Finite Group Databasebeta

← #46848402

Subgroups — #46848402

Order 34688.

402 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#31774617C211yesmaximal, characteristic, complemented
2#31774543C211yesmaximal, characteristic
3#31774257C211yesmaximal, characteristic, complemented
4#31774239C211yesmaximal, characteristic, complemented
5#31774222C211yesmaximal, characteristic, complemented
6#31774188C211yesmaximal, characteristic, complemented
7#31774155C211yesmaximal, characteristic, complemented
8#31774126C211yesmaximal, characteristic, complemented
9#31774120C211yesmaximal, characteristic, complemented
10#31774061C211yesmaximal, characteristic, complemented
11#31774050C211yesmaximal, characteristic, complemented
12#31773999C211yesmaximal, characteristic, complemented
13#31773903C211yesmaximal, characteristic, complemented
14#31773894C211yesmaximal, characteristic, complemented
15#435×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C22×D1084C2211yescharacteristic
18C2×Dic1084C2211yescharacteristic, complemented
19C2×#16864643C2211yescharacteristic
20C2×#16864643C2211yescharacteristic
21#22576160C2211yescharacteristic, complemented
22#22576153—21no—
23#22576153—21no—
24#22576148C2211yescharacteristic
25#22576147—21no—
26#22576145—21no—
27#22576139C2211yescharacteristic
28#22576139C2211yescharacteristic
29#22576139C2211yescharacteristic, complemented
30#22576134C2211yescharacteristic
31#22576133—21no—
32#22576128—21no—
33C4×Dic542C2211yescharacteristic, complemented
34#22576125—21no—
35C4×D1084C2211yescharacteristic
36#22576119C2211yescharacteristic
37#22576119C2211yescharacteristic, complemented
38C22⋊C4×D271—21no—
39#22576094C2211yescharacteristic, complemented
40#22576092C2211yescharacteristic, complemented
41#22576092C2211yescharacteristic, complemented
42#22576085C2211yescharacteristic, complemented
43#22576085C2211yescharacteristic, complemented
44#22576085C2211yescharacteristic, complemented
45#22576085C2211yescharacteristic, complemented
46#22576082C2211yescharacteristic
47#22576082C2211yescharacteristic
48#22576082C2211yescharacteristic
49#22576082C2211yescharacteristic
50#22576082C2211yescharacteristic
51#22576082C2211yescharacteristic
52C2×M16×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic, complemented
54C4×D4×C271C2211yescharacteristic
55C4⋊C8×C271C2211yescharacteristic, complemented
56C22⋊C8×C271C2211yescharacteristic
57C22⋊C8×C271C2211yescharacteristic, complemented
58#97×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60#16864654—22no—
61#16864654—41no—
62C2×Dic542—21no—
63C2×Dic542C2311yescharacteristic
64C2×Dic542C2311yescharacteristic
65C2×D1084—22no—
66C2×D1084—21no—
67C2×D1084—41no—
68C2×D1084C2311yescharacteristic
69C2×D1084C2311yescharacteristic
70C2×C22⋊D271—21no—
71C2×C22⋊D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271—21no—
74Dic1084—41no—
75#16864643—22no—
76#16864643—41no—
77C4⋊Dic271—21no—
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80C4⋊Dic271C2311yescharacteristic
81C4⋊Dic271C2311yescharacteristic
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88C22⋊Dic271—21no—
89C22⋊Dic271—21no—
90C22⋊Dic271—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95C4×Dic271—21no—
96C4×Dic271—21no—
97C4×Dic271—21no—
98C2×D4×C271D411yescharacteristic
99C22×C4×C271C2311yescharacteristic
100C22×C4×C271C2311yescharacteristic
101M16×C271—41no—
102C2×C8×C271—41no—
103C2×C8×C271C2311yescharacteristic
104C2×C8×C271C2311yescharacteristic
105C2×C8×C271C2311yescharacteristic
106C2×C8×C271C2311yescharacteristic
107C4⋊C4×C271D411yescharacteristic
108C22⋊C4×C271D411yescharacteristic
109C22⋊C4×C271D411yescharacteristic
110C42×C271C2311yescharacteristic
111C22×D271—21no—
112C22×D271—21no—
113C22×D271—42no—
114C22×D271—42no—
115C22×D271—81no—
116Dic542—21no—
117Dic542—21no—
118Dic542—21no—
119Dic542—21no—
120Dic542—42no—
121D1084—21no—
122D1084—21no—
123D1084—21no—
124D1084—42no—
125D1084#4112yes—
126C22⋊D271—42no—
127C22⋊D271—81no—
128C2×Dic271—21no—
129C2×Dic271—21no—
130C2×Dic271—21no—
131C2×Dic271—21no—
132C2×Dic271—21no—
133C2×Dic271—21no—
134C4×D271—42no—
135C4×D271—81no—
136C23×C271C2×D411yescharacteristic
137C23×C271C2×D411yescharacteristic
138D4×C271—41no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271C2×D411yescharacteristic
144C2×C4×C271C2×D411yescharacteristic
145C2×C4×C271C2×D411yescharacteristic
146C2×C4×C271C2×D411yescharacteristic
147C2×C4×C271C2411yescharacteristic
148C8×C271—21no—
149C8×C271—21no—
150C8×C271—21no—
151C8×C271—21no—
152C22×C271—21no—
153C22×C271—21no—
154C22×C271—21no—
155C22×C271—21no—
156C22×C271—21no—
157C22×C271—21no—
158C22×C271C22×D411yescharacteristic
159C2×D271—22no—
160C2×D271—41no—
161C2×D271—41no—
162C2×D271—41no—
163C2×D271—41no—
164C2×D271—41no—
165C2×D271—84no—
166C4×C271—21no—
167C4×C271—21no—
168C4×C271—21no—
169C4×C271—21no—
170C4×C271C2×#4111yescharacteristic
171C4×C271H(5,𝔽2)11yescharacteristic
172Dic271—41no—
173Dic271—41no—
174Dic271—41no—
175Dic271—41no—
176Dic271—41no—
177Dic271—41no—
178C2×C271—41no—
179C2×C271—41no—
180C2×C271#54611yescharacteristic
181C2×C271#57511yescharacteristic
182C2×C271#57711yescharacteristic
183D271—42no—
184D271—81no—
185C271#331111yescharacteristic, complemented, Sylow
186#3311—2711nomaximal, Sylow
187#556—2711no—
188#546—2711no—
189#489—2711no—
190#485—2711no—
191#482—2711no—
192#476—2711no—
193#470—2711no—
194#465—2711no—
195#464—2711no—
196#452—2711no—
197#450—2711no—
198#440—2711no—
199#435D27111yescharacteristic, complemented
200#418—2711no—
201#416—2711no—
202C2×#41—2711no—
203C22×D4—2711no—
204C2×Q16—2711no—
205C2×SD16—2711no—
206C2×SD16—2711no—
207C2×M16C2×D27111yescharacteristic
208C22×C8C2×D27111yescharacteristic, complemented
209C4⋊Q8—2711no—
210#126—5421no—
211#126—5421no—
212#124—2711no—
213#124—5421no—
214#123—5421no—
215C22⋊Q8—2711no—
216C22⋊Q8—2711no—
217C22⋊Q8—2711no—
218#121—2711no—
219#121—5421no—
220#120—5421no—
221C4×Q8—2711no—
222C4×Q8—5421no—
223C4×D4—2711no—
224C4×D4C2×D27111yescharacteristic
225#117—2711no—
226#117—2711no—
227C2×C22⋊C4—5421no—
228#107—2711no—
229#106—2711no—
230#106—2711no—
231C4⋊C8C2×D27111yescharacteristic, complemented
232#103—2711no—
233#103—2711no—
234#103—2711no—
235#103—2711no—
236#102—2711no—
237#102—2711no—
238#102—2711no—
239#102—2711no—
240#102—2711no—
241#102—2711no—
242C22⋊C8C2×D27111yescharacteristic
243C22⋊C8C2×D27111yescharacteristic, complemented
244#97C2×D27111yescharacteristic, complemented
245C24—5421no—
246#41—5422no—
247#41—10841no—
248C2×Q8—2711no—
249C2×Q8—2711no—
250C2×Q8—5421no—
251C2×D4—2711no—
252C2×D4—2711no—
253C2×D4—5422no—
254C2×D4—5421no—
255C2×D4—5421no—
256C2×D4—5421no—
257C2×D4—10841no—
258C2×D4D108411yescharacteristic
259C22×C4—5421no—
260C22×C4—5421no—
261C22×C4C22×D27111yescharacteristic
262C22×C4C22×D27111yescharacteristic
263Q16—10841no—
264SD16—5422no—
265SD16—10841no—
266M16—41no—
267C2×C8—41no—
268C2×C8C22×D27111yescharacteristic
269C2×C8C22×D27111yescharacteristic
270C2×C8C22×D27111yescharacteristic
271C2×C8C22×D27111yescharacteristic
272C4⋊C4—2711no—
273C4⋊C4—2711no—
274C4⋊C4—2711no—
275C4⋊C4—2711no—
276C4⋊C4—5421no—
277C4⋊C4—5421no—
278C4⋊C4—5421no—
279C4⋊C4—5421no—
280C4⋊C4—5421no—
281C4⋊C4D108411yescharacteristic
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4—5421no—
285C22⋊C4—5421no—
286C22⋊C4—5421no—
287C22⋊C4—5421no—
288C22⋊C4—5421no—
289C22⋊C4—5421no—
290C22⋊C4—5421no—
291C22⋊C4D108411yescharacteristic
292C22⋊C4D108411yescharacteristic
293C42—5421no—
294C42—5421no—
295C42—5421no—
296C42C22×D27111yescharacteristic
297C23—5421no—
298C23—5421no—
299C23—10842no—
300C23—10842no—
301C23—21681no—
302C23C2×D108411yescharacteristic
303C23C2×D108411yescharacteristic
304Q8—5421no—
305Q8—5421no—
306Q8—5421no—
307Q8—5421no—
308Q8—10842no—
309D4—41no—
310D4—2712no—
311D4—5421no—
312D4—5421no—
313D4—5421no—
314D4—10842no—
315D4—10842no—
316D4—21681no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—5421no—
322C2×C4—5421no—
323C2×C4—5421no—
324C2×C4—5421no—
325C2×C4—5421no—
326C2×C4—5421no—
327C2×C4—10842no—
328C2×C4—21681no—
329C2×C4C2×D108411yescharacteristic
330C2×C4C2×D108411yescharacteristic
331C2×C4C2×D108411yescharacteristic
332C2×C4C2×D108411yescharacteristic
333C2×C4C23×D27111yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—5422no—
345C22—10841no—
346C22—10841no—
347C22—10841no—
348C22—10841no—
349C22—10841no—
350C22—21684no—
351C22C22×D108411yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—10841no—
357C4—10841no—
358C4—10841no—
359C4—10841no—
360C4—10841no—
361C4—10841no—
362C4#41×D27111yescharacteristic
363C4#2257620511yescharacteristic
364C2—41no—
365C2—41no—
366C2—10842no—
367C2—21681no—
368C2#3177454211yescharacteristic
369C2#3177469711yescharacteristic
370C2#3177470911yescharacteristic
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#435×C271C2
socleC22×C271C22×D4
prime radical#31774543C2
hypercenter#435D271
hypocenterC271#3311
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #31774617C2semidirect
3. #31774257C2semidirect
4. #31774239C2semidirect
5. #31774222C2semidirect
6. #31774188C2semidirect
7. #31774155C2semidirect
8. #31774126C2semidirect
9. #31774120C2semidirect
10. #31774061C2semidirect
11. #31774050C2semidirect
12. #31773999C2semidirect
13. #31773903C2semidirect
14. #31773894C2semidirect
15. #435×C271C2semidirect
18. C2×Dic1084C22semidirect
21. #22576160C22semidirect
29. #22576139C22semidirect
33. C4×Dic542C22semidirect
37. #22576119C22semidirect
39. #22576094C22semidirect
40. #22576092C22semidirect
41. #22576092C22semidirect
42. #22576085C22semidirect
43. #22576085C22semidirect
44. #22576085C22semidirect
45. #22576085C22semidirect
53. C22×C8×C271C22semidirect
55. C4⋊C8×C271C22semidirect
57. C22⋊C8×C271C22semidirect
58. #97×C271C22semidirect
185. C271#3311semidirect
199. #435D271semidirect
208. C22×C8C2×D271semidirect
231. C4⋊C8C2×D271semidirect
243. C22⋊C8C2×D271semidirect
244. #97C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317746172
#317745432
#317742572
#317742392
#317742222
#317741882
#317741552
#317741262
#317741202
#317740612
#317740502
#317739992
#317739032
#317738942
#435×C2712
#3311271