Finite Group Databasebeta

← #44562335

Subgroups — #44562335

Order 32128.

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#30632342C211yesmaximal, characteristic, complemented
2#30632057C211yesmaximal, characteristic, complemented
3#30632028C211yesmaximal, characteristic, complemented
4#30632017C211yesmaximal, characteristic, complemented
5#30631984C211yesmaximal, characteristic, complemented
6#30631931C211yesmaximal, characteristic, complemented
7#30631914C211yesmaximal, characteristic, complemented
8#30631892C211yesmaximal, characteristic, complemented
9#30631830C211yesmaximal, characteristic, complemented
10#30631830C211yesmaximal, characteristic, complemented
11#30631797C211yesmaximal, characteristic, complemented
12#30631759C211yesmaximal, characteristic, complemented
13#30631686C211yesmaximal, characteristic, complemented
14#30631681C211yesmaximal, characteristic, complemented
15#553×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C2×#16860810C2211yescharacteristic
18C2×#16860798C2211yescharacteristic
19C2×#16860798C2211yescharacteristic
20C2×#16860795C2211yescharacteristic
21C2×#16860794C2211yescharacteristic
22C22×C251⋊C8C2211yescharacteristic
23#22548715C2211yescharacteristic
24#22548697—21no—
25#22548697—21no—
26#22548694C2211yescharacteristic
27#22548694C2211yescharacteristic
28#22548686—21no—
29C4×D1004C2211yescharacteristic
30C4⋊C4×D251—21no—
31#22548648C2211yescharacteristic
32#22548646C2211yescharacteristic
33#22548646C2211yescharacteristic
34C4⋊(C251⋊C8)C2211yescharacteristic
35#22548638C2211yescharacteristic
36#22548638C2211yescharacteristic
37#22548638C2211yescharacteristic
38#22548638C2211yescharacteristic
39#22548638C2211yescharacteristic
40#22548638C2211yescharacteristic
41#22548636C2211yescharacteristic
42#22548635C2211yescharacteristic
43#22548635C2211yescharacteristic
44#22548635C2211yescharacteristic
45C22⋊(C251⋊C8)C2211yescharacteristic
46C22⋊(C251⋊C8)C2211yescharacteristic
47#22548626C2211yescharacteristic
48#126×C251—21no—
49#125×C251C2211yescharacteristic
50#123×C251—21no—
51#121×C251—21no—
52#121×C251C2211yescharacteristic
53#121×C251C2211yescharacteristic
54C4×D4×C251—21no—
55C4×D4×C251C2211yescharacteristic
56C4×D4×C251C2211yescharacteristic
57#117×C251C2211yescharacteristic
58C2×C4⋊C4×C251C2211yescharacteristic
59#16860810—22no—
60#16860810—22no—
61#16860810—41no—
62#16860810—41no—
63C2×Dic502C2311yescharacteristic
64C2×Dic502C2311yescharacteristic
65C2×D1004C2311yescharacteristic
66C2×C22⋊D251—21no—
67C2×C22⋊D251—21no—
68C2×C4×D251—21no—
69C2×C4×D251—21no—
70C2×C4×D251—21no—
71#16860798—41no—
72#16860798—41no—
73#16860795—22no—
74#16860794—41no—
75C2×C251⋊C8—41no—
76C2×C251⋊C8C2311yescharacteristic
77C2×C251⋊C8C2311yescharacteristic
78C2×C251⋊C8C2311yescharacteristic
79C2×C251⋊C8C2311yescharacteristic
80C4⋊Dic251—21no—
81C4⋊Dic251—21no—
82C4⋊Dic251C2311yescharacteristic
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85C22⋊Dic251—21no—
86C22⋊Dic251—21no—
87(C2×D251)⋊C4—21no—
88C2×D4×C251—21no—
89C2×D4×C251—21no—
90C2×D4×C251D411yescharacteristic
91C2×D4×C251C2311yescharacteristic
92C22×C4×C251—21no—
93C22×C4×C251—21no—
94C22×C4×C251C2311yescharacteristic
95C22×C4×C251C2311yescharacteristic
96C4⋊C4×C251—21no—
97C4⋊C4×C251—21no—
98C4⋊C4×C251—21no—
99C4⋊C4×C251D411yescharacteristic
100C4⋊C4×C251C2311yescharacteristic
101C4⋊C4×C251C2311yescharacteristic
102C4⋊C4×C251C2311yescharacteristic
103C22⋊C4×C251—21no—
104C22⋊C4×C251—21no—
105C22⋊C4×C251—21no—
106C22⋊C4×C251—21no—
107C22⋊C4×C251D411yescharacteristic
108C22⋊C4×C251D411yescharacteristic
109C42×C251—21no—
110C42×C251C2311yescharacteristic
111C22×D251—21no—
112Dic502—21no—
113Dic502—21no—
114Dic502—21no—
115Dic502—21no—
116D1004—21no—
117D1004#4112yes—
118C22⋊D251—42no—
119C22⋊D251—42no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C4×D251—42no—
124C4×D251—42no—
125C4×D251—42no—
126C251⋊C8—21no—
127C251⋊C8—21no—
128C251⋊C8—21no—
129C251⋊C8—21no—
130C23×C251—21no—
131C23×C251C2×D411yescharacteristic
132C23×C251C2×D411yescharacteristic
133D4×C251—21no—
134D4×C251—21no—
135D4×C251—41no—
136D4×C251—42no—
137D4×C251—81no—
138C2×C4×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—81no—
146C2×C4×C251—81no—
147C2×C4×C251C2×D411yescharacteristic
148C2×C4×C251C2×D411yescharacteristic
149C2×C4×C251C2×D411yescharacteristic
150C2×C4×C251C2×D411yescharacteristic
151C2×C4×C251C2411yescharacteristic
152C22×C251—21no—
153C22×C251—21no—
154C22×C251—21no—
155C22×C251—21no—
156C22×C251—21no—
157C22×C251—21no—
158C22×C251—41no—
159C22×C251—41no—
160C22×C251—41no—
161C22×C251C22×D411yescharacteristic
162C2×D251—22no—
163C2×D251—41no—
164C2×D251—41no—
165C4×C251—21no—
166C4×C251—21no—
167C4×C251—21no—
168C4×C251—21no—
169C4×C251—41no—
170C4×C251—41no—
171C4×C251—41no—
172C4×C251C2×#4111yescharacteristic
173C4×C251H(5,𝔽2)11yescharacteristic
174Dic251—41no—
175Dic251—41no—
176Dic251—41no—
177C2×C251—41no—
178C2×C251—41no—
179C2×C251—81no—
180C2×C251#54611yescharacteristic
181C2×C251#57511yescharacteristic
182C2×C251#57711yescharacteristic
183D251—42no—
184C251#330711yescharacteristic, complemented, Sylow
185#3307—2511nomaximal, Sylow
186#553D25111yescharacteristic, complemented
187#547—2511no—
188#491—2511no—
189#485—2511no—
190#483—2511no—
191#478—2511no—
192#468—2511no—
193#465—2511no—
194#461—2511no—
195#449—2511no—
196#449—2511no—
197#442—2511no—
198#435—2511no—
199#417—2511no—
200#416—2511no—
201C2×#41—2511no—
202C2×#41—2511no—
203C2×SD16—2511no—
204C2×SD16—2511no—
205C2×D8—2511no—
206C2×M16—2511no—
207C22×C8—2511no—
208C4⋊Q8—2511no—
209#126—21no—
210#125C2×D25111yescharacteristic
211#123—21no—
212#123—5021no—
213#123—5021no—
214C22⋊Q8—2511no—
215C22⋊Q8—2511no—
216#121—21no—
217#121—5021no—
218#121C2×D25111yescharacteristic
219#121C2×D25111yescharacteristic
220C4×D4—21no—
221C4×D4—2511no—
222C4×D4C2×D25111yescharacteristic
223C4×D4C2×D25111yescharacteristic
224#117C2×D25111yescharacteristic
225C2×C4⋊C4—5021no—
226C2×C4⋊C4C2×D25111yescharacteristic
227#107—2511no—
228#106—2511no—
229#106—2511no—
230C4⋊C8—2511no—
231#103—2511no—
232#103—2511no—
233#103—2511no—
234#103—2511no—
235#103—2511no—
236#103—2511no—
237#102—2511no—
238#102—2511no—
239#102—2511no—
240#102—2511no—
241C22⋊C8—2511no—
242C22⋊C8—2511no—
243#97—2511no—
244#41—10041no—
245#41—10041no—
246#41—5022no—
247#41—5022no—
248C2×Q8—2511no—
249C2×Q8—2511no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—5021no—
253C2×D4—5021no—
254C2×D4—2511no—
255C2×D4C22⋊D25111yescharacteristic
256C2×D4C22×D25111yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—5021no—
260C22×C4—5021no—
261C22×C4—5021no—
262C22×C4C22×D25111yescharacteristic
263C22×C4C22×D25111yescharacteristic
264SD16—10041no—
265SD16—10041no—
266D8—5022no—
267M16—10041no—
268C2×C8—10041no—
269C2×C8—2511no—
270C2×C8—2511no—
271C2×C8—2511no—
272C2×C8—2511no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—5021no—
277C4⋊C4—5021no—
278C4⋊C4—5021no—
279C4⋊C4—5021no—
280C4⋊C4—2511no—
281C4⋊C4C22⋊D25111yescharacteristic
282C4⋊C4C22×D25111yescharacteristic
283C4⋊C4C22×D25111yescharacteristic
284C4⋊C4C22×D25111yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—5021no—
290C22⋊C4—5021no—
291C22⋊C4—5021no—
292C22⋊C4C22⋊D25111yescharacteristic
293C22⋊C4C22⋊D25111yescharacteristic
294C42—21no—
295C42C22×D25111yescharacteristic
296C23—21no—
297C23—5021no—
298C23C2×C22⋊D25111yescharacteristic
299C23C2×C22⋊D25111yescharacteristic
300Q8—5021no—
301Q8—5021no—
302Q8—5021no—
303Q8—5021no—
304D4—21no—
305D4—21no—
306D4—41no—
307D4—42no—
308D4—81no—
309D4—10042no—
310D4—10042no—
311D4—5021no—
312D4—2512no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—10042no—
323C2×C4—10042no—
324C2×C4—10042no—
325C2×C4—5021no—
326C2×C4—5021no—
327C2×C4—5021no—
328C2×C4C2×C22⋊D25111yescharacteristic
329C2×C4C2×C22⋊D25111yescharacteristic
330C2×C4C2×C22⋊D25111yescharacteristic
331C2×C4C2×C22⋊D25111yescharacteristic
332C2×C4C23×D25111yescharacteristic
333C8—5021no—
334C8—5021no—
335C8—5021no—
336C8—5021no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—10041no—
347C22—10041no—
348C22—5022no—
349C22C22×C22⋊D25111yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—10041no—
358C4—10041no—
359C4—10041no—
360C4#41×D25111yescharacteristic
361C4#2254876111yescharacteristic
362C2—41no—
363C2—41no—
364C2—81no—
365C2—10042no—
366C2#3063232511yescharacteristic
367C2#3063248511yescharacteristic
368C2#3063249811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#553×C251C2
socleC22×C251C22×D4
hypercenter#553D251
hypocenterC251#3307
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #30632342C2semidirect
2. #30632057C2semidirect
3. #30632028C2semidirect
4. #30632017C2semidirect
5. #30631984C2semidirect
6. #30631931C2semidirect
7. #30631914C2semidirect
8. #30631892C2semidirect
9. #30631830C2semidirect
10. #30631830C2semidirect
11. #30631797C2semidirect
12. #30631759C2semidirect
13. #30631686C2semidirect
14. #30631681C2semidirect
15. #553×C251C2semidirect
184. C251#3307semidirect
186. #553D251semidirect

Maximal subgroups

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

subgroupindex
#306323422
#306320572
#306320282
#306320172
#306319842
#306319312
#306319142
#306318922
#306318302
#306317972
#306317592
#306316862
#306316812
#553×C2512
#3307251