Finite Group Databasebeta

← C127×#8493 #44572166

Subgroups — C127×#8493 #44572166

Order 32512.

748 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
1C127×#3548C211yesmaximal, characteristic
2C127×#1963C211yesmaximal, characteristic, complemented
3C127×#1559C211yesmaximal, characteristic, complemented
4C127×#1557C211yesmaximal, characteristic, complemented
5C127×#1536C211yesmaximal, characteristic, complemented
6C127×#1506C211yesmaximal, characteristic, complemented
7C127×#1505C211yesmaximal, characteristic, complemented
8#562×C127—22no—
9#554×C127—21no—
10#554×C127C412yescomplemented
11#550×C127—21no—
12#550×C127C412yescomplemented
13#547×C127—21no—
14#547×C127—21no—
15#540×C127—22no—
16#537×C127—22no—
17#537×C127—22no—
18#533×C127C2211yescharacteristic
19#532×C127C2211yescharacteristic
20#532×C127C2211yescharacteristic
21#432×C127C2211yescharacteristic
22#432×C127C2211yescharacteristic
23#423×C127C2211yescharacteristic
24C2×#104×C127—22no—
25#376×C127C2211yescharacteristic
26#333×C127—21no—
27#332×C127—21no—
28#332×C127—21no—
29#331×C127—21no—
30#329×C127—22no—
31#328×C127—22no—
32#327×C127—22no—
33C2×#41×C127—22no—
34C2×#41×C127—22no—
35C2×#41×C127—21no—
36C2×#41×C127D412yes—
37C2×M16×C127—21no—
38C4⋊Q8×C127C2×C411yescharacteristic
39C4⋊Q8×C127D411yescharacteristic
40C4⋊Q8×C127D411yescharacteristic
41C4⋊Q8×C127D411yescharacteristic
42#127×C127—22no—
43#127×C127C2×C411yescharacteristic
44#127×C127D411yescharacteristic
45#125×C127—22no—
46#125×C127—21no—
47#124×C127—21no—
48#124×C127D412yes—
49#123×C127—24no—
50#123×C127—24no—
51C22⋊Q8×C127—22no—
52C22⋊Q8×C127—24no—
53C22⋊Q8×C127—22no—
54C22⋊Q8×C127—24no—
55C22⋊Q8×C127—21no—
56C22⋊Q8×C127D412yes—
57#121×C127—24no—
58#121×C127—24no—
59#121×C127—22no—
60#121×C127—24no—
61#121×C127—24no—
62#121×C127—21no—
63#121×C127—21no—
64#121×C127—21no—
65#121×C127—22no—
66#121×C127D412yes—
67C4×Q8×C127—21no—
68C4×Q8×C127C2×C412yes—
69C4×D4×C127—22no—
70C4×D4×C127—22no—
71C4×D4×C127—21no—
72C4×D4×C127—21no—
73C4×D4×C127—21no—
74C4×D4×C127C2×C412yes—
75#117×C127—21no—
76C2×C4⋊C4×C127—22no—
77C2×C4⋊C4×C127—22no—
78C2×C42×C127—21no—
79C2×C42×C127C2311yescharacteristic
80C4⋊C8×C127—21no—
81C4⋊C8×C127—21no—
82C4⋊C8×C127—21no—
83C4⋊C8×C127—21no—
84#104×C127—44no—
85C22⋊C8×C127—21no—
86C22⋊C8×C127—21no—
87#97×C127—21no—
88#97×C127—21no—
89#95×C127—21no—
90#95×C127—21no—
91#41×C127—42no—
92#41×C127—42no—
93#41×C127—82no—
94#41×C127—84no—
95#41×C127—82no—
96#41×C127—82no—
97C2×Q8×C127—21no—
98C2×Q8×C127—21no—
99C2×Q8×C127C22⋊C412yes—
100C2×Q8×C127C22⋊C412yes—
101C2×D4×C127—24no—
102C2×D4×C127—24no—
103C2×D4×C127—22no—
104C2×D4×C127—22no—
105C2×D4×C127—21no—
106C2×D4×C127—21no—
107C2×D4×C127—21no—
108C2×D4×C127—21no—
109C2×D4×C127C22⋊C412yes—
110C2×D4×C127C22⋊C412yes—
111C22×C4×C127—24no—
112C22×C4×C127—24no—
113C22×C4×C127—22no—
114C22×C4×C127—22no—
115C22×C4×C127—41no—
116C22×C4×C127C2×D411yescharacteristic
117C22×C4×C127C2×D411yescharacteristic
118C22×C4×C127C2×D411yescharacteristic
119M16×C127—42no—
120C2×C8×C127—41no—
121C2×C8×C127—41no—
122C2×C8×C127—41no—
123C4⋊C4×C127—24no—
124C4⋊C4×C127—24no—
125C4⋊C4×C127—21no—
126C4⋊C4×C127—21no—
127C4⋊C4×C127—22no—
128C4⋊C4×C127—21no—
129C4⋊C4×C127—21no—
130C4⋊C4×C127—21no—
131C4⋊C4×C127—21no—
132C4⋊C4×C127—22no—
133C4⋊C4×C127C22⋊C412yes—
134C4⋊C4×C127C22⋊C412yes—
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—22no—
142C42×C127—22no—
143C42×C127—22no—
144C42×C127—41no—
145C42×C127C22×C411yescharacteristic
146C42×C127C2×D411yescharacteristic
147C42×C127C2×D411yescharacteristic
148C42×C127C2×D411yescharacteristic
149C23×C127—22no—
150C23×C127—22no—
151C23×C127#12011yescharacteristic
152Q8×C127—42no—
153Q8×C127—42no—
154Q8×C127—41no—
155Q8×C127—41no—
156Q8×C127—81no—
157D4×C127—42no—
158D4×C127—42no—
159D4×C127—42no—
160D4×C127—42no—
161D4×C127—41no—
162D4×C127—41no—
163D4×C127—81no—
164D4×C127—82no—
165D4×C127—82no—
166D4×C127—84no—
167D4×C127—84no—
168C2×C4×C127—21no—
169C2×C4×C127—21no—
170C2×C4×C127—22no—
171C2×C4×C127—22no—
172C2×C4×C127—22no—
173C2×C4×C127—22no—
174C2×C4×C127—41no—
175C2×C4×C127—41no—
176C2×C4×C127—44no—
177C2×C4×C127—42no—
178C2×C4×C127—42no—
179C2×C4×C127—42no—
180C2×C4×C127—84no—
181C2×C4×C127—84no—
182C2×C4×C127—81no—
183C2×C4×C127—81no—
184C2×C4×C127C2×C22⋊C411yescharacteristic
185C2×C4×C127C2×C22⋊C411yescharacteristic
186C2×C4×C127C2×C22⋊C411yescharacteristic
187C2×C4×C127#12011yescharacteristic
188C2×C4×C127#12011yescharacteristic
189C2×C4×C127#12011yescharacteristic
190C8×C127—81no—
191C8×C127—81no—
192C8×C127—81no—
193C22×C127—21no—
194C22×C127—21no—
195C22×C127—21no—
196C22×C127—42no—
197C22×C127—42no—
198C22×C127—42no—
199C22×C127—42no—
200C22×C127—42no—
201C22×C127—42no—
202C22×C127#37911yescharacteristic
203C4×C127—21no—
204C4×C127—21no—
205C4×C127—21no—
206C4×C127—21no—
207C4×C127—21no—
208C4×C127—42no—
209C4×C127—42no—
210C4×C127—42no—
211C4×C127—42no—
212C4×C127—82no—
213C4×C127#45311yescharacteristic
214C4×C127#45511yescharacteristic
215#8493C12711yesmaximal, characteristic, direct product factor, Sylow
216C2×C127—41no—
217C2×C127—82no—
218C2×C127—82no—
219C2×C127#179211yescharacteristic
220C2×C127#179511yescharacteristic
221C2×C127#179611yescharacteristic
222#3548C2×C12711yescharacteristic
223#1963C2×C12711yescharacteristic, complemented
224#1559C2×C12711yescharacteristic, complemented
225#1557C2×C12711yescharacteristic, complemented
226#1536C2×C12711yescharacteristic, complemented
227#1506C2×C12711yescharacteristic, complemented
228#1505C2×C12711yescharacteristic, complemented
229C127#849311yescharacteristic, direct product factor, Sylow
230#562—22no—
231#554—21no—
232#554C4×C12712yescomplemented
233#550—21no—
234#550C4×C12712yescomplemented
235#547—21no—
236#547—21no—
237#540—22no—
238#537—22no—
239#537—22no—
240#533C22×C12711yescharacteristic
241#532C22×C12711yescharacteristic
242#532C22×C12711yescharacteristic
243#432C22×C12711yescharacteristic
244#432C22×C12711yescharacteristic
245#423C22×C12711yescharacteristic
246C2×#104—22no—
247#376C22×C12711yescharacteristic
248#333—21no—
249#332—21no—
250#332—21no—
251#331—21no—
252#329—22no—
253#328—22no—
254#327—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—21no—
258C2×#41D4×C12712yes—
259C2×M16—21no—
260C4⋊Q8C2×C4×C12711yescharacteristic
261C4⋊Q8D4×C12711yescharacteristic
262C4⋊Q8D4×C12711yescharacteristic
263C4⋊Q8D4×C12711yescharacteristic
264#127—22no—
265#127C2×C4×C12711yescharacteristic
266#127D4×C12711yescharacteristic
267#125—22no—
268#125—21no—
269#124—21no—
270#124D4×C12712yes—
271#123—24no—
272#123—24no—
273C22⋊Q8—22no—
274C22⋊Q8—24no—
275C22⋊Q8—22no—
276C22⋊Q8—24no—
277C22⋊Q8—21no—
278C22⋊Q8D4×C12712yes—
279#121—24no—
280#121—24no—
281#121—22no—
282#121—24no—
283#121—24no—
284#121—21no—
285#121—21no—
286#121—21no—
287#121—22no—
288#121D4×C12712yes—
289C4×Q8—21no—
290C4×Q8C2×C4×C12712yes—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—21no—
294C4×D4—21no—
295C4×D4—21no—
296C4×D4C2×C4×C12712yes—
297#117—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C42—21no—
301C2×C42C23×C12711yescharacteristic
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306#104—44no—
307C22⋊C8—21no—
308C22⋊C8—21no—
309#97—21no—
310#97—21no—
311#95—21no—
312#95—21no—
313#41—42no—
314#41—42no—
315#41—82no—
316#41—84no—
317#41—82no—
318#41—82no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8C22⋊C4×C12712yes—
322C2×Q8C22⋊C4×C12712yes—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4C22⋊C4×C12712yes—
332C2×D4C22⋊C4×C12712yes—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4C2×D4×C12711yescharacteristic
339C22×C4C2×D4×C12711yescharacteristic
340C22×C4C2×D4×C12711yescharacteristic
341M16—42no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4C22⋊C4×C12712yes—
356C4⋊C4C22⋊C4×C12712yes—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C42—22no—
365C42—22no—
366C42—41no—
367C42C22×C4×C12711yescharacteristic
368C42C2×D4×C12711yescharacteristic
369C42C2×D4×C12711yescharacteristic
370C42C2×D4×C12711yescharacteristic
371C23—22no—
372C23—22no—
373C23#120×C12711yescharacteristic
374Q8—42no—
375Q8—42no—
376Q8—41no—
377Q8—41no—
378Q8—81no—
379D4—42no—
380D4—42no—
381D4—42no—
382D4—42no—
383D4—41no—
384D4—41no—
385D4—81no—
386D4—82no—
387D4—82no—
388D4—84no—
389D4—84no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4C2×C22⋊C4×C12711yescharacteristic
407C2×C4C2×C22⋊C4×C12711yescharacteristic
408C2×C4C2×C22⋊C4×C12711yescharacteristic
409C2×C4#120×C12711yescharacteristic
410C2×C4#120×C12711yescharacteristic
411C2×C4#120×C12711yescharacteristic
412C8—81no—
413C8—81no—
414C8—81no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22#379×C12711yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—42no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—82no—
435C4#453×C12711yescharacteristic
436C4#455×C12711yescharacteristic
437C2—41no—
438C2—82no—
439C2—82no—
440C2C127×#179211yescharacteristic
441C2C127×#179511yescharacteristic
442C2C127×#179611yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC42C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC127×#3548C2
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×#1963C2semidirect
3. C127×#1559C2semidirect
4. C127×#1557C2semidirect
5. C127×#1536C2semidirect
6. C127×#1506C2semidirect
7. C127×#1505C2semidirect
10. #554×C127C4semidirect
12. #550×C127C4semidirect
215. #8493C127direct (N × H)
223. #1963C2×C127semidirect
224. #1559C2×C127semidirect
225. #1557C2×C127semidirect
226. #1536C2×C127semidirect
227. #1506C2×C127semidirect
228. #1505C2×C127semidirect
229. C127#8493direct (N × H)
232. #554C4×C127semidirect
234. #550C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#35482
C127×#19632
C127×#15592
C127×#15572
C127×#15362
C127×#15062
C127×#15052
#8493127