Finite Group Databasebeta

← #30648929

Subgroups — #30648929

Order 16256.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550481C211yesmaximal, characteristic, complemented
2#22550194C211yesmaximal, characteristic, complemented
3#22550189C211yesmaximal, characteristic, complemented
4#22550183C211yesmaximal, characteristic, complemented
5#22550176C211yesmaximal, characteristic, complemented
6#22550076C211yesmaximal, characteristic, complemented
7#22550071C211yesmaximal, characteristic, complemented
8#22550021C211yesmaximal, characteristic, complemented
9#22550021C211yesmaximal, characteristic, complemented
10#22549986C211yesmaximal, characteristic, complemented
11#22549974C211yesmaximal, characteristic, complemented
12C4×#14523235C211yesmaximal, characteristic, complemented
13C4×#14523233C211yesmaximal, characteristic, complemented
14#22549937C211yesmaximal, characteristic, complemented
15#520×C127C211yesmaximal, characteristic, complemented
16C2×#14523235C2211yescharacteristic
17C2×#14523235C2211yescharacteristic
18C2×#14523233C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20#16861206C2211yescharacteristic
21#16861201C2211yescharacteristic
22#16861194—21no—
23#16861194—21no—
24#16861183C2211yescharacteristic
25#16861183C2211yescharacteristic
26#16861182—21no—
27#16861182—21no—
28C4×D508C2211yescharacteristic
29C4×D508C2211yescharacteristic
30#16861142C2211yescharacteristic
31#16861142C2211yescharacteristic
32#16861140C2211yescharacteristic
33#16861140C2211yescharacteristic
34C4⋊(C127⋊C8)C2211yescharacteristic
35C4⋊(C127⋊C8)C2211yescharacteristic
36#16861133C2211yescharacteristic
37#16861133C2211yescharacteristic
38#16861130C2211yescharacteristic
39#16861130C2211yescharacteristic
40#16861129C2211yescharacteristic
41#16861129C2211yescharacteristic
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×D508—21no—
60C2×D508—21no—
61C2×D508C2311yescharacteristic
62C2×D508C2311yescharacteristic
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65C2×C4×D127—21no—
66C2×C4×D127—21no—
67#14523235—22no—
68#14523235—22no—
69#14523233—22no—
70#14523233—22no—
71C2×C127⋊C8C2311yescharacteristic
72C2×C127⋊C8C2311yescharacteristic
73C2×C127⋊C8C2311yescharacteristic
74C2×C127⋊C8C2311yescharacteristic
75C4⋊Dic127C2311yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77Dic127⋊C4—21no—
78Dic127⋊C4—21no—
79(C2×D127)⋊C4—21no—
80(C2×D127)⋊C4—21no—
81(C2×D127)⋊C4—21no—
82(C2×D127)⋊C4—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—
109C22×D127—21no—
110D508—21no—
111D508—21no—
112D508—21no—
113D508—21no—
114D508—81no—
115D508—81no—
116C22⋊D127—81no—
117C22⋊D127—81no—
118C2×Dic127—21no—
119C2×Dic127—21no—
120C4×D127—81no—
121C4×D127—81no—
122C127⋊C8—21no—
123C127⋊C8—21no—
124C127⋊C8—21no—
125C127⋊C8—21no—
126C23×C127—21no—
127C23×C127C2×D411yescharacteristic
128Q8×C127—21no—
129Q8×C127#4112yes—
130D4×C127—24no—
131D4×C127—21no—
132D4×C127#4112yes—
133C2×C4×C127—24no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—81no—
140C2×C4×C127—81no—
141C2×C4×C127—81no—
142C2×C4×C127C2×D411yescharacteristic
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2411yescharacteristic
148C22×C127—21no—
149C22×C127—22no—
150C22×C127—21no—
151C22×C127—21no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127C22×D411yescharacteristic
155C2×D127—41no—
156C2×D127—41no—
157C2×D127—41no—
158C2×D127—41no—
159C2×D127—41no—
160C2×D127—41no—
161C4×C127—21no—
162C4×C127—22no—
163C4×C127—21no—
164C4×C127—21no—
165C4×C127—21no—
166C4×C127—21no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127C2×#4111yescharacteristic
170C4×C127C2×#4111yescharacteristic
171Dic127—41no—
172Dic127—41no—
173C2×C127—41no—
174C2×C127—42no—
175C2×C127#52511yescharacteristic
176C2×C127#57611yescharacteristic
177C2×C127#57711yescharacteristic
178D127—81no—
179D127—81no—
180#3119—1271nomaximal, Sylow
181C127#311911yescharacteristic, complemented, Sylow
182#540—1271no—
183#520D12711yescharacteristic, complemented
184#481—1271no—
185#480—1271no—
186#479—1271no—
187#478—1271no—
188#460—1271no—
189#459—1271no—
190#449—1271no—
191#449—1271no—
192#442—1271no—
193#440—1271no—
194C4×SD16—1271no—
195C4×D8—1271no—
196#433—1271no—
197C2×#41C2×D12711yescharacteristic
198C2×SD16—1271no—
199C2×SD16—1271no—
200C2×D8—1271no—
201C2×D8—1271no—
202#127—1271no—
203#125—1271no—
204#123—2541no—
205#123—2541no—
206#121—1271no—
207#121—1271no—
208#121—2541no—
209#121—2541no—
210C4×Q8C2×D12711yescharacteristic
211C4×Q8C2×D12711yescharacteristic
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—1271no—
215C4×D4—1271no—
216C4×D4C2×D12711yescharacteristic
217C4×D4C2×D12711yescharacteristic
218#117—21no—
219#117C2×D12711yescharacteristic
220C2×C4⋊C4—21no—
221C2×C4⋊C4C2×D12711yescharacteristic
222#107—1271no—
223#107—1271no—
224#106—1271no—
225#106—1271no—
226C4⋊C8—1271no—
227C4⋊C8—1271no—
228#103—1271no—
229#103—1271no—
230#102—1271no—
231#102—1271no—
232#102—1271no—
233#102—1271no—
234#102—1271no—
235#102—1271no—
236C22⋊C8—1271no—
237C22⋊C8—1271no—
238#97—1271no—
239C4×C8—1271no—
240#41—22no—
241#41—22no—
242C2×Q8C22×D12711yescharacteristic
243C2×D4—21no—
244C2×D4—1271no—
245C2×D4—1271no—
246C2×D4—2541no—
247C2×D4—2541no—
248C2×D4—2541no—
249C2×D4—2541no—
250C2×D4C22×D12711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—2541no—
256C22×C4—2541no—
257C22×C4C22×D12711yescharacteristic
258SD16—2542no—
259SD16—2542no—
260D8—2542no—
261D8—2542no—
262C2×C8—1271no—
263C2×C8—1271no—
264C2×C8—1271no—
265C2×C8—1271no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—1271no—
270C4⋊C4—1271no—
271C4⋊C4—2541no—
272C4⋊C4—2541no—
273C4⋊C4C22⋊D12711yescharacteristic
274C4⋊C4C22⋊D12711yescharacteristic
275C4⋊C4C22×D12711yescharacteristic
276C4⋊C4C22×D12711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4C22⋊D12711yescharacteristic
284C22⋊C4C22⋊D12711yescharacteristic
285C42—21no—
286C42—21no—
287C42C22×D12711yescharacteristic
288C42C22×D12711yescharacteristic
289C23—21no—
290C23—2541no—
291C23—2541no—
292C23C2×C22⋊D12711yescharacteristic
293Q8—21no—
294Q8#1452324812yes—
295D4—24no—
296D4—21no—
297D4—10161no—
298D4—10161no—
299D4—10161no—
300D4—10161no—
301D4—2541no—
302D4—2541no—
303D4—2541no—
304D4—2541no—
305D4#1452324812yes—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—10161no—
316C2×C4—10161no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4C2×C22⋊D12711yescharacteristic
320C2×C4C2×C22⋊D12711yescharacteristic
321C2×C4C2×C22⋊D12711yescharacteristic
322C2×C4C2×C22⋊D12711yescharacteristic
323C2×C4C2×C22⋊D12711yescharacteristic
324C2×C4C23×D12711yescharacteristic
325C8—2541no—
326C8—2541no—
327C8—2541no—
328C8—2541no—
329C22—21no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—5081no—
336C22—5081no—
337C22—5081no—
338C22—5081no—
339C22—5081no—
340C22—5081no—
341C22C22×C22⋊D12711yescharacteristic
342C4—21no—
343C4—22no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—41no—
349C4—41no—
350C4—5081no—
351C4—5081no—
352C4C2×#1452324811yescharacteristic
353C4C2×#1452324811yescharacteristic
354C2—41no—
355C2—42no—
356C2—10161no—
357C2—10161no—
358C2#2255038711yescharacteristic
359C2#2255067911yescharacteristic
360C2#2255068411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#520×C127C2
socleC22×C127C22×D4
hypercenter#520D127
hypocenterC127#3119
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. #22550481C2semidirect
2. #22550194C2semidirect
3. #22550189C2semidirect
4. #22550183C2semidirect
5. #22550176C2semidirect
6. #22550076C2semidirect
7. #22550071C2semidirect
8. #22550021C2semidirect
9. #22550021C2semidirect
10. #22549986C2semidirect
11. #22549974C2semidirect
12. C4×#14523235C2semidirect
13. C4×#14523233C2semidirect
14. #22549937C2semidirect
15. #520×C127C2semidirect
181. C127#3119semidirect
183. #520D127semidirect

Maximal subgroups

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

subgroupindex
#225504812
#225501942
#225501892
#225501832
#225501762
#225500762
#225500712
#225500212
#225499862
#225499742
C4×#145232352
C4×#145232332
#225499372
#520×C1272
#3119127