Finite Group Databasebeta

← C127×#18528 #44582201

Subgroups — C127×#18528 #44582201

Order 32512.

1060 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3481C211yesmaximal, characteristic
2C127×#3152C211yesmaximal, characteristic, complemented
3C127×#3079C212yesmaximal, complemented
4C127×#1632C214yesmaximal, complemented
5C127×#1630C212yesmaximal, complemented
6C127×#1629C212yesmaximal, complemented
7C127×#1560C211yesmaximal, characteristic, complemented
8C127×#1489C212yesmaximal, complemented
9C2×#137×C127—22no—
10#559×C127C2211yescharacteristic
11#554×C127C2212yes—
12#553×C127—22no—
13#551×C127—22no—
14#546×C127—22no—
15D42×C127C2211yescharacteristic
16#532×C127C2211yescharacteristic
17C2×C4⋊Q8×C127C2211yescharacteristic, complemented
18#529×C127—22no—
19#525×C127—22no—
20C4×#41×C127C2212yes—
21C2×C4×D4×C127—22no—
22#497×C127—22no—
23#494×C127C2212yescomplemented
24#492×C127C2212yes—
25#477×C127C2214yescomplemented
26#476×C127C2214yes—
27#475×C127—22no—
28#474×C127—22no—
29#428×C127—22no—
30#423×C127C2212yescomplemented
31#417×C127—22no—
32C4×M16×C127C2211yescharacteristic, complemented
33#333×C127C2212yescomplemented
34#331×C127C2212yes—
35Q8⋊C8×C127C2214yescomplemented
36D4⋊C8×C127C2214yes—
37C2×#41×C127D412yes—
38C2×#41×C127D412yes—
39C22×Q8×C127—21no—
40C22×D4×C127—22no—
41C23×C4×C127—22no—
42#137×C127—48no—
43C2×Q16×C127—24no—
44C2×SD16×C127—24no—
45C2×M16×C127—21no—
46C2×M16×C127—22no—
47C4⋊Q8×C127—22no—
48C4⋊Q8×C127—21no—
49C4⋊Q8×C127C2312yes—
50C4⋊Q8×C127C2311yescharacteristic
51#127×C127C2311yescharacteristic
52#126×C127—24no—
53#125×C127—22no—
54#124×C127—22no—
55#124×C127D412yes—
56#123×C127—22no—
57#123×C127—24no—
58#123×C127—22no—
59C22⋊Q8×C127—24no—
60C22⋊Q8×C127—22no—
61#121×C127—24no—
62#121×C127—22no—
63#121×C127—22no—
64#120×C127—22no—
65#120×C127—22no—
66C4×Q8×C127D412yes—
67C4×Q8×C127C2312yes—
68C4×D4×C127—24no—
69C4×D4×C127—24no—
70C4×D4×C127—22no—
71C4×D4×C127—22no—
72C4×D4×C127—22no—
73C4×D4×C127—22no—
74C4×D4×C127—22no—
75C4×D4×C127D412yes—
76C4×D4×C127C2312yes—
77#117×C127—22no—
78#117×C127—22no—
79#117×C127D412yes—
80C2×C4⋊C4×C127—22no—
81C2×C4⋊C4×C127—22no—
82C2×C22⋊C4×C127—24no—
83C2×C42×C127—22no—
84C2×C42×C127C2311yescharacteristic
85#107×C127—24no—
86#106×C127—24no—
87C4⋊C8×C127—22no—
88C4⋊C8×C127C2314yes—
89#103×C127—24no—
90#102×C127—24no—
91#97×C127—21no—
92C4×C8×C127C2312yes—
93C24×C127—22no—
94#41×C127—24no—
95#41×C127—42no—
96#41×C127—44no—
97C2×Q8×C127—22no—
98C2×Q8×C127—44no—
99C2×Q8×C127—41no—
100C2×Q8×C127C2×D412yes—
101C2×Q8×C127C2×D412yes—
102C2×D4×C127—24no—
103C2×D4×C127—22no—
104C2×D4×C127—22no—
105C2×D4×C127—22no—
106C2×D4×C127—44no—
107C2×D4×C127—44no—
108C2×D4×C127C2×D412yes—
109C2×D4×C127C2×D412yes—
110C22×C4×C127—24no—
111C22×C4×C127—22no—
112C22×C4×C127—22no—
113C22×C4×C127—22no—
114C22×C4×C127—22no—
115C22×C4×C127—22no—
116C22×C4×C127—41no—
117C22×C4×C127—84no—
118C22×C4×C127C2×D412yes—
119C22×C4×C127C2×D411yescharacteristic
120Q16×C127—48no—
121SD16×C127—48no—
122M16×C127—24no—
123M16×C127—44no—
124C2×C8×C127—22no—
125C2×C8×C127—24no—
126C4⋊C4×C127—24no—
127C4⋊C4×C127—24no—
128C4⋊C4×C127—22no—
129C4⋊C4×C127—22no—
130C4⋊C4×C127—22no—
131C4⋊C4×C127—24no—
132C4⋊C4×C127—42no—
133C4⋊C4×C127C2×D412yes—
134C4⋊C4×C127C2×D412yes—
135C22⋊C4×C127—24no—
136C22⋊C4×C127—24no—
137C22⋊C4×C127—22no—
138C22⋊C4×C127—22no—
139C22⋊C4×C127—22no—
140C22⋊C4×C127—22no—
141C22⋊C4×C127—24no—
142C22⋊C4×C127—22no—
143C42×C127—22no—
144C42×C127—22no—
145C42×C127—22no—
146C42×C127C2×D411yescharacteristic
147C42×C127C2×D412yes—
148C42×C127C2411yescharacteristic
149C23×C127—22no—
150C23×C127—22no—
151C23×C127—48no—
152C23×C127—82no—
153C23×C127#12011yescharacteristic
154Q8×C127—22no—
155Q8×C127—22no—
156Q8×C127—42no—
157Q8×C127—44no—
158Q8×C127—82no—
159D4×C127—22no—
160D4×C127—42no—
161D4×C127—44no—
162D4×C127—48no—
163D4×C127—42no—
164D4×C127—42no—
165D4×C127—82no—
166D4×C127#13714yescomplemented
167C2×C4×C127—22no—
168C2×C4×C127—22no—
169C2×C4×C127—22no—
170C2×C4×C127—22no—
171C2×C4×C127—22no—
172C2×C4×C127—22no—
173C2×C4×C127—22no—
174C2×C4×C127—44no—
175C2×C4×C127—42no—
176C2×C4×C127—42no—
177C2×C4×C127—44no—
178C2×C4×C127—44no—
179C2×C4×C127—42no—
180C2×C4×C127—82no—
181C2×C4×C127—84no—
182C2×C4×C127—82no—
183C2×C4×C127#12012yes—
184C2×C4×C127#12011yescharacteristic
185C2×C4×C127C22×D411yescharacteristic
186C2×C4×C127C22×D412yes—
187C8×C127—24no—
188C8×C127—44no—
189C22×C127—22no—
190C22×C127—24no—
191C22×C127—21no—
192C22×C127—42no—
193C22×C127—42no—
194C22×C127—44no—
195C22×C127—42no—
196C22×C127—88no—
197C22×C127C2×#12011yescharacteristic
198C4×C127—22no—
199C4×C127—21no—
200C4×C127—42no—
201C4×C127—42no—
202C4×C127—42no—
203C4×C127—42no—
204C4×C127—82no—
205C4×C127#45312yes—
206C4×C127C2×#13712yes—
207C4×C127#57512yes—
208#18528C12711yesmaximal, characteristic, direct product factor, Sylow
209C2×C127—41no—
210C2×C127—44no—
211C2×C127—82no—
212C2×C127#300712yes—
213C2×C127C2×#45311yescharacteristic
214#3481C2×C12711yescharacteristic
215#3152C2×C12711yescharacteristic, complemented
216#3079C2×C12712yescomplemented
217#1632C2×C12714yescomplemented
218#1630C2×C12712yescomplemented
219#1629C2×C12712yescomplemented
220#1560C2×C12711yescharacteristic, complemented
221#1489C2×C12712yescomplemented
222C127#1852811yescharacteristic, direct product factor, Sylow
223C2×#137—22no—
224#559C22×C12711yescharacteristic
225#554C22×C12712yes—
226#553—22no—
227#551—22no—
228#546—22no—
229D42C22×C12711yescharacteristic
230#532C22×C12711yescharacteristic
231C2×C4⋊Q8C22×C12711yescharacteristic, complemented
232#529—22no—
233#525—22no—
234C4×#41C22×C12712yes—
235C2×C4×D4—22no—
236#497—22no—
237#494C22×C12712yescomplemented
238#492C22×C12712yes—
239#477C22×C12714yescomplemented
240#476C22×C12714yes—
241#475—22no—
242#474—22no—
243#428—22no—
244#423C22×C12712yescomplemented
245#417—22no—
246C4×M16C22×C12711yescharacteristic, complemented
247#333C22×C12712yescomplemented
248#331C22×C12712yes—
249Q8⋊C8C22×C12714yescomplemented
250D4⋊C8C22×C12714yes—
251C2×#41D4×C12712yes—
252C2×#41D4×C12712yes—
253C22×Q8—21no—
254C22×D4—22no—
255C23×C4—22no—
256#137—48no—
257C2×Q16—24no—
258C2×SD16—24no—
259C2×M16—21no—
260C2×M16—22no—
261C4⋊Q8—22no—
262C4⋊Q8—21no—
263C4⋊Q8C23×C12712yes—
264C4⋊Q8C23×C12711yescharacteristic
265#127C23×C12711yescharacteristic
266#126—24no—
267#125—22no—
268#124—22no—
269#124D4×C12712yes—
270#123—22no—
271#123—24no—
272#123—22no—
273C22⋊Q8—24no—
274C22⋊Q8—22no—
275#121—24no—
276#121—22no—
277#121—22no—
278#120—22no—
279#120—22no—
280C4×Q8D4×C12712yes—
281C4×Q8C23×C12712yes—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4D4×C12712yes—
290C4×D4C23×C12712yes—
291#117—22no—
292#117—22no—
293#117D4×C12712yes—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C42—22no—
298C2×C42C23×C12711yescharacteristic
299#107—24no—
300#106—24no—
301C4⋊C8—22no—
302C4⋊C8C23×C12714yes—
303#103—24no—
304#102—24no—
305#97—21no—
306C4×C8C23×C12712yes—
307C24—22no—
308#41—24no—
309#41—42no—
310#41—44no—
311C2×Q8—22no—
312C2×Q8—44no—
313C2×Q8—41no—
314C2×Q8C2×D4×C12712yes—
315C2×Q8C2×D4×C12712yes—
316C2×D4—24no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4C2×D4×C12712yes—
323C2×D4C2×D4×C12712yes—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—84no—
332C22×C4C2×D4×C12712yes—
333C22×C4C2×D4×C12711yescharacteristic
334Q16—48no—
335SD16—48no—
336M16—24no—
337M16—44no—
338C2×C8—22no—
339C2×C8—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—42no—
347C4⋊C4C2×D4×C12712yes—
348C4⋊C4C2×D4×C12712yes—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—24no—
356C22⋊C4—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42C2×D4×C12711yescharacteristic
361C42C2×D4×C12712yes—
362C42C24×C12711yescharacteristic
363C23—22no—
364C23—22no—
365C23—48no—
366C23—82no—
367C23#120×C12711yescharacteristic
368Q8—22no—
369Q8—22no—
370Q8—42no—
371Q8—44no—
372Q8—82no—
373D4—22no—
374D4—42no—
375D4—44no—
376D4—48no—
377D4—42no—
378D4—42no—
379D4—82no—
380D4#137×C12714yescomplemented
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—82no—
395C2×C4—84no—
396C2×C4—82no—
397C2×C4#120×C12712yes—
398C2×C4#120×C12711yescharacteristic
399C2×C4C22×D4×C12711yescharacteristic
400C2×C4C22×D4×C12712yes—
401C8—24no—
402C8—44no—
403C22—22no—
404C22—24no—
405C22—21no—
406C22—42no—
407C22—42no—
408C22—44no—
409C22—42no—
410C22—88no—
411C22C2×#120×C12711yescharacteristic
412C4—22no—
413C4—21no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—82no—
419C4#453×C12712yes—
420C4C2×#137×C12712yes—
421C4#575×C12712yes—
422C2—41no—
423C2—44no—
424C2—82no—
425C2C127×#300712yes—
426C2C2×#453×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutator, Frattini, cosocleC42C24×C127
prime radicalC127×#3481C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#3152C2semidirect
3. C127×#3079C2semidirect
4. C127×#1632C2semidirect
5. C127×#1630C2semidirect
6. C127×#1629C2semidirect
7. C127×#1560C2semidirect
8. C127×#1489C2semidirect
17. C2×C4⋊Q8×C127C22semidirect
23. #494×C127C22semidirect
25. #477×C127C22semidirect
30. #423×C127C22semidirect
32. C4×M16×C127C22semidirect
33. #333×C127C22semidirect
35. Q8⋊C8×C127C22semidirect
166. D4×C127#137semidirect
208. #18528C127direct (N × H)
215. #3152C2×C127semidirect
216. #3079C2×C127semidirect
217. #1632C2×C127semidirect
218. #1630C2×C127semidirect
219. #1629C2×C127semidirect
220. #1560C2×C127semidirect
221. #1489C2×C127semidirect
222. C127#18528direct (N × H)
231. C2×C4⋊Q8C22×C127semidirect
237. #494C22×C127semidirect
239. #477C22×C127semidirect
244. #423C22×C127semidirect
246. C4×M16C22×C127semidirect
247. #333C22×C127semidirect
249. Q8⋊C8C22×C127semidirect
380. D4#137×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34812
C127×#31522
C127×#30792
C127×#16322
C127×#16302
C127×#16292
C127×#15602
C127×#14892
#18528127