Finite Group Databasebeta

← #44560627

Subgroups — #44560627

Order 32128.

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#30632308C211yesmaximal, characteristic, complemented
2#30632022C211yesmaximal, characteristic, complemented
3#30632000C211yesmaximal, characteristic, complemented
4#30631982C211yesmaximal, characteristic, complemented
5#30631967C211yesmaximal, characteristic, complemented
6#30631909C211yesmaximal, characteristic, complemented
7#30631902C211yesmaximal, characteristic, complemented
8#30631849C211yesmaximal, characteristic, complemented
9#30631832C211yesmaximal, characteristic
10#30631787C211yesmaximal, characteristic, complemented
11#30631785C211yesmaximal, characteristic, complemented
12C4×#16860800C211yesmaximal, characteristic, complemented
13C4×#16860795C211yesmaximal, characteristic, complemented
14#30631748C211yesmaximal, characteristic, complemented
15#520×C251C211yesmaximal, characteristic, complemented
16C2×#16860800C2211yescharacteristic
17C2×#16860798C2211yescharacteristic
18C2×#16860797C2211yescharacteristic
19C2×#16860795C2211yescharacteristic
20#22548715C2211yescharacteristic
21#22548710—21no—
22#22548703C2211yescharacteristic
23#22548701—21no—
24#22548697—21no—
25#22548694C2211yescharacteristic
26#22548693—21no—
27#22548689C2211yescharacteristic
28C4×Dic502C2211yescharacteristic
29C4×D1004C2211yescharacteristic
30#22548648C2211yescharacteristic
31#22548648C2211yescharacteristic
32#22548648C2211yescharacteristic
33#22548648C2211yescharacteristic
34C4⋊(C251⋊C8)C2211yescharacteristic
35C4⋊(C251⋊C8)C2211yescharacteristic
36#22548639C2211yescharacteristic
37#22548639C2211yescharacteristic
38#22548638C2211yescharacteristic
39#22548638C2211yescharacteristic
40#22548636C2211yescharacteristic
41#22548636C2211yescharacteristic
42#22548635C2211yescharacteristic
43#22548635C2211yescharacteristic
44C22⋊(C251⋊C8)C2211yescharacteristic
45C22⋊(C251⋊C8)C2211yescharacteristic
46#22548626C2211yescharacteristic
47C4×C251⋊C8C2211yescharacteristic
48C2×#41×C251C2211yescharacteristic
49C4×Q8×C251C2211yescharacteristic
50C4×Q8×C251C2211yescharacteristic
51C4×D4×C251—21no—
52C4×D4×C251—21no—
53C4×D4×C251C2211yescharacteristic
54C4×D4×C251C2211yescharacteristic
55#117×C251—21no—
56#117×C251C2211yescharacteristic
57C2×C4⋊C4×C251—21no—
58C2×C4⋊C4×C251C2211yescharacteristic
59C2×Dic502—21no—
60C2×Dic502C2311yescharacteristic
61C2×D1004C2311yescharacteristic
62C2×C22⋊D251—21no—
63C2×C4×D251—21no—
64#16860800—22no—
65#16860798—22no—
66#16860797—22no—
67#16860795—22no—
68C2×C251⋊C8C2311yescharacteristic
69C2×C251⋊C8C2311yescharacteristic
70C2×C251⋊C8C2311yescharacteristic
71C2×C251⋊C8C2311yescharacteristic
72C4⋊Dic251—21no—
73C4⋊Dic251C2311yescharacteristic
74C4⋊Dic251C2311yescharacteristic
75Dic251⋊C4—21no—
76Dic251⋊C4—21no—
77C22⋊Dic251—21no—
78(C2×D251)⋊C4—21no—
79(C2×D251)⋊C4—21no—
80(C2×D251)⋊C4—21no—
81(C2×D251)⋊C4—21no—
82C4×Dic251—21no—
83#41×C251—22no—
84#41×C251—22no—
85C2×Q8×C251C2311yescharacteristic
86C2×D4×C251—21no—
87C2×D4×C251C2311yescharacteristic
88C22×C4×C251—21no—
89C22×C4×C251—21no—
90C22×C4×C251—21no—
91C22×C4×C251—21no—
92C22×C4×C251C2311yescharacteristic
93C4⋊C4×C251—21no—
94C4⋊C4×C251—21no—
95C4⋊C4×C251—21no—
96C4⋊C4×C251D411yescharacteristic
97C4⋊C4×C251D411yescharacteristic
98C4⋊C4×C251C2311yescharacteristic
99C4⋊C4×C251C2311yescharacteristic
100C22⋊C4×C251—21no—
101C22⋊C4×C251—21no—
102C22⋊C4×C251D411yescharacteristic
103C22⋊C4×C251D411yescharacteristic
104C42×C251—21no—
105C42×C251—21no—
106C42×C251C2311yescharacteristic
107C42×C251C2311yescharacteristic
108C22×D251—21no—
109Dic502—21no—
110Dic502—21no—
111Dic502—81no—
112D1004—21no—
113D1004—21no—
114C22⋊D251—81no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251—21no—
118C4×D251—81no—
119C251⋊C8—21no—
120C251⋊C8—21no—
121C251⋊C8—21no—
122C251⋊C8—21no—
123C23×C251—21no—
124C23×C251C2×D411yescharacteristic
125Q8×C251—21no—
126Q8×C251#4112yes—
127D4×C251—24no—
128D4×C251—21no—
129D4×C251#4112yes—
130C2×C4×C251—24no—
131C2×C4×C251—21no—
132C2×C4×C251—21no—
133C2×C4×C251—21no—
134C2×C4×C251—21no—
135C2×C4×C251—21no—
136C2×C4×C251—81no—
137C2×C4×C251—81no—
138C2×C4×C251—81no—
139C2×C4×C251C2×D411yescharacteristic
140C2×C4×C251C2×D411yescharacteristic
141C2×C4×C251C2×D411yescharacteristic
142C2×C4×C251C2×D411yescharacteristic
143C2×C4×C251C2×D411yescharacteristic
144C2×C4×C251C2411yescharacteristic
145C22×C251—21no—
146C22×C251—22no—
147C22×C251—21no—
148C22×C251—21no—
149C22×C251—41no—
150C22×C251—41no—
151C22×C251C22×D411yescharacteristic
152C2×D251—41no—
153C2×D251—41no—
154C2×D251—41no—
155C4×C251—21no—
156C4×C251—22no—
157C4×C251—21no—
158C4×C251—21no—
159C4×C251—21no—
160C4×C251—21no—
161C4×C251—41no—
162C4×C251—41no—
163C4×C251C2×#4111yescharacteristic
164C4×C251C2×#4111yescharacteristic
165Dic251—41no—
166Dic251—41no—
167Dic251—41no—
168C2×C251—41no—
169C2×C251—42no—
170C2×C251#52511yescharacteristic
171C2×C251#57611yescharacteristic
172C2×C251#57811yescharacteristic
173D251—81no—
174C251#312411yescharacteristic, complemented, Sylow
175#3124—2511nomaximal, Sylow
176#542—2511no—
177#520D25111yescharacteristic, complemented
178#484—2511no—
179#480—2511no—
180#477—2511no—
181#474—2511no—
182#464—2511no—
183#463—2511no—
184#452—2511no—
185#449—2511no—
186#440—2511no—
187#440—2511no—
188C4×Q16—2511no—
189C4×D8—2511no—
190#433—2511no—
191C2×#41C2×D25111yescharacteristic
192C2×Q16—2511no—
193C2×SD16—2511no—
194C2×SD16—2511no—
195C2×D8—2511no—
196C4⋊Q8—2511no—
197#126—5021no—
198#124—5021no—
199#124—2511no—
200#123—5021no—
201C22⋊Q8—5021no—
202C22⋊Q8—2511no—
203#121—2511no—
204C4×Q8—2511no—
205C4×Q8C2×D25111yescharacteristic
206C4×Q8C2×D25111yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—2511no—
210C4×D4C2×D25111yescharacteristic
211C4×D4C2×D25111yescharacteristic
212#117—21no—
213#117C2×D25111yescharacteristic
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×D25111yescharacteristic
216#107—2511no—
217#107—2511no—
218#107—2511no—
219#107—2511no—
220C4⋊C8—2511no—
221C4⋊C8—2511no—
222#103—2511no—
223#103—2511no—
224#103—2511no—
225#103—2511no—
226#102—2511no—
227#102—2511no—
228#102—2511no—
229#102—2511no—
230C22⋊C8—2511no—
231C22⋊C8—2511no—
232#97—2511no—
233C4×C8—2511no—
234#41—22no—
235#41—22no—
236C2×Q8—5021no—
237C2×Q8—2511no—
238C2×Q8C22×D25111yescharacteristic
239C2×D4—21no—
240C2×D4—5021no—
241C2×D4—2511no—
242C2×D4C22×D25111yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—5021no—
248C22×C4C22×D25111yescharacteristic
249Q16—5022no—
250SD16—5022no—
251SD16—5022no—
252D8—5022no—
253C2×C8—2511no—
254C2×C8—2511no—
255C2×C8—2511no—
256C2×C8—2511no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—5021no—
261C4⋊C4—5021no—
262C4⋊C4—5021no—
263C4⋊C4—2511no—
264C4⋊C4—2511no—
265C4⋊C4C22⋊D25111yescharacteristic
266C4⋊C4C22⋊D25111yescharacteristic
267C4⋊C4C22×D25111yescharacteristic
268C4⋊C4C22×D25111yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—5021no—
272C22⋊C4—5021no—
273C22⋊C4—5021no—
274C22⋊C4—5021no—
275C22⋊C4—5021no—
276C22⋊C4C22⋊D25111yescharacteristic
277C22⋊C4C22⋊D25111yescharacteristic
278C42—21no—
279C42—21no—
280C42—5021no—
281C42C22×D25111yescharacteristic
282C42C22×D25111yescharacteristic
283C23—21no—
284C23—5021no—
285C23C2×C22⋊D25111yescharacteristic
286Q8—21no—
287Q8—20081no—
288Q8—5021no—
289Q8—5021no—
290Q8#1686081012yes—
291D4—24no—
292D4—21no—
293D4—20081no—
294D4—5021no—
295D4—5021no—
296D4#1686081012yes—
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—20081no—
307C2×C4—5021no—
308C2×C4—5021no—
309C2×C4—5021no—
310C2×C4C2×C22⋊D25111yescharacteristic
311C2×C4C2×C22⋊D25111yescharacteristic
312C2×C4C2×C22⋊D25111yescharacteristic
313C2×C4C2×C22⋊D25111yescharacteristic
314C2×C4C2×C22⋊D25111yescharacteristic
315C2×C4C23×D25111yescharacteristic
316C8—5021no—
317C8—5021no—
318C8—5021no—
319C8—5021no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—10041no—
327C22—10041no—
328C22—10041no—
329C22C22×C22⋊D25111yescharacteristic
330C4—21no—
331C4—22no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—10041no—
339C4—10041no—
340C4—10041no—
341C4C2×#1686081011yescharacteristic
342C4C2×#1686081011yescharacteristic
343C2—41no—
344C2—42no—
345C2—20081no—
346C2#3063219811yescharacteristic
347C2#3063249011yescharacteristic
348C2#3063250211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#520×C251C2
socleC22×C251C22×D4
prime radical#30631832C2
hypercenter#520D251
hypocenterC251#3124
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #30632308C2semidirect
2. #30632022C2semidirect
3. #30632000C2semidirect
4. #30631982C2semidirect
5. #30631967C2semidirect
6. #30631909C2semidirect
7. #30631902C2semidirect
8. #30631849C2semidirect
10. #30631787C2semidirect
11. #30631785C2semidirect
12. C4×#16860800C2semidirect
13. C4×#16860795C2semidirect
14. #30631748C2semidirect
15. #520×C251C2semidirect
174. C251#3124semidirect
177. #520D251semidirect

Maximal subgroups

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

subgroupindex
#306323082
#306320222
#306320002
#306319822
#306319672
#306319092
#306319022
#306318492
#306318322
#306317872
#306317852
C4×#168608002
C4×#168607952
#306317482
#520×C2512
#3124251