Finite Group Databasebeta

← #46848333

Subgroups — #46848333

Order 34688.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774599C211yesmaximal, characteristic, complemented
2#31774546C211yesmaximal, characteristic, complemented
3#31774264C211yesmaximal, characteristic, complemented
4#31774236C211yesmaximal, characteristic, complemented
5#31774228C211yesmaximal, characteristic, complemented
6#31774199C211yesmaximal, characteristic, complemented
7#31774141C211yesmaximal, characteristic, complemented
8#31774123C211yesmaximal, characteristic, complemented
9#31774100C211yesmaximal, characteristic, complemented
10#31774039C211yesmaximal, characteristic, complemented
11#31774009C211yesmaximal, characteristic, complemented
12#31773971C211yesmaximal, characteristic, complemented
13#31773898C211yesmaximal, characteristic, complemented
14#31773890C211yesmaximal, characteristic, complemented
15#449×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×#16864642C2211yescharacteristic
18C2×#16864638C2211yescharacteristic
19C2×C8×D271C2211yescharacteristic
20#22576158C2211yescharacteristic
21#22576153—21no—
22#22576151C2211yescharacteristic
23#22576144—21no—
24#22576143—21no—
25#22576141—21no—
26#22576139C2211yescharacteristic
27#22576136C2211yescharacteristic
28#22576132—21no—
29#22576132—21no—
30#22576131C2211yescharacteristic
31D4×Dic271—21no—
32D4×Dic271C2211yescharacteristic
33D4×Dic271C2211yescharacteristic
34C4×C22⋊D271C2211yescharacteristic
35#22576118C2211yescharacteristic
36C2×C4⋊Dic271C2211yescharacteristic
37C2×Dic271⋊C4—21no—
38#22576093C2211yescharacteristic
39#22576092C2211yescharacteristic
40#22576092C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42#22576085C2211yescharacteristic
43#22576085C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576084C2211yescharacteristic
46#22576083C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic
49#22576081C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51#22576070C2211yescharacteristic
52C2×#41×C271C2211yescharacteristic
53C2×SD16×C271C2211yescharacteristic
54C2×D8×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59#16864653—22no—
60#16864653—41no—
61C2×Dic542C2311yescharacteristic
62C2×C22⋊D271—21no—
63C2×C22⋊D271—21no—
64C2×C22⋊D271D411yescharacteristic
65C22×Dic271—21no—
66C22×Dic271—21no—
67C22×Dic271—21no—
68C22×Dic271—21no—
69C2×C4×D271C2311yescharacteristic
70#16864642—41no—
71#16864638—41no—
72C2×C271⋊C8C2311yescharacteristic
73C2×C271⋊C8C2311yescharacteristic
74C8×D271—41no—
75C4⋊Dic271—21no—
76C4⋊Dic271—21no—
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4D411yescharacteristic
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88C22⋊Dic271—21no—
89C22⋊Dic271—21no—
90C22⋊Dic271D411yescharacteristic
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4D411yescharacteristic
94C4×Dic271—21no—
95C4×Dic271C2311yescharacteristic
96#41×C271—22no—
97#41×C271—41no—
98C2×Q8×C271C2311yescharacteristic
99C2×D4×C271—21no—
100C2×D4×C271—21no—
101C2×D4×C271C2311yescharacteristic
102C2×D4×C271C2311yescharacteristic
103C22×C4×C271—21no—
104C22×C4×C271C2311yescharacteristic
105SD16×C271—41no—
106D8×C271—22no—
107C2×C8×C271C2311yescharacteristic
108C2×C8×C271C2311yescharacteristic
109C4⋊C4×C271C2311yescharacteristic
110C22⋊C4×C271—21no—
111C22×D271C2×D411yescharacteristic
112Dic542—21no—
113Dic542—21no—
114C22⋊D271—41no—
115C22⋊D271—42no—
116C22⋊D271—81no—
117C2×Dic271—21no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—42no—
122C2×Dic271—42no—
123C2×Dic271—81no—
124C2×Dic271—81no—
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C2×Dic271C2×D411yescharacteristic
128C4×D271—21no—
129C4×D271—21no—
130C271⋊C8—21no—
131C271⋊C8—21no—
132C23×C271—21no—
133C23×C271—21no—
134C23×C271C2×D411yescharacteristic
135Q8×C271—21no—
136Q8×C271—21no—
137D4×C271—21no—
138D4×C271—21no—
139D4×C271—21no—
140D4×C271—42no—
141D4×C271—42no—
142D4×C271#4112yes—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—42no—
148C2×C4×C271C2×D411yescharacteristic
149C2×C4×C271C2411yescharacteristic
150C8×C271—21no—
151C8×C271—21no—
152C22×C271—21no—
153C22×C271—22no—
154C22×C271—21no—
155C22×C271—21no—
156C22×C271—41no—
157C22×C271—41no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271—41no—
161C22×C271C22×D411yescharacteristic
162C2×D271—21no—
163C2×D271—21no—
164C2×D271—21no—
165C4×C271—21no—
166C4×C271—41no—
167C4×C271—41no—
168C4×C271C2×#4111yescharacteristic
169C4×C271H(5,𝔽2)11yescharacteristic
170Dic271—21no—
171Dic271—21no—
172Dic271—21no—
173Dic271—41no—
174Dic271—41no—
175Dic271—41no—
176Dic271—41no—
177C2×C271—41no—
178C2×C271—42no—
179C2×C271—81no—
180C2×C271#54611yescharacteristic
181C2×C271#57511yescharacteristic
182C2×C271#57711yescharacteristic
183D271—41no—
184C271#330711yescharacteristic, complemented, Sylow
185#3307—2711nomaximal, Sylow
186#553—2711no—
187#547—2711no—
188#491—2711no—
189#485—2711no—
190#483—2711no—
191#478—2711no—
192#468—2711no—
193#465—2711no—
194#461—2711no—
195#449—2711no—
196#449D27111yescharacteristic, complemented
197#442—2711no—
198#435—2711no—
199#417—2711no—
200#416—2711no—
201C2×#41—2711no—
202C2×#41C2×D27111yescharacteristic
203C2×SD16—2711no—
204C2×SD16C2×D27111yescharacteristic
205C2×D8C2×D27111yescharacteristic
206C2×M16—2711no—
207C22×C8—2711no—
208C4⋊Q8—2711no—
209#126—5421no—
210#125—2711no—
211#123—5421no—
212#123—5421no—
213#123—5421no—
214C22⋊Q8—2711no—
215C22⋊Q8—2711no—
216#121—2711no—
217#121—5421no—
218#121—5421no—
219#121C2×D27111yescharacteristic
220C4×D4—2711no—
221C4×D4—2711no—
222C4×D4—2711no—
223C4×D4—5421no—
224#117—2711no—
225C2×C4⋊C4—2711no—
226C2×C4⋊C4—5421no—
227#107—2711no—
228#106—2711no—
229#106—2711no—
230C4⋊C8—2711no—
231#103—2711no—
232#103—2711no—
233#103—2711no—
234#103—2711no—
235#103—2711no—
236#103C2×D27111yescharacteristic
237#102—2711no—
238#102—2711no—
239#102—2711no—
240#102C2×D27111yescharacteristic
241C22⋊C8—2711no—
242C22⋊C8C2×D27111yescharacteristic
243#97—2711no—
244#41—22no—
245#41—41no—
246#41—5422no—
247#41—10841no—
248C2×Q8—2711no—
249C2×Q8C22×D27111yescharacteristic
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—2711no—
253C2×D4—5421no—
254C2×D4—5421no—
255C2×D4C22×D27111yescharacteristic
256C2×D4C22×D27111yescharacteristic
257C22×C4—21no—
258C22×C4—2711no—
259C22×C4—5421no—
260C22×C4—5421no—
261C22×C4—5421no—
262C22×C4—5421no—
263C22×C4C22×D27111yescharacteristic
264SD16—41no—
265SD16—10841no—
266D8—22no—
267M16—10841no—
268C2×C8—2711no—
269C2×C8—2711no—
270C2×C8—10841no—
271C2×C8C22×D27111yescharacteristic
272C2×C8C22×D27111yescharacteristic
273C4⋊C4—2711no—
274C4⋊C4—2711no—
275C4⋊C4—2711no—
276C4⋊C4—2711no—
277C4⋊C4—5421no—
278C4⋊C4—5421no—
279C4⋊C4—5421no—
280C4⋊C4—5421no—
281C4⋊C4—5421no—
282C4⋊C4—5421no—
283C4⋊C4—5421no—
284C4⋊C4C22×D27111yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—2711no—
287C22⋊C4—2711no—
288C22⋊C4—5421no—
289C22⋊C4—5421no—
290C22⋊C4—5421no—
291C22⋊C4—5421no—
292C22⋊C4—5421no—
293C22⋊C4—5421no—
294C42—2711no—
295C42—5421no—
296C23—21no—
297C23—21no—
298C23—2711no—
299C23D4×D27111yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—5421no—
303Q8—5421no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—42no—
309D4—10841no—
310D4—10842no—
311D4—21681no—
312D4#1686465312yes—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—2711no—
319C2×C4—2711no—
320C2×C4—2711no—
321C2×C4—5421no—
322C2×C4—5421no—
323C2×C4—5421no—
324C2×C4—5421no—
325C2×C4—5421no—
326C2×C4—5421no—
327C2×C4—10842no—
328C2×C4—10842no—
329C2×C4—21681no—
330C2×C4—21681no—
331C2×C4D4×D27111yescharacteristic
332C2×C4C23×D27111yescharacteristic
333C8—21no—
334C8—21no—
335C8—5421no—
336C8—5421no—
337C22—21no—
338C22—22no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—5421no—
347C22—5421no—
348C22—5421no—
349C22C2×D4×D27111yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—5421no—
354C4—5421no—
355C4—5421no—
356C4—10841no—
357C4—10841no—
358C4—10841no—
359C4—10841no—
360C4C2×#1686465311yescharacteristic
361C4#2257620511yescharacteristic
362C2—41no—
363C2—42no—
364C2—81no—
365C2—10841no—
366C2#3177453411yescharacteristic
367C2#3177469311yescharacteristic
368C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#449×C271C2
socleC22×C271C22×D4
hypercenter#449D271
hypocenterC271#3307
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #31774599C2semidirect
2. #31774546C2semidirect
3. #31774264C2semidirect
4. #31774236C2semidirect
5. #31774228C2semidirect
6. #31774199C2semidirect
7. #31774141C2semidirect
8. #31774123C2semidirect
9. #31774100C2semidirect
10. #31774039C2semidirect
11. #31774009C2semidirect
12. #31773971C2semidirect
13. #31773898C2semidirect
14. #31773890C2semidirect
15. #449×C271C2semidirect
184. C271#3307semidirect
196. #449D271semidirect

Maximal subgroups

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

subgroupindex
#317745992
#317745462
#317742642
#317742362
#317742282
#317741992
#317741412
#317741232
#317741002
#317740392
#317740092
#317739712
#317738982
#317738902
#449×C2712
#3307271