Finite Group Databasebeta

← #30648992

Subgroups — #30648992

Order 16256.

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#22550497C211yesmaximal, characteristic, complemented
2#22550211C211yesmaximal, characteristic, complemented
3#22550189C211yesmaximal, characteristic, complemented
4#22550171C211yesmaximal, characteristic, complemented
5#22550156C211yesmaximal, characteristic, complemented
6#22550098C211yesmaximal, characteristic, complemented
7#22550091C211yesmaximal, characteristic, complemented
8#22550038C211yesmaximal, characteristic, complemented
9#22550021C211yesmaximal, characteristic
10#22549976C211yesmaximal, characteristic, complemented
11#22549974C211yesmaximal, characteristic, complemented
12C4×#14523238C211yesmaximal, characteristic, complemented
13C4×#14523233C211yesmaximal, characteristic, complemented
14#22549937C211yesmaximal, characteristic, complemented
15#520×C127C211yesmaximal, characteristic, complemented
16C2×#14523238C2211yescharacteristic
17C2×#14523236C2211yescharacteristic
18C2×#14523235C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20#16861209C2211yescharacteristic
21#16861204—21no—
22#16861197C2211yescharacteristic
23#16861195—21no—
24#16861191—21no—
25#16861188C2211yescharacteristic
26#16861187—21no—
27#16861183C2211yescharacteristic
28C4×Dic254C2211yescharacteristic
29C4×D508C2211yescharacteristic
30#16861142C2211yescharacteristic
31#16861142C2211yescharacteristic
32#16861142C2211yescharacteristic
33#16861142C2211yescharacteristic
34C4⋊(C127⋊C8)C2211yescharacteristic
35C4⋊(C127⋊C8)C2211yescharacteristic
36#16861133C2211yescharacteristic
37#16861133C2211yescharacteristic
38#16861132C2211yescharacteristic
39#16861132C2211yescharacteristic
40#16861130C2211yescharacteristic
41#16861130C2211yescharacteristic
42#16861129C2211yescharacteristic
43#16861129C2211yescharacteristic
44C22⋊(C127⋊C8)C2211yescharacteristic
45C22⋊(C127⋊C8)C2211yescharacteristic
46#16861120C2211yescharacteristic
47C4×C127⋊C8C2211yescharacteristic
48C2×#41×C127C2211yescharacteristic
49C4×Q8×C127C2211yescharacteristic
50C4×Q8×C127C2211yescharacteristic
51C4×D4×C127—21no—
52C4×D4×C127—21no—
53C4×D4×C127C2211yescharacteristic
54C4×D4×C127C2211yescharacteristic
55#117×C127—21no—
56#117×C127C2211yescharacteristic
57C2×C4⋊C4×C127—21no—
58C2×C4⋊C4×C127C2211yescharacteristic
59C2×Dic254—21no—
60C2×Dic254C2311yescharacteristic
61C2×D508C2311yescharacteristic
62C2×C22⋊D127—21no—
63C2×C4×D127—21no—
64#14523238—22no—
65#14523236—22no—
66#14523235—22no—
67#14523233—22no—
68C2×C127⋊C8C2311yescharacteristic
69C2×C127⋊C8C2311yescharacteristic
70C2×C127⋊C8C2311yescharacteristic
71C2×C127⋊C8C2311yescharacteristic
72C4⋊Dic127—21no—
73C4⋊Dic127C2311yescharacteristic
74C4⋊Dic127C2311yescharacteristic
75Dic127⋊C4—21no—
76Dic127⋊C4—21no—
77C22⋊Dic127—21no—
78(C2×D127)⋊C4—21no—
79(C2×D127)⋊C4—21no—
80(C2×D127)⋊C4—21no—
81(C2×D127)⋊C4—21no—
82C4×Dic127—21no—
83#41×C127—22no—
84#41×C127—22no—
85C2×Q8×C127C2311yescharacteristic
86C2×D4×C127—21no—
87C2×D4×C127C2311yescharacteristic
88C22×C4×C127—21no—
89C22×C4×C127—21no—
90C22×C4×C127—21no—
91C22×C4×C127—21no—
92C22×C4×C127C2311yescharacteristic
93C4⋊C4×C127—21no—
94C4⋊C4×C127—21no—
95C4⋊C4×C127—21no—
96C4⋊C4×C127D411yescharacteristic
97C4⋊C4×C127D411yescharacteristic
98C4⋊C4×C127C2311yescharacteristic
99C4⋊C4×C127C2311yescharacteristic
100C22⋊C4×C127—21no—
101C22⋊C4×C127—21no—
102C22⋊C4×C127D411yescharacteristic
103C22⋊C4×C127D411yescharacteristic
104C42×C127—21no—
105C42×C127—21no—
106C42×C127C2311yescharacteristic
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109Dic254—21no—
110Dic254—21no—
111Dic254—81no—
112D508—21no—
113D508—21no—
114C22⋊D127—81no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C4×D127—81no—
119C127⋊C8—21no—
120C127⋊C8—21no—
121C127⋊C8—21no—
122C127⋊C8—21no—
123C23×C127—21no—
124C23×C127C2×D411yescharacteristic
125Q8×C127—21no—
126Q8×C127#4112yes—
127D4×C127—24no—
128D4×C127—21no—
129D4×C127#4112yes—
130C2×C4×C127—24no—
131C2×C4×C127—21no—
132C2×C4×C127—21no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—81no—
137C2×C4×C127—81no—
138C2×C4×C127—81no—
139C2×C4×C127C2×D411yescharacteristic
140C2×C4×C127C2×D411yescharacteristic
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2×D411yescharacteristic
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2411yescharacteristic
145C22×C127—21no—
146C22×C127—22no—
147C22×C127—21no—
148C22×C127—21no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127C22×D411yescharacteristic
152C2×D127—41no—
153C2×D127—41no—
154C2×D127—41no—
155C4×C127—21no—
156C4×C127—22no—
157C4×C127—21no—
158C4×C127—21no—
159C4×C127—21no—
160C4×C127—21no—
161C4×C127—41no—
162C4×C127—41no—
163C4×C127C2×#4111yescharacteristic
164C4×C127C2×#4111yescharacteristic
165Dic127—41no—
166Dic127—41no—
167Dic127—41no—
168C2×C127—41no—
169C2×C127—42no—
170C2×C127#52511yescharacteristic
171C2×C127#57611yescharacteristic
172C2×C127#57811yescharacteristic
173D127—81no—
174#3124—1271nomaximal, Sylow
175C127#312411yescharacteristic, complemented, Sylow
176#542—1271no—
177#520D12711yescharacteristic, complemented
178#484—1271no—
179#480—1271no—
180#477—1271no—
181#474—1271no—
182#464—1271no—
183#463—1271no—
184#452—1271no—
185#449—1271no—
186#440—1271no—
187#440—1271no—
188C4×Q16—1271no—
189C4×D8—1271no—
190#433—1271no—
191C2×#41C2×D12711yescharacteristic
192C2×Q16—1271no—
193C2×SD16—1271no—
194C2×SD16—1271no—
195C2×D8—1271no—
196C4⋊Q8—1271no—
197#126—2541no—
198#124—1271no—
199#124—2541no—
200#123—2541no—
201C22⋊Q8—1271no—
202C22⋊Q8—2541no—
203#121—1271no—
204C4×Q8—1271no—
205C4×Q8C2×D12711yescharacteristic
206C4×Q8C2×D12711yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—1271no—
210C4×D4C2×D12711yescharacteristic
211C4×D4C2×D12711yescharacteristic
212#117—21no—
213#117C2×D12711yescharacteristic
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×D12711yescharacteristic
216#107—1271no—
217#107—1271no—
218#107—1271no—
219#107—1271no—
220C4⋊C8—1271no—
221C4⋊C8—1271no—
222#103—1271no—
223#103—1271no—
224#103—1271no—
225#103—1271no—
226#102—1271no—
227#102—1271no—
228#102—1271no—
229#102—1271no—
230C22⋊C8—1271no—
231C22⋊C8—1271no—
232#97—1271no—
233C4×C8—1271no—
234#41—22no—
235#41—22no—
236C2×Q8—1271no—
237C2×Q8—2541no—
238C2×Q8C22×D12711yescharacteristic
239C2×D4—21no—
240C2×D4—1271no—
241C2×D4—2541no—
242C2×D4C22×D12711yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—2541no—
248C22×C4C22×D12711yescharacteristic
249Q16—2542no—
250SD16—2542no—
251SD16—2542no—
252D8—2542no—
253C2×C8—1271no—
254C2×C8—1271no—
255C2×C8—1271no—
256C2×C8—1271no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—1271no—
261C4⋊C4—1271no—
262C4⋊C4—2541no—
263C4⋊C4—2541no—
264C4⋊C4—2541no—
265C4⋊C4C22⋊D12711yescharacteristic
266C4⋊C4C22⋊D12711yescharacteristic
267C4⋊C4C22×D12711yescharacteristic
268C4⋊C4C22×D12711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—2541no—
272C22⋊C4—2541no—
273C22⋊C4—2541no—
274C22⋊C4—2541no—
275C22⋊C4—2541no—
276C22⋊C4C22⋊D12711yescharacteristic
277C22⋊C4C22⋊D12711yescharacteristic
278C42—21no—
279C42—21no—
280C42—2541no—
281C42C22×D12711yescharacteristic
282C42C22×D12711yescharacteristic
283C23—21no—
284C23—2541no—
285C23C2×C22⋊D12711yescharacteristic
286Q8—21no—
287Q8—10161no—
288Q8—2541no—
289Q8—2541no—
290Q8#1452324812yes—
291D4—24no—
292D4—21no—
293D4—10161no—
294D4—2541no—
295D4—2541no—
296D4#1452324812yes—
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—10161no—
307C2×C4—2541no—
308C2×C4—2541no—
309C2×C4—2541no—
310C2×C4C2×C22⋊D12711yescharacteristic
311C2×C4C2×C22⋊D12711yescharacteristic
312C2×C4C2×C22⋊D12711yescharacteristic
313C2×C4C2×C22⋊D12711yescharacteristic
314C2×C4C2×C22⋊D12711yescharacteristic
315C2×C4C23×D12711yescharacteristic
316C8—2541no—
317C8—2541no—
318C8—2541no—
319C8—2541no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—5081no—
327C22—5081no—
328C22—5081no—
329C22C22×C22⋊D12711yescharacteristic
330C4—21no—
331C4—22no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—5081no—
339C4—5081no—
340C4—5081no—
341C4C2×#1452324811yescharacteristic
342C4C2×#1452324811yescharacteristic
343C2—41no—
344C2—42no—
345C2—10161no—
346C2#2255038711yescharacteristic
347C2#2255067911yescharacteristic
348C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#520×C127C2
socleC22×C127C22×D4
prime radical#22550021C2
hypercenter#520D127
hypocenterC127#3124
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #22550497C2semidirect
2. #22550211C2semidirect
3. #22550189C2semidirect
4. #22550171C2semidirect
5. #22550156C2semidirect
6. #22550098C2semidirect
7. #22550091C2semidirect
8. #22550038C2semidirect
10. #22549976C2semidirect
11. #22549974C2semidirect
12. C4×#14523238C2semidirect
13. C4×#14523233C2semidirect
14. #22549937C2semidirect
15. #520×C127C2semidirect
175. C127#3124semidirect
177. #520D127semidirect

Maximal subgroups

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

subgroupindex
#225504972
#225502112
#225501892
#225501712
#225501562
#225500982
#225500912
#225500382
#225500212
#225499762
#225499742
C4×#145232382
C4×#145232332
#225499372
#520×C1272
#3124127