Finite Group Databasebeta

← C127×#12264 #44575937

Subgroups — C127×#12264 #44575937

Order 32512.

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

normal subgroups only

This group has 468 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
2C127×#2029C211yesmaximal, characteristic
3C127×#1608C211yesmaximal, characteristic, complemented
4C127×#1602C211yesmaximal, characteristic, complemented
5C127×#1351C211yesmaximal, characteristic, complemented
6C127×#1344C211yesmaximal, characteristic, complemented
7C127×#1316C211yesmaximal, characteristic
8#543×C127C2211yescharacteristic
9#483×C127—21no—
10#451×C127—21no—
11#397×C127—21no—
12#397×C127C2211yescharacteristic
13#393×C127—21no—
14#393×C127C2211yescharacteristic
15#388×C127—21no—
16#388×C127—21no—
17#388×C127—21no—
18#379×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×Q8×C127D411yescharacteristic
29C2×Q16×C127—41no—
30C22⋊Q8×C127—21no—
31C22⋊Q8×C127—21no—
32C22⋊Q8×C127—41no—
33C22⋊Q8×C127—41no—
34C2×C4⋊C4×C127—21no—
35C2×C4⋊C4×C127D411yescharacteristic
36C2×C22⋊C4×C127—21no—
37C2×C22⋊C4×C127—21no—
38C2×C22⋊C4×C127—21no—
39C2×C22⋊C4×C127—21no—
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—41no—
47C2×C22⋊C4×C127—41no—
48C2×C22⋊C4×C127D411yescharacteristic
49C2×C22⋊C4×C127C2311yescharacteristic
50C2×C42×C127—21no—
51#106×C127—41no—
52#103×C127—41no—
53#103×C127—41no—
54C22⋊C8×C127—21no—
55C22⋊C8×C127—21no—
56#95×C127—21no—
57#95×C127—21no—
58#95×C127—21no—
59#95×C127—21no—
60#95×C127D411yescharacteristic
61#95×C127D411yescharacteristic
62C24×C127—21no—
63C24×C127—21no—
64C24×C127—21no—
65C24×C127—41no—
66C24×C127—41no—
67C24×C127—42no—
68C24×C127—81no—
69C24×C127C2×D411yescharacteristic
70C2×Q8×C127—21no—
71C2×Q8×C127—21no—
72C2×Q8×C127—41no—
73C2×Q8×C127—81no—
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—21no—
81C22×C4×C127—41no—
82C22×C4×C127C2×D411yescharacteristic
83C22×C4×C127C2×D411yescharacteristic
84Q16×C127—82no—
85C2×C8×C127—41no—
86C2×C8×C127—41no—
87C4⋊C4×C127—21no—
88C4⋊C4×C127—21no—
89C4⋊C4×C127—41no—
90C4⋊C4×C127—41no—
91C4⋊C4×C127—41no—
92C4⋊C4×C127—81no—
93C22⋊C4×C127—41no—
94C22⋊C4×C127—42no—
95C22⋊C4×C127—42no—
96C22⋊C4×C127—41no—
97C22⋊C4×C127—41no—
98C22⋊C4×C127—41no—
99C22⋊C4×C127—41no—
100C22⋊C4×C127—41no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127—41no—
103C22⋊C4×C127—81no—
104C22⋊C4×C127—81no—
105C22⋊C4×C127—81no—
106C22⋊C4×C127—81no—
107C22⋊C4×C127—81no—
108C22⋊C4×C127—82no—
109C22⋊C4×C127—82no—
110C42×C127—81no—
111C23×C127—21no—
112C23×C127—21no—
113C23×C127—22no—
114C23×C127—41no—
115C23×C127—41no—
116C23×C127—41no—
117C23×C127—41no—
118C23×C127—41no—
119C23×C127—42no—
120C23×C127—42no—
121C23×C127—42no—
122C23×C127—41no—
123C23×C127—41no—
124C23×C127—41no—
125C23×C127—81no—
126C23×C127—82no—
127C23×C127—81no—
128C23×C127—82no—
129C23×C127—84no—
130C23×C127—81no—
131C23×C127#12011yescharacteristic
132C23×C127#13511yescharacteristic
133C23×C127#13711yescharacteristic
134Q8×C127—42no—
135Q8×C127—81no—
136Q8×C127—81no—
137Q8×C127—81no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
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—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—81no—
160C2×C4×C127—81no—
161C2×C4×C127—81no—
162C2×C4×C127—81no—
163C2×C4×C127—81no—
164C8×C127—81no—
165C8×C127—81no—
166C22×C127—21no—
167C22×C127—21no—
168C22×C127—21no—
169C22×C127—21no—
170C22×C127—41no—
171C22×C127—41no—
172C22×C127—42no—
173C22×C127—42no—
174C22×C127—41no—
175C22×C127—41no—
176C22×C127—44no—
177C22×C127—42no—
178C22×C127—81no—
179C22×C127—81no—
180C22×C127—81no—
181C22×C127—82no—
182C22×C127—82no—
183C22×C127—82no—
184C22×C127—81no—
185C22×C127—81no—
186C22×C127#45211yescharacteristic
187C22×C127#45511yescharacteristic
188C22×C127B(4,𝔽2)11yescharacteristic
189C4×C127—41no—
190C4×C127—41no—
191C4×C127—81no—
192C4×C127—81no—
193C4×C127—81no—
194C4×C127—81no—
195C4×C127—81no—
196C4×C127—81no—
197C4×C127—81no—
198C4×C127—161no—
199#12264C12711yesmaximal, characteristic, direct product factor, Sylow
200C2×C127—21no—
201C2×C127—21no—
202C2×C127—41no—
203C2×C127—41no—
204C2×C127—82no—
205C2×C127#160611yescharacteristic
206C2×C127#219711yescharacteristic
207C2×C127#219911yescharacteristic
208#2736C2×C12711yescharacteristic
209#2029C2×C12711yescharacteristic
210#1608C2×C12711yescharacteristic, complemented
211#1602C2×C12711yescharacteristic, complemented
212#1351C2×C12711yescharacteristic, complemented
213#1344C2×C12711yescharacteristic, complemented
214#1316C2×C12711yescharacteristic
215C127#1226411yescharacteristic, direct product factor, Sylow
216#543C22×C12711yescharacteristic
217#483—21no—
218#451—21no—
219#397—21no—
220#397C22×C12711yescharacteristic
221#393—21no—
222#393C22×C12711yescharacteristic
223#388—21no—
224#388—21no—
225#388—21no—
226#379—21no—
227#379—21no—
228#379C22×C12711yescharacteristic
229#342—21no—
230#342C22×C12711yescharacteristic
231#328—21no—
232#328—21no—
233#323C22×C12711yescharacteristic
234#323C22×C12711yescharacteristic
235C25D4×C12711yescharacteristic
236C22×Q8D4×C12711yescharacteristic
237C2×Q16—41no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—41no—
241C22⋊Q8—41no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4D4×C12711yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
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×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4D4×C12711yescharacteristic
257C2×C22⋊C4C23×C12711yescharacteristic
258C2×C42—21no—
259#106—41no—
260#103—41no—
261#103—41no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95D4×C12711yescharacteristic
269#95D4×C12711yescharacteristic
270C24—21no—
271C24—21no—
272C24—21no—
273C24—41no—
274C24—41no—
275C24—42no—
276C24—81no—
277C24C2×D4×C12711yescharacteristic
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—41no—
281C2×Q8—81no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4C2×D4×C12711yescharacteristic
291C22×C4C2×D4×C12711yescharacteristic
292Q16—82no—
293C2×C8—41no—
294C2×C8—41no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—81no—
301C22⋊C4—41no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—81no—
312C22⋊C4—81no—
313C22⋊C4—81no—
314C22⋊C4—81no—
315C22⋊C4—81no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C42—81no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—81no—
334C23—82no—
335C23—81no—
336C23—82no—
337C23—84no—
338C23—81no—
339C23#120×C12711yescharacteristic
340C23#135×C12711yescharacteristic
341C23#137×C12711yescharacteristic
342Q8—42no—
343Q8—81no—
344Q8—81no—
345Q8—81no—
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—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C8—81no—
373C8—81no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—42no—
381C22—42no—
382C22—41no—
383C22—41no—
384C22—44no—
385C22—42no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22—81no—
393C22—81no—
394C22#452×C12711yescharacteristic
395C22#455×C12711yescharacteristic
396C22B(4,𝔽2)×C12711yescharacteristic
397C4—41no—
398C4—41no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4—161no—
407C2—21no—
408C2—21no—
409C2—41no—
410C2—41no—
411C2—82no—
412C2C127×#160611yescharacteristic
413C2C127×#219711yescharacteristic
414C2C127×#219911yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC25×C127D4
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
3. C127×#1608C2semidirect
4. C127×#1602C2semidirect
5. C127×#1351C2semidirect
6. C127×#1344C2semidirect
199. #12264C127direct (N × H)
210. #1608C2×C127semidirect
211. #1602C2×C127semidirect
212. #1351C2×C127semidirect
213. #1344C2×C127semidirect
215. C127#12264direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#27362
C127×#20292
C127×#16082
C127×#16022
C127×#13512
C127×#13442
C127×#13162
#12264127