Finite Group Databasebeta

← C127×#12205 #44575878

Subgroups — C127×#12205 #44575878

Order 32512.

544 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2812C211yesmaximal, characteristic
2C127×#2029C211yesmaximal, characteristic, complemented
3C127×#1608C211yesmaximal, characteristic, complemented
4C127×#1596C211yesmaximal, characteristic, complemented
5C127×#1344C211yesmaximal, characteristic, complemented
6C127×#1343C211yesmaximal, characteristic, complemented
7C127×#1316C211yesmaximal, characteristic, complemented
8#543×C127C2211yescharacteristic
9C2×#120×C127—21no—
10#483×C127—21no—
11#450×C127—21no—
12#397×C127—21no—
13#397×C127C2211yescharacteristic
14#394×C127—21no—
15#394×C127—21no—
16#394×C127—21no—
17#392×C127—21no—
18#392×C127C2211yescharacteristic
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
32C2×SD16×C127—41no—
33C22⋊Q8×C127—21no—
34C22⋊Q8×C127—21no—
35C22⋊Q8×C127—41no—
36C22⋊Q8×C127—41no—
37#120×C127—42no—
38#120×C127—42no—
39C2×C4⋊C4×C127—21no—
40C2×C4⋊C4×C127D411yescharacteristic
41C2×C22⋊C4×C127—21no—
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#106×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—41no—
68C24×C127—41no—
69C24×C127—81no—
70C24×C127—81no—
71C24×C127SD1612yescomplemented
72C24×C127C2×D411yescharacteristic
73C2×Q8×C127—21no—
74C2×Q8×C127—21no—
75C2×D4×C127—41no—
76C2×D4×C127—42no—
77C2×D4×C127—42no—
78C2×D4×C127—41no—
79C2×D4×C127—41no—
80C2×D4×C127—81no—
81C2×D4×C127—81no—
82C2×D4×C127—81no—
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—41no—
94C22×C4×C127C2×D411yescharacteristic
95C22×C4×C127C2×D411yescharacteristic
96SD16×C127—82no—
97C2×C8×C127—41no—
98C2×C8×C127—41no—
99C4⋊C4×C127—21no—
100C4⋊C4×C127—21no—
101C4⋊C4×C127—41no—
102C4⋊C4×C127—41no—
103C4⋊C4×C127—41no—
104C4⋊C4×C127—81no—
105C22⋊C4×C127—41no—
106C22⋊C4×C127—42no—
107C22⋊C4×C127—42no—
108C22⋊C4×C127—41no—
109C22⋊C4×C127—41no—
110C22⋊C4×C127—41no—
111C22⋊C4×C127—41no—
112C22⋊C4×C127—41no—
113C22⋊C4×C127—41no—
114C22⋊C4×C127—41no—
115C22⋊C4×C127—81no—
116C22⋊C4×C127—81no—
117C22⋊C4×C127—81no—
118C22⋊C4×C127—81no—
119C22⋊C4×C127—81no—
120C22⋊C4×C127—82no—
121C22⋊C4×C127—82no—
122C23×C127—21no—
123C23×C127—21no—
124C23×C127—22no—
125C23×C127—41no—
126C23×C127—41no—
127C23×C127—42no—
128C23×C127—41no—
129C23×C127—41no—
130C23×C127—41no—
131C23×C127—41no—
132C23×C127—41no—
133C23×C127—41no—
134C23×C127—41no—
135C23×C127—41no—
136C23×C127—42no—
137C23×C127—42no—
138C23×C127—41no—
139C23×C127—41no—
140C23×C127—41no—
141C23×C127—41no—
142C23×C127—41no—
143C23×C127—81no—
144C23×C127—82no—
145C23×C127—82no—
146C23×C127—81no—
147C23×C127—81no—
148C23×C127—84no—
149C23×C127#12011yescharacteristic
150C23×C127C2×SD1611yescharacteristic
151C23×C127AGL(1,ℤ/8)11yescharacteristic
152Q8×C127—81no—
153Q8×C127—81no—
154D4×C127—42no—
155D4×C127—81no—
156D4×C127—82no—
157D4×C127—82no—
158D4×C127—84no—
159D4×C127—84no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—41no—
163C2×C4×C127—41no—
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—81no—
179C2×C4×C127—81no—
180C2×C4×C127—81no—
181C2×C4×C127—81no—
182C2×C4×C127—81no—
183C8×C127—81no—
184C8×C127—81no—
185C22×C127—21no—
186C22×C127—21no—
187C22×C127—21no—
188C22×C127—21no—
189C22×C127—41no—
190C22×C127—41no—
191C22×C127—42no—
192C22×C127—42no—
193C22×C127—41no—
194C22×C127—41no—
195C22×C127—44no—
196C22×C127—42no—
197C22×C127—81no—
198C22×C127—81no—
199C22×C127—81no—
200C22×C127—81no—
201C22×C127—81no—
202C22×C127—81no—
203C22×C127—81no—
204C22×C127—82no—
205C22×C127—82no—
206C22×C127—82no—
207C22×C127—81no—
208C22×C127—81no—
209C22×C127—81no—
210C22×C127—81no—
211C22×C127—81no—
212C22×C127#45011yescharacteristic
213C22×C127#45511yescharacteristic
214C22×C127B(4,𝔽2)11yescharacteristic
215C4×C127—41no—
216C4×C127—41no—
217C4×C127—81no—
218C4×C127—81no—
219C4×C127—81no—
220C4×C127—81no—
221C4×C127—81no—
222C4×C127—81no—
223C4×C127—161no—
224#12205C12711yesmaximal, characteristic, direct product factor, Sylow
225C2×C127—21no—
226C2×C127—21no—
227C2×C127—41no—
228C2×C127—41no—
229C2×C127—82no—
230C2×C127—161no—
231C2×C127#160111yescharacteristic
232C2×C127#219911yescharacteristic
233C2×C127#220011yescharacteristic
234#2812C2×C12711yescharacteristic
235#2029C2×C12711yescharacteristic, complemented
236#1608C2×C12711yescharacteristic, complemented
237#1596C2×C12711yescharacteristic, complemented
238#1344C2×C12711yescharacteristic, complemented
239#1343C2×C12711yescharacteristic, complemented
240#1316C2×C12711yescharacteristic, complemented
241C127#1220511yescharacteristic, direct product factor, Sylow
242#543C22×C12711yescharacteristic
243C2×#120—21no—
244#483—21no—
245#450—21no—
246#397—21no—
247#397C22×C12711yescharacteristic
248#394—21no—
249#394—21no—
250#394—21no—
251#392—21no—
252#392C22×C12711yescharacteristic
253#379—21no—
254#379C22×C12711yescharacteristic
255#342—21no—
256#342C22×C12711yescharacteristic
257#328—21no—
258#328—21no—
259#323C22×C12711yescharacteristic
260#323C22×C12711yescharacteristic
261C25D4×C12711yescharacteristic
262C22×D4—21no—
263C22×D4—21no—
264C22×D4—21no—
265C22×D4D4×C12711yescharacteristic
266C2×SD16—41no—
267C22⋊Q8—21no—
268C22⋊Q8—21no—
269C22⋊Q8—41no—
270C22⋊Q8—41no—
271#120—42no—
272#120—42no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4D4×C12711yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
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—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4D4×C12711yescharacteristic
288C2×C22⋊C4C23×C12711yescharacteristic
289#106—41no—
290#103—41no—
291#102—41no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294#95—21no—
295#95—21no—
296#95D4×C12711yescharacteristic
297#95D4×C12711yescharacteristic
298C24—21no—
299C24—21no—
300C24—21no—
301C24—41no—
302C24—41no—
303C24—81no—
304C24—81no—
305C24SD16×C12712yescomplemented
306C24C2×D4×C12711yescharacteristic
307C2×Q8—21no—
308C2×Q8—21no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—81no—
315C2×D4—81no—
316C2×D4—81no—
317C2×D4—81no—
318C2×D4—81no—
319C2×D4—81no—
320C2×D4—81no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4C2×D4×C12711yescharacteristic
329C22×C4C2×D4×C12711yescharacteristic
330SD16—82no—
331C2×C8—41no—
332C2×C8—41no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—81no—
339C22⋊C4—41no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—81no—
350C22⋊C4—81no—
351C22⋊C4—81no—
352C22⋊C4—81no—
353C22⋊C4—81no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C23—21no—
357C23—21no—
358C23—22no—
359C23—41no—
360C23—41no—
361C23—42no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—81no—
378C23—82no—
379C23—82no—
380C23—81no—
381C23—81no—
382C23—84no—
383C23#120×C12711yescharacteristic
384C23C2×SD16×C12711yescharacteristic
385C23AGL(1,ℤ/8)×C12711yescharacteristic
386Q8—81no—
387Q8—81no—
388D4—42no—
389D4—81no—
390D4—82no—
391D4—82no—
392D4—84no—
393D4—84no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
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—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C8—81no—
418C8—81no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—41no—
424C22—41no—
425C22—42no—
426C22—42no—
427C22—41no—
428C22—41no—
429C22—44no—
430C22—42no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—82no—
439C22—82no—
440C22—82no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22#450×C12711yescharacteristic
447C22#455×C12711yescharacteristic
448C22B(4,𝔽2)×C12711yescharacteristic
449C4—41no—
450C4—41no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—161no—
458C2—21no—
459C2—21no—
460C2—41no—
461C2—41no—
462C2—82no—
463C2—161no—
464C2C127×#160111yescharacteristic
465C2C127×#219911yescharacteristic
466C2C127×#220011yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC127×#2812C2
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×#1608C2semidirect
4. C127×#1596C2semidirect
5. C127×#1344C2semidirect
6. C127×#1343C2semidirect
7. C127×#1316C2semidirect
71. C24×C127SD16semidirect
224. #12205C127direct (N × H)
235. #2029C2×C127semidirect
236. #1608C2×C127semidirect
237. #1596C2×C127semidirect
238. #1344C2×C127semidirect
239. #1343C2×C127semidirect
240. #1316C2×C127semidirect
241. C127#12205direct (N × H)
305. C24SD16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#28122
C127×#20292
C127×#16082
C127×#15962
C127×#13442
C127×#13432
C127×#13162
#12205127