Finite Group Databasebeta

← C127×#12204 #44575877

Subgroups — C127×#12204 #44575877

Order 32512.

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

normal subgroups only

This group has 474 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×#2600C211yesmaximal, characteristic
2C127×#2030C211yesmaximal, characteristic, complemented
3C127×#1608C211yesmaximal, characteristic, complemented
4C127×#1596C211yesmaximal, characteristic
5C127×#1344C211yesmaximal, characteristic, complemented
6C127×#1343C211yesmaximal, characteristic
7C127×#1317C211yesmaximal, characteristic, complemented
8#543×C127C2211yescharacteristic
9C2×C4×D4×C127—21no—
10#483×C127—21no—
11#450×C127—21no—
12#397×C127C2211yescharacteristic
13#394×C127—21no—
14#392×C127C2211yescharacteristic
15#388×C127—21no—
16#387×C127—21no—
17#386×C127—21no—
18#385×C127—21no—
19#381×C127—21no—
20#381×C127C2211yescharacteristic
21#342×C127—21no—
22#342×C127C2211yescharacteristic
23#328×C127—21no—
24#328×C127—21no—
25#323×C127C2211yescharacteristic
26#323×C127C2211yescharacteristic
27C22×D4×C127—21no—
28C22×D4×C127D411yescharacteristic
29C23×C4×C127—21no—
30C23×C4×C127D411yescharacteristic
31C2×SD16×C127—41no—
32C22⋊Q8×C127—21no—
33C22⋊Q8×C127—21no—
34C22⋊Q8×C127—41no—
35C22⋊Q8×C127—41no—
36C4×D4×C127—42no—
37C4×D4×C127—42no—
38C2×C4⋊C4×C127—21no—
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×C127D411yescharacteristic
49C2×C22⋊C4×C127C2311yescharacteristic
50C2×C42×C127—21no—
51#106×C127—41no—
52#103×C127—41no—
53#102×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×C127—41no—
61#95×C127—41no—
62#95×C127D411yescharacteristic
63#95×C127D411yescharacteristic
64C24×C127—21no—
65C24×C127C2×D411yescharacteristic
66C2×Q8×C127—21no—
67C2×Q8×C127—21no—
68C2×D4×C127—42no—
69C2×D4×C127—41no—
70C2×D4×C127—42no—
71C2×D4×C127—41no—
72C2×D4×C127—41no—
73C2×D4×C127—42no—
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—21no—
82C22×C4×C127—42no—
83C22×C4×C127—41no—
84C22×C4×C127—41no—
85C22×C4×C127—41no—
86C22×C4×C127—81no—
87C22×C4×C127—81no—
88C22×C4×C127SD1612yescomplemented
89C22×C4×C127C2×D411yescharacteristic
90C22×C4×C127C2×D411yescharacteristic
91SD16×C127—82no—
92C2×C8×C127—41no—
93C2×C8×C127—41no—
94C4⋊C4×C127—21no—
95C4⋊C4×C127—21no—
96C4⋊C4×C127—41no—
97C4⋊C4×C127—42no—
98C4⋊C4×C127—42no—
99C4⋊C4×C127—41no—
100C4⋊C4×C127—41no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127—42no—
103C22⋊C4×C127—42no—
104C22⋊C4×C127—42no—
105C22⋊C4×C127—41no—
106C22⋊C4×C127—41no—
107C22⋊C4×C127—41no—
108C22⋊C4×C127—41no—
109C22⋊C4×C127—41no—
110C22⋊C4×C127—41no—
111C22⋊C4×C127—41no—
112C22⋊C4×C127—81no—
113C42×C127—42no—
114C23×C127—21no—
115C23×C127—21no—
116C23×C127—22no—
117C23×C127—41no—
118C23×C127—41no—
119C23×C127—41no—
120C23×C127—41no—
121C23×C127—41no—
122C23×C127—41no—
123C23×C127—41no—
124C23×C127—41no—
125C23×C127#12011yescharacteristic
126C23×C127C2×SD1611yescharacteristic
127C23×C127AGL(1,ℤ/8)11yescharacteristic
128Q8×C127—81no—
129Q8×C127—81no—
130D4×C127—42no—
131D4×C127—44no—
132D4×C127—44no—
133D4×C127—81no—
134C2×C4×C127—22no—
135C2×C4×C127—22no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—41no—
139C2×C4×C127—41no—
140C2×C4×C127—41no—
141C2×C4×C127—41no—
142C2×C4×C127—41no—
143C2×C4×C127—42no—
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—81no—
157C2×C4×C127—84no—
158C2×C4×C127—81no—
159C2×C4×C127—81no—
160C2×C4×C127—81no—
161C2×C4×C127—81no—
162C2×C4×C127—81no—
163C8×C127—81no—
164C8×C127—81no—
165C22×C127—21no—
166C22×C127—21no—
167C22×C127—21no—
168C22×C127—21no—
169C22×C127—41no—
170C22×C127—41no—
171C22×C127—42no—
172C22×C127—42no—
173C22×C127—41no—
174C22×C127—41no—
175C22×C127—42no—
176C22×C127—81no—
177C22×C127—81no—
178C22×C127—81no—
179C22×C127—81no—
180C22×C127—81no—
181C22×C127#45011yescharacteristic
182C22×C127#45511yescharacteristic
183C22×C127B(4,𝔽2)11yescharacteristic
184C4×C127—42no—
185C4×C127—42no—
186C4×C127—41no—
187C4×C127—41no—
188C4×C127—81no—
189C4×C127—81no—
190C4×C127—81no—
191C4×C127—81no—
192C4×C127—81no—
193C4×C127—161no—
194#12204C12711yesmaximal, characteristic, direct product factor, Sylow
195C2×C127—21no—
196C2×C127—21no—
197C2×C127—41no—
198C2×C127—41no—
199C2×C127—82no—
200C2×C127#160111yescharacteristic
201C2×C127#219711yescharacteristic
202C2×C127#219811yescharacteristic
203#2600C2×C12711yescharacteristic
204#2030C2×C12711yescharacteristic, complemented
205#1608C2×C12711yescharacteristic, complemented
206#1596C2×C12711yescharacteristic
207#1344C2×C12711yescharacteristic, complemented
208#1343C2×C12711yescharacteristic
209#1317C2×C12711yescharacteristic, complemented
210C127#1220411yescharacteristic, direct product factor, Sylow
211#543C22×C12711yescharacteristic
212C2×C4×D4—21no—
213#483—21no—
214#450—21no—
215#397C22×C12711yescharacteristic
216#394—21no—
217#392C22×C12711yescharacteristic
218#388—21no—
219#387—21no—
220#386—21no—
221#385—21no—
222#381—21no—
223#381C22×C12711yescharacteristic
224#342—21no—
225#342C22×C12711yescharacteristic
226#328—21no—
227#328—21no—
228#323C22×C12711yescharacteristic
229#323C22×C12711yescharacteristic
230C22×D4—21no—
231C22×D4D4×C12711yescharacteristic
232C23×C4—21no—
233C23×C4D4×C12711yescharacteristic
234C2×SD16—41no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—41no—
238C22⋊Q8—41no—
239C4×D4—42no—
240C4×D4—42no—
241C2×C4⋊C4—21no—
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⋊C4D4×C12711yescharacteristic
252C2×C22⋊C4C23×C12711yescharacteristic
253C2×C42—21no—
254#106—41no—
255#103—41no—
256#102—41no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—41no—
264#95—41no—
265#95D4×C12711yescharacteristic
266#95D4×C12711yescharacteristic
267C24—21no—
268C24C2×D4×C12711yescharacteristic
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—42no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—42no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4SD16×C12712yescomplemented
292C22×C4C2×D4×C12711yescharacteristic
293C22×C4C2×D4×C12711yescharacteristic
294SD16—82no—
295C2×C8—41no—
296C2×C8—41no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—81no—
316C42—42no—
317C23—21no—
318C23—21no—
319C23—22no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23#120×C12711yescharacteristic
329C23C2×SD16×C12711yescharacteristic
330C23AGL(1,ℤ/8)×C12711yescharacteristic
331Q8—81no—
332Q8—81no—
333D4—42no—
334D4—44no—
335D4—44no—
336D4—81no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
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—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—84no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C8—81no—
367C8—81no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—41no—
378C22—42no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22#450×C12711yescharacteristic
385C22#455×C12711yescharacteristic
386C22B(4,𝔽2)×C12711yescharacteristic
387C4—42no—
388C4—42no—
389C4—41no—
390C4—41no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—161no—
397C2—21no—
398C2—21no—
399C2—41no—
400C2—41no—
401C2—82no—
402C2C127×#160111yescharacteristic
403C2C127×#219711yescharacteristic
404C2C127×#219811yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C127
prime radical#392×C127C22
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×#2030C2semidirect
3. C127×#1608C2semidirect
5. C127×#1344C2semidirect
7. C127×#1317C2semidirect
88. C22×C4×C127SD16semidirect
194. #12204C127direct (N × H)
204. #2030C2×C127semidirect
205. #1608C2×C127semidirect
207. #1344C2×C127semidirect
209. #1317C2×C127semidirect
210. C127#12204direct (N × H)
291. C22×C4SD16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#26002
C127×#20302
C127×#16082
C127×#15962
C127×#13442
C127×#13432
C127×#13172
#12204127