Finite Group Databasebeta

← C127×#12227 #44575900

Subgroups — C127×#12227 #44575900

Order 32512.

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

normal subgroups only

This group has 520 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×#2463C211yesmaximal, characteristic
2C127×#2062C211yesmaximal, characteristic, complemented
3C127×#1602C211yesmaximal, characteristic, complemented
4C127×#1596C211yesmaximal, characteristic
5C127×#1351C211yesmaximal, characteristic, complemented
6C127×#1343C211yesmaximal, characteristic
7C127×#1326C211yesmaximal, characteristic, complemented
8#543×C127C2211yescharacteristic
9C2×C22⋊Q8×C127—21no—
10C2×C4×D4×C127—21no—
11#451×C127—21no—
12#450×C127—21no—
13#394×C127—21no—
14#393×C127C2211yescharacteristic
15#392×C127C2211yescharacteristic
16#388×C127—21no—
17#387×C127—21no—
18#386×C127—21no—
19#385×C127—21no—
20#385×C127C2211yescharacteristic
21#342×C127—21no—
22#342×C127C2211yescharacteristic
23#328×C127—21no—
24#328×C127—21no—
25#323×C127C2211yescharacteristic
26#323×C127C2211yescharacteristic
27C22×Q8×C127D411yescharacteristic
28C22×D4×C127—21no—
29C22×D4×C127D411yescharacteristic
30C23×C4×C127—21no—
31C23×C4×C127D411yescharacteristic
32C2×Q16×C127—41no—
33C2×SD16×C127—41no—
34C22⋊Q8×C127—21no—
35C22⋊Q8×C127—21no—
36C22⋊Q8×C127—41no—
37C22⋊Q8×C127—41no—
38C22⋊Q8×C127—44no—
39C4×D4×C127—44no—
40C2×C4⋊C4×C127—21no—
41C2×C4⋊C4×C127—21no—
42C2×C4⋊C4×C127—21no—
43C2×C4⋊C4×C127—41no—
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×C22⋊C4×C127D411yescharacteristic
52C2×C22⋊C4×C127C2311yescharacteristic
53C2×C42×C127—21no—
54#103×C127—41no—
55#102×C127—41no—
56C22⋊C8×C127—21no—
57C22⋊C8×C127—21no—
58#95×C127—21no—
59#95×C127—21no—
60#95×C127—21no—
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×Q8×C127—42no—
69C2×Q8×C127—41no—
70C2×D4×C127—42no—
71C2×D4×C127—41no—
72C2×D4×C127—42no—
73C2×D4×C127—41no—
74C2×D4×C127—41no—
75C2×D4×C127—42no—
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—21no—
83C22×C4×C127—42no—
84C22×C4×C127—42no—
85C22×C4×C127—42no—
86C22×C4×C127—41no—
87C22×C4×C127—41no—
88C22×C4×C127—81no—
89C22×C4×C127C2×D411yescharacteristic
90C22×C4×C127C2×D411yescharacteristic
91Q16×C127—82no—
92SD16×C127—82no—
93C2×C8×C127—41no—
94C2×C8×C127—41no—
95C4⋊C4×C127—24no—
96C4⋊C4×C127—21no—
97C4⋊C4×C127—21no—
98C4⋊C4×C127—42no—
99C4⋊C4×C127—42no—
100C4⋊C4×C127—41no—
101C4⋊C4×C127—41no—
102C4⋊C4×C127—82no—
103C22⋊C4×C127—22no—
104C22⋊C4×C127—21no—
105C22⋊C4×C127—21no—
106C22⋊C4×C127—41no—
107C22⋊C4×C127—42no—
108C22⋊C4×C127—42no—
109C22⋊C4×C127—42no—
110C22⋊C4×C127—42no—
111C22⋊C4×C127—42no—
112C22⋊C4×C127—41no—
113C22⋊C4×C127—41no—
114C22⋊C4×C127—41no—
115C42×C127—42no—
116C23×C127—21no—
117C23×C127—21no—
118C23×C127—22no—
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#12011yescharacteristic
128C23×C127#13511yescharacteristic
129C23×C127#13711yescharacteristic
130Q8×C127—42no—
131Q8×C127—81no—
132Q8×C127—81no—
133Q8×C127—81no—
134D4×C127—42no—
135D4×C127—44no—
136D4×C127—44no—
137D4×C127—81no—
138C2×C4×C127—22no—
139C2×C4×C127—22no—
140C2×C4×C127—22no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
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—42no—
152C2×C4×C127—41no—
153C2×C4×C127—42no—
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—41no—
161C2×C4×C127—81no—
162C2×C4×C127—84no—
163C2×C4×C127—81no—
164C2×C4×C127—81no—
165C2×C4×C127—81no—
166C2×C4×C127—81no—
167C8×C127—81no—
168C8×C127—81no—
169C22×C127—21no—
170C22×C127—21no—
171C22×C127—21no—
172C22×C127—21no—
173C22×C127—41no—
174C22×C127—41no—
175C22×C127—42no—
176C22×C127—42no—
177C22×C127—41no—
178C22×C127—41no—
179C22×C127—42no—
180C22×C127—81no—
181C22×C127—81no—
182C22×C127—81no—
183C22×C127—81no—
184C22×C127—81no—
185C22×C127#45211yescharacteristic
186C22×C127#45611yescharacteristic
187C22×C127B(4,𝔽2)11yescharacteristic
188C4×C127—42no—
189C4×C127—41no—
190C4×C127—42no—
191C4×C127—42no—
192C4×C127—41no—
193C4×C127—81no—
194C4×C127—81no—
195C4×C127—82no—
196C4×C127—81no—
197C4×C127—81no—
198#12227C12711yesmaximal, characteristic, direct product factor, Sylow
199C2×C127—21no—
200C2×C127—21no—
201C2×C127—41no—
202C2×C127—41no—
203C2×C127—82no—
204C2×C127#160311yescharacteristic
205C2×C127#219611yescharacteristic
206C2×C127#219711yescharacteristic
207#2463C2×C12711yescharacteristic
208#2062C2×C12711yescharacteristic, complemented
209#1602C2×C12711yescharacteristic, complemented
210#1596C2×C12711yescharacteristic
211#1351C2×C12711yescharacteristic, complemented
212#1343C2×C12711yescharacteristic
213#1326C2×C12711yescharacteristic, complemented
214C127#1222711yescharacteristic, direct product factor, Sylow
215#543C22×C12711yescharacteristic
216C2×C22⋊Q8—21no—
217C2×C4×D4—21no—
218#451—21no—
219#450—21no—
220#394—21no—
221#393C22×C12711yescharacteristic
222#392C22×C12711yescharacteristic
223#388—21no—
224#387—21no—
225#386—21no—
226#385—21no—
227#385C22×C12711yescharacteristic
228#342—21no—
229#342C22×C12711yescharacteristic
230#328—21no—
231#328—21no—
232#323C22×C12711yescharacteristic
233#323C22×C12711yescharacteristic
234C22×Q8D4×C12711yescharacteristic
235C22×D4—21no—
236C22×D4D4×C12711yescharacteristic
237C23×C4—21no—
238C23×C4D4×C12711yescharacteristic
239C2×Q16—41no—
240C2×SD16—41no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8—41no—
244C22⋊Q8—41no—
245C22⋊Q8—44no—
246C4×D4—44no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—41no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4D4×C12711yescharacteristic
259C2×C22⋊C4C23×C12711yescharacteristic
260C2×C42—21no—
261#103—41no—
262#102—41no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—41no—
269#95D4×C12711yescharacteristic
270#95D4×C12711yescharacteristic
271C24—21no—
272C24C2×D4×C12711yescharacteristic
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—42no—
276C2×Q8—41no—
277C2×D4—42no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—81no—
296C22×C4C2×D4×C12711yescharacteristic
297C22×C4C2×D4×C12711yescharacteristic
298Q16—82no—
299SD16—82no—
300C2×C8—41no—
301C2×C8—41no—
302C4⋊C4—24no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—82no—
310C22⋊C4—22no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—41no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C42—42no—
323C23—21no—
324C23—21no—
325C23—22no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23#120×C12711yescharacteristic
335C23#135×C12711yescharacteristic
336C23#137×C12711yescharacteristic
337Q8—42no—
338Q8—81no—
339Q8—81no—
340Q8—81no—
341D4—42no—
342D4—44no—
343D4—44no—
344D4—81no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
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—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—81no—
369C2×C4—84no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C8—81no—
375C8—81no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—41no—
381C22—41no—
382C22—42no—
383C22—42no—
384C22—41no—
385C22—41no—
386C22—42no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22#452×C12711yescharacteristic
393C22#456×C12711yescharacteristic
394C22B(4,𝔽2)×C12711yescharacteristic
395C4—42no—
396C4—41no—
397C4—42no—
398C4—42no—
399C4—41no—
400C4—81no—
401C4—81no—
402C4—82no—
403C4—81no—
404C4—81no—
405C2—21no—
406C2—21no—
407C2—41no—
408C2—41no—
409C2—82no—
410C2C127×#160311yescharacteristic
411C2C127×#219611yescharacteristic
412C2C127×#219711yescharacteristic
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×#2062C2semidirect
3. C127×#1602C2semidirect
5. C127×#1351C2semidirect
7. C127×#1326C2semidirect
198. #12227C127direct (N × H)
208. #2062C2×C127semidirect
209. #1602C2×C127semidirect
211. #1351C2×C127semidirect
213. #1326C2×C127semidirect
214. C127#12227direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#24632
C127×#20622
C127×#16022
C127×#15962
C127×#13512
C127×#13432
C127×#13262
#12227127