Finite Group Databasebeta

← C127×#11779 #44575452

Subgroups — C127×#11779 #44575452

Order 32512.

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

normal subgroups only

This group has 768 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×#2408C211yesmaximal, characteristic
2C4×#100×C127C211yesmaximal, characteristic, complemented
3C127×#1465C211yesmaximal, characteristic, complemented
4C127×#1345C214yesmaximal, complemented
5C23×D4×C127C411yescharacteristic
6C2×#121×C127—21no—
7C2×#120×C127—21no—
8C2×C4×D4×C127C2211yescharacteristic
9C2×#100×C127—21no—
10#407×C127—21no—
11C4×M16×C127—21no—
12#394×C127C2212yes—
13#392×C127C412yes—
14#390×C127—21no—
15#380×C127C411yescharacteristic
16#379×C127—21no—
17#343×C127C2212yescomplemented
18#324×C127C2212yescomplemented
19#323×C127—21no—
20C25×C127—21no—
21C22×D4×C127—22no—
22C22×D4×C127—21no—
23C22×D4×C127—41no—
24C22×D4×C127—41no—
25C22×D4×C127—44no—
26C22×D4×C127C2×C412yes—
27C22×D4×C127D411yescharacteristic
28C23×C4×C127C2×C411yescharacteristic
29C23×C4×C127D411yescharacteristic
30C2×M16×C127—22no—
31C2×M16×C127—41no—
32#121×C127—44no—
33#120×C127—44no—
34C4×D4×C127—41no—
35C4×D4×C127—41no—
36C2×C4⋊C4×C127—21no—
37C2×C4⋊C4×C127C2311yescharacteristic
38C2×C22⋊C4×C127—21no—
39C2×C22⋊C4×C127—22no—
40C2×C22⋊C4×C127—22no—
41C2×C22⋊C4×C127—22no—
42C2×C22⋊C4×C127C2×C411yescharacteristic
43C2×C22⋊C4×C127D411yescharacteristic
44C2×C42×C127D411yescharacteristic
45#100×C127—24no—
46C22⋊C8×C127—22no—
47C22⋊C8×C127—41no—
48#97×C127—41no—
49C4×C8×C127—41no—
50#95×C127C2×C412yes—
51C24×C127—22no—
52C24×C127—42no—
53C24×C127—44no—
54C24×C127—42no—
55C24×C127—42no—
56C24×C127—82no—
57C24×C127C22⋊C411yescharacteristic
58C24×C127C22⋊C411yescharacteristic
59C2×D4×C127—24no—
60C2×D4×C127—22no—
61C2×D4×C127—22no—
62C2×D4×C127—41no—
63C2×D4×C127—42no—
64C2×D4×C127—42no—
65C2×D4×C127—48no—
66C2×D4×C127—44no—
67C2×D4×C127—44no—
68C2×D4×C127—44no—
69C2×D4×C127—44no—
70C2×D4×C127—44no—
71C2×D4×C127—41no—
72C2×D4×C127—41no—
73C2×D4×C127—82no—
74C22×C4×C127—21no—
75C22×C4×C127—21no—
76C22×C4×C127—21no—
77C22×C4×C127—21no—
78C22×C4×C127—22no—
79C22×C4×C127—42no—
80C22×C4×C127—42no—
81C22×C4×C127C22⋊C411yescharacteristic
82C22×C4×C127C22⋊C411yescharacteristic
83C22×C4×C127C22×C411yescharacteristic
84C22×C4×C127C2×D411yescharacteristic
85C22×C4×C127C2×D411yescharacteristic
86M16×C127—24no—
87M16×C127—42no—
88M16×C127—82no—
89C2×C8×C127—42no—
90C2×C8×C127—42no—
91C4⋊C4×C127—41no—
92C4⋊C4×C127—42no—
93C22⋊C4×C127—41no—
94C22⋊C4×C127—41no—
95C22⋊C4×C127—42no—
96C22⋊C4×C127—44no—
97C22⋊C4×C127—44no—
98C22⋊C4×C127—82no—
99C42×C127—21no—
100C42×C127—21no—
101C23×C127—21no—
102C23×C127—21no—
103C23×C127—24no—
104C23×C127—21no—
105C23×C127—21no—
106C23×C127—21no—
107C23×C127—41no—
108C23×C127—41no—
109C23×C127—41no—
110C23×C127—41no—
111C23×C127—42no—
112C23×C127—42no—
113C23×C127—42no—
114C23×C127—48no—
115C23×C127—42no—
116C23×C127—42no—
117C23×C127—42no—
118C23×C127—84no—
119C23×C127—88no—
120C23×C127—88no—
121C23×C127—84no—
122C23×C127#10412yes—
123C23×C127C2×C22⋊C411yescharacteristic
124D4×C127—44no—
125D4×C127—44no—
126D4×C127—44no—
127D4×C127—44no—
128D4×C127—44no—
129D4×C127—88no—
130C2×C4×C127—21no—
131C2×C4×C127—21no—
132C2×C4×C127—22no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—42no—
137C2×C4×C127—42no—
138C2×C4×C127—42no—
139C2×C4×C127—41no—
140C2×C4×C127—41no—
141C2×C4×C127—41no—
142C2×C4×C127—42no—
143C2×C4×C127—42no—
144C2×C4×C127—42no—
145C2×C4×C127—81no—
146C2×C4×C127#9911yescharacteristic
147C2×C4×C127#9911yescharacteristic
148C2×C4×C127AGL(1,ℤ/8)12yes—
149C8×C127—44no—
150C8×C127—82no—
151C22×C127—21no—
152C22×C127—21no—
153C22×C127—22no—
154C22×C127—42no—
155C22×C127—42no—
156C22×C127—42no—
157C22×C127—44no—
158C22×C127—44no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127—41no—
162C22×C127—44no—
163C22×C127—41no—
164C22×C127—41no—
165C22×C127—81no—
166C22×C127—82no—
167C22×C127—82no—
168C22×C127—816no—
169C22×C127—82no—
170C22×C127—82no—
171C22×C127C2×#9911yescharacteristic
172C22×C127#41811yescharacteristic
173C22×C127C2×#10411yescharacteristic
174C4×C127—21no—
175C4×C127—22no—
176C4×C127—41no—
177C4×C127—41no—
178C4×C127—41no—
179C4×C127—82no—
180C4×C127AΣL(1,𝔽16)12yes—
181#11779C12711yesmaximal, characteristic, direct product factor, Sylow
182C2×C127—21no—
183C2×C127—21no—
184C2×C127—42no—
185C2×C127—81no—
186C2×C127—84no—
187C2×C127#150711yescharacteristic
188C2×C127C2×AΣL(1,𝔽16)11yescharacteristic
189C2×C127#213411yescharacteristic
190#2408C2×C12711yescharacteristic
191C4×#100C2×C12711yescharacteristic, complemented
192#1465C2×C12711yescharacteristic, complemented
193#1345C2×C12714yescomplemented
194C127#1177911yescharacteristic, direct product factor, Sylow
195C23×D4C4×C12711yescharacteristic
196C2×#121—21no—
197C2×#120—21no—
198C2×C4×D4C22×C12711yescharacteristic
199C2×#100—21no—
200#407—21no—
201C4×M16—21no—
202#394C22×C12712yes—
203#392C4×C12712yes—
204#390—21no—
205#380C4×C12711yescharacteristic
206#379—21no—
207#343C22×C12712yescomplemented
208#324C22×C12712yescomplemented
209#323—21no—
210C25—21no—
211C22×D4—22no—
212C22×D4—21no—
213C22×D4—41no—
214C22×D4—41no—
215C22×D4—44no—
216C22×D4C2×C4×C12712yes—
217C22×D4D4×C12711yescharacteristic
218C23×C4C2×C4×C12711yescharacteristic
219C23×C4D4×C12711yescharacteristic
220C2×M16—22no—
221C2×M16—41no—
222#121—44no—
223#120—44no—
224C4×D4—41no—
225C4×D4—41no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4C23×C12711yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4C2×C4×C12711yescharacteristic
233C2×C22⋊C4D4×C12711yescharacteristic
234C2×C42D4×C12711yescharacteristic
235#100—24no—
236C22⋊C8—22no—
237C22⋊C8—41no—
238#97—41no—
239C4×C8—41no—
240#95C2×C4×C12712yes—
241C24—22no—
242C24—42no—
243C24—44no—
244C24—42no—
245C24—42no—
246C24—82no—
247C24C22⋊C4×C12711yescharacteristic
248C24C22⋊C4×C12711yescharacteristic
249C2×D4—24no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—48no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—82no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4C22⋊C4×C12711yescharacteristic
272C22×C4C22⋊C4×C12711yescharacteristic
273C22×C4C22×C4×C12711yescharacteristic
274C22×C4C2×D4×C12711yescharacteristic
275C22×C4C2×D4×C12711yescharacteristic
276M16—24no—
277M16—42no—
278M16—82no—
279C2×C8—42no—
280C2×C8—42no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—42no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—82no—
289C42—21no—
290C42—21no—
291C23—21no—
292C23—21no—
293C23—24no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—48no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—84no—
309C23—88no—
310C23—88no—
311C23—84no—
312C23#104×C12712yes—
313C23C2×C22⋊C4×C12711yescharacteristic
314D4—44no—
315D4—44no—
316D4—44no—
317D4—44no—
318D4—44no—
319D4—88no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4#99×C12711yescharacteristic
337C2×C4#99×C12711yescharacteristic
338C2×C4AGL(1,ℤ/8)×C12712yes—
339C8—44no—
340C8—82no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—44no—
348C22—44no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—44no—
353C22—41no—
354C22—41no—
355C22—81no—
356C22—82no—
357C22—82no—
358C22—816no—
359C22—82no—
360C22—82no—
361C22C2×#99×C12711yescharacteristic
362C22#418×C12711yescharacteristic
363C22C2×#104×C12711yescharacteristic
364C4—21no—
365C4—22no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—82no—
370C4AΣL(1,𝔽16)×C12712yes—
371C2—21no—
372C2—21no—
373C2—42no—
374C2—81no—
375C2—84no—
376C2C127×#150711yescharacteristic
377C2C2×AΣL(1,𝔽16)×C12711yescharacteristic
378C2C127×#213411yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC127×#2408C2
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×#100×C127C2semidirect
3. C127×#1465C2semidirect
4. C127×#1345C2semidirect
17. #343×C127C22semidirect
18. #324×C127C22semidirect
181. #11779C127direct (N × H)
191. C4×#100C2×C127semidirect
192. #1465C2×C127semidirect
193. #1345C2×C127semidirect
194. C127#11779direct (N × H)
207. #343C22×C127semidirect
208. #324C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#24082
C4×#100×C1272
C127×#14652
C127×#13452
#11779127