Finite Group Databasebeta

← C127×#11776 #44575449

Subgroups — C127×#11776 #44575449

Order 32512.

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

normal subgroups only

This group has 652 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×#2541C211yesmaximal, characteristic
2C4×#99×C127C211yesmaximal, characteristic, complemented
3C127×#1465C211yesmaximal, characteristic, complemented
4C127×#1343C214yesmaximal, complemented
5C2×#127×C127—21no—
6C2×#127×C127C411yescharacteristic, complemented
7C2×#121×C127—21no—
8C2×#121×C127—21no—
9C2×C4×D4×C127C2211yescharacteristic
10C2×#99×C127—21no—
11#407×C127—21no—
12#392×C127C412yescomplemented
13#392×C127C2212yes—
14#390×C127—21no—
15C4×C4⋊C4×C127C411yescharacteristic, complemented
16C4×C22⋊C4×C127—21no—
17#342×C127C2212yes—
18#324×C127—21no—
19#323×C127C2212yes—
20C22×D4×C127—22no—
21C22×D4×C127—22no—
22C22×D4×C127—21no—
23C22×D4×C127—21no—
24C22×D4×C127C2×C412yes—
25C22×D4×C127D411yescharacteristic
26C23×C4×C127D411yescharacteristic
27C23×C4×C127D411yescharacteristic
28C2×M16×C127—41no—
29#127×C127—41no—
30#127×C127—41no—
31#127×C127—44no—
32#121×C127—44no—
33#121×C127—44no—
34C4×D4×C127—41no—
35C4×D4×C127—41no—
36C2×C4⋊C4×C127—21no—
37C2×C4⋊C4×C127D411yescharacteristic
38C2×C22⋊C4×C127—22no—
39C2×C22⋊C4×C127—22no—
40C2×C22⋊C4×C127—22no—
41C2×C22⋊C4×C127C2×C411yescharacteristic
42C2×C22⋊C4×C127C2311yescharacteristic
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—41no—
50#95×C127C2×C412yes—
51C24×C127—22no—
52C24×C127C22⋊C411yescharacteristic
53C24×C127C2×D411yescharacteristic
54C2×D4×C127—22no—
55C2×D4×C127—22no—
56C2×D4×C127—41no—
57C2×D4×C127—42no—
58C2×D4×C127—42no—
59C2×D4×C127—42no—
60C2×D4×C127—44no—
61C2×D4×C127—44no—
62C2×D4×C127—42no—
63C2×D4×C127—42no—
64C2×D4×C127—44no—
65C2×D4×C127—44no—
66C2×D4×C127—41no—
67C2×D4×C127—41no—
68C2×D4×C127—42no—
69C2×D4×C127—82no—
70C2×D4×C127—82no—
71C2×D4×C127—82no—
72C22×C4×C127—22no—
73C22×C4×C127—21no—
74C22×C4×C127—21no—
75C22×C4×C127—21no—
76C22×C4×C127—21no—
77C22×C4×C127—42no—
78C22×C4×C127—42no—
79C22×C4×C127—42no—
80C22×C4×C127C22⋊C411yescharacteristic
81C22×C4×C127C22⋊C411yescharacteristic
82C22×C4×C127C22⋊C411yescharacteristic
83C22×C4×C127C22×C411yescharacteristic
84C22×C4×C127C2×D411yescharacteristic
85M16×C127—82no—
86C2×C8×C127—42no—
87C4⋊C4×C127—24no—
88C4⋊C4×C127—41no—
89C22⋊C4×C127—24no—
90C22⋊C4×C127—41no—
91C22⋊C4×C127—41no—
92C22⋊C4×C127—44no—
93C22⋊C4×C127—44no—
94C22⋊C4×C127—42no—
95C42×C127—21no—
96C42×C127—21no—
97C42×C127—42no—
98C42×C127—82no—
99C23×C127—21no—
100C23×C127—21no—
101C23×C127—21no—
102C23×C127—21no—
103C23×C127—21no—
104C23×C127—41no—
105C23×C127—41no—
106C23×C127—41no—
107C23×C127—41no—
108C23×C127—42no—
109C23×C127—42no—
110C23×C127—42no—
111C23×C127—42no—
112C23×C127—42no—
113C23×C127—42no—
114C23×C127—42no—
115C23×C127C2×C22⋊C411yescharacteristic
116C23×C127AGL(1,ℤ/8)12yes—
117D4×C127—44no—
118D4×C127—82no—
119D4×C127—84no—
120D4×C127—88no—
121D4×C127—88no—
122D4×C127—84no—
123D4×C127—82no—
124C2×C4×C127—21no—
125C2×C4×C127—21no—
126C2×C4×C127—22no—
127C2×C4×C127—24no—
128C2×C4×C127—21no—
129C2×C4×C127—21no—
130C2×C4×C127—21no—
131C2×C4×C127—42no—
132C2×C4×C127—42no—
133C2×C4×C127—42no—
134C2×C4×C127—42no—
135C2×C4×C127—41no—
136C2×C4×C127—41no—
137C2×C4×C127—41no—
138C2×C4×C127—44no—
139C2×C4×C127—81no—
140C2×C4×C127—81no—
141C2×C4×C127—82no—
142C2×C4×C127—82no—
143C2×C4×C127#9911yescharacteristic
144C2×C4×C127#9911yescharacteristic
145C2×C4×C127#10412yes—
146C8×C127—82no—
147C22×C127—21no—
148C22×C127—21no—
149C22×C127—22no—
150C22×C127—42no—
151C22×C127—42no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127—81no—
158C22×C127—81no—
159C22×C127—82no—
160C22×C127—82no—
161C22×C127—82no—
162C22×C127—82no—
163C22×C127—82no—
164C22×C127—82no—
165C22×C127—82no—
166C22×C127C2×#9911yescharacteristic
167C22×C127#41811yescharacteristic
168C22×C127C2×#10411yescharacteristic
169C4×C127—21no—
170C4×C127—22no—
171C4×C127—44no—
172C4×C127—41no—
173C4×C127—41no—
174C4×C127—41no—
175C4×C127—84no—
176C4×C127#35312yes—
177#11776C12711yesmaximal, characteristic, direct product factor, Sylow
178C2×C127—21no—
179C2×C127—21no—
180C2×C127—42no—
181C2×C127—81no—
182C2×C127—162no—
183C2×C127#150711yescharacteristic
184C2×C127#212211yescharacteristic
185C2×C127C2×#35311yescharacteristic
186#2541C2×C12711yescharacteristic
187C4×#99C2×C12711yescharacteristic, complemented
188#1465C2×C12711yescharacteristic, complemented
189#1343C2×C12714yescomplemented
190C127#1177611yescharacteristic, direct product factor, Sylow
191C2×#127—21no—
192C2×#127C4×C12711yescharacteristic, complemented
193C2×#121—21no—
194C2×#121—21no—
195C2×C4×D4C22×C12711yescharacteristic
196C2×#99—21no—
197#407—21no—
198#392C4×C12712yescomplemented
199#392C22×C12712yes—
200#390—21no—
201C4×C4⋊C4C4×C12711yescharacteristic, complemented
202C4×C22⋊C4—21no—
203#342C22×C12712yes—
204#324—21no—
205#323C22×C12712yes—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4C2×C4×C12712yes—
211C22×D4D4×C12711yescharacteristic
212C23×C4D4×C12711yescharacteristic
213C23×C4D4×C12711yescharacteristic
214C2×M16—41no—
215#127—41no—
216#127—41no—
217#127—44no—
218#121—44no—
219#121—44no—
220C4×D4—41no—
221C4×D4—41no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4D4×C12711yescharacteristic
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4C2×C4×C12711yescharacteristic
228C2×C22⋊C4C23×C12711yescharacteristic
229C2×C42—21no—
230C2×C42—41no—
231C2×C42C2×C4×C12711yescharacteristic
232#99—24no—
233C22⋊C8—22no—
234C22⋊C8—41no—
235#95—41no—
236#95C2×C4×C12712yes—
237C24—22no—
238C24C22⋊C4×C12711yescharacteristic
239C24C2×D4×C12711yescharacteristic
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—82no—
256C2×D4—82no—
257C2×D4—82no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4C22⋊C4×C12711yescharacteristic
267C22×C4C22⋊C4×C12711yescharacteristic
268C22×C4C22⋊C4×C12711yescharacteristic
269C22×C4C22×C4×C12711yescharacteristic
270C22×C4C2×D4×C12711yescharacteristic
271M16—82no—
272C2×C8—42no—
273C4⋊C4—24no—
274C4⋊C4—41no—
275C22⋊C4—24no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—42no—
281C42—21no—
282C42—21no—
283C42—42no—
284C42—82no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23C2×C22⋊C4×C12711yescharacteristic
302C23AGL(1,ℤ/8)×C12712yes—
303D4—44no—
304D4—82no—
305D4—84no—
306D4—88no—
307D4—88no—
308D4—84no—
309D4—82no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—44no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4#99×C12711yescharacteristic
330C2×C4#99×C12711yescharacteristic
331C2×C4#104×C12712yes—
332C8—82no—
333C22—21no—
334C22—21no—
335C22—22no—
336C22—42no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22—82no—
346C22—82no—
347C22—82no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—82no—
352C22C2×#99×C12711yescharacteristic
353C22#418×C12711yescharacteristic
354C22C2×#104×C12711yescharacteristic
355C4—21no—
356C4—22no—
357C4—44no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—84no—
362C4#353×C12712yes—
363C2—21no—
364C2—21no—
365C2—42no—
366C2—81no—
367C2—162no—
368C2C127×#150711yescharacteristic
369C2C127×#212211yescharacteristic
370C2C2×#353×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC127×#2541C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×#99×C127C2semidirect
3. C127×#1465C2semidirect
4. C127×#1343C2semidirect
6. C2×#127×C127C4semidirect
12. #392×C127C4semidirect
15. C4×C4⋊C4×C127C4semidirect
177. #11776C127direct (N × H)
187. C4×#99C2×C127semidirect
188. #1465C2×C127semidirect
189. #1343C2×C127semidirect
190. C127#11776direct (N × H)
192. C2×#127C4×C127semidirect
198. #392C4×C127semidirect
201. C4×C4⋊C4C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#25412
C4×#99×C1272
C127×#14652
C127×#13432
#11776127