Finite Group Databasebeta

← C127×#11824 #44575497

Subgroups — C127×#11824 #44575497

Order 32512.

464 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2635C211yesmaximal, characteristic
2C127×#1799C211yesmaximal, characteristic, complemented
3C127×#1466C211yesmaximal, characteristic, complemented
4C127×#1352C212yesmaximal, complemented
5C127×#1343C212yesmaximal, complemented
6C2×#121×C127C411yescharacteristic, complemented
7C2×C4×D4×C127C2211yescharacteristic
8#432×C127—21no—
9C2×#99×C127—21no—
10#400×C127C411yescharacteristic, complemented
11#399×C127C2211yescharacteristic
12#394×C127—21no—
13#394×C127C411yescharacteristic, complemented
14#392×C127—21no—
15#392×C127C2211yescharacteristic
16#388×C127—21no—
17#386×C127—21no—
18#381×C127C411yescharacteristic, complemented
19#380×C127—21no—
20#342×C127C2212yes—
21#323×C127—21no—
22#323×C127C2212yes—
23C22×D4×C127—21no—
24C22×D4×C127—21no—
25C22×D4×C127C2×C411yescharacteristic
26C22×D4×C127D411yescharacteristic
27C23×C4×C127C2×C411yescharacteristic
28C23×C4×C127D411yescharacteristic
29#121×C127—41no—
30#121×C127—41no—
31C4×D4×C127—41no—
32C4×D4×C127—41no—
33C2×C4⋊C4×C127—21no—
34C2×C4⋊C4×C127—41no—
35C2×C4⋊C4×C127C2×C411yescharacteristic
36C2×C4⋊C4×C127C2×C411yescharacteristic
37C2×C22⋊C4×C127—21no—
38C2×C22⋊C4×C127—21no—
39C2×C22⋊C4×C127—22no—
40C2×C22⋊C4×C127—21no—
41C2×C22⋊C4×C127—21no—
42C2×C22⋊C4×C127—21no—
43C2×C22⋊C4×C127—21no—
44C2×C22⋊C4×C127D411yescharacteristic
45C2×C22⋊C4×C127C2311yescharacteristic
46C2×C42×C127D411yescharacteristic
47C4⋊C8×C127—41no—
48#99×C127—42no—
49C22⋊C8×C127—22no—
50#97×C127—41no—
51#95×C127—21no—
52#95×C127—21no—
53#95×C127—21no—
54#95×C127—41no—
55#95×C127C2×C411yescharacteristic
56#95×C127C2×C411yescharacteristic
57C24×C127—21no—
58C24×C127C22⋊C411yescharacteristic
59C24×C127C2×D411yescharacteristic
60C2×D4×C127—21no—
61C2×D4×C127—21no—
62C2×D4×C127—41no—
63C2×D4×C127—41no—
64C2×D4×C127—41no—
65C2×D4×C127—42no—
66C2×D4×C127—41no—
67C2×D4×C127—41no—
68C2×D4×C127—81no—
69C2×D4×C127—81no—
70C2×D4×C127—81no—
71C2×D4×C127—81no—
72C2×D4×C127—81no—
73C22×C4×C127—21no—
74C22×C4×C127—21no—
75C22×C4×C127—21no—
76C22×C4×C127—21no—
77C22×C4×C127—21no—
78C22×C4×C127—21no—
79C22×C4×C127—21no—
80C22×C4×C127—42no—
81C22×C4×C127—81no—
82C22×C4×C127—81no—
83C22×C4×C127C22⋊C411yescharacteristic
84C22×C4×C127C22⋊C411yescharacteristic
85C22×C4×C127C22⋊C411yescharacteristic
86C22×C4×C127C22×C411yescharacteristic
87C22×C4×C127C2×D411yescharacteristic
88C2×C8×C127—42no—
89C4⋊C4×C127—41no—
90C4⋊C4×C127—41no—
91C4⋊C4×C127—81no—
92C4⋊C4×C127—82no—
93C22⋊C4×C127—41no—
94C22⋊C4×C127—41no—
95C22⋊C4×C127—42no—
96C22⋊C4×C127—41no—
97C22⋊C4×C127—41no—
98C22⋊C4×C127—42no—
99C22⋊C4×C127—42no—
100C22⋊C4×C127—81no—
101C22⋊C4×C127—81no—
102C22⋊C4×C127—81no—
103C22⋊C4×C127—81no—
104C42×C127—21no—
105C42×C127—21no—
106C23×C127—21no—
107C23×C127—21no—
108C23×C127—21no—
109C23×C127—21no—
110C23×C127—21no—
111C23×C127—41no—
112C23×C127—41no—
113C23×C127—41no—
114C23×C127—41no—
115C23×C127—41no—
116C23×C127—41no—
117C23×C127—41no—
118C23×C127—41no—
119C23×C127—41no—
120C23×C127—41no—
121C23×C127—41no—
122C23×C127C2×C22⋊C411yescharacteristic
123C23×C127AGL(1,ℤ/8)11yescharacteristic
124C23×C127#13711yescharacteristic
125D4×C127—42no—
126D4×C127—42no—
127D4×C127—82no—
128D4×C127—82no—
129D4×C127—82no—
130D4×C127—84no—
131C2×C4×C127—21no—
132C2×C4×C127—21no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—41no—
138C2×C4×C127—42no—
139C2×C4×C127—41no—
140C2×C4×C127—41no—
141C2×C4×C127—41no—
142C2×C4×C127—41no—
143C2×C4×C127—41no—
144C2×C4×C127—41no—
145C2×C4×C127—41no—
146C2×C4×C127—41no—
147C2×C4×C127—41no—
148C2×C4×C127—41no—
149C2×C4×C127—41no—
150C2×C4×C127—41no—
151C2×C4×C127—81no—
152C2×C4×C127—81no—
153C2×C4×C127—81no—
154C2×C4×C127—82no—
155C2×C4×C127—82no—
156C2×C4×C127—82no—
157C2×C4×C127#9911yescharacteristic
158C2×C4×C127#9911yescharacteristic
159C8×C127—82no—
160C22×C127—21no—
161C22×C127—21no—
162C22×C127—21no—
163C22×C127—21no—
164C22×C127—41no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127—41no—
169C22×C127—41no—
170C22×C127—41no—
171C22×C127—81no—
172C22×C127—81no—
173C22×C127—81no—
174C22×C127—81no—
175C22×C127—81no—
176C22×C127—81no—
177C22×C127—81no—
178C22×C127—81no—
179C22×C127—81no—
180C22×C127—81no—
181C22×C127C2×#9911yescharacteristic
182C22×C127#41711yescharacteristic
183C22×C127#42111yescharacteristic
184C4×C127—21no—
185C4×C127—21no—
186C4×C127—41no—
187C4×C127—41no—
188C4×C127—41no—
189C4×C127—41no—
190C4×C127—81no—
191C4×C127—81no—
192C4×C127—81no—
193C4×C127—162no—
194#11824C12711yesmaximal, characteristic, direct product factor, Sylow
195C2×C127—21no—
196C2×C127—21no—
197C2×C127—41no—
198C2×C127—41no—
199C2×C127—81no—
200C2×C127—161no—
201C2×C127#151311yescharacteristic
202C2×C127#212111yescharacteristic
203C2×C127#212811yescharacteristic
204#2635C2×C12711yescharacteristic
205#1799C2×C12711yescharacteristic, complemented
206#1466C2×C12711yescharacteristic, complemented
207#1352C2×C12712yescomplemented
208#1343C2×C12712yescomplemented
209C127#1182411yescharacteristic, direct product factor, Sylow
210C2×#121C4×C12711yescharacteristic, complemented
211C2×C4×D4C22×C12711yescharacteristic
212#432—21no—
213C2×#99—21no—
214#400C4×C12711yescharacteristic, complemented
215#399C22×C12711yescharacteristic
216#394—21no—
217#394C4×C12711yescharacteristic, complemented
218#392—21no—
219#392C22×C12711yescharacteristic
220#388—21no—
221#386—21no—
222#381C4×C12711yescharacteristic, complemented
223#380—21no—
224#342C22×C12712yes—
225#323—21no—
226#323C22×C12712yes—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C2×C4×C12711yescharacteristic
230C22×D4D4×C12711yescharacteristic
231C23×C4C2×C4×C12711yescharacteristic
232C23×C4D4×C12711yescharacteristic
233#121—41no—
234#121—41no—
235C4×D4—41no—
236C4×D4—41no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4C2×C4×C12711yescharacteristic
240C2×C4⋊C4C2×C4×C12711yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4D4×C12711yescharacteristic
249C2×C22⋊C4C23×C12711yescharacteristic
250C2×C42D4×C12711yescharacteristic
251C4⋊C8—41no—
252#99—42no—
253C22⋊C8—22no—
254#97—41no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—41no—
259#95C2×C4×C12711yescharacteristic
260#95C2×C4×C12711yescharacteristic
261C24—21no—
262C24C22⋊C4×C12711yescharacteristic
263C24C2×D4×C12711yescharacteristic
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—81no—
273C2×D4—81no—
274C2×D4—81no—
275C2×D4—81no—
276C2×D4—81no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4C22⋊C4×C12711yescharacteristic
288C22×C4C22⋊C4×C12711yescharacteristic
289C22×C4C22⋊C4×C12711yescharacteristic
290C22×C4C22×C4×C12711yescharacteristic
291C22×C4C2×D4×C12711yescharacteristic
292C2×C8—42no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—81no—
296C4⋊C4—82no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—81no—
305C22⋊C4—81no—
306C22⋊C4—81no—
307C22⋊C4—81no—
308C42—21no—
309C42—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23C2×C22⋊C4×C12711yescharacteristic
327C23AGL(1,ℤ/8)×C12711yescharacteristic
328C23#137×C12711yescharacteristic
329D4—42no—
330D4—42no—
331D4—82no—
332D4—82no—
333D4—82no—
334D4—84no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4#99×C12711yescharacteristic
362C2×C4#99×C12711yescharacteristic
363C8—82no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22C2×#99×C12711yescharacteristic
386C22#417×C12711yescharacteristic
387C22#421×C12711yescharacteristic
388C4—21no—
389C4—21no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—162no—
398C2—21no—
399C2—21no—
400C2—41no—
401C2—41no—
402C2—81no—
403C2—161no—
404C2C127×#151311yescharacteristic
405C2C127×#212111yescharacteristic
406C2C127×#212811yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC127×#2635C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1799C2semidirect
3. C127×#1466C2semidirect
4. C127×#1352C2semidirect
5. C127×#1343C2semidirect
6. C2×#121×C127C4semidirect
10. #400×C127C4semidirect
13. #394×C127C4semidirect
18. #381×C127C4semidirect
194. #11824C127direct (N × H)
205. #1799C2×C127semidirect
206. #1466C2×C127semidirect
207. #1352C2×C127semidirect
208. #1343C2×C127semidirect
209. C127#11824direct (N × H)
210. C2×#121C4×C127semidirect
214. #400C4×C127semidirect
217. #394C4×C127semidirect
222. #381C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#26352
C127×#17992
C127×#14662
C127×#13522
C127×#13432
#11824127