Finite Group Databasebeta

← #30648105

Subgroups — #30648105

Order 16256.

704 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550711C211yesmaximal, characteristic
2#137×D127C212yesmaximal, complemented
3#22550446C211yesmaximal, characteristic, complemented
4#22550060C212yesmaximal, complemented
5#22550057C212yesmaximal, complemented
6#22550056C211yesmaximal, characteristic, complemented
7#22550054C211yesmaximal, characteristic, complemented
8#22550053C211yesmaximal, characteristic, complemented
9#22549891C212yesmaximal, complemented
10#22549857C211yesmaximal, characteristic, complemented
11#455×C127C211yesmaximal, characteristic, complemented
12#16861256—22no—
13#16861256C2212yes—
14#16861254—22no—
15#16861254—22no—
16#16861254C2211yescharacteristic
17C2×#14523247—22no—
18C2×#14523247—22no—
19C2×#14523247C2211yescharacteristic
20#41×D127—22no—
21#41×D127C2212yes—
22C2×Q8×D127C2211yescharacteristic, complemented
23#16861242C2212yescomplemented
24#16861241—22no—
25#16861240C2212yescomplemented
26#16861239C2212yescomplemented
27#16861238—22no—
28Q16×D127—22no—
29SD16×D127—22no—
30M16×D127C2212yescomplemented
31#16861209C2211yescharacteristic, complemented
32#16861207C2211yescharacteristic, complemented
33#16861196C2211yescharacteristic, complemented
34#16861191—22no—
35#16861186—22no—
36#16861166C2211yescharacteristic, complemented
37#16861137C2212yescomplemented
38#16861136C2212yescomplemented
39#16861135C2212yescomplemented
40#16861127C2212yescomplemented
41#16861126C2211yescharacteristic, complemented
42H(5,𝔽2)×C127C2211yescharacteristic
43#137×C127C2212yescomplemented
44#124×C127C2211yescharacteristic, complemented
45#104×C127C2212yescomplemented
46#100×C127C2211yescharacteristic, complemented
47#14523249D412yes—
48#14523248—24no—
49#14523248—22no—
50#14523248—22no—
51#14523248C2312yes—
52#14523247—24no—
53#14523247—24no—
54#14523247—24no—
55#14523247—22no—
56#14523247—22no—
57#14523247—24no—
58#14523247—22no—
59#14523247—22no—
60#14523247—22no—
61#14523247D412yes—
62C2×Dic254—22no—
63C2×Dic254—22no—
64C2×Dic254—21no—
65C2×Dic254C2311yescharacteristic
66C2×Dic254C2311yescharacteristic
67Q8×D127—22no—
68Q8×D127—22no—
69Q8×D127—22no—
70Q8×D127D412yes—
71C2×C22⋊D127—24no—
72C2×C22⋊D127—22no—
73C2×C22⋊D127—22no—
74C2×C22⋊D127D412yes—
75D4×D127—24no—
76D4×D127—22no—
77D4×D127—22no—
78D4×D127D412yes—
79C22×Dic127—22no—
80C22×Dic127D412yes—
81C2×C4×D127—22no—
82C2×C4×D127—22no—
83C2×C4×D127—21no—
84C2×C4×D127C2311yescharacteristic
85Dic508—22no—
86#14523238—22no—
87#14523238—22no—
88#14523237—22no—
89#14523236—22no—
90#14523235—22no—
91#14523232C2312yes—
92#14523231—22no—
93C8×D127—22no—
94C4⋊Dic127—22no—
95Dic127⋊C4—21no—
96Dic127⋊C4—22no—
97C22⋊Dic127—22no—
98(C2×D127)⋊C4—21no—
99C4×Dic127C2311yescharacteristic
100#41×C127—22no—
101#41×C127C2312yes—
102C2×Q8×C127C2311yescharacteristic
103C2×D4×C127—22no—
104C2×D4×C127—22no—
105C2×D4×C127C2311yescharacteristic
106Q16×C127—22no—
107SD16×C127—22no—
108M16×C127C2312yes—
109C22⋊C4×C127—22no—
110C42×C127C2311yescharacteristic
111C22×D127—22no—
112C22×D127—22no—
113C22×D127C2×D411yescharacteristic
114Dic254—24no—
115Dic254—22no—
116Dic254—22no—
117Dic254—22no—
118Dic254—41no—
119Dic254—41no—
120Dic254C2×D412yes—
121D508—22no—
122D508C2×D412yes—
123C22⋊D127—24no—
124C22⋊D127—24no—
125C22⋊D127—22no—
126C22⋊D127—22no—
127C22⋊D127—24no—
128C22⋊D127—22no—
129C2×Dic127—22no—
130C2×Dic127—24no—
131C2×Dic127—22no—
132C2×Dic127—22no—
133C2×Dic127—22no—
134C2×Dic127—21no—
135C2×Dic127—21no—
136C2×Dic127C2×D412yes—
137C2×Dic127C2×D411yescharacteristic
138C4×D127—24no—
139C4×D127—22no—
140C4×D127—22no—
141C4×D127—22no—
142C4×D127—42no—
143C4×D127C2×D412yes—
144C4×D127C2×D412yes—
145C127⋊C8—22no—
146C23×C127—22no—
147C23×C127C2×D412yes—
148Q8×C127—22no—
149Q8×C127C2×D412yes—
150D4×C127—24no—
151D4×C127—22no—
152D4×C127—22no—
153D4×C127C2×D412yes—
154C2×C4×C127—22no—
155C2×C4×C127—22no—
156C2×C4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127C2411yescharacteristic
159C8×C127—22no—
160C22×C127—22no—
161C22×C127—22no—
162C22×C127—42no—
163C22×C127—82no—
164C22×C127C22×D411yescharacteristic
165C2×D127—22no—
166C2×D127—41no—
167C2×D127—82no—
168C2×D127—82no—
169C2×D127#12012yes—
170C4×C127—22no—
171C4×C127—41no—
172C4×C127—41no—
173C4×C127C22×D412yes—
174Dic127—22no—
175Dic127—22no—
176Dic127—41no—
177Dic127—41no—
178Dic127#12011yescharacteristic
179Dic127#12011yescharacteristic
180C2×C127—21no—
181C2×C127—42no—
182C2×C127—42no—
183C2×C127C2×#12011yescharacteristic
184D127—22no—
185D127—42no—
186#3022—1271nomaximal, Sylow
187C127#302211yescharacteristic, complemented, Sylow
188#585—1271no—
189C2×#137—1272no—
190#536—1271no—
191#456—1272no—
192#456—1272no—
193#455—1271no—
194#455—1271no—
195#455—1271no—
196#455D12711yescharacteristic, complemented
197#421—1272no—
198#413—1271no—
1992−1+4—1272no—
2002−1+4—2542no—
201H(5,𝔽2)—1271no—
202H(5,𝔽2)—2542no—
203H(5,𝔽2)—2542no—
204H(5,𝔽2)C2×D12711yescharacteristic
205C2×#41—1271no—
206C2×#41—1272no—
207C2×#41—2542no—
208C2×#41—2542no—
209C2×#41—2542no—
210C22×Q8—1271no—
211#137—1272no—
212#137—1272no—
213#137—1272no—
214#137—2542no—
215#137—2542no—
216#137C2×D12712yescomplemented
217C2×Q16—2542no—
218C2×SD16—2542no—
219C2×M16—1272no—
220C4⋊Q8—1271no—
221C4⋊Q8—1271no—
222#124—1271no—
223#124C2×D12711yescharacteristic, complemented
224#123—2542no—
225C22⋊Q8—2542no—
226#117—1271no—
227#104—1272no—
228#104—1272no—
229#104—1272no—
230#104C2×D12712yescomplemented
231#101—1272no—
232#100—1271no—
233#100C2×D12711yescharacteristic, complemented
234#41—22no—
235#41—1272no—
236#41—1272no—
237#41—1272no—
238#41—2542no—
239#41—2542no—
240#41—2544no—
241#41—2542no—
242#41—2542no—
243#41—2542no—
244#41—2544no—
245#41—2542no—
246#41—2544no—
247#41—2544no—
248#41—2544no—
249#41—2542no—
250#41C22×D12712yes—
251C2×Q8—1271no—
252C2×Q8—1272no—
253C2×Q8—1271no—
254C2×Q8—2541no—
255C2×Q8—2542no—
256C2×Q8—2542no—
257C2×Q8—2542no—
258C2×Q8—2542no—
259C2×Q8—2542no—
260C2×Q8C22×D12711yescharacteristic
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—1272no—
264C2×D4—1272no—
265C2×D4—2542no—
266C2×D4—2542no—
267C2×D4—2544no—
268C2×D4—2542no—
269C2×D4—2544no—
270C2×D4—2542no—
271C2×D4C22×D12711yescharacteristic
272C22×C4—1271no—
273C22×C4—1272no—
274C22×C4—2542no—
275C22×C4—2541no—
276C22×C4—2542no—
277C22×C4—2542no—
278Q16—22no—
279Q16—2542no—
280Q16—2542no—
281Q16—2542no—
282SD16—22no—
283SD16—2542no—
284SD16—2542no—
285SD16—2542no—
286M16—1272no—
287M16—2542no—
288M16C22×D12712yes—
289C2×C8—2542no—
290C4⋊C4—2541no—
291C4⋊C4—2542no—
292C4⋊C4—2542no—
293C22⋊C4—22no—
294C22⋊C4—2541no—
295C22⋊C4—2542no—
296C42—1271no—
297C42C22×D12711yescharacteristic
298C23—22no—
299C23—1271no—
300C23—2542no—
301C23—2542no—
302C23D4×D12712yes—
303Q8—22no—
304Q8—1272no—
305Q8—5081no—
306Q8—5081no—
307Q8—2542no—
308Q8—2544no—
309Q8—2542no—
310Q8—2542no—
311Q8D4×D12712yes—
312D4—24no—
313D4—22no—
314D4—22no—
315D4—1272no—
316D4—2542no—
317D4—2542no—
318D4—2544no—
319D4—2542no—
320D4—2544no—
321D4—2544no—
322D4—2542no—
323D4D4×D12712yes—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—1271no—
329C2×C4—1272no—
330C2×C4—1272no—
331C2×C4—1272no—
332C2×C4—5082no—
333C2×C4—2542no—
334C2×C4—2542no—
335C2×C4—2544no—
336C2×C4—2542no—
337C2×C4—2542no—
338C2×C4—2542no—
339C2×C4—2541no—
340C2×C4—2541no—
341C2×C4—2544no—
342C2×C4—2542no—
343C2×C4—2542no—
344C2×C4C23×D12711yescharacteristic
345C8—22no—
346C8—2542no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—82no—
351C22—1272no—
352C22—10162no—
353C22—10162no—
354C22—5081no—
355C22—2542no—
356C22C2×D4×D12711yescharacteristic
357C4—22no—
358C4—41no—
359C4—41no—
360C4—1271no—
361C4—1271no—
362C4—5081no—
363C4—5081no—
364C4—2542no—
365C4—2542no—
366C4C2×D4×D12712yes—
367C2—21no—
368C2—42no—
369C2—42no—
370C2—5082no—
371C2—2542no—
372C2#120×D12711yescharacteristic
subgroupG/H
centerC2#120×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#455×C127C2
socleC2×C127C2×#120
prime radical#22550711C2
hypercenter#455D127
hypocenterC127#3022
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
2. #137×D127C2semidirect
3. #22550446C2semidirect
4. #22550060C2semidirect
5. #22550057C2semidirect
6. #22550056C2semidirect
7. #22550054C2semidirect
8. #22550053C2semidirect
9. #22549891C2semidirect
10. #22549857C2semidirect
11. #455×C127C2semidirect
22. C2×Q8×D127C22semidirect
23. #16861242C22semidirect
25. #16861240C22semidirect
26. #16861239C22semidirect
30. M16×D127C22semidirect
31. #16861209C22semidirect
32. #16861207C22semidirect
33. #16861196C22semidirect
36. #16861166C22semidirect
37. #16861137C22semidirect
38. #16861136C22semidirect
39. #16861135C22semidirect
40. #16861127C22semidirect
41. #16861126C22semidirect
43. #137×C127C22semidirect
44. #124×C127C22semidirect
45. #104×C127C22semidirect
46. #100×C127C22semidirect
187. C127#3022semidirect
196. #455D127semidirect
216. #137C2×D127semidirect
223. #124C2×D127semidirect
230. #104C2×D127semidirect
233. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225507112
#137×D1272
#225504462
#225500602
#225500572
#225500562
#225500542
#225500532
#225498912
#225498572
#455×C1272
#3022127