
265C2×D4—1071no—
266C2×D4—4284no—
267C2×D4—4281no—
268C2×D4—2141no—
269C2×D4—2141no—
270C2×D4—2141no—
271C2×D4—2141no—
272C2×D4—2141no—
273C2×D4—2142no—
274C2×D4C22×D10711yescharacteristic
275C2×D4C22×D10711yescharacteristic
276C22×C4—1071no—
277C22×C4—2141no—
278C22×C4—2141no—
279C22×C4—2141no—
280C22×C4C22×D10711yescharacteristic
281SD16—22no—
282SD16—41no—
283SD16—4281no—
284C2×C8—1071no—
285C2×C8—1071no—
286C2×C8—4281no—
287C2×C8—4281no—
288C2×C8C22×D10711yescharacteristic
289C2×C8C22×D10711yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—1071no—
292C4⋊C4—1071no—
293C4⋊C4—1071no—
294C4⋊C4—1071no—
295C4⋊C4—2141no—
296C4⋊C4—2141no—
297C4⋊C4—2141no—
298C4⋊C4—2141no—
299C4⋊C4—2141no—
300C4⋊C4C22×D10711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—1071no—
303C22⋊C4—1071no—
304C22⋊C4—2141no—
305C22⋊C4—2141no—
306C22⋊C4—2141no—
307C22⋊C4—2141no—
308C42—1071no—
309C42—2141no—
310C23—21no—
311C23—21no—
312C23—42no—
313C23—42no—
314C23—81no—
315C23—8561no—
316C23—1071no—
317C23—4282no—
318C23—4282no—
319C23—2141no—
320C23D4×D10711yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—8561no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—8561no—
329D4—1074no—
330D4—4282no—
331D4—4282no—
332D4—4282no—
333D4—4282no—
334D4—2141no—
335D4—2141no—
336D4#1313381712yes—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—8561no—
342C2×C4—8561no—
343C2×C4—1071no—
344C2×C4—1071no—
345C2×C4—1071no—
346C2×C4—4282no—
347C2×C4—2141no—
348C2×C4—2141no—
349C2×C4—2141no—
350C2×C4—2141no—
351C2×C4—2141no—
352C2×C4D4×D10711yescharacteristic
353C2×C4C23×D10711yescharacteristic
354C8—21no—
355C8—21no—
356C8—2141no—
357C8—2141no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22—8564no—
368C22—1072no—
369C22—4281no—
370C22—4281no—
371C22—4281no—
372C22—2141no—
373C22—2141no—
374C22SD16×D10712yes—
375C22C2×D4×D10711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—1072no—
380C4—4281no—
381C4—4281no—
382C4—4281no—
383C4—2141no—
384C4—2141no—
385C4C2×#1313381711yescharacteristic
386C4#1681215411yescharacteristic
387C2—22no—
388C2—42no—
389C2—81no—
390C2—8561no—
391C2—2142no—
392C2#2029326411yescharacteristic
393C2C2×SD16×D10711yescharacteristic
394C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#450×C107C2
socleC22×C107C22×D4
prime radicalD4×C22⋊D107C2
hypercenter#450D107
hypocenterC107#3295
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #20293335C2semidirect
3. #20292972C2semidirect
4. #20292946C2semidirect
5. #20292944C2semidirect
6. #20292915C2semidirect
7. #20292858C2semidirect
8. #20292853C2semidirect
9. #20292825C2semidirect
10. #20292776C2semidirect
11. SD16×Dic107C2semidirect
12. (C22⋊D107)⋊C8C2semidirect
13. C2×#16812030C2semidirect
14. #102×D107C2semidirect
15. #450×C107C2semidirect
18. C2×#13133805C22semidirect
19. C22×C107⋊C8C22semidirect
20. C2×C8×D107C22semidirect
21. #16812108C22semidirect
26. #16812085C22semidirect
32. Q8×Dic107C22semidirect
36. C2×C4⋊Dic107C22semidirect
37. C4⋊C4×D107C22semidirect
38. #16812041C22semidirect
39. #16812041C22semidirect
40. #16812040C22semidirect
41. Dic107⋊C8C22semidirect
42. #16812033C22semidirect
43. #16812031C22semidirect
44. #16812031C22semidirect
47. #16812030C22semidirect
48. #16812030C22semidirect
49. #16812029C22semidirect
50. (C2×D107)⋊C8C22semidirect
51. C8×Dic107C22semidirect
54. C2×SD16×C107C22semidirect
55. C22⋊Q8×C107C22semidirect
57. #102×C107C22semidirect
58. C22⋊C8×C107C22semidirect
141. C22⋊D107SD16semidirect
198. C107#3295semidirect
209. #450D107semidirect
219. C2×SD16C2×D107semidirect
228. C22⋊Q8C2×D107semidirect
253. #102C2×D107semidirect
255. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202933352
D4×C22⋊D1072
#202929722
#202929462
#202929442
#202929152
#202928582
#202928532
#202928252
#202927762
SD16×Dic1072
(C22⋊D107)⋊C82
C2×#168120302
#102×D1072
#450×C1072
#3295107