Finite Group Databasebeta

← #46848370

Subgroups — #46848370

Order 34688.

404 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774593C211yesmaximal, characteristic, complemented
2#31774261C211yesmaximal, characteristic, complemented
3#31774218C211yesmaximal, characteristic, complemented
4#31774208C211yesmaximal, characteristic, complemented
5#31774204C211yesmaximal, characteristic, complemented
6#31774148C211yesmaximal, characteristic, complemented
7#31774131C211yesmaximal, characteristic, complemented
8#31774089C211yesmaximal, characteristic, complemented
9#31774042C211yesmaximal, characteristic, complemented
10#31774042C211yesmaximal, characteristic, complemented
11#31773995C211yesmaximal, characteristic, complemented
12#31773969C211yesmaximal, characteristic, complemented
13#31773898C211yesmaximal, characteristic, complemented
14#31773893C211yesmaximal, characteristic, complemented
15#550×C271C211yesmaximal, characteristic, complemented
16C2×#16864641C2211yescharacteristic
17C2×#16864639C2211yescharacteristic
18C2×#16864639C2211yescharacteristic
19C2×#16864638C2211yescharacteristic
20C22×C271⋊C8C2211yescharacteristic
21#22576155—21no—
22#22576152C2211yescharacteristic
23#22576142—21no—
24#22576134C2211yescharacteristic
25#22576134C2211yescharacteristic
26#22576132—21no—
27C4×D1084C2211yescharacteristic
28(C22⋊D271)⋊C4—21no—
29#22576119C2211yescharacteristic
30C2×C4⋊Dic271C2211yescharacteristic
31#22576093C2211yescharacteristic
32#22576093C2211yescharacteristic
33#22576091C2211yescharacteristic
34C4⋊(C271⋊C8)C2211yescharacteristic
35#22576084C2211yescharacteristic
36#22576084C2211yescharacteristic
37#22576084C2211yescharacteristic
38#22576081C2211yescharacteristic
39#22576081C2211yescharacteristic
40#22576081C2211yescharacteristic
41#22576081C2211yescharacteristic
42#22576081C2211yescharacteristic
43#22576081C2211yescharacteristic
44#22576080C2211yescharacteristic
45C22⋊(C271⋊C8)C2211yescharacteristic
46C22⋊(C271⋊C8)C2211yescharacteristic
47#22576071C2211yescharacteristic
48C2×#41×C271C2211yescharacteristic
49C2×#41×C271C2211yescharacteristic
50#127×C271—21no—
51#127×C271C2211yescharacteristic
52#121×C271—21no—
53#121×C271—21no—
54#121×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56C4×Q8×C271C2211yescharacteristic
57C4×D4×C271—21no—
58C4×D4×C271C2211yescharacteristic
59C2×D1084C2311yescharacteristic
60C2×C22⋊D271—21no—
61C2×C22⋊D271—21no—
62C22×Dic271—21no—
63C2×C4×D271—21no—
64#16864641—22no—
65#16864639—41no—
66#16864639—41no—
67#16864638—41no—
68C2×C271⋊C8—41no—
69C2×C271⋊C8C2311yescharacteristic
70C2×C271⋊C8C2311yescharacteristic
71C2×C271⋊C8C2311yescharacteristic
72C2×C271⋊C8C2311yescharacteristic
73C4⋊Dic271—21no—
74C4⋊Dic271C2311yescharacteristic
75C4⋊Dic271C2311yescharacteristic
76C4⋊Dic271C2311yescharacteristic
77Dic271⋊C4—21no—
78Dic271⋊C4—21no—
79C22⋊Dic271—21no—
80(C2×D271)⋊C4—21no—
81(C2×D271)⋊C4—21no—
82(C2×D271)⋊C4—21no—
83C4×Dic271—21no—
84#41×C271—22no—
85#41×C271—22no—
86#41×C271—41no—
87#41×C271—41no—
88C2×Q8×C271C2311yescharacteristic
89C2×D4×C271—21no—
90C2×D4×C271—21no—
91C2×D4×C271—21no—
92C2×D4×C271—21no—
93C2×D4×C271—21no—
94C2×D4×C271—21no—
95C2×D4×C271D411yescharacteristic
96C2×D4×C271C2311yescharacteristic
97C2×D4×C271C2311yescharacteristic
98C22×C4×C271—21no—
99C22×C4×C271—21no—
100C22×C4×C271C2311yescharacteristic
101C22×C4×C271C2311yescharacteristic
102C4⋊C4×C271—21no—
103C4⋊C4×C271D411yescharacteristic
104C4⋊C4×C271C2311yescharacteristic
105C22⋊C4×C271—21no—
106C22⋊C4×C271—21no—
107C22⋊C4×C271D411yescharacteristic
108C22⋊C4×C271D411yescharacteristic
109C42×C271—21no—
110C42×C271C2311yescharacteristic
111C22×D271—21no—
112D1084—21no—
113D1084—21no—
114C22⋊D271—42no—
115C22⋊D271—81no—
116C2×Dic271—21no—
117C2×Dic271—21no—
118C2×Dic271—21no—
119C2×Dic271—81no—
120C4×D271—81no—
121C271⋊C8—21no—
122C271⋊C8—21no—
123C271⋊C8—21no—
124C271⋊C8—21no—
125C23×C271—21no—
126C23×C271—21no—
127C23×C271C2×D411yescharacteristic
128C23×C271C2×D411yescharacteristic
129Q8×C271—21no—
130Q8×C271#4112yes—
131D4×C271—21no—
132D4×C271—21no—
133D4×C271—21no—
134D4×C271—21no—
135D4×C271—41no—
136D4×C271—42no—
137D4×C271—42no—
138D4×C271—42no—
139D4×C271—81no—
140D4×C271—81no—
141D4×C271—81no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—42no—
149C2×C4×C271—42no—
150C2×C4×C271C2×D411yescharacteristic
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2×D411yescharacteristic
153C2×C4×C271C2×D411yescharacteristic
154C2×C4×C271C2411yescharacteristic
155C22×C271—21no—
156C22×C271—21no—
157C22×C271—21no—
158C22×C271—21no—
159C22×C271—21no—
160C22×C271—21no—
161C22×C271—41no—
162C22×C271—41no—
163C22×C271—41no—
164C22×C271—41no—
165C22×C271—41no—
166C22×C271—41no—
167C22×C271C22×D411yescharacteristic
168C2×D271—41no—
169C2×D271—41no—
170C2×D271—41no—
171C4×C271—21no—
172C4×C271—21no—
173C4×C271—22no—
174C4×C271—21no—
175C4×C271—21no—
176C4×C271—41no—
177C4×C271C2×#4111yescharacteristic
178C4×C271H(5,𝔽2)11yescharacteristic
179Dic271—41no—
180Dic271—41no—
181Dic271—41no—
182C2×C271—41no—
183C2×C271—41no—
184C2×C271—81no—
185C2×C271—81no—
186C2×C271#54611yescharacteristic
187C2×C271#57511yescharacteristic
188C2×C271#57611yescharacteristic
189D271—81no—
190C271#330911yescharacteristic, complemented, Sylow
191#3309—2711nomaximal, Sylow
192#553—2711no—
193#550D27111yescharacteristic, complemented
194#490—2711no—
195#482—2711no—
196#480—2711no—
197#479—2711no—
198#469—2711no—
199#466—2711no—
200#459—2711no—
201#449—2711no—
202#449—2711no—
203#440—2711no—
204#435—2711no—
205#417—2711no—
206#416—2711no—
207C2×#41C2×D27111yescharacteristic
208C2×#41C2×D27111yescharacteristic
209C2×SD16—2711no—
210C2×D8—2711no—
211C2×D8—2711no—
212C2×M16—2711no—
213C22×C8—2711no—
214#127—21no—
215#127C2×D27111yescharacteristic
216#126—5421no—
217#125—2711no—
218#123—5421no—
219#121—21no—
220#121—21no—
221#121—2711no—
222#121—2711no—
223#121—5421no—
224#121C2×D27111yescharacteristic
225#121C2×D27111yescharacteristic
226C4×Q8C2×D27111yescharacteristic
227C4×D4—21no—
228C4×D4—2711no—
229C4×D4—5421no—
230C4×D4C2×D27111yescharacteristic
231#117—2711no—
232C2×C4⋊C4—2711no—
233#107—2711no—
234#107—2711no—
235#106—2711no—
236C4⋊C8—2711no—
237#103—2711no—
238#103—2711no—
239#103—2711no—
240#102—2711no—
241#102—2711no—
242#102—2711no—
243#102—2711no—
244#102—2711no—
245#102—2711no—
246#102—2711no—
247C22⋊C8—2711no—
248C22⋊C8—2711no—
249#97—2711no—
250#41—22no—
251#41—22no—
252#41—41no—
253#41—41no—
254C2×Q8C22×D27111yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—2711no—
262C2×D4—5421no—
263C2×D4—5421no—
264C2×D4C22⋊D27111yescharacteristic
265C2×D4C22×D27111yescharacteristic
266C2×D4C22×D27111yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—5421no—
270C22×C4—5421no—
271C22×C4C22×D27111yescharacteristic
272C22×C4C22×D27111yescharacteristic
273SD16—5422no—
274D8—10841no—
275D8—10841no—
276M16—10841no—
277C2×C8—2711no—
278C2×C8—2711no—
279C2×C8—2711no—
280C2×C8—2711no—
281C2×C8—10841no—
282C4⋊C4—21no—
283C4⋊C4—2711no—
284C4⋊C4—2711no—
285C4⋊C4—2711no—
286C4⋊C4—5421no—
287C4⋊C4—5421no—
288C4⋊C4—5421no—
289C4⋊C4C22⋊D27111yescharacteristic
290C4⋊C4C22×D27111yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—5421no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4—5421no—
297C22⋊C4C22⋊D27111yescharacteristic
298C22⋊C4C22⋊D27111yescharacteristic
299C42—21no—
300C42—5421no—
301C42C22×D27111yescharacteristic
302C23—21no—
303C23—21no—
304C23—5421no—
305C23C2×C22⋊D27111yescharacteristic
306C23C2×C22⋊D27111yescharacteristic
307Q8—21no—
308Q8#1686465312yes—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—42no—
317D4—81no—
318D4—81no—
319D4—81no—
320D4—5421no—
321D4—5421no—
322D4—10842no—
323D4—21681no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—5421no—
333C2×C4—5421no—
334C2×C4—5421no—
335C2×C4—21681no—
336C2×C4—21681no—
337C2×C4C2×C22⋊D27111yescharacteristic
338C2×C4C2×C22⋊D27111yescharacteristic
339C2×C4C2×C22⋊D27111yescharacteristic
340C2×C4C2×C22⋊D27111yescharacteristic
341C2×C4C23×D27111yescharacteristic
342C8—5421no—
343C8—5421no—
344C8—5421no—
345C8—5421no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—10841no—
359C22—10841no—
360C22—10841no—
361C22C22×C22⋊D27111yescharacteristic
362C4—21no—
363C4—21no—
364C4—22no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—10841no—
369C4—10841no—
370C4—10841no—
371C4C2×#1686465311yescharacteristic
372C4#2257620511yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—81no—
377C2—21681no—
378C2#3177454111yescharacteristic
379C2#3177469511yescharacteristic
380C2#3177470011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#550×C271C2
socleC22×C271C22×D4
hypercenter#550D271
hypocenterC271#3309
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #31774593C2semidirect
2. #31774261C2semidirect
3. #31774218C2semidirect
4. #31774208C2semidirect
5. #31774204C2semidirect
6. #31774148C2semidirect
7. #31774131C2semidirect
8. #31774089C2semidirect
9. #31774042C2semidirect
10. #31774042C2semidirect
11. #31773995C2semidirect
12. #31773969C2semidirect
13. #31773898C2semidirect
14. #31773893C2semidirect
15. #550×C271C2semidirect
190. C271#3309semidirect
193. #550D271semidirect

Maximal subgroups

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

subgroupindex
#317745932
#317742612
#317742182
#317742082
#317742042
#317741482
#317741312
#317740892
#317740422
#317739952
#317739692
#317738982
#317738932
#550×C2712
#3309271