Finite Group Databasebeta

← C127×#11728 #44575401

Subgroups — C127×#11728 #44575401

Order 32512.

470 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2398C211yesmaximal, characteristic
2C4×#99×C127C211yesmaximal, characteristic
3C127×#1459C211yesmaximal, characteristic
4C127×#1351C212yesmaximal, complemented
5C127×#1344C212yesmaximal, complemented
6C2×C4×Q8×C127C411yescharacteristic, complemented
7C2×C4×D4×C127C2211yescharacteristic
8C2×#99×C127—21no—
9C2×C22⋊C8×C127—21no—
10#400×C127—21no—
11#397×C127C411yescharacteristic, complemented
12#397×C127C2211yescharacteristic
13#394×C127—21no—
14#393×C127C411yescharacteristic, complemented
15#393×C127C2211yescharacteristic
16C4×C22⋊C4×C127—21no—
17C4×C22⋊C4×C127—21no—
18C4×C22⋊C4×C127—21no—
19#376×C127C411yescharacteristic, complemented
20#342×C127C2212yes—
21#324×C127—21no—
22#323×C127C2212yes—
23C22×Q8×C127C2×C411yescharacteristic
24C22×D4×C127D411yescharacteristic
25C23×C4×C127D411yescharacteristic
26C23×C4×C127D411yescharacteristic
27C22×C8×C127—41no—
28C4×Q8×C127—41no—
29C4×Q8×C127—41no—
30C4×D4×C127—41no—
31C4×D4×C127—41no—
32C2×C4⋊C4×C127—21no—
33C2×C4⋊C4×C127C2×C411yescharacteristic
34C2×C4⋊C4×C127D411yescharacteristic
35C2×C22⋊C4×C127—22no—
36C2×C22⋊C4×C127—21no—
37C2×C22⋊C4×C127—21no—
38C2×C22⋊C4×C127—21no—
39C2×C22⋊C4×C127—21no—
40C2×C22⋊C4×C127C2×C411yescharacteristic
41C2×C22⋊C4×C127C2311yescharacteristic
42C2×C42×C127—21no—
43C2×C42×C127—21no—
44C2×C42×C127—41no—
45C2×C42×C127C2×C411yescharacteristic
46#99×C127—24no—
47C22⋊C8×C127—22no—
48C22⋊C8×C127—41no—
49#95×C127—21no—
50#95×C127—21no—
51#95×C127—21no—
52#95×C127—21no—
53#95×C127—21no—
54#95×C127—41no—
55#95×C127C2×C411yescharacteristic
56#95×C127C2×C411yescharacteristic
57C24×C127C22⋊C411yescharacteristic
58C24×C127C2×D411yescharacteristic
59C2×Q8×C127—42no—
60C2×Q8×C127—41no—
61C2×D4×C127—22no—
62C2×D4×C127—22no—
63C2×D4×C127—41no—
64C22×C4×C127—21no—
65C22×C4×C127—21no—
66C22×C4×C127—21no—
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127—21no—
70C22×C4×C127—21no—
71C22×C4×C127—21no—
72C22×C4×C127—42no—
73C22×C4×C127—42no—
74C22×C4×C127—42no—
75C22×C4×C127C22⋊C411yescharacteristic
76C22×C4×C127C22⋊C411yescharacteristic
77C22×C4×C127C22⋊C411yescharacteristic
78C22×C4×C127C22×C411yescharacteristic
79C22×C4×C127C2×D411yescharacteristic
80C2×C8×C127—42no—
81C2×C8×C127—82no—
82C4⋊C4×C127—41no—
83C4⋊C4×C127—41no—
84C4⋊C4×C127—42no—
85C22⋊C4×C127—24no—
86C22⋊C4×C127—41no—
87C22⋊C4×C127—41no—
88C22⋊C4×C127—42no—
89C22⋊C4×C127—42no—
90C22⋊C4×C127—42no—
91C22⋊C4×C127—42no—
92C22⋊C4×C127—42no—
93C42×C127—21no—
94C42×C127—21no—
95C42×C127—42no—
96C42×C127—41no—
97C42×C127—41no—
98C42×C127—82no—
99C23×C127—21no—
100C23×C127—21no—
101C23×C127—21no—
102C23×C127—21no—
103C23×C127—41no—
104C23×C127—41no—
105C23×C127—41no—
106C23×C127—41no—
107C23×C127#10412yes—
108C23×C127C2×C22⋊C411yescharacteristic
109C23×C127#13512yes—
110Q8×C127—42no—
111Q8×C127—42no—
112D4×C127—44no—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127—21no—
116C2×C4×C127—21no—
117C2×C4×C127—21no—
118C2×C4×C127—21no—
119C2×C4×C127—21no—
120C2×C4×C127—21no—
121C2×C4×C127—41no—
122C2×C4×C127—41no—
123C2×C4×C127—42no—
124C2×C4×C127—42no—
125C2×C4×C127—41no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—41no—
129C2×C4×C127—41no—
130C2×C4×C127—44no—
131C2×C4×C127—41no—
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—81no—
139C2×C4×C127—81no—
140C2×C4×C127—82no—
141C2×C4×C127—82no—
142C2×C4×C127#9911yescharacteristic
143C2×C4×C127#9911yescharacteristic
144C8×C127—82no—
145C22×C127—21no—
146C22×C127—21no—
147C22×C127—22no—
148C22×C127—22no—
149C22×C127—42no—
150C22×C127—42no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—81no—
156C22×C127—81no—
157C22×C127C2×#9911yescharacteristic
158C22×C127#41611yescharacteristic
159C22×C127C2×#10411yescharacteristic
160C4×C127—21no—
161C4×C127—42no—
162C4×C127—41no—
163C4×C127—41no—
164C4×C127—41no—
165C4×C127—41no—
166C4×C127—81no—
167C4×C127—81no—
168C4×C127—84no—
169C4×C127—81no—
170C4×C127#35412yes—
171#11728C12711yesmaximal, characteristic, direct product factor, Sylow
172C2×C127—21no—
173C2×C127—21no—
174C2×C127—42no—
175C2×C127—42no—
176C2×C127#150111yescharacteristic
177C2×C127#212011yescharacteristic
178C2×C127C2×#35411yescharacteristic
179#2398C2×C12711yescharacteristic
180C4×#99C2×C12711yescharacteristic
181#1459C2×C12711yescharacteristic
182#1351C2×C12712yescomplemented
183#1344C2×C12712yescomplemented
184C127#1172811yescharacteristic, direct product factor, Sylow
185C2×C4×Q8C4×C12711yescharacteristic, complemented
186C2×C4×D4C22×C12711yescharacteristic
187C2×#99—21no—
188C2×C22⋊C8—21no—
189#400—21no—
190#397C4×C12711yescharacteristic, complemented
191#397C22×C12711yescharacteristic
192#394—21no—
193#393C4×C12711yescharacteristic, complemented
194#393C22×C12711yescharacteristic
195C4×C22⋊C4—21no—
196C4×C22⋊C4—21no—
197C4×C22⋊C4—21no—
198#376C4×C12711yescharacteristic, complemented
199#342C22×C12712yes—
200#324—21no—
201#323C22×C12712yes—
202C22×Q8C2×C4×C12711yescharacteristic
203C22×D4D4×C12711yescharacteristic
204C23×C4D4×C12711yescharacteristic
205C23×C4D4×C12711yescharacteristic
206C22×C8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C2×C4×C12711yescharacteristic
213C2×C4⋊C4D4×C12711yescharacteristic
214C2×C22⋊C4—22no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4C2×C4×C12711yescharacteristic
220C2×C22⋊C4C23×C12711yescharacteristic
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—41no—
224C2×C42C2×C4×C12711yescharacteristic
225#99—24no—
226C22⋊C8—22no—
227C22⋊C8—41no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—41no—
234#95C2×C4×C12711yescharacteristic
235#95C2×C4×C12711yescharacteristic
236C24C22⋊C4×C12711yescharacteristic
237C24C2×D4×C12711yescharacteristic
238C2×Q8—42no—
239C2×Q8—41no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4C22⋊C4×C12711yescharacteristic
255C22×C4C22⋊C4×C12711yescharacteristic
256C22×C4C22⋊C4×C12711yescharacteristic
257C22×C4C22×C4×C12711yescharacteristic
258C22×C4C2×D4×C12711yescharacteristic
259C2×C8—42no—
260C2×C8—82no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C22⋊C4—24no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C42—21no—
273C42—21no—
274C42—42no—
275C42—41no—
276C42—41no—
277C42—82no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#104×C12712yes—
287C23C2×C22⋊C4×C12711yescharacteristic
288C23#135×C12712yes—
289Q8—42no—
290Q8—42no—
291D4—44no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—44no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4#99×C12711yescharacteristic
322C2×C4#99×C12711yescharacteristic
323C8—82no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22—81no—
336C22C2×#99×C12711yescharacteristic
337C22#416×C12711yescharacteristic
338C22C2×#104×C12711yescharacteristic
339C4—21no—
340C4—42no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—81no—
346C4—81no—
347C4—84no—
348C4—81no—
349C4#354×C12712yes—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—42no—
354C2C127×#150111yescharacteristic
355C2C127×#212011yescharacteristic
356C2C2×#354×C12711yescharacteristic
subgroupG/H
centerC2×C4×C127#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C22⋊C4C23×C127
socleC22×C127C2×#99
prime radicalC22×D4×C127D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C22×C4×C127 ◁ C2×C4×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1351C2semidirect
5. C127×#1344C2semidirect
6. C2×C4×Q8×C127C4semidirect
11. #397×C127C4semidirect
14. #393×C127C4semidirect
19. #376×C127C4semidirect
171. #11728C127direct (N × H)
182. #1351C2×C127semidirect
183. #1344C2×C127semidirect
184. C127#11728direct (N × H)
185. C2×C4×Q8C4×C127semidirect
190. #397C4×C127semidirect
193. #393C4×C127semidirect
198. #376C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#23982
C4×#99×C1272
C127×#14592
C127×#13512
C127×#13442
#11728127