Finite Group Databasebeta

← C127×#12267 #44575940

Subgroups — C127×#12267 #44575940

Order 32512.

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

normal subgroups only

This group has 414 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×#2752C211yesmaximal, characteristic
2C127×#2030C211yesmaximal, characteristic, complemented
3C127×#1614C211yesmaximal, characteristic, complemented
4C127×#1596C211yesmaximal, characteristic
5C127×#1352C211yesmaximal, characteristic, complemented
6C127×#1343C211yesmaximal, characteristic
7C127×#1317C211yesmaximal, characteristic, complemented
8#543×C127C2211yescharacteristic
9#484×C127—21no—
10#450×C127—21no—
11#399×C127C2211yescharacteristic
12#397×C127—21no—
13#394×C127—21no—
14#392×C127C2211yescharacteristic
15#390×C127—21no—
16#388×C127—21no—
17#387×C127—21no—
18#386×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×C127D411yescharacteristic
30C2×SD16×C127—41no—
31C22⋊Q8×C127—21no—
32C22⋊Q8×C127—21no—
33C22⋊Q8×C127—41no—
34C22⋊Q8×C127—41no—
35C2×C4⋊C4×C127—21no—
36C2×C4⋊C4×C127—21no—
37C2×C4⋊C4×C127D411yescharacteristic
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×C127D411yescharacteristic
47C2×C22⋊C4×C127C2311yescharacteristic
48C2×C42×C127—21no—
49#107×C127—41no—
50#103×C127—41no—
51#102×C127—41no—
52C22⋊C8×C127—21no—
53C22⋊C8×C127—21no—
54#95×C127—21no—
55#95×C127—21no—
56#95×C127—21no—
57#95×C127—21no—
58#95×C127—41no—
59#95×C127—41no—
60#95×C127D411yescharacteristic
61#95×C127D411yescharacteristic
62C24×C127—21no—
63C24×C127C2×D411yescharacteristic
64C2×Q8×C127—21no—
65C2×Q8×C127—21no—
66C2×D4×C127—41no—
67C2×D4×C127—42no—
68C2×D4×C127—41no—
69C2×D4×C127—41no—
70C2×D4×C127—81no—
71C2×D4×C127—81no—
72C22×C4×C127—21no—
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—21no—
81C22×C4×C127—41no—
82C22×C4×C127—41no—
83C22×C4×C127—41no—
84C22×C4×C127C2×D411yescharacteristic
85C22×C4×C127C2×D411yescharacteristic
86SD16×C127—82no—
87C2×C8×C127—41no—
88C2×C8×C127—41no—
89C4⋊C4×C127—21no—
90C4⋊C4×C127—21no—
91C4⋊C4×C127—41no—
92C4⋊C4×C127—42no—
93C4⋊C4×C127—41no—
94C4⋊C4×C127—41no—
95C4⋊C4×C127—81no—
96C22⋊C4×C127—41no—
97C22⋊C4×C127—42no—
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—
108C42×C127—81no—
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#12011yescharacteristic
121C23×C127#13511yescharacteristic
122C23×C127#13711yescharacteristic
123Q8×C127—81no—
124Q8×C127—81no—
125D4×C127—42no—
126D4×C127—82no—
127D4×C127—82no—
128D4×C127—81no—
129C2×C4×C127—22no—
130C2×C4×C127—21no—
131C2×C4×C127—21no—
132C2×C4×C127—41no—
133C2×C4×C127—41no—
134C2×C4×C127—41no—
135C2×C4×C127—41no—
136C2×C4×C127—41no—
137C2×C4×C127—41no—
138C2×C4×C127—41no—
139C2×C4×C127—41no—
140C2×C4×C127—41no—
141C2×C4×C127—41no—
142C2×C4×C127—42no—
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—81no—
155C2×C4×C127—81no—
156C2×C4×C127—81no—
157C8×C127—81no—
158C8×C127—81no—
159C22×C127—21no—
160C22×C127—21no—
161C22×C127—21no—
162C22×C127—21no—
163C22×C127—41no—
164C22×C127—41no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—81no—
168C22×C127—81no—
169C22×C127—81no—
170C22×C127—81no—
171C22×C127—81no—
172C22×C127—81no—
173C22×C127—81no—
174C22×C127—81no—
175C22×C127#45211yescharacteristic
176C22×C127#45511yescharacteristic
177C22×C127B(4,𝔽2)11yescharacteristic
178C4×C127—42no—
179C4×C127—41no—
180C4×C127—41no—
181C4×C127—81no—
182C4×C127—81no—
183C4×C127—81no—
184C4×C127—81no—
185C4×C127—81no—
186C4×C127—81no—
187C4×C127—161no—
188#12267C12711yesmaximal, characteristic, direct product factor, Sylow
189C2×C127—21no—
190C2×C127—21no—
191C2×C127—41no—
192C2×C127—41no—
193C2×C127—161no—
194C2×C127#160611yescharacteristic
195C2×C127#219811yescharacteristic
196C2×C127#220011yescharacteristic
197#2752C2×C12711yescharacteristic
198#2030C2×C12711yescharacteristic, complemented
199#1614C2×C12711yescharacteristic, complemented
200#1596C2×C12711yescharacteristic
201#1352C2×C12711yescharacteristic, complemented
202#1343C2×C12711yescharacteristic
203#1317C2×C12711yescharacteristic, complemented
204C127#1226711yescharacteristic, direct product factor, Sylow
205#543C22×C12711yescharacteristic
206#484—21no—
207#450—21no—
208#399C22×C12711yescharacteristic
209#397—21no—
210#394—21no—
211#392C22×C12711yescharacteristic
212#390—21no—
213#388—21no—
214#387—21no—
215#386—21no—
216#381—21no—
217#381C22×C12711yescharacteristic
218#342—21no—
219#342C22×C12711yescharacteristic
220#328—21no—
221#328—21no—
222#323C22×C12711yescharacteristic
223#323C22×C12711yescharacteristic
224C22×D4—21no—
225C22×D4D4×C12711yescharacteristic
226C23×C4D4×C12711yescharacteristic
227C2×SD16—41no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C22⋊Q8—41no—
231C22⋊Q8—41no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4D4×C12711yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4D4×C12711yescharacteristic
244C2×C22⋊C4C23×C12711yescharacteristic
245C2×C42—21no—
246#107—41no—
247#103—41no—
248#102—41no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—41no—
256#95—41no—
257#95D4×C12711yescharacteristic
258#95D4×C12711yescharacteristic
259C24—21no—
260C24C2×D4×C12711yescharacteristic
261C2×Q8—21no—
262C2×Q8—21no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—81no—
268C2×D4—81no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4C2×D4×C12711yescharacteristic
282C22×C4C2×D4×C12711yescharacteristic
283SD16—82no—
284C2×C8—41no—
285C2×C8—41no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—81no—
293C22⋊C4—41no—
294C22⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C22⋊C4—81no—
303C22⋊C4—81no—
304C22⋊C4—81no—
305C42—81no—
306C23—21no—
307C23—21no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23#120×C12711yescharacteristic
318C23#135×C12711yescharacteristic
319C23#137×C12711yescharacteristic
320Q8—81no—
321Q8—81no—
322D4—42no—
323D4—82no—
324D4—82no—
325D4—81no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C8—81no—
355C8—81no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22#452×C12711yescharacteristic
373C22#455×C12711yescharacteristic
374C22B(4,𝔽2)×C12711yescharacteristic
375C4—42no—
376C4—41no—
377C4—41no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4—161no—
385C2—21no—
386C2—21no—
387C2—41no—
388C2—41no—
389C2—161no—
390C2C127×#160611yescharacteristic
391C2C127×#219811yescharacteristic
392C2C127×#220011yescharacteristic
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×#1614C2semidirect
5. C127×#1352C2semidirect
7. C127×#1317C2semidirect
188. #12267C127direct (N × H)
198. #2030C2×C127semidirect
199. #1614C2×C127semidirect
201. #1352C2×C127semidirect
203. #1317C2×C127semidirect
204. C127#12267direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#27522
C127×#20302
C127×#16142
C127×#15962
C127×#13522
C127×#13432
C127×#13172
#12267127