Finite Group Databasebeta

← #454×D127 #30648063

Subgroups — #454×D127 #30648063

Order 16256.

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

normal subgroups only

This group has 728 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
12−1+4×D127C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D127C212yesmaximal, complemented
3#124×D127C211yesmaximal, characteristic, complemented
4#22550051C211yesmaximal, characteristic, complemented
5#22550050C212yesmaximal, complemented
6#22550049C211yesmaximal, characteristic, complemented
7#22550048C211yesmaximal, characteristic, complemented
8#22550047C212yesmaximal, complemented
9#104×D127C212yesmaximal, complemented
10#101×D127C211yesmaximal, characteristic, complemented
11#454×C127C211yesmaximal, characteristic, complemented
12#16861257—22no—
13#16861257C2211yescharacteristic, complemented
14#16861256—22no—
15#16861256—22no—
16#16861256C2212yescomplemented
17#41×D127—22no—
18#41×D127—22no—
19#41×D127C2212yes—
20C2×Q8×D127—22no—
21C2×Q8×D127C2211yescharacteristic
22C2×D4×D127C2211yescharacteristic
23#16861237C2212yescomplemented
24#16861236—22no—
25#16861235C2212yescomplemented
26#16861234C2212yescomplemented
27#16861233—22no—
28SD16×D127—22no—
29D8×D127—22no—
30M16×D127C2212yes—
31#16861198C2211yescharacteristic, complemented
32#16861197C2211yescharacteristic, complemented
33#16861196C2211yescharacteristic, complemented
34#16861195—22no—
35C22⋊C4×D127—22no—
36C42×D127C2211yescharacteristic
37#16861137C2212yescomplemented
38#16861136C2212yescomplemented
39#16861135C2212yescomplemented
40#16861128C2211yescharacteristic, complemented
41#16861127C2212yescomplemented
422−1+4×C127C2211yescharacteristic, complemented
43AGL(1,ℤ/8)×C127C2212yescomplemented
44#124×C127C2211yescharacteristic, complemented
45#104×C127C2212yescomplemented
46#101×C127C2211yescharacteristic, complemented
47C23×D127—21no—
48#14523249—22no—
49#14523249—22no—
50#14523249D412yes—
51#14523248—24no—
52#14523248—24no—
53#14523248—22no—
54#14523248—22no—
55#14523248—22no—
56#14523248C2312yes—
57#14523247—24no—
58#14523247—22no—
59#14523247—24no—
60#14523247—22no—
61#14523247—22no—
62#14523247D412yes—
63C2×Dic254—24no—
64C2×Dic254—22no—
65C2×Dic254D412yes—
66C2×Dic254C2311yescharacteristic
67Q8×D127—22no—
68Q8×D127—24no—
69Q8×D127—22no—
70Q8×D127—22no—
71Q8×D127—22no—
72Q8×D127—22no—
73Q8×D127D412yes—
74C2×D508C2311yescharacteristic
75C2×C22⋊D127—21no—
76D4×D127—22no—
77D4×D127—22no—
78D4×D127—22no—
79D4×D127—22no—
80D4×D127D412yes—
81C2×C4×D127—24no—
82C2×C4×D127—22no—
83C2×C4×D127—21no—
84C2×C4×D127D412yes—
85C2×C4×D127C2311yescharacteristic
86#14523237—22no—
87#14523236—22no—
88#14523235—22no—
89D1016—22no—
90#14523233—22no—
91#14523233—22no—
92#14523232C2312yes—
93#14523231—22no—
94C8×D127—22no—
95C22⋊Dic127—22no—
96(C2×D127)⋊C4—22no—
97(C2×D127)⋊C4—22no—
98C4×Dic127—21no—
99C4×Dic127C2311yescharacteristic
100#41×C127—22no—
101#41×C127—22no—
102#41×C127C2312yes—
103C2×Q8×C127—22no—
104C2×Q8×C127C2311yescharacteristic
105C2×D4×C127C2311yescharacteristic
106SD16×C127—22no—
107D8×C127—22no—
108M16×C127C2312yes—
109C22⋊C4×C127—22no—
110C42×C127C2311yescharacteristic
111C22×D127—22no—
112C22×D127—21no—
113C22×D127—44no—
114C22×D127—41no—
115C22×D127—41no—
116C22×D127C2×D411yescharacteristic
117Dic254—22no—
118Dic254—24no—
119Dic254—22no—
120Dic254—24no—
121Dic254—22no—
122Dic254C2×D412yes—
123D508—22no—
124D508—22no—
125D508—22no—
126D508C2×D412yes—
127C22⋊D127—24no—
128C22⋊D127—22no—
129C22⋊D127—22no—
130C22⋊D127—22no—
131C22⋊D127—42no—
132C2×Dic127—24no—
133C2×Dic127—22no—
134C2×Dic127—21no—
135C2×Dic127C2×D412yes—
136C2×Dic127C2×D411yescharacteristic
137C4×D127—24no—
138C4×D127—24no—
139C4×D127—22no—
140C4×D127—22no—
141C4×D127—22no—
142C4×D127—22no—
143C4×D127—22no—
144C4×D127—42no—
145C4×D127C2×D412yes—
146C4×D127C2×D412yes—
147C127⋊C8—22no—
148C23×C127—21no—
149Q8×C127—22no—
150Q8×C127—22no—
151Q8×C127C2×D412yes—
152D4×C127—22no—
153D4×C127—22no—
154D4×C127—22no—
155D4×C127C2×D412yes—
156C2×C4×C127—24no—
157C2×C4×C127—22no—
158C2×C4×C127—21no—
159C2×C4×C127C2×D412yes—
160C2×C4×C127C2411yescharacteristic
161C8×C127—22no—
162C22×C127—22no—
163C22×C127—42no—
164C22×C127—41no—
165C22×C127C22×D411yescharacteristic
166C2×D127—22no—
167C2×D127—22no—
168C2×D127—44no—
169C2×D127—42no—
170C2×D127—41no—
171C2×D127—82no—
172C2×D127—82no—
173C2×D127#12011yescharacteristic
174C2×D127#12011yescharacteristic
175C4×C127—22no—
176C4×C127—22no—
177C4×C127—41no—
178C4×C127C22×D412yes—
179Dic127—22no—
180Dic127—22no—
181Dic127—41no—
182Dic127#12012yes—
183C2×C127—21no—
184C2×C127—42no—
185C2×C127—81no—
186C2×C127C2×#12011yescharacteristic
187D127—21no—
188D127—42no—
189D127—81no—
190D127#45412yesdirect product factor
191C2×#454—1271nomaximal, Sylow
192C127C2×#45411yescharacteristic, complemented, Sylow
193C2×2−1+4—1271no—
194C2×AGL(1,ℤ/8)—1272no—
195C2×#124—1271no—
196#454—1271no—
197#454—1272no—
198#454—1271no—
199#454—1272no—
200#454—1271no—
201#454D12711yescharacteristic, direct product factor
202C2×#104—1272no—
203C2×#101—1271no—
2042−1+4—1272no—
2052−1+4—1271no—
2062−1+4—2542no—
2072−1+4—2542no—
2082−1+4—2542no—
2092−1+4C2×D12711yescharacteristic, complemented
210C2×#41—1272no—
211C2×#41—2542no—
212C2×#41—2542no—
213C22×Q8—1271no—
214C22×Q8—2542no—
215C22×D4—1271no—
216AGL(1,ℤ/8)—1272no—
217AGL(1,ℤ/8)—1272no—
218AGL(1,ℤ/8)—1272no—
219AGL(1,ℤ/8)—2542no—
220AGL(1,ℤ/8)—2542no—
221AGL(1,ℤ/8)C2×D12712yescomplemented
222C2×SD16—2542no—
223C2×D8—2542no—
224C2×M16—1272no—
225#124—1271no—
226#124—1271no—
227#124—1271no—
228#124—2542no—
229#124C2×D12711yescharacteristic, complemented
230C2×C22⋊C4—2542no—
231C2×C42—1271no—
232#104—1272no—
233#104—1272no—
234#104—1272no—
235#104C2×D12712yescomplemented
236#101—1271no—
237#101—1272no—
238#101C2×D12711yescharacteristic, complemented
239C24—2541no—
240#41—22no—
241#41—22no—
242#41—1272no—
243#41—1272no—
244#41—1272no—
245#41—2542no—
246#41—2542no—
247#41—2544no—
248#41—2544no—
249#41—2542no—
250#41—2544no—
251#41—2542no—
252#41—2544no—
253#41—2542no—
254#41—2542no—
255#41—2542no—
256#41—2542no—
257#41C22×D12712yes—
258C2×Q8—22no—
259C2×Q8—1272no—
260C2×Q8—1271no—
261C2×Q8—1272no—
262C2×Q8—2542no—
263C2×Q8—2544no—
264C2×Q8—2542no—
265C2×Q8—2542no—
266C2×Q8—2542no—
267C2×Q8—2542no—
268C2×Q8—2544no—
269C2×Q8—2542no—
270C2×Q8C22×D12711yescharacteristic
271C2×D4—1272no—
272C2×D4—1271no—
273C2×D4—2542no—
274C2×D4—2542no—
275C2×D4—2541no—
276C2×D4—2542no—
277C2×D4—2542no—
278C2×D4C22×D12711yescharacteristic
279C22×C4—1271no—
280C22×C4—1272no—
281C22×C4—2542no—
282C22×C4—2541no—
283C22×C4—2544no—
284SD16—22no—
285SD16—2542no—
286SD16—2542no—
287SD16—2542no—
288D8—22no—
289D8—2542no—
290D8—2542no—
291D8—2542no—
292M16—1272no—
293M16—2542no—
294M16C22×D12712yes—
295C2×C8—2542no—
296C22⋊C4—22no—
297C22⋊C4—2542no—
298C22⋊C4—2542no—
299C22⋊C4—2542no—
300C42—1271no—
301C42—2541no—
302C42C22×D12711yescharacteristic
303C23—21no—
304C23—1271no—
305C23—5081no—
306C23—5081no—
307C23—5084no—
308C23—2542no—
309C23—2541no—
310Q8—22no—
311Q8—22no—
312Q8—1272no—
313Q8—2542no—
314Q8—2544no—
315Q8—2544no—
316Q8—2542no—
317Q8—2542no—
318Q8D4×D12712yes—
319D4—22no—
320D4—22no—
321D4—22no—
322D4—1272no—
323D4—5082no—
324D4—2542no—
325D4—2542no—
326D4—2542no—
327D4—2542no—
328D4—2544no—
329D4—2542no—
330D4—2542no—
331D4D4×D12712yes—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—1271no—
336C2×C4—1272no—
337C2×C4—1272no—
338C2×C4—1272no—
339C2×C4—5082no—
340C2×C4—2542no—
341C2×C4—2542no—
342C2×C4—2542no—
343C2×C4—2542no—
344C2×C4—2544no—
345C2×C4—2544no—
346C2×C4—2542no—
347C2×C4—2541no—
348C2×C4—2544no—
349C2×C4—2542no—
350C2×C4D4×D12712yes—
351C2×C4C23×D12711yescharacteristic
352C8—22no—
353C8—2542no—
354C22—22no—
355C22—42no—
356C22—41no—
357C22—1271no—
358C22—1271no—
359C22—10162no—
360C22—10162no—
361C22—5082no—
362C22—5084no—
363C22—5081no—
364C22—2542no—
365C22—2542no—
366C22C2×D4×D12711yescharacteristic
367C4—22no—
368C4—22no—
369C4—41no—
370C4—1272no—
371C4—5081no—
372C4—2542no—
373C4—2542no—
374C4C2×D4×D12712yes—
375C2—21no—
376C2—42no—
377C2—81no—
378C2—1272no—
379C2—10161no—
380C2—5082no—
381C2—2541no—
382C2#120×D12711yescharacteristic
subgroupG/H
centerC2#120×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#454×C127C2
socleC2×C127C2×#120
hypercenter#454D127
hypocenterC127C2×#454
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. 2−1+4×D127C2semidirect
2. AGL(1,ℤ/8)×D127C2semidirect
3. #124×D127C2semidirect
4. #22550051C2semidirect
5. #22550050C2semidirect
6. #22550049C2semidirect
7. #22550048C2semidirect
8. #22550047C2semidirect
9. #104×D127C2semidirect
10. #101×D127C2semidirect
11. #454×C127C2semidirect
13. #16861257C22semidirect
16. #16861256C22semidirect
23. #16861237C22semidirect
25. #16861235C22semidirect
26. #16861234C22semidirect
31. #16861198C22semidirect
32. #16861197C22semidirect
33. #16861196C22semidirect
37. #16861137C22semidirect
38. #16861136C22semidirect
39. #16861135C22semidirect
40. #16861128C22semidirect
41. #16861127C22semidirect
42. 2−1+4×C127C22semidirect
43. AGL(1,ℤ/8)×C127C22semidirect
44. #124×C127C22semidirect
45. #104×C127C22semidirect
46. #101×C127C22semidirect
190. D127#454direct (N × H)
192. C127C2×#454semidirect
201. #454D127direct (N × H)
209. 2−1+4C2×D127semidirect
221. AGL(1,ℤ/8)C2×D127semidirect
229. #124C2×D127semidirect
235. #104C2×D127semidirect
238. #101C2×D127semidirect

Maximal subgroups

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

subgroupindex
2−1+4×D1272
AGL(1,ℤ/8)×D1272
#124×D1272
#225500512
#225500502
#225500492
#225500482
#225500472
#104×D1272
#101×D1272
#454×C1272
C2×#454127