Finite Group Databasebeta

← #453×D127 #30648051

Subgroups — #453×D127 #30648051

Order 16256.

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

normal subgroups only

This group has 844 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, complemented
2AGL(1,ℤ/8)×D127C212yesmaximal, complemented
3#127×D127C211yesmaximal, characteristic, complemented
4#22550046C211yesmaximal, characteristic, complemented
5#22550045C212yesmaximal, complemented
6#22550044C211yesmaximal, characteristic, complemented
7#22550043C211yesmaximal, characteristic, complemented
8#22550042C212yesmaximal, complemented
9#104×D127C212yesmaximal, complemented
10#100×D127C211yesmaximal, characteristic, complemented
11#453×C127C211yesmaximal, characteristic, complemented
12#16861255—22no—
13#16861255C2212yescomplemented
14#16861254—22no—
15#16861254—22no—
16#16861254C2211yescharacteristic, complemented
17#41×D127—22no—
18#41×D127C2212yes—
19C2×D4×D127—22no—
20C2×D4×D127—22no—
21C2×D4×D127—22no—
22C2×D4×D127C2211yescharacteristic
23C2×D4×D127C2211yescharacteristic, complemented
24#16861237C2212yescomplemented
25#16861236—22no—
26#16861235C2212yescomplemented
27#16861234C2212yescomplemented
28#16861233—22no—
29SD16×D127—22no—
30D8×D127—22no—
31M16×D127C2212yescomplemented
32#16861206C2211yescharacteristic, complemented
33#16861205—22no—
34#16861205C2211yescharacteristic, complemented
35#16861205C2211yescharacteristic, complemented
36C42×D127C2211yescharacteristic, complemented
37#16861137C2212yescomplemented
38#16861136C2212yescomplemented
39#16861135C2212yescomplemented
40#16861126C2211yescharacteristic, complemented
41#16861125C2212yescomplemented
42H(5,𝔽2)×C127C2211yescharacteristic, complemented
43AGL(1,ℤ/8)×C127C2212yescomplemented
44#127×C127C2211yescharacteristic, complemented
45#104×C127C2212yescomplemented
46#100×C127C2211yescharacteristic, complemented
47C23×D127—22no—
48C23×D127—21no—
49C23×D127D412yes—
50#14523249—22no—
51#14523249D412yescomplemented
52#14523248—24no—
53#14523248—22no—
54#14523248—22no—
55#14523248C2312yes—
56#14523247—24no—
57#14523247—22no—
58#14523247—22no—
59#14523247D412yescomplemented
60Q8×D127D412yescomplemented
61C2×D508—22no—
62C2×D508—22no—
63C2×D508—22no—
64C2×D508C2311yescharacteristic
65C2×D508C2311yescharacteristic, complemented
66C2×C22⋊D127—22no—
67C2×C22⋊D127—24no—
68C2×C22⋊D127—22no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127—22no—
71C2×C22⋊D127—22no—
72C2×C22⋊D127D412yes—
73D4×D127—24no—
74D4×D127—24no—
75D4×D127—24no—
76D4×D127—22no—
77D4×D127—24no—
78D4×D127—22no—
79D4×D127—22no—
80D4×D127—22no—
81D4×D127—22no—
82D4×D127—22no—
83D4×D127—22no—
84D4×D127—48no—
85D4×D127D412yescomplemented
86C2×C4×D127—22no—
87C2×C4×D127—21no—
88C2×C4×D127—22no—
89C2×C4×D127C2311yescharacteristic
90#14523237—22no—
91#14523236—22no—
92#14523235—22no—
93D1016—22no—
94#14523233—22no—
95#14523233—22no—
96#14523232C2312yescomplemented
97#14523231—22no—
98C8×D127—22no—
99C4×Dic127—21no—
100C4×Dic127C2311yescharacteristic, complemented
101#41×C127—22no—
102#41×C127C2312yes—
103C2×D4×C127—22no—
104C2×D4×C127—22no—
105C2×D4×C127—22no—
106C2×D4×C127C2311yescharacteristic
107C2×D4×C127C2311yescharacteristic, complemented
108SD16×C127—22no—
109D8×C127—22no—
110M16×C127C2312yescomplemented
111C42×C127C2311yescharacteristic, complemented
112C22×D127—24no—
113C22×D127—22no—
114C22×D127—22no—
115C22×D127—22no—
116C22×D127—22no—
117C22×D127—21no—
118C22×D127—44no—
119C22×D127—44no—
120C22×D127—41no—
121C22×D127—41no—
122C22×D127—84no—
123C22×D127C2×D411yescharacteristic
124C22×D127C2×D412yes—
125Dic254—22no—
126Dic254C2×D412yescomplemented
127D508—24no—
128D508—22no—
129D508—22no—
130D508—22no—
131D508—44no—
132D508C2×D412yescomplemented
133C22⋊D127—24no—
134C22⋊D127—24no—
135C22⋊D127—22no—
136C22⋊D127—22no—
137C22⋊D127—24no—
138C22⋊D127—22no—
139C22⋊D127—44no—
140C22⋊D127—44no—
141C22⋊D127—42no—
142C2×Dic127—22no—
143C2×Dic127—21no—
144C2×Dic127—22no—
145C2×Dic127C2×D411yescharacteristic
146C4×D127—24no—
147C4×D127—24no—
148C4×D127—22no—
149C4×D127—22no—
150C4×D127—22no—
151C4×D127C2×D412yes—
152C4×D127C2×D412yes—
153C127⋊C8—22no—
154C23×C127—22no—
155C23×C127—21no—
156C23×C127C2×D412yes—
157Q8×C127C2×D412yescomplemented
158D4×C127—24no—
159D4×C127—22no—
160D4×C127—22no—
161D4×C127—22no—
162D4×C127—44no—
163D4×C127C2×D412yescomplemented
164C2×C4×C127—22no—
165C2×C4×C127—21no—
166C2×C4×C127—22no—
167C2×C4×C127C2411yescharacteristic
168C8×C127—22no—
169C22×C127—22no—
170C22×C127—22no—
171C22×C127—42no—
172C22×C127—42no—
173C22×C127—41no—
174C22×C127—82no—
175C22×C127C22×D411yescharacteristic
176C2×D127—22no—
177C2×D127—22no—
178C2×D127—22no—
179C2×D127—44no—
180C2×D127—44no—
181C2×D127—42no—
182C2×D127—42no—
183C2×D127—41no—
184C2×D127—82no—
185C2×D127—82no—
186C2×D127—84no—
187C2×D127—82no—
188C2×D127—82no—
189C2×D127#12011yescharacteristic
190C2×D127#12011yescharacteristic
191C4×C127—22no—
192C4×C127—22no—
193C4×C127C22×D412yes—
194Dic127—22no—
195Dic127—22no—
196Dic127#12012yes—
197C2×C127—21no—
198C2×C127—42no—
199C2×C127—42no—
200C2×C127—81no—
201C2×C127C2×#12011yescharacteristic
202D127—21no—
203D127—42no—
204D127—42no—
205D127—81no—
206D127#45312yesdirect product factor
207C2×#453—1271nomaximal, Sylow
208C127C2×#45311yescharacteristic, complemented, Sylow
209C2×H(5,𝔽2)—1271no—
210C2×AGL(1,ℤ/8)—1272no—
211C2×#127—1271no—
212#453—1271no—
213#453—1272no—
214#453—1271no—
215#453—1272no—
216#453—1271no—
217#453D12711yescharacteristic, direct product factor
218C2×#104—1272no—
219C2×#100—1271no—
220H(5,𝔽2)—1272no—
221H(5,𝔽2)—1271no—
222H(5,𝔽2)—2542no—
223H(5,𝔽2)—2542no—
224H(5,𝔽2)—2542no—
225H(5,𝔽2)C2×D12711yescharacteristic, complemented
226C2×#41—1272no—
227C2×#41—2542no—
228C22×D4—1271no—
229C22×D4—1271no—
230C22×D4—2542no—
231C22×D4—2542no—
232C22×D4—2542no—
233AGL(1,ℤ/8)—1272no—
234AGL(1,ℤ/8)—1272no—
235AGL(1,ℤ/8)—1272no—
236AGL(1,ℤ/8)—2542no—
237AGL(1,ℤ/8)—2542no—
238AGL(1,ℤ/8)C2×D12712yescomplemented
239C2×SD16—2542no—
240C2×D8—2542no—
241C2×M16—1272no—
242#127—1271no—
243#127—1271no—
244#127—1271no—
245#127—2542no—
246#127C2×D12711yescharacteristic, complemented
247C2×C42—1271no—
248#104—1272no—
249#104—1272no—
250#104—1272no—
251#104C2×D12712yescomplemented
252#100—1271no—
253#100—1272no—
254#100C2×D12711yescharacteristic, complemented
255C24—1272no—
256C24—2541no—
257C24—2542no—
258#41—22no—
259#41—1272no—
260#41—1272no—
261#41—1272no—
262#41—2542no—
263#41—2542no—
264#41—2544no—
265#41—2542no—
266#41—2542no—
267#41—2544no—
268#41—2542no—
269#41C22×D12712yes—
270C2×Q8—1272no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—1272no—
275C2×D4—1272no—
276C2×D4—1271no—
277C2×D4—1271no—
278C2×D4—5088no—
279C2×D4—2542no—
280C2×D4—2542no—
281C2×D4—2542no—
282C2×D4—2542no—
283C2×D4—2542no—
284C2×D4—2542no—
285C2×D4—2542no—
286C2×D4—2541no—
287C2×D4—2542no—
288C2×D4—2542no—
289C2×D4—2544no—
290C2×D4—2544no—
291C2×D4—2544no—
292C2×D4—2544no—
293C2×D4—2542no—
294C2×D4—2542no—
295C2×D4—2544no—
296C2×D4—2542no—
297C2×D4—2542no—
298C2×D4—2542no—
299C2×D4C22×D12711yescharacteristic
300C2×D4C22×D12711yescharacteristic, complemented
301C22×C4—1271no—
302C22×C4—2542no—
303C22×C4—2541no—
304C22×C4—2542no—
305SD16—22no—
306SD16—2542no—
307SD16—2542no—
308SD16—2542no—
309D8—22no—
310D8—2542no—
311D8—2542no—
312D8—2542no—
313M16—1272no—
314M16—2542no—
315M16C22×D12712yescomplemented
316C2×C8—2542no—
317C42—1271no—
318C42—2541no—
319C42C22×D12711yescharacteristic, complemented
320C23—22no—
321C23—21no—
322C23—1271no—
323C23—1272no—
324C23—10164no—
325C23—5081no—
326C23—5081no—
327C23—5084no—
328C23—5084no—
329C23—2542no—
330C23—2542no—
331C23—2542no—
332C23—2542no—
333C23—2544no—
334C23—2541no—
335C23D4×D12712yes—
336Q8—1272no—
337Q8—2542no—
338Q8D4×D12712yescomplemented
339D4—24no—
340D4—22no—
341D4—22no—
342D4—22no—
343D4—44no—
344D4—1272no—
345D4—5084no—
346D4—5082no—
347D4—5084no—
348D4—5084no—
349D4—2542no—
350D4—2544no—
351D4—2544no—
352D4—2542no—
353D4—2542no—
354D4—2544no—
355D4—2544no—
356D4—2542no—
357D4—2542no—
358D4—2542no—
359D4D4×D12712yescomplemented
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—1271no—
364C2×C4—1272no—
365C2×C4—1272no—
366C2×C4—2542no—
367C2×C4—2542no—
368C2×C4—2544no—
369C2×C4—2544no—
370C2×C4—2542no—
371C2×C4—2542no—
372C2×C4—2541no—
373C2×C4—2542no—
374C2×C4C23×D12711yescharacteristic
375C8—22no—
376C8—2542no—
377C22—22no—
378C22—22no—
379C22—42no—
380C22—42no—
381C22—41no—
382C22—82no—
383C22—1271no—
384C22—1271no—
385C22—10162no—
386C22—10164no—
387C22—10162no—
388C22—10162no—
389C22—10162no—
390C22—5082no—
391C22—5082no—
392C22—5084no—
393C22—5084no—
394C22—5081no—
395C22—2542no—
396C22—2542no—
397C22—2542no—
398C22C2×D4×D12711yescharacteristic
399C4—22no—
400C4—22no—
401C4—1272no—
402C4—2542no—
403C4—2542no—
404C4C2×D4×D12712yes—
405C2—21no—
406C2—42no—
407C2—42no—
408C2—81no—
409C2—1272no—
410C2—10161no—
411C2—5082no—
412C2—5082no—
413C2—2541no—
414C2#120×D12711yescharacteristic
subgroupG/H
centerC2#120×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#453×C127C2
socleC2×C127C2×#120
hypercenter#453D127
hypocenterC127C2×#453
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. H(5,𝔽2)×D127C2semidirect
2. AGL(1,ℤ/8)×D127C2semidirect
3. #127×D127C2semidirect
4. #22550046C2semidirect
5. #22550045C2semidirect
6. #22550044C2semidirect
7. #22550043C2semidirect
8. #22550042C2semidirect
9. #104×D127C2semidirect
10. #100×D127C2semidirect
11. #453×C127C2semidirect
13. #16861255C22semidirect
16. #16861254C22semidirect
23. C2×D4×D127C22semidirect
24. #16861237C22semidirect
26. #16861235C22semidirect
27. #16861234C22semidirect
31. M16×D127C22semidirect
32. #16861206C22semidirect
34. #16861205C22semidirect
35. #16861205C22semidirect
36. C42×D127C22semidirect
37. #16861137C22semidirect
38. #16861136C22semidirect
39. #16861135C22semidirect
40. #16861126C22semidirect
41. #16861125C22semidirect
42. H(5,𝔽2)×C127C22semidirect
43. AGL(1,ℤ/8)×C127C22semidirect
44. #127×C127C22semidirect
45. #104×C127C22semidirect
46. #100×C127C22semidirect
51. #14523249D4semidirect
59. #14523247D4semidirect
60. Q8×D127D4semidirect
65. C2×D508C23semidirect
85. D4×D127D4semidirect
96. #14523232C23semidirect
100. C4×Dic127C23semidirect
107. C2×D4×C127C23semidirect
110. M16×C127C23semidirect
111. C42×C127C23semidirect
126. Dic254C2×D4semidirect
132. D508C2×D4semidirect
157. Q8×C127C2×D4semidirect
163. D4×C127C2×D4semidirect
206. D127#453direct (N × H)
208. C127C2×#453semidirect
217. #453D127direct (N × H)
225. H(5,𝔽2)C2×D127semidirect
238. AGL(1,ℤ/8)C2×D127semidirect
246. #127C2×D127semidirect
251. #104C2×D127semidirect
254. #100C2×D127semidirect
300. C2×D4C22×D127semidirect
315. M16C22×D127semidirect
319. C42C22×D127semidirect
338. Q8D4×D127semidirect
359. D4D4×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
AGL(1,ℤ/8)×D1272
#127×D1272
#225500462
#225500452
#225500442
#225500432
#225500422
#104×D1272
#100×D1272
#453×C1272
C2×#453127