Finite Group Databasebeta

← C4×B(4,𝔽2)×C127 #44585152

Subgroups — C4×B(4,𝔽2)×C127 #44585152

Order 32512.

1910 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×H(5,𝔽2)×C127C211yesmaximal, characteristic, complemented
2C2×B(4,𝔽2)×C127C211yesmaximal, characteristic
3C4×#120×C127C213yesmaximal, complemented
4C127×#1889C213yesmaximal, complemented
5C127×#1867C213yesmaximal, complemented
6C127×#1790C211yesmaximal, characteristic, complemented
7C4×#99×C127C213yesmaximal, complemented
8C2×H(5,𝔽2)×C127C2211yescharacteristic
9C24×C4×C127C2211yescharacteristic, complemented
10C2×#120×C127C2213yes—
11#520×C127—23no—
12#518×C127—23no—
13#518×C127C2213yes—
14C4×#41×C127—23no—
15C2×C4×D4×C127—23no—
16C2×C4×D4×C127—23no—
17C2×C4×D4×C127C2213yes—
18B(4,𝔽2)×C127C418yesdirect product factor
19C2×#99×C127C2213yes—
20C2×#99×C127C2213yes—
21#386×C127—23no—
22#386×C127C2213yes—
23#385×C127—23no—
24#385×C127C2213yescomplemented
25#379×C127C2213yescomplemented
26C4×C22⋊C4×C127—23no—
27C4×C22⋊C4×C127C2213yescomplemented
28#342×C127C2216yescomplemented
29C25×C127C2311yescharacteristic
30H(5,𝔽2)×C127—26no—
31H(5,𝔽2)×C127C2×C414yescomplemented
32C2×#41×C127—23no—
33C22×D4×C127—23no—
34C22×D4×C127—23no—
35C22×D4×C127C2313yes—
36C23×C4×C127—21no—
37C23×C4×C127—23no—
38C23×C4×C127—23no—
39C23×C4×C127—44no—
40C23×C4×C127D413yescomplemented
41C23×C4×C127C2311yescharacteristic
42C23×C4×C127C2311yescharacteristic
43#120×C127—26no—
44#120×C127C2×C4112yescomplemented
45C4×Q8×C127—21no—
46C4×Q8×C127—23no—
47C4×D4×C127—23no—
48C4×D4×C127—26no—
49C4×D4×C127—23no—
50C4×D4×C127—26no—
51C4×D4×C127—26no—
52C4×D4×C127—26no—
53C4×D4×C127—26no—
54C4×D4×C127—412no—
55#117×C127—26no—
56#117×C127—23no—
57C2×C4⋊C4×C127—23no—
58C2×C4⋊C4×C127—23no—
59C2×C4⋊C4×C127D413yescomplemented
60C2×C22⋊C4×C127—23no—
61C2×C22⋊C4×C127—23no—
62C2×C22⋊C4×C127—23no—
63C2×C22⋊C4×C127—23no—
64C2×C22⋊C4×C127—23no—
65C2×C22⋊C4×C127—23no—
66C2×C22⋊C4×C127D413yescomplemented
67C2×C22⋊C4×C127C2313yes—
68C2×C22⋊C4×C127C2313yes—
69C2×C22⋊C4×C127C2313yes—
70C2×C42×C127—23no—
71C2×C42×C127—23no—
72C2×C42×C127D413yescomplemented
73#99×C127—26no—
74#99×C127C2×C4112yescomplemented
75#95×C127—23no—
76#95×C127—23no—
77#95×C127—23no—
78C24×C127—21no—
79C24×C127—23no—
80C24×C127—23no—
81C24×C127—42no—
82C24×C127—42no—
83C24×C127C22×C412yescomplemented
84C24×C127C2×D413yes—
85C24×C127C2411yescharacteristic
86#41×C127—212no—
87#41×C127—212no—
88C2×Q8×C127—21no—
89C2×D4×C127—23no—
90C2×D4×C127—212no—
91C2×D4×C127—212no—
92C2×D4×C127—26no—
93C2×D4×C127—26no—
94C2×D4×C127—212no—
95C2×D4×C127—46no—
96C2×D4×C127—46no—
97C2×D4×C127C22×C416yes—
98C2×D4×C127#4116yes—
99C22×C4×C127—23no—
100C22×C4×C127—212no—
101C22×C4×C127—23no—
102C22×C4×C127—23no—
103C22×C4×C127—23no—
104C22×C4×C127—23no—
105C22×C4×C127—23no—
106C22×C4×C127—23no—
107C22×C4×C127—23no—
108C22×C4×C127—42no—
109C22×C4×C127—42no—
110C22×C4×C127—46no—
111C22×C4×C127—46no—
112C22×C4×C127—46no—
113C22×C4×C127—46no—
114C22×C4×C127—43no—
115C22×C4×C127—41no—
116C22×C4×C127—82no—
117C22×C4×C127C2×D413yes—
118C22×C4×C127C2×D413yes—
119C22×C4×C127C2×D413yes—
120C22×C4×C127C2×D413yes—
121C22×C4×C127C2×D413yes—
122C4⋊C4×C127—23no—
123C4⋊C4×C127—23no—
124C4⋊C4×C127—26no—
125C4⋊C4×C127—46no—
126C22⋊C4×C127—26no—
127C22⋊C4×C127—26no—
128C22⋊C4×C127—23no—
129C22⋊C4×C127—212no—
130C22⋊C4×C127—26no—
131C22⋊C4×C127—26no—
132C22⋊C4×C127—46no—
133C22⋊C4×C127—46no—
134C22⋊C4×C127—46no—
135C22⋊C4×C127C22×C416yescomplemented
136C42×C127—26no—
137C42×C127—23no—
138C42×C127—23no—
139C42×C127—46no—
140C23×C127—26no—
141C23×C127—26no—
142C23×C127—24no—
143C23×C127—26no—
144C23×C127—23no—
145C23×C127—23no—
146C23×C127—41no—
147C23×C127—44no—
148C23×C127—41no—
149C23×C127—43no—
150C23×C127—43no—
151C23×C127—412no—
152C23×C127—43no—
153C23×C127—43no—
154C23×C127—41no—
155C23×C127—43no—
156C23×C127—81no—
157C23×C127—81no—
158C23×C127C4×D416yescomplemented
159C23×C127C23×C411yescharacteristic
160C23×C127C22×D413yes—
161C23×C127C2×#4113yes—
162Q8×C127—24no—
163D4×C127—212no—
164D4×C127—212no—
165D4×C127—212no—
166D4×C127—424no—
167C2×C4×C127—212no—
168C2×C4×C127—23no—
169C2×C4×C127—23no—
170C2×C4×C127—26no—
171C2×C4×C127—26no—
172C2×C4×C127—26no—
173C2×C4×C127—23no—
174C2×C4×C127—23no—
175C2×C4×C127—43no—
176C2×C4×C127—412no—
177C2×C4×C127—43no—
178C2×C4×C127—43no—
179C2×C4×C127—43no—
180C2×C4×C127—41no—
181C2×C4×C127—44no—
182C2×C4×C127—46no—
183C2×C4×C127—43no—
184C2×C4×C127—43no—
185C2×C4×C127—43no—
186C2×C4×C127—43no—
187C2×C4×C127—43no—
188C2×C4×C127—83no—
189C2×C4×C127—83no—
190C2×C4×C127—83no—
191C2×C4×C127C4×D416yescomplemented
192C2×C4×C127#12013yes—
193C2×C4×C127#12011yescharacteristic
194C22×C127—23no—
195C22×C127—23no—
196C22×C127—212no—
197C22×C127—26no—
198C22×C127—41no—
199C22×C127—43no—
200C22×C127—46no—
201C22×C127—46no—
202C22×C127—46no—
203C22×C127—46no—
204C22×C127—42no—
205C22×C127—43no—
206C22×C127—43no—
207C22×C127—42no—
208C22×C127—42no—
209C22×C127—83no—
210C22×C127—84no—
211C22×C127C2×C4×D413yes—
212C22×C127C2×#12011yescharacteristic
213C22×C127#52513yes—
214C4×C127—23no—
215C4×C127—26no—
216C4×C127—46no—
217C4×C127—42no—
218C4×C127—43no—
219C4×C127—43no—
220C4×C127—83no—
221C4×C127B(4,𝔽2)12yesdirect product factor
222C4×B(4,𝔽2)C12711yesmaximal, characteristic, direct product factor, Sylow
223C2×C127—23no—
224C2×C127—23no—
225C2×C127—44no—
226C2×C127—46no—
227C2×C127C4×#12011yescharacteristic
228C2×C127C2×B(4,𝔽2)11yescharacteristic
229C2×C127#302511yescharacteristic
230C4×H(5,𝔽2)C2×C12711yescharacteristic, complemented
231C2×B(4,𝔽2)C2×C12711yescharacteristic
232C4×#120C2×C12713yescomplemented
233#1889C2×C12713yescomplemented
234#1867C2×C12713yescomplemented
235#1790C2×C12711yescharacteristic, complemented
236C4×#99C2×C12713yescomplemented
237C127C4×B(4,𝔽2)11yescharacteristic, direct product factor, Sylow
238C2×H(5,𝔽2)C22×C12711yescharacteristic
239C24×C4C22×C12711yescharacteristic, complemented
240C2×#120C22×C12713yes—
241#520—23no—
242#518—23no—
243#518C22×C12713yes—
244C4×#41—23no—
245C2×C4×D4—23no—
246C2×C4×D4—23no—
247C2×C4×D4C22×C12713yes—
248B(4,𝔽2)C4×C12718yesdirect product factor
249C2×#99C22×C12713yes—
250C2×#99C22×C12713yes—
251#386—23no—
252#386C22×C12713yes—
253#385—23no—
254#385C22×C12713yescomplemented
255#379C22×C12713yescomplemented
256C4×C22⋊C4—23no—
257C4×C22⋊C4C22×C12713yescomplemented
258#342C22×C12716yescomplemented
259C25C23×C12711yescharacteristic
260H(5,𝔽2)—26no—
261H(5,𝔽2)C2×C4×C12714yescomplemented
262C2×#41—23no—
263C22×D4—23no—
264C22×D4—23no—
265C22×D4C23×C12713yes—
266C23×C4—21no—
267C23×C4—23no—
268C23×C4—23no—
269C23×C4—44no—
270C23×C4D4×C12713yescomplemented
271C23×C4C23×C12711yescharacteristic
272C23×C4C23×C12711yescharacteristic
273#120—26no—
274#120C2×C4×C127112yescomplemented
275C4×Q8—21no—
276C4×Q8—23no—
277C4×D4—23no—
278C4×D4—26no—
279C4×D4—23no—
280C4×D4—26no—
281C4×D4—26no—
282C4×D4—26no—
283C4×D4—26no—
284C4×D4—412no—
285#117—26no—
286#117—23no—
287C2×C4⋊C4—23no—
288C2×C4⋊C4—23no—
289C2×C4⋊C4D4×C12713yescomplemented
290C2×C22⋊C4—23no—
291C2×C22⋊C4—23no—
292C2×C22⋊C4—23no—
293C2×C22⋊C4—23no—
294C2×C22⋊C4—23no—
295C2×C22⋊C4—23no—
296C2×C22⋊C4D4×C12713yescomplemented
297C2×C22⋊C4C23×C12713yes—
298C2×C22⋊C4C23×C12713yes—
299C2×C22⋊C4C23×C12713yes—
300C2×C42—23no—
301C2×C42—23no—
302C2×C42D4×C12713yescomplemented
303#99—26no—
304#99C2×C4×C127112yescomplemented
305#95—23no—
306#95—23no—
307#95—23no—
308C24—21no—
309C24—23no—
310C24—23no—
311C24—42no—
312C24—42no—
313C24C22×C4×C12712yescomplemented
314C24C2×D4×C12713yes—
315C24C24×C12711yescharacteristic
316#41—212no—
317#41—212no—
318C2×Q8—21no—
319C2×D4—23no—
320C2×D4—212no—
321C2×D4—212no—
322C2×D4—26no—
323C2×D4—26no—
324C2×D4—212no—
325C2×D4—46no—
326C2×D4—46no—
327C2×D4C22×C4×C12716yes—
328C2×D4#41×C12716yes—
329C22×C4—23no—
330C22×C4—212no—
331C22×C4—23no—
332C22×C4—23no—
333C22×C4—23no—
334C22×C4—23no—
335C22×C4—23no—
336C22×C4—23no—
337C22×C4—23no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—46no—
341C22×C4—46no—
342C22×C4—46no—
343C22×C4—46no—
344C22×C4—43no—
345C22×C4—41no—
346C22×C4—82no—
347C22×C4C2×D4×C12713yes—
348C22×C4C2×D4×C12713yes—
349C22×C4C2×D4×C12713yes—
350C22×C4C2×D4×C12713yes—
351C22×C4C2×D4×C12713yes—
352C4⋊C4—23no—
353C4⋊C4—23no—
354C4⋊C4—26no—
355C4⋊C4—46no—
356C22⋊C4—26no—
357C22⋊C4—26no—
358C22⋊C4—23no—
359C22⋊C4—212no—
360C22⋊C4—26no—
361C22⋊C4—26no—
362C22⋊C4—46no—
363C22⋊C4—46no—
364C22⋊C4—46no—
365C22⋊C4C22×C4×C12716yescomplemented
366C42—26no—
367C42—23no—
368C42—23no—
369C42—46no—
370C23—26no—
371C23—26no—
372C23—24no—
373C23—26no—
374C23—23no—
375C23—23no—
376C23—41no—
377C23—44no—
378C23—41no—
379C23—43no—
380C23—43no—
381C23—412no—
382C23—43no—
383C23—43no—
384C23—41no—
385C23—43no—
386C23—81no—
387C23—81no—
388C23C4×D4×C12716yescomplemented
389C23C23×C4×C12711yescharacteristic
390C23C22×D4×C12713yes—
391C23C2×#41×C12713yes—
392Q8—24no—
393D4—212no—
394D4—212no—
395D4—212no—
396D4—424no—
397C2×C4—212no—
398C2×C4—23no—
399C2×C4—23no—
400C2×C4—26no—
401C2×C4—26no—
402C2×C4—26no—
403C2×C4—23no—
404C2×C4—23no—
405C2×C4—43no—
406C2×C4—412no—
407C2×C4—43no—
408C2×C4—43no—
409C2×C4—43no—
410C2×C4—41no—
411C2×C4—44no—
412C2×C4—46no—
413C2×C4—43no—
414C2×C4—43no—
415C2×C4—43no—
416C2×C4—43no—
417C2×C4—43no—
418C2×C4—83no—
419C2×C4—83no—
420C2×C4—83no—
421C2×C4C4×D4×C12716yescomplemented
422C2×C4#120×C12713yes—
423C2×C4#120×C12711yescharacteristic
424C22—23no—
425C22—23no—
426C22—212no—
427C22—26no—
428C22—41no—
429C22—43no—
430C22—46no—
431C22—46no—
432C22—46no—
433C22—46no—
434C22—42no—
435C22—43no—
436C22—43no—
437C22—42no—
438C22—42no—
439C22—83no—
440C22—84no—
441C22C2×C4×D4×C12713yes—
442C22C2×#120×C12711yescharacteristic
443C22#525×C12713yes—
444C4—23no—
445C4—26no—
446C4—46no—
447C4—42no—
448C4—43no—
449C4—43no—
450C4—83no—
451C4B(4,𝔽2)×C12712yesdirect product factor
452C2—23no—
453C2—23no—
454C2—44no—
455C2—46no—
456C2C4×#120×C12711yescharacteristic
457C2C2×B(4,𝔽2)×C12711yescharacteristic
458C2C127×#302511yescharacteristic
subgroupG/H
centerC2×C4×C127#120
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
socleC22×C127C2×#120
prime radicalC2×B(4,𝔽2)×C127C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C4×H(5,𝔽2)×C127C2semidirect
3. C4×#120×C127C2semidirect
4. C127×#1889C2semidirect
5. C127×#1867C2semidirect
6. C127×#1790C2semidirect
7. C4×#99×C127C2semidirect
9. C24×C4×C127C22semidirect
18. B(4,𝔽2)×C127C4direct (N × H)
24. #385×C127C22semidirect
25. #379×C127C22semidirect
27. C4×C22⋊C4×C127C22semidirect
28. #342×C127C22semidirect
31. H(5,𝔽2)×C127C2×C4semidirect
40. C23×C4×C127D4semidirect
44. #120×C127C2×C4semidirect
59. C2×C4⋊C4×C127D4semidirect
66. C2×C22⋊C4×C127D4semidirect
72. C2×C42×C127D4semidirect
74. #99×C127C2×C4semidirect
83. C24×C127C22×C4semidirect
135. C22⋊C4×C127C22×C4semidirect
158. C23×C127C4×D4semidirect
191. C2×C4×C127C4×D4semidirect
221. C4×C127B(4,𝔽2)direct (N × H)
222. C4×B(4,𝔽2)C127direct (N × H)
230. C4×H(5,𝔽2)C2×C127semidirect
232. C4×#120C2×C127semidirect
233. #1889C2×C127semidirect
234. #1867C2×C127semidirect
235. #1790C2×C127semidirect
236. C4×#99C2×C127semidirect
237. C127C4×B(4,𝔽2)direct (N × H)
239. C24×C4C22×C127semidirect
248. B(4,𝔽2)C4×C127direct (N × H)
254. #385C22×C127semidirect
255. #379C22×C127semidirect
257. C4×C22⋊C4C22×C127semidirect
258. #342C22×C127semidirect
261. H(5,𝔽2)C2×C4×C127semidirect
270. C23×C4D4×C127semidirect
274. #120C2×C4×C127semidirect
289. C2×C4⋊C4D4×C127semidirect
296. C2×C22⋊C4D4×C127semidirect
302. C2×C42D4×C127semidirect
304. #99C2×C4×C127semidirect
313. C24C22×C4×C127semidirect
365. C22⋊C4C22×C4×C127semidirect
388. C23C4×D4×C127semidirect
421. C2×C4C4×D4×C127semidirect
451. C4B(4,𝔽2)×C127direct (N × H)

Maximal subgroups

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

subgroupindex
C4×H(5,𝔽2)×C1272
C2×B(4,𝔽2)×C1272
C4×#120×C1272
C127×#18892
C127×#18672
C127×#17902
C4×#99×C1272
C4×B(4,𝔽2)127