Finite Group Databasebeta

← #455×D127 #30648057

Subgroups — #455×D127 #30648057

Order 16256.

760 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 135 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
1H(5,𝔽2)×D127C211yesmaximal, characteristic
2#137×D127C212yesmaximal, complemented
3#124×D127C211yesmaximal, characteristic, complemented
4#22550056C211yesmaximal, characteristic, complemented
5#22550055C212yesmaximal, complemented
6#22550054C211yesmaximal, characteristic, complemented
7#22550053C211yesmaximal, characteristic, complemented
8#22550052C212yesmaximal, complemented
9#104×D127C212yesmaximal, complemented
10#100×D127C211yesmaximal, characteristic, complemented
11#455×C127C211yesmaximal, characteristic, complemented
12#16861255—22no—
13#16861255C2212yes—
14#16861254—22no—
15#16861254—22no—
16#16861254C2211yescharacteristic
17#41×D127—22no—
18#41×D127C2212yes—
19C2×Q8×D127C2211yescharacteristic, complemented
20C2×D4×D127—22no—
21C2×D4×D127—22no—
22C2×D4×D127C2211yescharacteristic
23#16861242C2212yescomplemented
24#16861241—22no—
25#16861240C2212yescomplemented
26#16861239C2212yescomplemented
27#16861238—22no—
28Q16×D127—22no—
29SD16×D127—22no—
30M16×D127C2212yescomplemented
31#16861198C2211yescharacteristic, complemented
32#16861197C2211yescharacteristic, complemented
33#16861196C2211yescharacteristic, complemented
34#16861195—22no—
35C22⋊C4×D127—22no—
36C42×D127C2211yescharacteristic, complemented
37#16861137C2212yescomplemented
38#16861136C2212yescomplemented
39#16861135C2212yescomplemented
40#16861126C2211yescharacteristic, complemented
41#16861125C2212yescomplemented
42H(5,𝔽2)×C127C2211yescharacteristic
43#137×C127C2212yescomplemented
44#124×C127C2211yescharacteristic, complemented
45#104×C127C2212yescomplemented
46#100×C127C2211yescharacteristic, complemented
47C23×D127—22no—
48C23×D127D412yes—
49#14523249—22no—
50#14523249D412yes—
51#14523248—24no—
52#14523248—22no—
53#14523248—22no—
54#14523248C2312yes—
55#14523247—24no—
56#14523247—22no—
57#14523247—22no—
58#14523247D412yes—
59C2×Dic254—21no—
60C2×Dic254C2311yescharacteristic, complemented
61Q8×D127—22no—
62Q8×D127—22no—
63Q8×D127D412yes—
64C2×D508—22no—
65C2×D508—22no—
66C2×D508C2311yescharacteristic
67C2×C22⋊D127—22no—
68C2×C22⋊D127—24no—
69C2×C22⋊D127—22no—
70C2×C22⋊D127D412yes—
71D4×D127—24no—
72D4×D127—24no—
73D4×D127—24no—
74D4×D127—22no—
75D4×D127—24no—
76D4×D127—22no—
77D4×D127—22no—
78D4×D127—22no—
79D4×D127—22no—
80D4×D127D412yes—
81C2×C4×D127—22no—
82C2×C4×D127—22no—
83C2×C4×D127—21no—
84C2×C4×D127—21no—
85C2×C4×D127C2311yescharacteristic
86Dic508—22no—
87#14523238—22no—
88#14523238—22no—
89#14523237—22no—
90#14523236—22no—
91#14523235—22no—
92#14523232C2312yescomplemented
93#14523231—22no—
94C8×D127—22no—
95C22⋊Dic127—22no—
96(C2×D127)⋊C4—22no—
97(C2×D127)⋊C4—22no—
98C4×Dic127—21no—
99C4×Dic127C2311yescharacteristic, complemented
100#41×C127—22no—
101#41×C127C2312yes—
102C2×Q8×C127C2311yescharacteristic, complemented
103C2×D4×C127—22no—
104C2×D4×C127—22no—
105C2×D4×C127C2311yescharacteristic
106Q16×C127—22no—
107SD16×C127—22no—
108M16×C127C2312yescomplemented
109C22⋊C4×C127—22no—
110C42×C127C2311yescharacteristic, complemented
111C22×D127—24no—
112C22×D127—22no—
113C22×D127—22no—
114C22×D127—22no—
115C22×D127—22no—
116C22×D127—44no—
117C22×D127—84no—
118C22×D127C2×D411yescharacteristic
119C22×D127C2×D412yes—
120Dic254—22no—
121Dic254—22no—
122Dic254—42no—
123Dic254C2×D412yes—
124D508—24no—
125D508—22no—
126D508—22no—
127D508C2×D412yes—
128C22⋊D127—24no—
129C22⋊D127—24no—
130C22⋊D127—22no—
131C22⋊D127—22no—
132C22⋊D127—24no—
133C22⋊D127—22no—
134C2×Dic127—22no—
135C2×Dic127—22no—
136C2×Dic127—21no—
137C2×Dic127—21no—
138C2×Dic127C2×D411yescharacteristic
139C4×D127—24no—
140C4×D127—22no—
141C4×D127—22no—
142C4×D127—22no—
143C4×D127—42no—
144C4×D127—41no—
145C4×D127—41no—
146C4×D127C2×D412yes—
147C4×D127C2×D412yes—
148C127⋊C8—22no—
149C23×C127—22no—
150C23×C127C2×D412yes—
151Q8×C127—22no—
152Q8×C127C2×D412yes—
153D4×C127—24no—
154D4×C127—22no—
155D4×C127—22no—
156D4×C127C2×D412yes—
157C2×C4×C127—22no—
158C2×C4×C127—22no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127C2411yescharacteristic
162C8×C127—22no—
163C22×C127—22no—
164C22×C127—22no—
165C22×C127—42no—
166C22×C127—82no—
167C22×C127C22×D411yescharacteristic
168C2×D127—22no—
169C2×D127—22no—
170C2×D127—22no—
171C2×D127—44no—
172C2×D127—44no—
173C2×D127—42no—
174C2×D127—82no—
175C2×D127—84no—
176C2×D127—82no—
177C2×D127#12011yescharacteristic
178C2×D127#12011yescharacteristic
179C4×C127—22no—
180C4×C127—41no—
181C4×C127—41no—
182C4×C127C22×D412yes—
183Dic127—22no—
184Dic127—41no—
185Dic127—41no—
186Dic127#12012yes—
187C2×C127—21no—
188C2×C127—42no—
189C2×C127—42no—
190C2×C127C2×#12011yescharacteristic
191D127—21no—
192D127—42no—
193D127—42no—
194D127#45512yesdirect product factor
195C2×#455—1271nomaximal, Sylow
196C127C2×#45511yescharacteristic, complemented, Sylow
197C2×H(5,𝔽2)—1271no—
198C2×#137—1272no—
199C2×#124—1271no—
200#455—1271no—
201#455—1272no—
202#455—1271no—
203#455—1272no—
204#455—1271no—
205#455D12711yescharacteristic, direct product factor
206C2×#104—1272no—
207C2×#100—1271no—
208H(5,𝔽2)—1272no—
209H(5,𝔽2)—1271no—
210H(5,𝔽2)—2542no—
211H(5,𝔽2)—2542no—
212H(5,𝔽2)—2542no—
213H(5,𝔽2)C2×D12711yescharacteristic
214C2×#41—1272no—
215C2×#41—2542no—
216C22×Q8—1271no—
217C22×D4—1271no—
218C22×D4—2542no—
219C22×D4—2542no—
220#137—1272no—
221#137—1272no—
222#137—1272no—
223#137—2542no—
224#137—2542no—
225#137C2×D12712yescomplemented
226C2×Q16—2542no—
227C2×SD16—2542no—
228C2×M16—1272no—
229#124—1271no—
230#124—1271no—
231#124—1271no—
232#124—2542no—
233#124C2×D12711yescharacteristic, complemented
234C2×C22⋊C4—2542no—
235C2×C42—1271no—
236#104—1272no—
237#104—1272no—
238#104—1272no—
239#104C2×D12712yescomplemented
240#100—1271no—
241#100—1272no—
242#100C2×D12711yescharacteristic, complemented
243C24—1272no—
244C24—2542no—
245#41—22no—
246#41—1272no—
247#41—1272no—
248#41—1272no—
249#41—2542no—
250#41—2542no—
251#41—2544no—
252#41—2542no—
253#41—2542no—
254#41—2544no—
255#41—2542no—
256#41C22×D12712yes—
257C2×Q8—1272no—
258C2×Q8—1271no—
259C2×Q8—2541no—
260C2×Q8—2542no—
261C2×Q8—2542no—
262C2×Q8C22×D12711yescharacteristic, complemented
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—1272no—
266C2×D4—1272no—
267C2×D4—1271no—
268C2×D4—2542no—
269C2×D4—2542no—
270C2×D4—2542no—
271C2×D4—2542no—
272C2×D4—2542no—
273C2×D4—2542no—
274C2×D4—2544no—
275C2×D4—2544no—
276C2×D4—2544no—
277C2×D4—2544no—
278C2×D4—2544no—
279C2×D4—2542no—
280C2×D4—2542no—
281C2×D4—2542no—
282C2×D4C22×D12711yescharacteristic
283C22×C4—1271no—
284C22×C4—2542no—
285C22×C4—2541no—
286C22×C4—2541no—
287C22×C4—2542no—
288Q16—22no—
289Q16—2542no—
290Q16—2542no—
291Q16—2542no—
292SD16—22no—
293SD16—2542no—
294SD16—2542no—
295SD16—2542no—
296M16—1272no—
297M16—2542no—
298M16C22×D12712yescomplemented
299C2×C8—2542no—
300C22⋊C4—22no—
301C22⋊C4—2542no—
302C22⋊C4—2542no—
303C22⋊C4—2542no—
304C42—1271no—
305C42—2541no—
306C42C22×D12711yescharacteristic, complemented
307C23—22no—
308C23—1271no—
309C23—1272no—
310C23—10164no—
311C23—5084no—
312C23—2542no—
313C23—2542no—
314C23—2542no—
315C23—2542no—
316C23—2544no—
317C23D4×D12712yes—
318Q8—22no—
319Q8—1272no—
320Q8—5082no—
321Q8—2542no—
322Q8—2542no—
323Q8D4×D12712yes—
324D4—24no—
325D4—22no—
326D4—22no—
327D4—1272no—
328D4—2542no—
329D4—2544no—
330D4—2544no—
331D4—2542no—
332D4—2542no—
333D4—2544no—
334D4—2544no—
335D4—2542no—
336D4—2542no—
337D4D4×D12712yes—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—1271no—
343C2×C4—1272no—
344C2×C4—1272no—
345C2×C4—5081no—
346C2×C4—5081no—
347C2×C4—5082no—
348C2×C4—2542no—
349C2×C4—2542no—
350C2×C4—2544no—
351C2×C4—2542no—
352C2×C4—2541no—
353C2×C4—2541no—
354C2×C4—2542no—
355C2×C4—2542no—
356C2×C4C23×D12711yescharacteristic
357C8—22no—
358C8—2542no—
359C22—22no—
360C22—22no—
361C22—42no—
362C22—82no—
363C22—1271no—
364C22—1271no—
365C22—10162no—
366C22—10164no—
367C22—10162no—
368C22—5082no—
369C22—5084no—
370C22—5084no—
371C22—2542no—
372C22—2542no—
373C22—2542no—
374C22C2×D4×D12711yescharacteristic
375C4—22no—
376C4—41no—
377C4—41no—
378C4—1272no—
379C4—5081no—
380C4—5081no—
381C4—2542no—
382C4C2×D4×D12712yes—
383C2—21no—
384C2—42no—
385C2—42no—
386C2—1272no—
387C2—5082no—
388C2—5082no—
389C2—2541no—
390C2#120×D12711yescharacteristic
subgroupG/H
centerC2#120×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#455×C127C2
socleC2×C127C2×#120
prime radicalH(5,𝔽2)×D127C2
hypercenter#455D127
hypocenterC127C2×#455
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. #124×D127C2semidirect
4. #22550056C2semidirect
5. #22550055C2semidirect
6. #22550054C2semidirect
7. #22550053C2semidirect
8. #22550052C2semidirect
9. #104×D127C2semidirect
10. #100×D127C2semidirect
11. #455×C127C2semidirect
19. C2×Q8×D127C22semidirect
23. #16861242C22semidirect
25. #16861240C22semidirect
26. #16861239C22semidirect
30. M16×D127C22semidirect
31. #16861198C22semidirect
32. #16861197C22semidirect
33. #16861196C22semidirect
36. C42×D127C22semidirect
37. #16861137C22semidirect
38. #16861136C22semidirect
39. #16861135C22semidirect
40. #16861126C22semidirect
41. #16861125C22semidirect
43. #137×C127C22semidirect
44. #124×C127C22semidirect
45. #104×C127C22semidirect
46. #100×C127C22semidirect
60. C2×Dic254C23semidirect
92. #14523232C23semidirect
99. C4×Dic127C23semidirect
102. C2×Q8×C127C23semidirect
108. M16×C127C23semidirect
110. C42×C127C23semidirect
194. D127#455direct (N × H)
196. C127C2×#455semidirect
205. #455D127direct (N × H)
225. #137C2×D127semidirect
233. #124C2×D127semidirect
239. #104C2×D127semidirect
242. #100C2×D127semidirect
262. C2×Q8C22×D127semidirect
298. M16C22×D127semidirect
306. C42C22×D127semidirect

Maximal subgroups

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

subgroupindex
H(5,𝔽2)×D1272
#137×D1272
#124×D1272
#225500562
#225500552
#225500542
#225500532
#225500522
#104×D1272
#100×D1272
#455×C1272
C2×#455127