Finite Group Databasebeta

← C127×#21482 #44585155

Subgroups — C127×#21482 #44585155

Order 32512.

1630 conjugacy classes of subgroups.

The subgroup lattice has 444 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
3C127×#2321C213yesmaximal, complemented
4C127×#1889C213yesmaximal, complemented
5C127×#1867C213yesmaximal, complemented
6C127×#1790C211yesmaximal, characteristic, complemented
7C4×#99×C127C213yesmaximal, complemented
8C2×H(5,𝔽2)×C127C2211yescharacteristic
9C2×#120×C127C2213yes—
10#520×C127—23no—
11#518×C127—23no—
12#518×C127C2213yes—
13C4×#41×C127—23no—
14C2×C4×D4×C127—23no—
15C2×C4×D4×C127C2213yes—
16C22×C22⋊C4×C127C2211yescharacteristic, complemented
17B(4,𝔽2)×C127C418yescomplemented
18C2×#99×C127C2213yes—
19C2×#99×C127C2213yes—
20#390×C127—23no—
21#388×C127—23no—
22#388×C127—23no—
23#386×C127—23no—
24#386×C127C2213yes—
25#385×C127C2213yescomplemented
26#379×C127C2213yescomplemented
27C4×C22⋊C4×C127C2213yescomplemented
28#342×C127C2216yescomplemented
29C25×C127C2311yescharacteristic
30H(5,𝔽2)×C127—26no—
31H(5,𝔽2)×C127C2×C414yes—
32C2×#41×C127—23no—
33C22×D4×C127—23no—
34C22×D4×C127—23no—
35C22×D4×C127C2313yes—
36C23×C4×C127—21no—
37C23×C4×C127D413yescomplemented
38C23×C4×C127C2311yescharacteristic
39C23×C4×C127C2311yescharacteristic
40#120×C127—26no—
41#120×C127C2×C4112yescomplemented
42C4×Q8×C127—21no—
43C4×Q8×C127—23no—
44C4×D4×C127—23no—
45C4×D4×C127—26no—
46C4×D4×C127—23no—
47C4×D4×C127—26no—
48C4×D4×C127—26no—
49C4×D4×C127—26no—
50C4×D4×C127—26no—
51#117×C127—26no—
52#117×C127—23no—
53C2×C4⋊C4×C127—23no—
54C2×C4⋊C4×C127—23no—
55C2×C4⋊C4×C127D413yescomplemented
56C2×C22⋊C4×C127—23no—
57C2×C22⋊C4×C127—23no—
58C2×C22⋊C4×C127—23no—
59C2×C22⋊C4×C127—23no—
60C2×C22⋊C4×C127—23no—
61C2×C22⋊C4×C127—23no—
62C2×C22⋊C4×C127—23no—
63C2×C22⋊C4×C127—23no—
64C2×C22⋊C4×C127D413yescomplemented
65C2×C22⋊C4×C127C2313yes—
66C2×C22⋊C4×C127C2313yes—
67C2×C22⋊C4×C127C2313yes—
68C2×C42×C127—23no—
69C2×C42×C127—23no—
70C2×C42×C127D413yescomplemented
71#99×C127—26no—
72#99×C127C2×C4112yescomplemented
73#95×C127—23no—
74#95×C127—23no—
75#95×C127—23no—
76C24×C127—21no—
77C24×C127—23no—
78C24×C127—23no—
79C24×C127—81no—
80C24×C127—81no—
81C24×C127C22×C412yescomplemented
82C24×C127C2×D413yes—
83C24×C127C2411yescharacteristic
84#41×C127—212no—
85#41×C127—212no—
86C2×Q8×C127—21no—
87C2×D4×C127—23no—
88C2×D4×C127—212no—
89C2×D4×C127—212no—
90C2×D4×C127—26no—
91C2×D4×C127—26no—
92C2×D4×C127—212no—
93C2×D4×C127—83no—
94C2×D4×C127—83no—
95C2×D4×C127C22×C416yes—
96C2×D4×C127#4116yes—
97C22×C4×C127—23no—
98C22×C4×C127—23no—
99C22×C4×C127—23no—
100C22×C4×C127—23no—
101C22×C4×C127—23no—
102C22×C4×C127—23no—
103C22×C4×C127—23no—
104C22×C4×C127—23no—
105C22×C4×C127—81no—
106C22×C4×C127—81no—
107C22×C4×C127—83no—
108C22×C4×C127—82no—
109C22×C4×C127C2×D413yes—
110C22×C4×C127C2×D413yes—
111C22×C4×C127C2×D413yes—
112C22×C4×C127C2×D413yes—
113C22×C4×C127C2×D413yes—
114C4⋊C4×C127—23no—
115C4⋊C4×C127—23no—
116C4⋊C4×C127—26no—
117C4⋊C4×C127—83no—
118C22⋊C4×C127—26no—
119C22⋊C4×C127—26no—
120C22⋊C4×C127—23no—
121C22⋊C4×C127—212no—
122C22⋊C4×C127—26no—
123C22⋊C4×C127—26no—
124C22⋊C4×C127—41no—
125C22⋊C4×C127—43no—
126C22⋊C4×C127—83no—
127C22⋊C4×C127—83no—
128C22⋊C4×C127—83no—
129C22⋊C4×C127C22×C416yescomplemented
130C42×C127—26no—
131C42×C127—23no—
132C42×C127—23no—
133C42×C127—83no—
134C23×C127—24no—
135C23×C127—23no—
136C23×C127—23no—
137C23×C127—26no—
138C23×C127—41no—
139C23×C127—41no—
140C23×C127—43no—
141C23×C127—43no—
142C23×C127—43no—
143C23×C127—43no—
144C23×C127—43no—
145C23×C127—43no—
146C23×C127—41no—
147C23×C127—43no—
148C23×C127—82no—
149C23×C127—81no—
150C23×C127—81no—
151C23×C127—86no—
152C23×C127C4×D416yes—
153C23×C127C23×C411yescharacteristic
154C23×C127C22×D413yes—
155C23×C127C2×#4113yes—
156Q8×C127—24no—
157D4×C127—212no—
158D4×C127—212no—
159D4×C127—212no—
160D4×C127—812no—
161C2×C4×C127—212no—
162C2×C4×C127—23no—
163C2×C4×C127—23no—
164C2×C4×C127—26no—
165C2×C4×C127—23no—
166C2×C4×C127—23no—
167C2×C4×C127—43no—
168C2×C4×C127—43no—
169C2×C4×C127—46no—
170C2×C4×C127—43no—
171C2×C4×C127—43no—
172C2×C4×C127—43no—
173C2×C4×C127—43no—
174C2×C4×C127—43no—
175C2×C4×C127—43no—
176C2×C4×C127—43no—
177C2×C4×C127—43no—
178C2×C4×C127—41no—
179C2×C4×C127—43no—
180C2×C4×C127—83no—
181C2×C4×C127—83no—
182C2×C4×C127—83no—
183C2×C4×C127C4×D416yes—
184C2×C4×C127#12011yescharacteristic
185C2×C4×C127#12013yes—
186C22×C127—23no—
187C22×C127—23no—
188C22×C127—26no—
189C22×C127—41no—
190C22×C127—43no—
191C22×C127—46no—
192C22×C127—46no—
193C22×C127—46no—
194C22×C127—43no—
195C22×C127—43no—
196C22×C127—42no—
197C22×C127—83no—
198C22×C127—83no—
199C22×C127—83no—
200C22×C127—84no—
201C22×C127—81no—
202C22×C127—81no—
203C22×C127C2×C4×D413yes—
204C22×C127C2×#12011yescharacteristic
205C22×C127#52513yes—
206C4×C127—21no—
207C4×C127—23no—
208C4×C127—26no—
209C4×C127—46no—
210C4×C127—43no—
211C4×C127—43no—
212C4×C127—83no—
213C4×C127—81no—
214#21482C12711yesmaximal, characteristic, direct product factor, Sylow
215C2×C127—23no—
216C2×C127—23no—
217C2×C127—46no—
218C2×C127—82no—
219C2×C127C4×#12011yescharacteristic
220C2×C127#302611yescharacteristic
221C2×C127#302711yescharacteristic
222C4×H(5,𝔽2)C2×C12711yescharacteristic, complemented
223C2×B(4,𝔽2)C2×C12711yescharacteristic
224#2321C2×C12713yescomplemented
225#1889C2×C12713yescomplemented
226#1867C2×C12713yescomplemented
227#1790C2×C12711yescharacteristic, complemented
228C4×#99C2×C12713yescomplemented
229C127#2148211yescharacteristic, direct product factor, Sylow
230C2×H(5,𝔽2)C22×C12711yescharacteristic
231C2×#120C22×C12713yes—
232#520—23no—
233#518—23no—
234#518C22×C12713yes—
235C4×#41—23no—
236C2×C4×D4—23no—
237C2×C4×D4C22×C12713yes—
238C22×C22⋊C4C22×C12711yescharacteristic, complemented
239B(4,𝔽2)C4×C12718yescomplemented
240C2×#99C22×C12713yes—
241C2×#99C22×C12713yes—
242#390—23no—
243#388—23no—
244#388—23no—
245#386—23no—
246#386C22×C12713yes—
247#385C22×C12713yescomplemented
248#379C22×C12713yescomplemented
249C4×C22⋊C4C22×C12713yescomplemented
250#342C22×C12716yescomplemented
251C25C23×C12711yescharacteristic
252H(5,𝔽2)—26no—
253H(5,𝔽2)C2×C4×C12714yes—
254C2×#41—23no—
255C22×D4—23no—
256C22×D4—23no—
257C22×D4C23×C12713yes—
258C23×C4—21no—
259C23×C4D4×C12713yescomplemented
260C23×C4C23×C12711yescharacteristic
261C23×C4C23×C12711yescharacteristic
262#120—26no—
263#120C2×C4×C127112yescomplemented
264C4×Q8—21no—
265C4×Q8—23no—
266C4×D4—23no—
267C4×D4—26no—
268C4×D4—23no—
269C4×D4—26no—
270C4×D4—26no—
271C4×D4—26no—
272C4×D4—26no—
273#117—26no—
274#117—23no—
275C2×C4⋊C4—23no—
276C2×C4⋊C4—23no—
277C2×C4⋊C4D4×C12713yescomplemented
278C2×C22⋊C4—23no—
279C2×C22⋊C4—23no—
280C2×C22⋊C4—23no—
281C2×C22⋊C4—23no—
282C2×C22⋊C4—23no—
283C2×C22⋊C4—23no—
284C2×C22⋊C4—23no—
285C2×C22⋊C4—23no—
286C2×C22⋊C4D4×C12713yescomplemented
287C2×C22⋊C4C23×C12713yes—
288C2×C22⋊C4C23×C12713yes—
289C2×C22⋊C4C23×C12713yes—
290C2×C42—23no—
291C2×C42—23no—
292C2×C42D4×C12713yescomplemented
293#99—26no—
294#99C2×C4×C127112yescomplemented
295#95—23no—
296#95—23no—
297#95—23no—
298C24—21no—
299C24—23no—
300C24—23no—
301C24—81no—
302C24—81no—
303C24C22×C4×C12712yescomplemented
304C24C2×D4×C12713yes—
305C24C24×C12711yescharacteristic
306#41—212no—
307#41—212no—
308C2×Q8—21no—
309C2×D4—23no—
310C2×D4—212no—
311C2×D4—212no—
312C2×D4—26no—
313C2×D4—26no—
314C2×D4—212no—
315C2×D4—83no—
316C2×D4—83no—
317C2×D4C22×C4×C12716yes—
318C2×D4#41×C12716yes—
319C22×C4—23no—
320C22×C4—23no—
321C22×C4—23no—
322C22×C4—23no—
323C22×C4—23no—
324C22×C4—23no—
325C22×C4—23no—
326C22×C4—23no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4—83no—
330C22×C4—82no—
331C22×C4C2×D4×C12713yes—
332C22×C4C2×D4×C12713yes—
333C22×C4C2×D4×C12713yes—
334C22×C4C2×D4×C12713yes—
335C22×C4C2×D4×C12713yes—
336C4⋊C4—23no—
337C4⋊C4—23no—
338C4⋊C4—26no—
339C4⋊C4—83no—
340C22⋊C4—26no—
341C22⋊C4—26no—
342C22⋊C4—23no—
343C22⋊C4—212no—
344C22⋊C4—26no—
345C22⋊C4—26no—
346C22⋊C4—41no—
347C22⋊C4—43no—
348C22⋊C4—83no—
349C22⋊C4—83no—
350C22⋊C4—83no—
351C22⋊C4C22×C4×C12716yescomplemented
352C42—26no—
353C42—23no—
354C42—23no—
355C42—83no—
356C23—24no—
357C23—23no—
358C23—23no—
359C23—26no—
360C23—41no—
361C23—41no—
362C23—43no—
363C23—43no—
364C23—43no—
365C23—43no—
366C23—43no—
367C23—43no—
368C23—41no—
369C23—43no—
370C23—82no—
371C23—81no—
372C23—86no—
373C23—81no—
374C23C4×D4×C12716yes—
375C23C23×C4×C12711yescharacteristic
376C23C22×D4×C12713yes—
377C23C2×#41×C12713yes—
378Q8—24no—
379D4—212no—
380D4—212no—
381D4—212no—
382D4—812no—
383C2×C4—212no—
384C2×C4—23no—
385C2×C4—23no—
386C2×C4—26no—
387C2×C4—23no—
388C2×C4—23no—
389C2×C4—43no—
390C2×C4—43no—
391C2×C4—46no—
392C2×C4—43no—
393C2×C4—43no—
394C2×C4—43no—
395C2×C4—43no—
396C2×C4—43no—
397C2×C4—43no—
398C2×C4—43no—
399C2×C4—43no—
400C2×C4—41no—
401C2×C4—43no—
402C2×C4—83no—
403C2×C4—83no—
404C2×C4—83no—
405C2×C4C4×D4×C12716yes—
406C2×C4#120×C12711yescharacteristic
407C2×C4#120×C12713yes—
408C22—23no—
409C22—23no—
410C22—26no—
411C22—41no—
412C22—43no—
413C22—46no—
414C22—46no—
415C22—46no—
416C22—43no—
417C22—43no—
418C22—42no—
419C22—83no—
420C22—83no—
421C22—83no—
422C22—84no—
423C22—81no—
424C22—81no—
425C22C2×C4×D4×C12713yes—
426C22C2×#120×C12711yescharacteristic
427C22#525×C12713yes—
428C4—21no—
429C4—23no—
430C4—26no—
431C4—46no—
432C4—43no—
433C4—43no—
434C4—83no—
435C4—81no—
436C2—23no—
437C2—23no—
438C2—46no—
439C2—82no—
440C2C4×#120×C12711yescharacteristic
441C2C127×#302611yescharacteristic
442C2C127×#302711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
prime radicalC2×B(4,𝔽2)×C127C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×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. C127×#2321C2semidirect
4. C127×#1889C2semidirect
5. C127×#1867C2semidirect
6. C127×#1790C2semidirect
7. C4×#99×C127C2semidirect
16. C22×C22⋊C4×C127C22semidirect
17. B(4,𝔽2)×C127C4semidirect
25. #385×C127C22semidirect
26. #379×C127C22semidirect
27. C4×C22⋊C4×C127C22semidirect
28. #342×C127C22semidirect
37. C23×C4×C127D4semidirect
41. #120×C127C2×C4semidirect
55. C2×C4⋊C4×C127D4semidirect
64. C2×C22⋊C4×C127D4semidirect
70. C2×C42×C127D4semidirect
72. #99×C127C2×C4semidirect
81. C24×C127C22×C4semidirect
129. C22⋊C4×C127C22×C4semidirect
214. #21482C127direct (N × H)
222. C4×H(5,𝔽2)C2×C127semidirect
224. #2321C2×C127semidirect
225. #1889C2×C127semidirect
226. #1867C2×C127semidirect
227. #1790C2×C127semidirect
228. C4×#99C2×C127semidirect
229. C127#21482direct (N × H)
238. C22×C22⋊C4C22×C127semidirect
239. B(4,𝔽2)C4×C127semidirect
247. #385C22×C127semidirect
248. #379C22×C127semidirect
249. C4×C22⋊C4C22×C127semidirect
250. #342C22×C127semidirect
259. C23×C4D4×C127semidirect
263. #120C2×C4×C127semidirect
277. C2×C4⋊C4D4×C127semidirect
286. C2×C22⋊C4D4×C127semidirect
292. C2×C42D4×C127semidirect
294. #99C2×C4×C127semidirect
303. C24C22×C4×C127semidirect
351. C22⋊C4C22×C4×C127semidirect

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
C127×#23212
C127×#18892
C127×#18672
C127×#17902
C4×#99×C1272
#21482127