Finite Group Databasebeta

← C127×#8357 #44572030

Subgroups — C127×#8357 #44572030

Order 32512.

710 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3507C211yesmaximal, characteristic
2C127×#1984C211yesmaximal, characteristic, complemented
3C127×#1801C211yesmaximal, characteristic, complemented
4C127×#1550C211yesmaximal, characteristic, complemented
5C127×#1547C211yesmaximal, characteristic
6C127×#1493C211yesmaximal, characteristic, complemented
7C127×#1492C211yesmaximal, characteristic
8#559×C127—21no—
9#557×C127—21no—
10#556×C127—21no—
11#556×C127—22no—
12#554×C127—21no—
13#552×C127—22no—
14#539×C127C414yes—
15#536×C127—21no—
16#533×C127C2211yescharacteristic
17#532×C127C2211yescharacteristic
18#529×C127—22no—
19#520×C127—21no—
20C2×C4×Q8×C127C2211yescharacteristic
21#423×C127C2211yescharacteristic
22#423×C127C2211yescharacteristic
23C2×C4⋊C8×C127C2211yescharacteristic
24C2×#101×C127—21no—
25C2×C4×C8×C127C2211yescharacteristic
26#343×C127—21no—
27#336×C127—22no—
28#333×C127—21no—
29#332×C127—21no—
30#332×C127—21no—
31#331×C127—21no—
32Q8⋊C8×C127—22no—
33D4⋊C8×C127—22no—
34C2×#41×C127—21no—
35C22×Q8×C127D411yescharacteristic
36C2×M16×C127—21no—
37C2×M16×C127—21no—
38C22×C8×C127—21no—
39C22×C8×C127—21no—
40C4⋊Q8×C127—22no—
41C4⋊Q8×C127D411yescharacteristic
42C4⋊Q8×C127D411yescharacteristic
43C4⋊Q8×C127D411yescharacteristic
44#127×C127D411yescharacteristic
45#126×C127—24no—
46#126×C127—24no—
47#125×C127—22no—
48#125×C127—24no—
49#125×C127—21no—
50#124×C127—22no—
51#124×C127—22no—
52#124×C127—21no—
53#123×C127—24no—
54#123×C127—24no—
55C22⋊Q8×C127—22no—
56C22⋊Q8×C127—22no—
57C22⋊Q8×C127—24no—
58C22⋊Q8×C127—21no—
59C22⋊Q8×C127C2×C412yes—
60#121×C127—21no—
61#121×C127C2×C412yes—
62C4×Q8×C127—22no—
63C4×Q8×C127—22no—
64C4×Q8×C127—21no—
65C4×Q8×C127—21no—
66C4×Q8×C127—21no—
67C4×Q8×C127—21no—
68C4×Q8×C127—21no—
69C4×Q8×C127D412yes—
70C4×D4×C127—22no—
71C4×D4×C127—22no—
72C4×D4×C127—24no—
73C4×D4×C127—21no—
74C4×D4×C127D412yes—
75#117×C127—22no—
76#117×C127—24no—
77#117×C127—21no—
78C2×C4⋊C4×C127—22no—
79C2×C4⋊C4×C127C2×C412yes—
80C2×C4⋊C4×C127D411yescharacteristic
81C2×C42×C127D412yes—
82C2×C42×C127C2311yescharacteristic
83C4⋊C8×C127—21no—
84C4⋊C8×C127—21no—
85C4⋊C8×C127—21no—
86C4⋊C8×C127—21no—
87C4⋊C8×C127—21no—
88C4⋊C8×C127—21no—
89#101×C127—42no—
90C4×C8×C127—21no—
91C4×C8×C127—21no—
92#41×C127—41no—
93#41×C127—41no—
94#41×C127—41no—
95#41×C127—41no—
96C2×Q8×C127—21no—
97C2×Q8×C127—21no—
98C2×Q8×C127—21no—
99C2×Q8×C127—21no—
100C2×Q8×C127—21no—
101C2×Q8×C127—21no—
102C2×Q8×C127—21no—
103C2×Q8×C127—21no—
104C2×Q8×C127C22⋊C412yes—
105C2×D4×C127—22no—
106C2×D4×C127—21no—
107C2×D4×C127—21no—
108C2×D4×C127C22⋊C412yes—
109C22×C4×C127—24no—
110C22×C4×C127—22no—
111C22×C4×C127C22⋊C412yes—
112C22×C4×C127C22⋊C412yes—
113C22×C4×C127C22×C411yescharacteristic
114C22×C4×C127C2×D411yescharacteristic
115C22×C4×C127C2×D411yescharacteristic
116M16×C127—41no—
117M16×C127—41no—
118M16×C127—41no—
119M16×C127—41no—
120C2×C8×C127—22no—
121C2×C8×C127—41no—
122C2×C8×C127—41no—
123C2×C8×C127—42no—
124C2×C8×C127—41no—
125C2×C8×C127—41no—
126C2×C8×C127—41no—
127C4⋊C4×C127—24no—
128C4⋊C4×C127—24no—
129C4⋊C4×C127—24no—
130C4⋊C4×C127—22no—
131C4⋊C4×C127—22no—
132C4⋊C4×C127—22no—
133C4⋊C4×C127—21no—
134C4⋊C4×C127—21no—
135C4⋊C4×C127—21no—
136C4⋊C4×C127—21no—
137C4⋊C4×C127—22no—
138C4⋊C4×C127—21no—
139C4⋊C4×C127—21no—
140C4⋊C4×C127C22⋊C412yes—
141C4⋊C4×C127C22⋊C412yes—
142C22⋊C4×C127—24no—
143C22⋊C4×C127—24no—
144C22⋊C4×C127—24no—
145C22⋊C4×C127—22no—
146C22⋊C4×C127—22no—
147C22⋊C4×C127—22no—
148C42×C127—24no—
149C42×C127—22no—
150C42×C127—22no—
151C42×C127—22no—
152C42×C127C2×D411yescharacteristic
153C42×C127C2×D411yescharacteristic
154C42×C127C2×D411yescharacteristic
155C42×C127C2×D411yescharacteristic
156C23×C127—22no—
157C23×C127C2×C22⋊C411yescharacteristic
158Q8×C127—42no—
159Q8×C127—41no—
160Q8×C127—41no—
161Q8×C127—41no—
162Q8×C127—41no—
163Q8×C127—41no—
164Q8×C127—41no—
165Q8×C127—41no—
166Q8×C127—41no—
167D4×C127—42no—
168D4×C127—42no—
169D4×C127—42no—
170D4×C127—41no—
171D4×C127—41no—
172D4×C127—41no—
173D4×C127—41no—
174C2×C4×C127—21no—
175C2×C4×C127—21no—
176C2×C4×C127—22no—
177C2×C4×C127—22no—
178C2×C4×C127—22no—
179C2×C4×C127—22no—
180C2×C4×C127—22no—
181C2×C4×C127—22no—
182C2×C4×C127—22no—
183C2×C4×C127—41no—
184C2×C4×C127—42no—
185C2×C4×C127—42no—
186C2×C4×C127—42no—
187C2×C4×C127—84no—
188C2×C4×C127C2×C22⋊C411yescharacteristic
189C2×C4×C127C2×C22⋊C411yescharacteristic
190C2×C4×C127#12011yescharacteristic
191C2×C4×C127#12011yescharacteristic
192C2×C4×C127#12011yescharacteristic
193C2×C4×C127#12011yescharacteristic
194C2×C4×C127#13511yescharacteristic
195C2×C4×C127#13511yescharacteristic
196C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
197C2×C4×C127#13711yescharacteristic
198C8×C127—42no—
199C8×C127—81no—
200C8×C127—81no—
201C8×C127—81no—
202C22×C127—21no—
203C22×C127—21no—
204C22×C127—42no—
205C22×C127—42no—
206C22×C127—42no—
207C22×C127#37911yescharacteristic
208C22×C127#41611yescharacteristic
209C22×C127#41711yescharacteristic
210C4×C127—21no—
211C4×C127—21no—
212C4×C127—21no—
213C4×C127—42no—
214C4×C127—42no—
215C4×C127—42no—
216C4×C127—42no—
217C4×C127—42no—
218C4×C127#44911yescharacteristic
219C4×C127#44911yescharacteristic
220C4×C127#45211yescharacteristic
221C4×C127#45211yescharacteristic
222C4×C127#45411yescharacteristic
223C4×C127#45611yescharacteristic
224#8357C12711yesmaximal, characteristic, direct product factor, Sylow
225C2×C127—21no—
226C2×C127—21no—
227C2×C127—82no—
228C2×C127#178811yescharacteristic
229C2×C127#179111yescharacteristic
230C2×C127#179311yescharacteristic
231#3507C2×C12711yescharacteristic
232#1984C2×C12711yescharacteristic, complemented
233#1801C2×C12711yescharacteristic, complemented
234#1550C2×C12711yescharacteristic, complemented
235#1547C2×C12711yescharacteristic
236#1493C2×C12711yescharacteristic, complemented
237#1492C2×C12711yescharacteristic
238C127#835711yescharacteristic, direct product factor, Sylow
239#559—21no—
240#557—21no—
241#556—21no—
242#556—22no—
243#554—21no—
244#552—22no—
245#539C4×C12714yes—
246#536—21no—
247#533C22×C12711yescharacteristic
248#532C22×C12711yescharacteristic
249#529—22no—
250#520—21no—
251C2×C4×Q8C22×C12711yescharacteristic
252#423C22×C12711yescharacteristic
253#423C22×C12711yescharacteristic
254C2×C4⋊C8C22×C12711yescharacteristic
255C2×#101—21no—
256C2×C4×C8C22×C12711yescharacteristic
257#343—21no—
258#336—22no—
259#333—21no—
260#332—21no—
261#332—21no—
262#331—21no—
263Q8⋊C8—22no—
264D4⋊C8—22no—
265C2×#41—21no—
266C22×Q8D4×C12711yescharacteristic
267C2×M16—21no—
268C2×M16—21no—
269C22×C8—21no—
270C22×C8—21no—
271C4⋊Q8—22no—
272C4⋊Q8D4×C12711yescharacteristic
273C4⋊Q8D4×C12711yescharacteristic
274C4⋊Q8D4×C12711yescharacteristic
275#127D4×C12711yescharacteristic
276#126—24no—
277#126—24no—
278#125—22no—
279#125—24no—
280#125—21no—
281#124—22no—
282#124—22no—
283#124—21no—
284#123—24no—
285#123—24no—
286C22⋊Q8—22no—
287C22⋊Q8—22no—
288C22⋊Q8—24no—
289C22⋊Q8—21no—
290C22⋊Q8C2×C4×C12712yes—
291#121—21no—
292#121C2×C4×C12712yes—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8D4×C12712yes—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—24no—
304C4×D4—21no—
305C4×D4D4×C12712yes—
306#117—22no—
307#117—24no—
308#117—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4C2×C4×C12712yes—
311C2×C4⋊C4D4×C12711yescharacteristic
312C2×C42D4×C12712yes—
313C2×C42C23×C12711yescharacteristic
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320#101—42no—
321C4×C8—21no—
322C4×C8—21no—
323#41—41no—
324#41—41no—
325#41—41no—
326#41—41no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8C22⋊C4×C12712yes—
336C2×D4—22no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4C22⋊C4×C12712yes—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4C22⋊C4×C12712yes—
343C22×C4C22⋊C4×C12712yes—
344C22×C4C22×C4×C12711yescharacteristic
345C22×C4C2×D4×C12711yescharacteristic
346C22×C4C2×D4×C12711yescharacteristic
347M16—41no—
348M16—41no—
349M16—41no—
350M16—41no—
351C2×C8—22no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4C22⋊C4×C12712yes—
372C4⋊C4C22⋊C4×C12712yes—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C42—24no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42C2×D4×C12711yescharacteristic
384C42C2×D4×C12711yescharacteristic
385C42C2×D4×C12711yescharacteristic
386C42C2×D4×C12711yescharacteristic
387C23—22no—
388C23C2×C22⋊C4×C12711yescharacteristic
389Q8—42no—
390Q8—41no—
391Q8—41no—
392Q8—41no—
393Q8—41no—
394Q8—41no—
395Q8—41no—
396Q8—41no—
397Q8—41no—
398D4—42no—
399D4—42no—
400D4—42no—
401D4—41no—
402D4—41no—
403D4—41no—
404D4—41no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—84no—
419C2×C4C2×C22⋊C4×C12711yescharacteristic
420C2×C4C2×C22⋊C4×C12711yescharacteristic
421C2×C4#120×C12711yescharacteristic
422C2×C4#120×C12711yescharacteristic
423C2×C4#120×C12711yescharacteristic
424C2×C4#120×C12711yescharacteristic
425C2×C4#135×C12711yescharacteristic
426C2×C4#135×C12711yescharacteristic
427C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
428C2×C4#137×C12711yescharacteristic
429C8—42no—
430C8—81no—
431C8—81no—
432C8—81no—
433C22—21no—
434C22—21no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22#379×C12711yescharacteristic
439C22#416×C12711yescharacteristic
440C22#417×C12711yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4#449×C12711yescharacteristic
450C4#449×C12711yescharacteristic
451C4#452×C12711yescharacteristic
452C4#452×C12711yescharacteristic
453C4#454×C12711yescharacteristic
454C4#456×C12711yescharacteristic
455C2—21no—
456C2—21no—
457C2—82no—
458C2C127×#178811yescharacteristic
459C2C127×#179111yescharacteristic
460C2C127×#179311yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radical#532×C127C22
LCSG ▷ C22×C4 ▷ 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×#1984C2semidirect
3. C127×#1801C2semidirect
4. C127×#1550C2semidirect
6. C127×#1493C2semidirect
224. #8357C127direct (N × H)
232. #1984C2×C127semidirect
233. #1801C2×C127semidirect
234. #1550C2×C127semidirect
236. #1493C2×C127semidirect
238. C127#8357direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#35072
C127×#19842
C127×#18012
C127×#15502
C127×#15472
C127×#14932
C127×#14922
#8357127