Finite Group Databasebeta

← C127×#12211 #44575884

Subgroups — C127×#12211 #44575884

Order 32512.

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

normal subgroups only

This group has 464 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×#2699C211yesmaximal, characteristic
2C127×#2027C211yesmaximal, characteristic, complemented
3C127×#1611C211yesmaximal, characteristic, complemented
4C127×#1599C211yesmaximal, characteristic, complemented
5C127×#1346C211yesmaximal, characteristic, complemented
6C127×#1345C211yesmaximal, characteristic, complemented
7C127×#1318C211yesmaximal, characteristic, complemented
8#544×C127C2211yescharacteristic
9C2×#121×C127—21no—
10#483×C127—21no—
11#450×C127—21no—
12#418×C127—21no—
13#398×C127C2211yescharacteristic
14#394×C127—21no—
15#394×C127C2211yescharacteristic
16#392×C127—21no—
17#390×C127C2211yescharacteristic
18#388×C127—21no—
19#386×C127—21no—
20#381×C127—21no—
21#343×C127C2211yescharacteristic
22#339×C127—21no—
23#328×C127—21no—
24#328×C127—21no—
25#324×C127C2211yescharacteristic
26#324×C127C2211yescharacteristic
27C22×D4×C127—21no—
28C22×D4×C127—21no—
29C22×D4×C127D411yescharacteristic
30C22×D4×C127D411yescharacteristic
31C23×C4×C127—21no—
32C2×SD16×C127—41no—
33C2×M16×C127—21no—
34C4⋊Q8×C127—41no—
35#125×C127—41no—
36C22⋊Q8×C127—21no—
37C22⋊Q8×C127—21no—
38#121×C127—42no—
39#121×C127—42no—
40#117×C127—21no—
41C2×C4⋊C4×C127—21no—
42C2×C4⋊C4×C127D411yescharacteristic
43C2×C4⋊C4×C127C2311yescharacteristic
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×C42×C127D411yescharacteristic
52#106×C127—41no—
53#103×C127—41no—
54#102×C127—41no—
55#102×C127—41no—
56#102×C127—41no—
57C22⋊C8×C127—21no—
58C22⋊C8×C127—21no—
59#95×C127—21no—
60#95×C127—21no—
61#95×C127—21no—
62#95×C127D411yescharacteristic
63#95×C127D411yescharacteristic
64C24×C127—21no—
65C24×C127—21no—
66C2×Q8×C127—21no—
67C2×Q8×C127—21no—
68C2×D4×C127—21no—
69C2×D4×C127—41no—
70C2×D4×C127—41no—
71C2×D4×C127—41no—
72C2×D4×C127—42no—
73C2×D4×C127—42no—
74C2×D4×C127—42no—
75C2×D4×C127—81no—
76C2×D4×C127—81no—
77C2×D4×C127—81no—
78C2×D4×C127—81no—
79C2×D4×C127SD1612yescomplemented
80C22×C4×C127—21no—
81C22×C4×C127—21no—
82C22×C4×C127—21no—
83C22×C4×C127—21no—
84C22×C4×C127—42no—
85C22×C4×C127—42no—
86C22×C4×C127—81no—
87C22×C4×C127C2×D411yescharacteristic
88C22×C4×C127C2×D411yescharacteristic
89C22×C4×C127C2×D411yescharacteristic
90SD16×C127—82no—
91M16×C127—41no—
92M16×C127—41no—
93C2×C8×C127—41no—
94C2×C8×C127—41no—
95C2×C8×C127—41no—
96C4⋊C4×C127—21no—
97C4⋊C4×C127—21no—
98C4⋊C4×C127—21no—
99C4⋊C4×C127—21no—
100C4⋊C4×C127—41no—
101C4⋊C4×C127—41no—
102C4⋊C4×C127—41no—
103C4⋊C4×C127—41no—
104C4⋊C4×C127—41no—
105C4⋊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—81no—
112C22⋊C4×C127—81no—
113C22⋊C4×C127—81no—
114C22⋊C4×C127—81no—
115C42×C127—41no—
116C42×C127—41no—
117C23×C127—22no—
118C23×C127—41no—
119C23×C127—41no—
120C23×C127—41no—
121C23×C127—41no—
122C23×C127—41no—
123C23×C127—41no—
124C23×C127—41no—
125C23×C127—41no—
126C23×C127—41no—
127C23×C127—41no—
128C23×C127—41no—
129C23×C127—41no—
130C23×C127—41no—
131C23×C127#12011yescharacteristic
132Q8×C127—81no—
133Q8×C127—81no—
134D4×C127—42no—
135D4×C127—42no—
136D4×C127—81no—
137D4×C127—84no—
138D4×C127—81no—
139D4×C127—84no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—22no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
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—41no—
152C2×C4×C127—41no—
153C2×C4×C127—41no—
154C2×C4×C127—41no—
155C2×C4×C127—41no—
156C2×C4×C127—41no—
157C2×C4×C127—41no—
158C2×C4×C127—41no—
159C2×C4×C127—41no—
160C2×C4×C127—84no—
161C2×C4×C127C2×SD1611yescharacteristic
162C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
163C8×C127—81no—
164C8×C127—81no—
165C8×C127—81no—
166C22×C127—21no—
167C22×C127—21no—
168C22×C127—42no—
169C22×C127—42no—
170C22×C127—42no—
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×C127—81no—
182C22×C127#45011yescharacteristic
183C22×C127#45511yescharacteristic
184C22×C127B(4,𝔽2)11yescharacteristic
185C4×C127—21no—
186C4×C127—21no—
187C4×C127—42no—
188C4×C127—41no—
189C4×C127—41no—
190C4×C127—81no—
191C4×C127—81no—
192C4×C127—81no—
193C4×C127—81no—
194C4×C127—81no—
195#12211C12711yesmaximal, characteristic, direct product factor, Sylow
196C2×C127—21no—
197C2×C127—21no—
198C2×C127—82no—
199C2×C127—161no—
200C2×C127#160111yescharacteristic
201C2×C127#220011yescharacteristic
202C2×C127#220411yescharacteristic
203#2699C2×C12711yescharacteristic
204#2027C2×C12711yescharacteristic, complemented
205#1611C2×C12711yescharacteristic, complemented
206#1599C2×C12711yescharacteristic, complemented
207#1346C2×C12711yescharacteristic, complemented
208#1345C2×C12711yescharacteristic, complemented
209#1318C2×C12711yescharacteristic, complemented
210C127#1221111yescharacteristic, direct product factor, Sylow
211#544C22×C12711yescharacteristic
212C2×#121—21no—
213#483—21no—
214#450—21no—
215#418—21no—
216#398C22×C12711yescharacteristic
217#394—21no—
218#394C22×C12711yescharacteristic
219#392—21no—
220#390C22×C12711yescharacteristic
221#388—21no—
222#386—21no—
223#381—21no—
224#343C22×C12711yescharacteristic
225#339—21no—
226#328—21no—
227#328—21no—
228#324C22×C12711yescharacteristic
229#324C22×C12711yescharacteristic
230C22×D4—21no—
231C22×D4—21no—
232C22×D4D4×C12711yescharacteristic
233C22×D4D4×C12711yescharacteristic
234C23×C4—21no—
235C2×SD16—41no—
236C2×M16—21no—
237C4⋊Q8—41no—
238#125—41no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241#121—42no—
242#121—42no—
243#117—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4D4×C12711yescharacteristic
246C2×C4⋊C4C23×C12711yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C42D4×C12711yescharacteristic
255#106—41no—
256#103—41no—
257#102—41no—
258#102—41no—
259#102—41no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95D4×C12711yescharacteristic
266#95D4×C12711yescharacteristic
267C24—21no—
268C24—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—21no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—81no—
279C2×D4—81no—
280C2×D4—81no—
281C2×D4—81no—
282C2×D4SD16×C12712yescomplemented
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—81no—
290C22×C4C2×D4×C12711yescharacteristic
291C22×C4C2×D4×C12711yescharacteristic
292C22×C4C2×D4×C12711yescharacteristic
293SD16—82no—
294M16—41no—
295M16—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—81no—
315C22⋊C4—81no—
316C22⋊C4—81no—
317C22⋊C4—81no—
318C42—41no—
319C42—41no—
320C23—22no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23#120×C12711yescharacteristic
335Q8—81no—
336Q8—81no—
337D4—42no—
338D4—42no—
339D4—81no—
340D4—84no—
341D4—81no—
342D4—84no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—84no—
364C2×C4C2×SD16×C12711yescharacteristic
365C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
366C8—81no—
367C8—81no—
368C8—81no—
369C22—21no—
370C22—21no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22#450×C12711yescharacteristic
386C22#455×C12711yescharacteristic
387C22B(4,𝔽2)×C12711yescharacteristic
388C4—21no—
389C4—21no—
390C4—42no—
391C4—41no—
392C4—41no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—81no—
398C2—21no—
399C2—21no—
400C2—82no—
401C2—161no—
402C2C127×#160111yescharacteristic
403C2C127×#220011yescharacteristic
404C2C127×#220411yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC127×#2699C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C4⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2027C2semidirect
3. C127×#1611C2semidirect
4. C127×#1599C2semidirect
5. C127×#1346C2semidirect
6. C127×#1345C2semidirect
7. C127×#1318C2semidirect
79. C2×D4×C127SD16semidirect
195. #12211C127direct (N × H)
204. #2027C2×C127semidirect
205. #1611C2×C127semidirect
206. #1599C2×C127semidirect
207. #1346C2×C127semidirect
208. #1345C2×C127semidirect
209. #1318C2×C127semidirect
210. C127#12211direct (N × H)
282. C2×D4SD16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#26992
C127×#20272
C127×#16112
C127×#15992
C127×#13462
C127×#13452
C127×#13182
#12211127