Finite Group Databasebeta

← C127×#12188 #44575861

Subgroups — C127×#12188 #44575861

Order 32512.

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

normal subgroups only

This group has 554 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×#2736C211yesmaximal, characteristic, complemented
2C127×#2021C211yesmaximal, characteristic
3C127×#1607C211yesmaximal, characteristic, complemented
4C127×#1601C211yesmaximal, characteristic, complemented
5C127×#1351C211yesmaximal, characteristic, complemented
6C127×#1344C211yesmaximal, characteristic, complemented
7C127×#1316C211yesmaximal, characteristic, complemented
8#534×C127C2211yescharacteristic
9C2×#120×C127—21no—
10#480×C127—21no—
11#448×C127—21no—
12C2×#99×C127—21no—
13#397×C127—21no—
14#397×C127C2211yescharacteristic
15#393×C127—21no—
16#393×C127C2211yescharacteristic
17#388×C127—21no—
18#388×C127—21no—
19#388×C127—21no—
20#379×C127—21no—
21#379×C127C2211yescharacteristic
22#342×C127C2211yescharacteristic
23#327×C127—21no—
24#327×C127—21no—
25#323×C127C2211yescharacteristic
26#323×C127C2211yescharacteristic
27C25×C127D411yescharacteristic
28C22×Q8×C127D411yescharacteristic, complemented
29C22×D4×C127—21no—
30C22×D4×C127—41no—
31C2×SD16×C127—41no—
32#123×C127—41no—
33#121×C127—21no—
34#121×C127—21no—
35#120×C127—42no—
36#120×C127—41no—
37#120×C127—42no—
38C2×C4⋊C4×C127—21no—
39C2×C4⋊C4×C127D411yescharacteristic, complemented
40C2×C22⋊C4×C127—21no—
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—41no—
50C2×C22⋊C4×C127D411yescharacteristic
51C2×C22⋊C4×C127C2311yescharacteristic
52C2×C42×C127—21no—
53#107×C127—41no—
54#103×C127—41no—
55#102×C127—41no—
56#99×C127—42no—
57C22⋊C8×C127—21no—
58C22⋊C8×C127—21no—
59#95×C127—21no—
60#95×C127—21no—
61#95×C127—21no—
62#95×C127—21no—
63#95×C127D411yescharacteristic, complemented
64#95×C127D411yescharacteristic, complemented
65C24×C127—21no—
66C24×C127—21no—
67C24×C127—21no—
68C24×C127—21no—
69C24×C127—41no—
70C24×C127—41no—
71C24×C127—42no—
72C24×C127—81no—
73C24×C127C2×D411yescharacteristic
74C2×Q8×C127—41no—
75C2×Q8×C127—81no—
76C2×D4×C127—21no—
77C2×D4×C127—21no—
78C2×D4×C127—41no—
79C2×D4×C127—41no—
80C2×D4×C127—41no—
81C2×D4×C127—44no—
82C2×D4×C127—41no—
83C2×D4×C127—41no—
84C2×D4×C127—41no—
85C2×D4×C127—82no—
86C2×D4×C127—82no—
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—81no—
103C22⋊C4×C127—22no—
104C22⋊C4×C127—41no—
105C22⋊C4×C127—41no—
106C22⋊C4×C127—41no—
107C22⋊C4×C127—42no—
108C22⋊C4×C127—42no—
109C22⋊C4×C127—44no—
110C22⋊C4×C127—41no—
111C22⋊C4×C127—41no—
112C22⋊C4×C127—41no—
113C22⋊C4×C127—41no—
114C22⋊C4×C127—81no—
115C22⋊C4×C127—81no—
116C22⋊C4×C127—81no—
117C22⋊C4×C127—81no—
118C22⋊C4×C127—81no—
119C42×C127—81no—
120C23×C127—21no—
121C23×C127—21no—
122C23×C127—22no—
123C23×C127—41no—
124C23×C127—41no—
125C23×C127—42no—
126C23×C127—41no—
127C23×C127—41no—
128C23×C127—41no—
129C23×C127—41no—
130C23×C127—41no—
131C23×C127—42no—
132C23×C127—42no—
133C23×C127—41no—
134C23×C127—41no—
135C23×C127—41no—
136C23×C127—41no—
137C23×C127—81no—
138C23×C127—82no—
139C23×C127—81no—
140C23×C127—82no—
141C23×C127—84no—
142C23×C127—81no—
143C23×C127—81no—
144C23×C127—81no—
145C23×C127#12011yescharacteristic
146C23×C127#13511yescharacteristic
147C23×C127AGL(1,ℤ/8)11yescharacteristic
148Q8×C127—42no—
149Q8×C127—81no—
150D4×C127—81no—
151D4×C127—81no—
152D4×C127—81no—
153D4×C127—81no—
154D4×C127—88no—
155D4×C127—81no—
156D4×C127—81no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—41no—
160C2×C4×C127—41no—
161C2×C4×C127—42no—
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—41no—
179C2×C4×C127—41no—
180C2×C4×C127—81no—
181C2×C4×C127—81no—
182C8×C127—81no—
183C8×C127—81no—
184C22×C127—21no—
185C22×C127—21no—
186C22×C127—21no—
187C22×C127—21no—
188C22×C127—41no—
189C22×C127—41no—
190C22×C127—42no—
191C22×C127—42no—
192C22×C127—41no—
193C22×C127—41no—
194C22×C127—44no—
195C22×C127—42no—
196C22×C127—81no—
197C22×C127—81no—
198C22×C127—81no—
199C22×C127—82no—
200C22×C127—82no—
201C22×C127—82no—
202C22×C127—82no—
203C22×C127—81no—
204C22×C127—81no—
205C22×C127—81no—
206C22×C127—81no—
207C22×C127—81no—
208C22×C127—161no—
209C22×C127#44911yescharacteristic
210C22×C127#45311yescharacteristic
211C22×C127B(4,𝔽2)11yescharacteristic
212C4×C127—41no—
213C4×C127—41no—
214C4×C127—81no—
215C4×C127—81no—
216C4×C127—81no—
217C4×C127—82no—
218C4×C127—81no—
219C4×C127—81no—
220C4×C127—81no—
221#12188C12711yesmaximal, characteristic, direct product factor, Sylow
222C2×C127—21no—
223C2×C127—21no—
224C2×C127—41no—
225C2×C127—41no—
226C2×C127—82no—
227C2×C127—161no—
228C2×C127#160011yescharacteristic
229C2×C127#219711yescharacteristic
230C2×C127#219911yescharacteristic
231#2736C2×C12711yescharacteristic, complemented
232#2021C2×C12711yescharacteristic
233#1607C2×C12711yescharacteristic, complemented
234#1601C2×C12711yescharacteristic, complemented
235#1351C2×C12711yescharacteristic, complemented
236#1344C2×C12711yescharacteristic, complemented
237#1316C2×C12711yescharacteristic, complemented
238C127#1218811yescharacteristic, direct product factor, Sylow
239#534C22×C12711yescharacteristic
240C2×#120—21no—
241#480—21no—
242#448—21no—
243C2×#99—21no—
244#397—21no—
245#397C22×C12711yescharacteristic
246#393—21no—
247#393C22×C12711yescharacteristic
248#388—21no—
249#388—21no—
250#388—21no—
251#379—21no—
252#379C22×C12711yescharacteristic
253#342C22×C12711yescharacteristic
254#327—21no—
255#327—21no—
256#323C22×C12711yescharacteristic
257#323C22×C12711yescharacteristic
258C25D4×C12711yescharacteristic
259C22×Q8D4×C12711yescharacteristic, complemented
260C22×D4—21no—
261C22×D4—41no—
262C2×SD16—41no—
263#123—41no—
264#121—21no—
265#121—21no—
266#120—42no—
267#120—41no—
268#120—42no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4D4×C12711yescharacteristic, complemented
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4D4×C12711yescharacteristic
282C2×C22⋊C4C23×C12711yescharacteristic
283C2×C42—21no—
284#107—41no—
285#103—41no—
286#102—41no—
287#99—42no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95D4×C12711yescharacteristic, complemented
295#95D4×C12711yescharacteristic, complemented
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—41no—
301C24—41no—
302C24—42no—
303C24—81no—
304C24C2×D4×C12711yescharacteristic
305C2×Q8—41no—
306C2×Q8—81no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—44no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—82no—
317C2×D4—82no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—41no—
325C22×C4C2×D4×C12711yescharacteristic
326C22×C4C2×D4×C12711yescharacteristic
327SD16—82no—
328C2×C8—41no—
329C2×C8—41no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—81no—
334C22⋊C4—22no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—44no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—81no—
346C22⋊C4—81no—
347C22⋊C4—81no—
348C22⋊C4—81no—
349C22⋊C4—81no—
350C42—81no—
351C23—21no—
352C23—21no—
353C23—22no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—81no—
369C23—82no—
370C23—81no—
371C23—82no—
372C23—84no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23#120×C12711yescharacteristic
377C23#135×C12711yescharacteristic
378C23AGL(1,ℤ/8)×C12711yescharacteristic
379Q8—42no—
380Q8—81no—
381D4—81no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4—88no—
386D4—81no—
387D4—81no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
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—81no—
412C2×C4—81no—
413C8—81no—
414C8—81no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—41no—
420C22—41no—
421C22—42no—
422C22—42no—
423C22—41no—
424C22—41no—
425C22—44no—
426C22—42no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—161no—
440C22#449×C12711yescharacteristic
441C22#453×C12711yescharacteristic
442C22B(4,𝔽2)×C12711yescharacteristic
443C4—41no—
444C4—41no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—82no—
449C4—81no—
450C4—81no—
451C4—81no—
452C2—21no—
453C2—21no—
454C2—41no—
455C2—41no—
456C2—82no—
457C2—161no—
458C2C127×#160011yescharacteristic
459C2C127×#219711yescharacteristic
460C2C127×#219911yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC127×#2021C2
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
1. C127×#2736C2semidirect
3. C127×#1607C2semidirect
4. C127×#1601C2semidirect
5. C127×#1351C2semidirect
6. C127×#1344C2semidirect
7. C127×#1316C2semidirect
28. C22×Q8×C127D4semidirect
39. C2×C4⋊C4×C127D4semidirect
63. #95×C127D4semidirect
64. #95×C127D4semidirect
221. #12188C127direct (N × H)
231. #2736C2×C127semidirect
233. #1607C2×C127semidirect
234. #1601C2×C127semidirect
235. #1351C2×C127semidirect
236. #1344C2×C127semidirect
237. #1316C2×C127semidirect
238. C127#12188direct (N × H)
259. C22×Q8D4×C127semidirect
270. C2×C4⋊C4D4×C127semidirect
294. #95D4×C127semidirect
295. #95D4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#27362
C127×#20212
C127×#16072
C127×#16012
C127×#13512
C127×#13442
C127×#13162
#12188127