Finite Group Databasebeta

← #16831331

Subgroups — #16831331

Order 3712.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14271855C211yesmaximal, characteristic, complemented
2#14271807C211yesmaximal, characteristic
3#14271516C211yesmaximal, characteristic, complemented
4#14271494C211yesmaximal, characteristic, complemented
5#14271486C211yesmaximal, characteristic, complemented
6#14271465C211yesmaximal, characteristic, complemented
7#14271402C211yesmaximal, characteristic, complemented
8#14271389C211yesmaximal, characteristic, complemented
9#14271383C211yesmaximal, characteristic, complemented
10#14271322C211yesmaximal, characteristic, complemented
11SD16×Dic29C211yesmaximal, characteristic, complemented
12#14271226C211yesmaximal, characteristic, complemented
13#14271156C211yesmaximal, characteristic, complemented
14C2×#11745180C211yesmaximal, characteristic, complemented
15C29×#450C211yesmaximal, characteristic, complemented
16C22×D4×C29C2211yescharacteristic
17C2×#90807C2211yescharacteristic
18C2×SD16×C29C2211yescharacteristic
19C2×SD16×C29C2211yescharacteristic
20C29×C22⋊Q8C2211yescharacteristic
21#11745315—21no—
22C2×C22⋊Dic29—21no—
23#11745308C2211yescharacteristic, complemented
24Q8×Dic29C2211yescharacteristic, complemented
25C2×#90785C2211yescharacteristic, complemented
26#11745301—21no—
27#11745300C2211yescharacteristic
28#11745298C2211yescharacteristic
29#11745297—21no—
30D4×Dic29C2211yescharacteristic
31C4×C22⋊D29C2211yescharacteristic
32C2×C4⋊Dic29C2211yescharacteristic
33#11745286C2211yescharacteristic
34C22×#6597C2211yescharacteristic
35C2×C8×D29C2211yescharacteristic, complemented
36#11745260—21no—
37#11745254C2211yescharacteristic, complemented
38#11745252—21no—
39#11745252C2211yescharacteristic, complemented
40#11745248—21no—
41#11745239—21no—
42C29×#102C2211yescharacteristic
43C29×#102C2211yescharacteristic
44C29×C22⋊C8C2211yescharacteristic
45#11745183C2211yescharacteristic, complemented
46#11745180C2211yescharacteristic
47#11745180C2211yescharacteristic
48#11745180C2211yescharacteristic
49#11745180C2211yescharacteristic
50#11745168C2211yescharacteristic, complemented
51#11745165C2211yescharacteristic, complemented
52#11745165C2211yescharacteristic, complemented
53#11745164C2211yescharacteristic, complemented
54#11745164C2211yescharacteristic, complemented
55#11745162C2211yescharacteristic, complemented
56C8×Dic29C2211yescharacteristic, complemented
57#11745158C2211yescharacteristic, complemented
58#11745155C2211yescharacteristic, complemented
59C24×C29—21no—
60C2×Q8×C29C2311yescharacteristic
61C2×D4×C29—22no—
62C2×D4×C29—21no—
63C2×D4×C29—41no—
64C2×D4×C29C2311yescharacteristic
65C2×D4×C29C2311yescharacteristic
66C22×C4×C29C2311yescharacteristic
67C2×C22⋊D29—21no—
68C2×C22⋊D29—21no—
69C2×C22⋊D29D411yescharacteristic
70C22×Dic29—21no—
71C22×Dic29—21no—
72C22×Dic29—21no—
73#90807—22no—
74#90807—41no—
75C2×C4×D29C2311yescharacteristic
76C2×Dic58C2311yescharacteristic
77SD16×C29—22no—
78SD16×C29—41no—
79C2×C8×C29C2311yescharacteristic
80C2×C8×C29C2311yescharacteristic
81C4⋊C4×C29—21no—
82C4⋊C4×C29C2311yescharacteristic
83C22⋊C4×C29—21no—
84C22⋊Dic29—21no—
85C22⋊Dic29—21no—
86C22⋊Dic29—21no—
87C22⋊Dic29—21no—
88C22⋊Dic29—21no—
89C22⋊Dic29D411yescharacteristic
90#90785—41no—
91#90781—21no—
92#90781—21no—
93#90781D411yescharacteristic
94C4⋊Dic29—21no—
95C4⋊Dic29—21no—
96C4⋊Dic29—21no—
97C4⋊Dic29C2311yescharacteristic
98C4⋊Dic29C2311yescharacteristic
99C4⋊Dic29C2311yescharacteristic
100#90779—21no—
101#90779—21no—
102#90779—21no—
103#90779D411yescharacteristic
104C4×Dic29—21no—
105C4×Dic29—21no—
106C4×Dic29C2311yescharacteristic
107C2×#6597—41no—
108C2×#6597C2311yescharacteristic
109C2×#6597C2311yescharacteristic
110C8×D29—41no—
111C23×C29—21no—
112C23×C29—21no—
113C23×C29—42no—
114C23×C29—42no—
115C23×C29—81no—
116C23×C29C2×D411yescharacteristic
117C22×D29C2×D411yescharacteristic
118Q8×C29—21no—
119Q8×C29—21no—
120D4×C29—21no—
121D4×C29—21no—
122D4×C29—21no—
123D4×C29—42no—
124D4×C29#4112yes—
125C2×C4×C29—21no—
126C2×C4×C29—21no—
127C2×C4×C29—21no—
128C2×C4×C29—21no—
129C2×C4×C29C2×D411yescharacteristic
130C2×C4×C29C2411yescharacteristic
131C22⋊D29—22no—
132C22⋊D29—42no—
133C22⋊D29—81no—
134C2×Dic29—21no—
135C2×Dic29—21no—
136C2×Dic29—21no—
137C2×Dic29—21no—
138C2×Dic29—42no—
139C2×Dic29—42no—
140C2×Dic29—81no—
141C2×Dic29C2×D411yescharacteristic
142C2×Dic29C2×D411yescharacteristic
143C2×Dic29C2×D411yescharacteristic
144C4×D29—21no—
145C4×D29—21no—
146Dic58—21no—
147Dic58—21no—
148C8×C29—21no—
149C8×C29—21no—
150#6597—21no—
151#6597—21no—
152#3296—291nomaximal, Sylow
153C22×C29—21no—
154C22×C29—21no—
155C22×C29—22no—
156C22×C29—41no—
157C22×C29—41no—
158C22×C29—41no—
159C22×C29—41no—
160C22×C29—41no—
161C22×C29—84no—
162C22×C29#13512yes—
163C22×C29C22×D411yescharacteristic
164C2×D29—21no—
165C2×D29—21no—
166C2×D29—21no—
167C4×C29—21no—
168C4×C29—41no—
169C4×C29—41no—
170C4×C29C2×#4111yescharacteristic
171C4×C29H(5,𝔽2)11yescharacteristic
172Dic29—21no—
173Dic29—21no—
174Dic29—21no—
175Dic29—41no—
176Dic29—41no—
177Dic29—41no—
178Dic29—41no—
179#552—291no—
180#546—291no—
181#488—291no—
182#483—291no—
183#482—291no—
184#478—291no—
185#467—291no—
186#465—291no—
187#464—291no—
188#452—291no—
189#450D2911yescharacteristic, complemented
190C4×SD16—291no—
191C8×D4—291no—
192#416—291no—
193C2×#102—291no—
194C2×C29—22no—
195C2×C29—42no—
196C2×C29—81no—
197C2×C29#54611yescharacteristic
198C2×C29C2×#13511yescharacteristic
199C2×C29#57711yescharacteristic
200D29—41no—
201C2×#41—291no—
202C22×D4C2×D2911yescharacteristic
203C2×Q16—291no—
204C2×SD16C2×D2911yescharacteristic
205C2×SD16C2×D2911yescharacteristic
206C22×C8—291no—
207C22×C8—291no—
208#126—581no—
209#125—291no—
210#125—581no—
211#124—291no—
212#123—581no—
213#123—581no—
214C22⋊Q8—291no—
215C22⋊Q8—291no—
216C22⋊Q8C2×D2911yescharacteristic
217#121—291no—
218#121—581no—
219#120—581no—
220C4×Q8—291no—
221C4×D4—291no—
222C4×D4—291no—
223#117—291no—
224#117—581no—
225C2×C4⋊C4—291no—
226C2×C22⋊C4—581no—
227#107—291no—
228#106—291no—
229#106—291no—
230C4⋊C8—291no—
231#103—291no—
232#103—291no—
233#103—291no—
234#103—291no—
235#102—291no—
236#102—291no—
237#102—291no—
238#102—291no—
239#102C2×D2911yescharacteristic
240#102C2×D2911yescharacteristic
241C22⋊C8—291no—
242C22⋊C8C2×D2911yescharacteristic
243C4×C8—291no—
244C29#329611yescharacteristic, complemented, Sylow
245C24—21no—
246#41—582no—
247#41—1161no—
248C2×Q8—291no—
249C2×Q8C22×D2911yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—291no—
254C2×D4—581no—
255C2×D4—581no—
256C2×D4C22×D2911yescharacteristic
257C2×D4C22×D2911yescharacteristic
258C22×C4—291no—
259C22×C4—581no—
260C22×C4—581no—
261C22×C4—581no—
262C22×C4C22×D2911yescharacteristic
263Q16—1161no—
264SD16—22no—
265SD16—41no—
266C2×C8—291no—
267C2×C8—291no—
268C2×C8—1161no—
269C2×C8—1161no—
270C2×C8C22×D2911yescharacteristic
271C2×C8C22×D2911yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—291no—
274C4⋊C4—291no—
275C4⋊C4—291no—
276C4⋊C4—291no—
277C4⋊C4—581no—
278C4⋊C4—581no—
279C4⋊C4—581no—
280C4⋊C4—581no—
281C4⋊C4—581no—
282C4⋊C4—581no—
283C4⋊C4C22×D2911yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—291no—
286C22⋊C4—291no—
287C22⋊C4—581no—
288C22⋊C4—581no—
289C22⋊C4—581no—
290C22⋊C4—581no—
291C22⋊C4—581no—
292C22⋊C4—581no—
293C22⋊C4—581no—
294C42—291no—
295C42—581no—
296C42—581no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—291no—
303C23D4×D2911yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—581no—
307Q8—581no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—582no—
313D4—1162no—
314D4—2321no—
315D4#9080712yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—291no—
321C2×C4—291no—
322C2×C4—291no—
323C2×C4—581no—
324C2×C4—581no—
325C2×C4—581no—
326C2×C4—581no—
327C2×C4—581no—
328C2×C4—581no—
329C2×C4—1162no—
330C2×C4—1162no—
331C2×C4—2321no—
332C2×C4D4×D2911yescharacteristic
333C2×C4C23×D2911yescharacteristic
334C8—21no—
335C8—21no—
336C8—581no—
337C8—581no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—84no—
347C22—581no—
348C22—581no—
349C22—581no—
350C22#1174527812yes—
351C22C2×D4×D2911yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—581no—
356C4—581no—
357C4—581no—
358C4—1161no—
359C4—1161no—
360C4—1161no—
361C4—1161no—
362C4C2×#9080711yescharacteristic
363C4#1174535911yescharacteristic
364C2—22no—
365C2—42no—
366C2—81no—
367C2—1161no—
368C2#1427180011yescharacteristic
369C2C2×#1174527811yescharacteristic
370C2#1427197311yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#450C2
socleC22×C29C22×D4
prime radical#14271807C2
hypercenter#450D29
hypocenterC29#3296
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #14271855C2semidirect
3. #14271516C2semidirect
4. #14271494C2semidirect
5. #14271486C2semidirect
6. #14271465C2semidirect
7. #14271402C2semidirect
8. #14271389C2semidirect
9. #14271383C2semidirect
10. #14271322C2semidirect
11. SD16×Dic29C2semidirect
12. #14271226C2semidirect
13. #14271156C2semidirect
14. C2×#11745180C2semidirect
15. C29×#450C2semidirect
23. #11745308C22semidirect
24. Q8×Dic29C22semidirect
25. C2×#90785C22semidirect
35. C2×C8×D29C22semidirect
37. #11745254C22semidirect
39. #11745252C22semidirect
45. #11745183C22semidirect
50. #11745168C22semidirect
51. #11745165C22semidirect
52. #11745165C22semidirect
53. #11745164C22semidirect
54. #11745164C22semidirect
55. #11745162C22semidirect
56. C8×Dic29C22semidirect
57. #11745158C22semidirect
58. #11745155C22semidirect
189. #450D29semidirect
244. C29#3296semidirect

Maximal subgroups

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

subgroupindex
#142718552
#142718072
#142715162
#142714942
#142714862
#142714652
#142714022
#142713892
#142713832
#142713222
SD16×Dic292
#142712262
#142711562
C2×#117451802
C29×#4502
#329629