Finite Group Databasebeta

← C127×#12262 #44575935

Subgroups — C127×#12262 #44575935

Order 32512.

596 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#2668C211yesmaximal, characteristic
2C127×#2029C211yesmaximal, characteristic, complemented
3C127×#1614C211yesmaximal, characteristic, complemented
4C127×#1596C211yesmaximal, characteristic, complemented
5C127×#1352C211yesmaximal, characteristic, complemented
6C127×#1343C211yesmaximal, characteristic, complemented
7C127×#1316C211yesmaximal, characteristic, complemented
8#543×C127C2211yescharacteristic
9C2×#121×C127—21no—
10C2×#120×C127—21no—
11#484×C127—21no—
12#450×C127—21no—
13#399×C127C2211yescharacteristic
14#397×C127—21no—
15#394×C127—21no—
16#392×C127C2211yescharacteristic
17#386×C127—21no—
18#386×C127—21no—
19#379×C127—21no—
20#379×C127C2211yescharacteristic
21#342×C127—21no—
22#342×C127C2211yescharacteristic
23#328×C127—21no—
24#328×C127—21no—
25#323×C127C2211yescharacteristic
26#323×C127C2211yescharacteristic
27C25×C127D411yescharacteristic
28C22×D4×C127—21no—
29C22×D4×C127—21no—
30C22×D4×C127—21no—
31C22×D4×C127D411yescharacteristic
32C23×C4×C127—21no—
33C2×SD16×C127—41no—
34C22⋊Q8×C127—21no—
35C22⋊Q8×C127—21no—
36C22⋊Q8×C127—41no—
37C22⋊Q8×C127—41no—
38#121×C127—44no—
39#120×C127—44no—
40C2×C4⋊C4×C127—21no—
41C2×C4⋊C4×C127D411yescharacteristic
42C2×C22⋊C4×C127—21no—
43C2×C22⋊C4×C127—21no—
44C2×C22⋊C4×C127—21no—
45C2×C22⋊C4×C127—21no—
46C2×C22⋊C4×C127—21no—
47C2×C22⋊C4×C127—21no—
48C2×C22⋊C4×C127—21no—
49C2×C22⋊C4×C127—21no—
50C2×C22⋊C4×C127—21no—
51C2×C22⋊C4×C127—41no—
52C2×C22⋊C4×C127—41no—
53C2×C22⋊C4×C127D411yescharacteristic
54C2×C22⋊C4×C127C2311yescharacteristic
55#107×C127—41no—
56#103×C127—41no—
57#102×C127—41no—
58C22⋊C8×C127—21no—
59C22⋊C8×C127—21no—
60#95×C127—21no—
61#95×C127—21no—
62#95×C127D411yescharacteristic
63#95×C127D411yescharacteristic
64C24×C127—21no—
65C24×C127—21no—
66C24×C127—21no—
67C24×C127—21no—
68C24×C127—41no—
69C24×C127—42no—
70C24×C127—41no—
71C24×C127—81no—
72C24×C127C2×D411yescharacteristic
73C2×Q8×C127—21no—
74C2×Q8×C127—21no—
75C2×D4×C127—24no—
76C2×D4×C127—41no—
77C2×D4×C127—42no—
78C2×D4×C127—42no—
79C2×D4×C127—42no—
80C2×D4×C127—42no—
81C2×D4×C127—41no—
82C2×D4×C127—41no—
83C2×D4×C127—81no—
84C2×D4×C127—81no—
85C2×D4×C127—81no—
86C2×D4×C127—81no—
87C22×C4×C127—21no—
88C22×C4×C127—21no—
89C22×C4×C127—21no—
90C22×C4×C127—21no—
91C22×C4×C127—21no—
92C22×C4×C127—21no—
93C22×C4×C127—42no—
94C22×C4×C127—41no—
95C22×C4×C127—81no—
96C22×C4×C127—81no—
97C22×C4×C127C2×D411yescharacteristic
98C22×C4×C127C2×D411yescharacteristic
99SD16×C127—82no—
100C2×C8×C127—41no—
101C2×C8×C127—41no—
102C4⋊C4×C127—21no—
103C4⋊C4×C127—21no—
104C4⋊C4×C127—41no—
105C4⋊C4×C127—42no—
106C4⋊C4×C127—41no—
107C4⋊C4×C127—41no—
108C22⋊C4×C127—41no—
109C22⋊C4×C127—42no—
110C22⋊C4×C127—42no—
111C22⋊C4×C127—42no—
112C22⋊C4×C127—42no—
113C22⋊C4×C127—42no—
114C22⋊C4×C127—41no—
115C22⋊C4×C127—41no—
116C22⋊C4×C127—41no—
117C22⋊C4×C127—41no—
118C22⋊C4×C127—41no—
119C22⋊C4×C127—81no—
120C22⋊C4×C127—81no—
121C22⋊C4×C127—82no—
122C22⋊C4×C127—82no—
123C23×C127—21no—
124C23×C127—21no—
125C23×C127—22no—
126C23×C127—22no—
127C23×C127—41no—
128C23×C127—41no—
129C23×C127—41no—
130C23×C127—41no—
131C23×C127—41no—
132C23×C127—41no—
133C23×C127—41no—
134C23×C127—41no—
135C23×C127—42no—
136C23×C127—42no—
137C23×C127—42no—
138C23×C127—41no—
139C23×C127—41no—
140C23×C127—41no—
141C23×C127—41no—
142C23×C127—41no—
143C23×C127—41no—
144C23×C127—81no—
145C23×C127—82no—
146C23×C127—81no—
147C23×C127—82no—
148C23×C127—84no—
149C23×C127—81no—
150C23×C127#12011yescharacteristic
151C23×C127#13511yescharacteristic
152C23×C127#13711yescharacteristic
153Q8×C127—81no—
154Q8×C127—81no—
155D4×C127—42no—
156D4×C127—84no—
157D4×C127—81no—
158D4×C127—82no—
159D4×C127—82no—
160D4×C127—84no—
161C2×C4×C127—22no—
162C2×C4×C127—21no—
163C2×C4×C127—21no—
164C2×C4×C127—41no—
165C2×C4×C127—41no—
166C2×C4×C127—41no—
167C2×C4×C127—41no—
168C2×C4×C127—41no—
169C2×C4×C127—41no—
170C2×C4×C127—41no—
171C2×C4×C127—41no—
172C2×C4×C127—41no—
173C2×C4×C127—41no—
174C2×C4×C127—41no—
175C2×C4×C127—41no—
176C2×C4×C127—41no—
177C2×C4×C127—41no—
178C2×C4×C127—41no—
179C2×C4×C127—84no—
180C2×C4×C127—81no—
181C2×C4×C127—81no—
182C2×C4×C127—81no—
183C2×C4×C127—81no—
184C2×C4×C127—81no—
185C8×C127—81no—
186C8×C127—81no—
187C22×C127—21no—
188C22×C127—21no—
189C22×C127—21no—
190C22×C127—21no—
191C22×C127—41no—
192C22×C127—41no—
193C22×C127—42no—
194C22×C127—42no—
195C22×C127—42no—
196C22×C127—42no—
197C22×C127—41no—
198C22×C127—41no—
199C22×C127—44no—
200C22×C127—42no—
201C22×C127—42no—
202C22×C127—81no—
203C22×C127—81no—
204C22×C127—81no—
205C22×C127—81no—
206C22×C127—81no—
207C22×C127—82no—
208C22×C127—82no—
209C22×C127—82no—
210C22×C127—81no—
211C22×C127—81no—
212C22×C127—81no—
213C22×C127—81no—
214C22×C127#45211yescharacteristic
215C22×C127#45511yescharacteristic
216C22×C127B(4,𝔽2)11yescharacteristic
217C4×C127—42no—
218C4×C127—41no—
219C4×C127—41no—
220C4×C127—81no—
221C4×C127—81no—
222C4×C127—81no—
223C4×C127—81no—
224C4×C127—81no—
225C4×C127—161no—
226#12262C12711yesmaximal, characteristic, direct product factor, Sylow
227C2×C127—21no—
228C2×C127—21no—
229C2×C127—41no—
230C2×C127—41no—
231C2×C127—82no—
232C2×C127—82no—
233C2×C127#160611yescharacteristic
234C2×C127#219611yescharacteristic
235C2×C127#220011yescharacteristic
236#2668C2×C12711yescharacteristic
237#2029C2×C12711yescharacteristic, complemented
238#1614C2×C12711yescharacteristic, complemented
239#1596C2×C12711yescharacteristic, complemented
240#1352C2×C12711yescharacteristic, complemented
241#1343C2×C12711yescharacteristic, complemented
242#1316C2×C12711yescharacteristic, complemented
243C127#1226211yescharacteristic, direct product factor, Sylow
244#543C22×C12711yescharacteristic
245C2×#121—21no—
246C2×#120—21no—
247#484—21no—
248#450—21no—
249#399C22×C12711yescharacteristic
250#397—21no—
251#394—21no—
252#392C22×C12711yescharacteristic
253#386—21no—
254#386—21no—
255#379—21no—
256#379C22×C12711yescharacteristic
257#342—21no—
258#342C22×C12711yescharacteristic
259#328—21no—
260#328—21no—
261#323C22×C12711yescharacteristic
262#323C22×C12711yescharacteristic
263C25D4×C12711yescharacteristic
264C22×D4—21no—
265C22×D4—21no—
266C22×D4—21no—
267C22×D4D4×C12711yescharacteristic
268C23×C4—21no—
269C2×SD16—41no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272C22⋊Q8—41no—
273C22⋊Q8—41no—
274#121—44no—
275#120—44no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4D4×C12711yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4D4×C12711yescharacteristic
290C2×C22⋊C4C23×C12711yescharacteristic
291#107—41no—
292#103—41no—
293#102—41no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296#95—21no—
297#95—21no—
298#95D4×C12711yescharacteristic
299#95D4×C12711yescharacteristic
300C24—21no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—41no—
305C24—42no—
306C24—41no—
307C24—81no—
308C24C2×D4×C12711yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×D4—24no—
312C2×D4—41no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—81no—
320C2×D4—81no—
321C2×D4—81no—
322C2×D4—81no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—81no—
332C22×C4—81no—
333C22×C4C2×D4×C12711yescharacteristic
334C22×C4C2×D4×C12711yescharacteristic
335SD16—82no—
336C2×C8—41no—
337C2×C8—41no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—81no—
356C22⋊C4—81no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C23—21no—
360C23—21no—
361C23—22no—
362C23—22no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—81no—
381C23—82no—
382C23—81no—
383C23—82no—
384C23—84no—
385C23—81no—
386C23#120×C12711yescharacteristic
387C23#135×C12711yescharacteristic
388C23#137×C12711yescharacteristic
389Q8—81no—
390Q8—81no—
391D4—42no—
392D4—84no—
393D4—81no—
394D4—82no—
395D4—82no—
396D4—84no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—84no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C8—81no—
422C8—81no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—41no—
428C22—41no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—41no—
434C22—41no—
435C22—44no—
436C22—42no—
437C22—42no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22#452×C12711yescharacteristic
451C22#455×C12711yescharacteristic
452C22B(4,𝔽2)×C12711yescharacteristic
453C4—42no—
454C4—41no—
455C4—41no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4—81no—
461C4—161no—
462C2—21no—
463C2—21no—
464C2—41no—
465C2—41no—
466C2—82no—
467C2—82no—
468C2C127×#160611yescharacteristic
469C2C127×#219611yescharacteristic
470C2C127×#220011yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC127×#2668C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C22⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2029C2semidirect
3. C127×#1614C2semidirect
4. C127×#1596C2semidirect
5. C127×#1352C2semidirect
6. C127×#1343C2semidirect
7. C127×#1316C2semidirect
226. #12262C127direct (N × H)
237. #2029C2×C127semidirect
238. #1614C2×C127semidirect
239. #1596C2×C127semidirect
240. #1352C2×C127semidirect
241. #1343C2×C127semidirect
242. #1316C2×C127semidirect
243. C127#12262direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#26682
C127×#20292
C127×#16142
C127×#15962
C127×#13522
C127×#13432
C127×#13162
#12262127