Finite Group Databasebeta

← #46846641

Subgroups — #46846641

Order 34688.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774518C211yesmaximal, characteristic, complemented
2#31774227C211yesmaximal, characteristic, complemented
3#31774215C211yesmaximal, characteristic, complemented
4#31774187C211yesmaximal, characteristic, complemented
5#31774182C211yesmaximal, characteristic, complemented
6#31774119C211yesmaximal, characteristic, complemented
7#31774112C211yesmaximal, characteristic, complemented
8#31774059C211yesmaximal, characteristic, complemented
9#31774042C211yesmaximal, characteristic
10#31774007C211yesmaximal, characteristic, complemented
11#31774002C211yesmaximal, characteristic, complemented
12C4×#16864641C211yesmaximal, characteristic, complemented
13C4×#16864642C211yesmaximal, characteristic, complemented
14#31773958C211yesmaximal, characteristic, complemented
15#520×C271C211yesmaximal, characteristic, complemented
16C2×#16864644C2211yescharacteristic
17C2×#16864642C2211yescharacteristic
18C2×#16864641C2211yescharacteristic
19C2×#16864639C2211yescharacteristic
20#22576160C2211yescharacteristic
21#22576155—21no—
22#22576148C2211yescharacteristic
23#22576146—21no—
24#22576142—21no—
25#22576139C2211yescharacteristic
26#22576138—21no—
27#22576134C2211yescharacteristic
28C4×Dic542C2211yescharacteristic
29C4×D1084C2211yescharacteristic
30#22576091C2211yescharacteristic
31#22576091C2211yescharacteristic
32#22576091C2211yescharacteristic
33#22576091C2211yescharacteristic
34C4⋊(C271⋊C8)C2211yescharacteristic
35C4⋊(C271⋊C8)C2211yescharacteristic
36#22576084C2211yescharacteristic
37#22576084C2211yescharacteristic
38#22576083C2211yescharacteristic
39#22576083C2211yescharacteristic
40#22576081C2211yescharacteristic
41#22576081C2211yescharacteristic
42#22576080C2211yescharacteristic
43#22576080C2211yescharacteristic
44C22⋊(C271⋊C8)C2211yescharacteristic
45C22⋊(C271⋊C8)C2211yescharacteristic
46#22576071C2211yescharacteristic
47C4×C271⋊C8C2211yescharacteristic
48C2×#41×C271C2211yescharacteristic
49C4×Q8×C271C2211yescharacteristic
50C4×Q8×C271C2211yescharacteristic
51C4×D4×C271—21no—
52C4×D4×C271—21no—
53C4×D4×C271C2211yescharacteristic
54C4×D4×C271C2211yescharacteristic
55#117×C271—21no—
56#117×C271C2211yescharacteristic
57C2×C4⋊C4×C271—21no—
58C2×C4⋊C4×C271C2211yescharacteristic
59C2×Dic542—21no—
60C2×Dic542C2311yescharacteristic
61C2×D1084C2311yescharacteristic
62C2×C22⋊D271—21no—
63C2×C4×D271—21no—
64#16864644—22no—
65#16864642—22no—
66#16864641—22no—
67#16864639—22no—
68C2×C271⋊C8C2311yescharacteristic
69C2×C271⋊C8C2311yescharacteristic
70C2×C271⋊C8C2311yescharacteristic
71C2×C271⋊C8C2311yescharacteristic
72C4⋊Dic271—21no—
73C4⋊Dic271C2311yescharacteristic
74C4⋊Dic271C2311yescharacteristic
75Dic271⋊C4—21no—
76Dic271⋊C4—21no—
77C22⋊Dic271—21no—
78(C2×D271)⋊C4—21no—
79(C2×D271)⋊C4—21no—
80(C2×D271)⋊C4—21no—
81(C2×D271)⋊C4—21no—
82C4×Dic271—21no—
83#41×C271—22no—
84#41×C271—22no—
85C2×Q8×C271C2311yescharacteristic
86C2×D4×C271—21no—
87C2×D4×C271C2311yescharacteristic
88C22×C4×C271—21no—
89C22×C4×C271—21no—
90C22×C4×C271—21no—
91C22×C4×C271—21no—
92C22×C4×C271C2311yescharacteristic
93C4⋊C4×C271—21no—
94C4⋊C4×C271—21no—
95C4⋊C4×C271—21no—
96C4⋊C4×C271D411yescharacteristic
97C4⋊C4×C271D411yescharacteristic
98C4⋊C4×C271C2311yescharacteristic
99C4⋊C4×C271C2311yescharacteristic
100C22⋊C4×C271—21no—
101C22⋊C4×C271—21no—
102C22⋊C4×C271D411yescharacteristic
103C22⋊C4×C271D411yescharacteristic
104C42×C271—21no—
105C42×C271—21no—
106C42×C271C2311yescharacteristic
107C42×C271C2311yescharacteristic
108C22×D271—21no—
109Dic542—21no—
110Dic542—21no—
111Dic542—81no—
112D1084—21no—
113D1084—21no—
114C22⋊D271—81no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C2×Dic271—21no—
118C4×D271—81no—
119C271⋊C8—21no—
120C271⋊C8—21no—
121C271⋊C8—21no—
122C271⋊C8—21no—
123C23×C271—21no—
124C23×C271C2×D411yescharacteristic
125Q8×C271—21no—
126Q8×C271#4112yes—
127D4×C271—24no—
128D4×C271—21no—
129D4×C271#4112yes—
130C2×C4×C271—24no—
131C2×C4×C271—21no—
132C2×C4×C271—21no—
133C2×C4×C271—21no—
134C2×C4×C271—21no—
135C2×C4×C271—21no—
136C2×C4×C271—81no—
137C2×C4×C271—81no—
138C2×C4×C271—81no—
139C2×C4×C271C2×D411yescharacteristic
140C2×C4×C271C2×D411yescharacteristic
141C2×C4×C271C2×D411yescharacteristic
142C2×C4×C271C2×D411yescharacteristic
143C2×C4×C271C2×D411yescharacteristic
144C2×C4×C271C2411yescharacteristic
145C22×C271—21no—
146C22×C271—22no—
147C22×C271—21no—
148C22×C271—21no—
149C22×C271—41no—
150C22×C271—41no—
151C22×C271C22×D411yescharacteristic
152C2×D271—41no—
153C2×D271—41no—
154C2×D271—41no—
155C4×C271—21no—
156C4×C271—22no—
157C4×C271—21no—
158C4×C271—21no—
159C4×C271—21no—
160C4×C271—21no—
161C4×C271—41no—
162C4×C271—41no—
163C4×C271C2×#4111yescharacteristic
164C4×C271C2×#4111yescharacteristic
165Dic271—41no—
166Dic271—41no—
167Dic271—41no—
168C2×C271—41no—
169C2×C271—42no—
170C2×C271#52511yescharacteristic
171C2×C271#57711yescharacteristic
172C2×C271#57711yescharacteristic
173D271—81no—
174C271#312511yescharacteristic, complemented, Sylow
175#3125—2711nomaximal, Sylow
176#542—2711no—
177#520D27111yescharacteristic, complemented
178#483—2711no—
179#481—2711no—
180#476—2711no—
181#475—2711no—
182#464—2711no—
183#463—2711no—
184#452—2711no—
185#449—2711no—
186#442—2711no—
187#441—2711no—
188C4×SD16—2711no—
189C4×SD16—2711no—
190#433—2711no—
191C2×#41C2×D27111yescharacteristic
192C2×Q16—2711no—
193C2×SD16—2711no—
194C2×SD16—2711no—
195C2×D8—2711no—
196C4⋊Q8—2711no—
197#126—5421no—
198#124—2711no—
199#124—5421no—
200#123—5421no—
201C22⋊Q8—2711no—
202C22⋊Q8—5421no—
203#121—2711no—
204C4×Q8—2711no—
205C4×Q8C2×D27111yescharacteristic
206C4×Q8C2×D27111yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—2711no—
210C4×D4C2×D27111yescharacteristic
211C4×D4C2×D27111yescharacteristic
212#117—21no—
213#117C2×D27111yescharacteristic
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×D27111yescharacteristic
216#106—2711no—
217#106—2711no—
218#106—2711no—
219#106—2711no—
220C4⋊C8—2711no—
221C4⋊C8—2711no—
222#103—2711no—
223#103—2711no—
224#103—2711no—
225#103—2711no—
226#102—2711no—
227#102—2711no—
228#102—2711no—
229#102—2711no—
230C22⋊C8—2711no—
231C22⋊C8—2711no—
232#97—2711no—
233C4×C8—2711no—
234#41—22no—
235#41—22no—
236C2×Q8—2711no—
237C2×Q8—5421no—
238C2×Q8C22×D27111yescharacteristic
239C2×D4—21no—
240C2×D4—2711no—
241C2×D4—5421no—
242C2×D4C22×D27111yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—5421no—
248C22×C4C22×D27111yescharacteristic
249Q16—5422no—
250SD16—5422no—
251SD16—5422no—
252D8—5422no—
253C2×C8—2711no—
254C2×C8—2711no—
255C2×C8—2711no—
256C2×C8—2711no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—2711no—
261C4⋊C4—2711no—
262C4⋊C4—5421no—
263C4⋊C4—5421no—
264C4⋊C4—5421no—
265C4⋊C4C22⋊D27111yescharacteristic
266C4⋊C4C22⋊D27111yescharacteristic
267C4⋊C4C22×D27111yescharacteristic
268C4⋊C4C22×D27111yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—5421no—
272C22⋊C4—5421no—
273C22⋊C4—5421no—
274C22⋊C4—5421no—
275C22⋊C4—5421no—
276C22⋊C4C22⋊D27111yescharacteristic
277C22⋊C4C22⋊D27111yescharacteristic
278C42—21no—
279C42—21no—
280C42—5421no—
281C42C22×D27111yescharacteristic
282C42C22×D27111yescharacteristic
283C23—21no—
284C23—5421no—
285C23C2×C22⋊D27111yescharacteristic
286Q8—21no—
287Q8—5421no—
288Q8—5421no—
289Q8—21681no—
290Q8#1686465412yes—
291D4—24no—
292D4—21no—
293D4—5421no—
294D4—5421no—
295D4—21681no—
296D4#1686465412yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—5421no—
307C2×C4—5421no—
308C2×C4—5421no—
309C2×C4—21681no—
310C2×C4C2×C22⋊D27111yescharacteristic
311C2×C4C2×C22⋊D27111yescharacteristic
312C2×C4C2×C22⋊D27111yescharacteristic
313C2×C4C2×C22⋊D27111yescharacteristic
314C2×C4C2×C22⋊D27111yescharacteristic
315C2×C4C23×D27111yescharacteristic
316C8—5421no—
317C8—5421no—
318C8—5421no—
319C8—5421no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—10841no—
327C22—10841no—
328C22—10841no—
329C22C22×C22⋊D27111yescharacteristic
330C4—21no—
331C4—22no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—10841no—
339C4—10841no—
340C4—10841no—
341C4C2×#1686465411yescharacteristic
342C4C2×#1686465411yescharacteristic
343C2—41no—
344C2—42no—
345C2—21681no—
346C2#3177440811yescharacteristic
347C2#3177470511yescharacteristic
348C2#3177470811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#520×C271C2
socleC22×C271C22×D4
prime radical#31774042C2
hypercenter#520D271
hypocenterC271#3125
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #31774518C2semidirect
2. #31774227C2semidirect
3. #31774215C2semidirect
4. #31774187C2semidirect
5. #31774182C2semidirect
6. #31774119C2semidirect
7. #31774112C2semidirect
8. #31774059C2semidirect
10. #31774007C2semidirect
11. #31774002C2semidirect
12. C4×#16864641C2semidirect
13. C4×#16864642C2semidirect
14. #31773958C2semidirect
15. #520×C271C2semidirect
174. C271#3125semidirect
177. #520D271semidirect

Maximal subgroups

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

subgroupindex
#317745182
#317742272
#317742152
#317741872
#317741822
#317741192
#317741122
#317740592
#317740422
#317740072
#317740022
C4×#168646412
C4×#168646422
#317739582
#520×C2712
#3125271