Finite Group Databasebeta

← #30648088

Subgroups — #30648088

Order 16256.

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

normal subgroups only

This group has 762 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
1H(5,𝔽2)×D127C211yesmaximal, characteristic, complemented
2#22550675C212yesmaximal, complemented
3#22550444C211yesmaximal, characteristic, complemented
4#22550053C211yesmaximal, characteristic, complemented
5#22550052C212yesmaximal, complemented
6#22550046C211yesmaximal, characteristic, complemented
7#22550045C212yesmaximal, complemented
8#22550044C211yesmaximal, characteristic, complemented
9#22549891C212yesmaximal, complemented
10#100×D127C211yesmaximal, characteristic, complemented
11#455×C127C211yesmaximal, characteristic, complemented
12#16861255—22no—
13#16861255C2212yes—
14#16861254—22no—
15#16861254—22no—
16#16861254C2211yescharacteristic
17C2×#14523249C2211yescharacteristic, complemented
18#41×D127—22no—
19#41×D127C2212yes—
20C2×D4×D127—22no—
21C2×D4×D127—22no—
22C2×D4×D127C2211yescharacteristic
23#16861241—22no—
24#16861240C2212yescomplemented
25#16861237C2212yescomplemented
26#16861236—22no—
27#16861234C2212yescomplemented
28#16861232—22no—
29#16861230—22no—
30M16×D127C2212yescomplemented
31#16861206C2211yescharacteristic, complemented
32#16861205C2211yescharacteristic, complemented
33#16861198C2211yescharacteristic, complemented
34#16861182—22no—
35#16861179—22no—
36#16861166C2211yescharacteristic, 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#14523249—22no—
51#14523249—22no—
52#14523249D412yescomplemented
53#14523248—24no—
54#14523248—22no—
55#14523248—22no—
56#14523248C2312yes—
57#14523247—24no—
58#14523247—22no—
59#14523247—22no—
60#14523247D412yescomplemented
61Q8×D127D412yescomplemented
62C2×D508—22no—
63C2×D508—22no—
64C2×D508—21no—
65C2×D508—22no—
66C2×D508C2311yescharacteristic
67C2×D508C2311yescharacteristic, complemented
68C2×C22⋊D127—22no—
69C2×C22⋊D127—24no—
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×D127D412yescomplemented
83C2×C4×D127—22no—
84C2×C4×D127—22no—
85C2×C4×D127—21no—
86C2×C4×D127C2311yescharacteristic
87#14523238—22no—
88#14523237—22no—
89#14523235—22no—
90#14523235—22no—
91D1016—22no—
92#14523233—22no—
93#14523232C2312yescomplemented
94#14523231—22no—
95C8×D127—22no—
96Dic127⋊C4—21no—
97(C2×D127)⋊C4—22no—
98(C2×D127)⋊C4—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—21no—
117C22×D127—82no—
118C22×D127—84no—
119C22×D127C2×D412yes—
120C22×D127C2×D411yescharacteristic
121Dic254—22no—
122Dic254C2×D412yes—
123D508—24no—
124D508—22no—
125D508—22no—
126D508—22no—
127D508—41no—
128D508—41no—
129D508—82no—
130D508C2×D412yes—
131C22⋊D127—24no—
132C22⋊D127—24no—
133C22⋊D127—22no—
134C22⋊D127—22no—
135C22⋊D127—24no—
136C22⋊D127—22no—
137C22⋊D127—82no—
138C2×Dic127—22no—
139C2×Dic127—22no—
140C2×Dic127—21no—
141C2×Dic127C2×D411yescharacteristic
142C4×D127—24no—
143C4×D127—22no—
144C4×D127—22no—
145C4×D127—22no—
146C4×D127—42no—
147C4×D127C2×D412yes—
148C4×D127C2×D412yes—
149C127⋊C8—22no—
150C23×C127—22no—
151C23×C127C2×D412yes—
152Q8×C127—22no—
153Q8×C127C2×D412yes—
154D4×C127—24no—
155D4×C127—22no—
156D4×C127—22no—
157D4×C127C2×D412yes—
158C2×C4×C127—22no—
159C2×C4×C127—22no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127C2411yescharacteristic
163C8×C127—22no—
164C22×C127—22no—
165C22×C127—22no—
166C22×C127—42no—
167C22×C127—82no—
168C22×C127C22×D411yescharacteristic
169C2×D127—22no—
170C2×D127—22no—
171C2×D127—41no—
172C2×D127—44no—
173C2×D127—44no—
174C2×D127—42no—
175C2×D127—42no—
176C2×D127—41no—
177C2×D127—82no—
178C2×D127—84no—
179C2×D127—82no—
180C2×D127#12011yescharacteristic
181C2×D127#12011yescharacteristic
182C4×C127—22no—
183C4×C127—41no—
184C4×C127—41no—
185C4×C127C22×D412yes—
186Dic127—22no—
187Dic127—41no—
188Dic127#12012yes—
189C2×C127—21no—
190C2×C127—42no—
191C2×C127—42no—
192C2×C127C2×#12011yescharacteristic
193D127—21no—
194D127—21no—
195D127—42no—
196D127—42no—
197D127—81no—
198#3019—1271nomaximal, Sylow
199C127#301911yescharacteristic, complemented, Sylow
200C2×H(5,𝔽2)—1271no—
201#575—1272no—
202#535—1271no—
203#455—1272no—
204#455—1271no—
205#455D12711yescharacteristic, complemented
206#453—1271no—
207#453—1272no—
208#453—1271no—
209#421—1272no—
210C2×#100—1271no—
211H(5,𝔽2)—1272no—
212H(5,𝔽2)—1271no—
213H(5,𝔽2)—2542no—
214H(5,𝔽2)—2542no—
215H(5,𝔽2)—2542no—
216H(5,𝔽2)C2×D12711yescharacteristic
217C2×#41—1272no—
218C2×#41—1271no—
219C2×#41—2542no—
220C22×D4—1271no—
221C22×D4—2542no—
222C22×D4—2542no—
223#137—1272no—
224#137—2542no—
225#137C2×D12712yescomplemented
226AGL(1,ℤ/8)—1272no—
227AGL(1,ℤ/8)—1272no—
228AGL(1,ℤ/8)—2542no—
229#135—2542no—
230#135—2542no—
231C2×M16—1272no—
232#127—1271no—
233#127—1271no—
234#124—1271no—
235#124C2×D12711yescharacteristic, complemented
236#121—2542no—
237#120—2542no—
238#117—1271no—
239#104—1272no—
240#104—1272no—
241#104—1272no—
242#104C2×D12712yescomplemented
243#100—1271no—
244#100—1272no—
245#100C2×D12711yescharacteristic, complemented
246C24—1272no—
247C24—2542no—
248#41—22no—
249#41—1272no—
250#41—1272no—
251#41—1272no—
252#41—2542no—
253#41—2542no—
254#41—2544no—
255#41—2542no—
256#41—2542no—
257#41—2542no—
258#41—2542no—
259#41—2544no—
260#41—2542no—
261#41C22×D12712yes—
262C2×Q8—1272no—
263C2×Q8C22×D12711yescharacteristic, complemented
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—1272no—
267C2×D4—1272no—
268C2×D4—1271no—
269C2×D4—1271no—
270C2×D4—2542no—
271C2×D4—2542no—
272C2×D4—2541no—
273C2×D4—2542no—
274C2×D4—2542no—
275C2×D4—2542no—
276C2×D4—2542no—
277C2×D4—2542no—
278C2×D4—2544no—
279C2×D4—2544no—
280C2×D4—2544no—
281C2×D4—2544no—
282C2×D4—2542no—
283C2×D4—2544no—
284C2×D4—2542no—
285C2×D4—2542no—
286C2×D4—2542no—
287C2×D4C22×D12711yescharacteristic
288C22×C4—1271no—
289C22×C4—2541no—
290C22×C4—2542no—
291C22×C4—2542no—
292Q16—22no—
293Q16—2542no—
294SD16—22no—
295SD16—2542no—
296SD16—2542no—
297SD16—2542no—
298D8—2542no—
299D8—2542no—
300M16—1272no—
301M16—2542no—
302M16C22×D12712yescomplemented
303C2×C8—2542no—
304C4⋊C4—2541no—
305C22⋊C4—22no—
306C22⋊C4—2541no—
307C22⋊C4—2542no—
308C42—1271no—
309C42C22×D12711yescharacteristic, complemented
310C23—22no—
311C23—1271no—
312C23—1272no—
313C23—10162no—
314C23—10164no—
315C23—2542no—
316C23—2542no—
317C23—2542no—
318C23—2542no—
319C23—2544no—
320C23—2541no—
321C23D4×D12712yes—
322Q8—22no—
323Q8—1272no—
324Q8—2542no—
325Q8D4×D12712yes—
326D4—24no—
327D4—22no—
328D4—22no—
329D4—1272no—
330D4—10162no—
331D4—10162no—
332D4—5081no—
333D4—5081no—
334D4—2542no—
335D4—2544no—
336D4—2544no—
337D4—2542no—
338D4—2542no—
339D4—2544no—
340D4—2544no—
341D4—2542no—
342D4—2542no—
343D4—2542no—
344D4D4×D12712yes—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—1271no—
350C2×C4—1272no—
351C2×C4—1272no—
352C2×C4—5082no—
353C2×C4—2542no—
354C2×C4—2542no—
355C2×C4—2544no—
356C2×C4—2542no—
357C2×C4—2541no—
358C2×C4—2542no—
359C2×C4—2542no—
360C2×C4C23×D12711yescharacteristic
361C8—22no—
362C8—2542no—
363C22—22no—
364C22—22no—
365C22—42no—
366C22—82no—
367C22—1271no—
368C22—1271no—
369C22—10162no—
370C22—10164no—
371C22—10162no—
372C22—5082no—
373C22—5082no—
374C22—5081no—
375C22—5084no—
376C22—5084no—
377C22—5081no—
378C22—2542no—
379C22—2542no—
380C22C2×D4×D12711yescharacteristic
381C4—22no—
382C4—41no—
383C4—41no—
384C4—1272no—
385C4—5081no—
386C4—2542no—
387C4C2×D4×D12712yes—
388C2—21no—
389C2—42no—
390C2—42no—
391C2—10161no—
392C2—5082no—
393C2—5082no—
394C2—2541no—
395C2—2541no—
396C2#120×D12711yescharacteristic
subgroupG/H
centerC2#120×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#455×C127C2
socleC2×C127C2×#120
hypercenter#455D127
hypocenterC127#3019
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
1. H(5,𝔽2)×D127C2semidirect
2. #22550675C2semidirect
3. #22550444C2semidirect
4. #22550053C2semidirect
5. #22550052C2semidirect
6. #22550046C2semidirect
7. #22550045C2semidirect
8. #22550044C2semidirect
9. #22549891C2semidirect
10. #100×D127C2semidirect
11. #455×C127C2semidirect
17. C2×#14523249C22semidirect
24. #16861240C22semidirect
25. #16861237C22semidirect
27. #16861234C22semidirect
30. M16×D127C22semidirect
31. #16861206C22semidirect
32. #16861205C22semidirect
33. #16861198C22semidirect
36. #16861166C22semidirect
37. #16861137C22semidirect
38. #16861136C22semidirect
39. #16861135C22semidirect
40. #16861126C22semidirect
41. #16861125C22semidirect
43. #137×C127C22semidirect
44. #124×C127C22semidirect
45. #104×C127C22semidirect
46. #100×C127C22semidirect
52. #14523249D4semidirect
60. #14523247D4semidirect
61. Q8×D127D4semidirect
67. C2×D508C23semidirect
82. D4×D127D4semidirect
93. #14523232C23semidirect
99. C4×Dic127C23semidirect
102. C2×Q8×C127C23semidirect
108. M16×C127C23semidirect
110. C42×C127C23semidirect
199. C127#3019semidirect
205. #455D127semidirect
225. #137C2×D127semidirect
235. #124C2×D127semidirect
242. #104C2×D127semidirect
245. #100C2×D127semidirect
263. C2×Q8C22×D127semidirect
302. M16C22×D127semidirect
309. 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
#225506752
#225504442
#225500532
#225500522
#225500462
#225500452
#225500442
#225498912
#100×D1272
#455×C1272
#3019127