Finite Group Databasebeta

← C127×#11774 #44575447

Subgroups — C127×#11774 #44575447

Order 32512.

544 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2535C211yesmaximal, characteristic
2C4×#99×C127C211yesmaximal, characteristic, complemented
3C127×#1465C211yesmaximal, characteristic, complemented
4C127×#1344C212yesmaximal, complemented
5C127×#1343C212yesmaximal, complemented
6C2×#121×C127—21no—
7C2×C4×D4×C127C411yescharacteristic, complemented
8C2×C4×D4×C127C2211yescharacteristic
9C2×#99×C127—21no—
10#407×C127—21no—
11#397×C127C411yescharacteristic, complemented
12#397×C127C2211yescharacteristic
13#392×C127C411yescharacteristic, complemented
14#392×C127C2211yescharacteristic
15#386×C127—21no—
16#386×C127—21no—
17#382×C127C411yescharacteristic, complemented
18#380×C127—21no—
19C4×C22⋊C4×C127—21no—
20#342×C127C2212yes—
21#324×C127—21no—
22#323×C127C2212yes—
23C22×D4×C127—21no—
24C22×D4×C127—21no—
25C22×D4×C127C2×C411yescharacteristic
26C22×D4×C127D411yescharacteristic
27C23×C4×C127—21no—
28C23×C4×C127D411yescharacteristic
29C23×C4×C127D411yescharacteristic
30C2×M16×C127—41no—
31#121×C127—44no—
32C4×D4×C127—41no—
33C4×D4×C127—41no—
34C4×D4×C127—41no—
35C4×D4×C127—41no—
36C2×C4⋊C4×C127—21no—
37C2×C4⋊C4×C127C2×C411yescharacteristic
38C2×C4⋊C4×C127D411yescharacteristic
39C2×C22⋊C4×C127—22no—
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×C127C2×C411yescharacteristic
46C2×C22⋊C4×C127C2311yescharacteristic
47C2×C42×C127—41no—
48C2×C42×C127C2×C411yescharacteristic
49#99×C127—24no—
50C22⋊C8×C127—22no—
51C22⋊C8×C127—41no—
52#95×C127—21no—
53#95×C127—21no—
54#95×C127—21no—
55#95×C127—41no—
56#95×C127C2×C411yescharacteristic
57#95×C127C2×C411yescharacteristic
58C24×C127—21no—
59C24×C127C22⋊C411yescharacteristic
60C24×C127C2×D411yescharacteristic
61C2×D4×C127—22no—
62C2×D4×C127—22no—
63C2×D4×C127—41no—
64C2×D4×C127—41no—
65C2×D4×C127—42no—
66C2×D4×C127—42no—
67C2×D4×C127—42no—
68C2×D4×C127—42no—
69C2×D4×C127—42no—
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—42no—
80C22×C4×C127—42no—
81C22×C4×C127—42no—
82C22×C4×C127—42no—
83C22×C4×C127—81no—
84C22×C4×C127—81no—
85C22×C4×C127C22⋊C411yescharacteristic
86C22×C4×C127C22⋊C411yescharacteristic
87C22×C4×C127C22⋊C411yescharacteristic
88C22×C4×C127C22×C411yescharacteristic
89C22×C4×C127C2×D411yescharacteristic
90M16×C127—82no—
91C2×C8×C127—42no—
92C4⋊C4×C127—41no—
93C4⋊C4×C127—41no—
94C4⋊C4×C127—42no—
95C22⋊C4×C127—24no—
96C22⋊C4×C127—41no—
97C22⋊C4×C127—41no—
98C22⋊C4×C127—42no—
99C22⋊C4×C127—42no—
100C22⋊C4×C127—42no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127—41no—
103C22⋊C4×C127—81no—
104C22⋊C4×C127—81no—
105C42×C127—21no—
106C42×C127—21no—
107C42×C127—82no—
108C23×C127—21no—
109C23×C127—21no—
110C23×C127—22no—
111C23×C127—21no—
112C23×C127—21no—
113C23×C127—21no—
114C23×C127—41no—
115C23×C127—41no—
116C23×C127—41no—
117C23×C127—41no—
118C23×C127—41no—
119C23×C127—41no—
120C23×C127—41no—
121C23×C127—41no—
122C23×C127—41no—
123C23×C127—41no—
124C23×C127C2×C22⋊C411yescharacteristic
125C23×C127AGL(1,ℤ/8)12yes—
126D4×C127—44no—
127D4×C127—42no—
128D4×C127—42no—
129D4×C127—84no—
130D4×C127—84no—
131C2×C4×C127—21no—
132C2×C4×C127—21no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—41no—
137C2×C4×C127—42no—
138C2×C4×C127—42no—
139C2×C4×C127—41no—
140C2×C4×C127—41no—
141C2×C4×C127—41no—
142C2×C4×C127—44no—
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—41no—
152C2×C4×C127—81no—
153C2×C4×C127—81no—
154C2×C4×C127—84no—
155C2×C4×C127—82no—
156C2×C4×C127—82no—
157C2×C4×C127#9911yescharacteristic
158C2×C4×C127#9911yescharacteristic
159C2×C4×C127#10412yes—
160C8×C127—82no—
161C22×C127—21no—
162C22×C127—21no—
163C22×C127—22no—
164C22×C127—42no—
165C22×C127—42no—
166C22×C127—42no—
167C22×C127—42no—
168C22×C127—41no—
169C22×C127—41no—
170C22×C127—42no—
171C22×C127—41no—
172C22×C127—41no—
173C22×C127—41no—
174C22×C127—81no—
175C22×C127—81no—
176C22×C127—81no—
177C22×C127—81no—
178C22×C127—81no—
179C22×C127—81no—
180C22×C127C2×#9911yescharacteristic
181C22×C127#41811yescharacteristic
182C22×C127C2×#10411yescharacteristic
183C4×C127—21no—
184C4×C127—21no—
185C4×C127—22no—
186C4×C127—41no—
187C4×C127—41no—
188C4×C127—41no—
189C4×C127—81no—
190C4×C127—81no—
191C4×C127—84no—
192C4×C127—81no—
193#11774C12711yesmaximal, characteristic, direct product factor, Sylow
194C2×C127—21no—
195C2×C127—21no—
196C2×C127—42no—
197C2×C127—81no—
198C2×C127—82no—
199C2×C127#150711yescharacteristic
200C2×C127#212011yescharacteristic
201C2×C127#212711yescharacteristic
202#2535C2×C12711yescharacteristic
203C4×#99C2×C12711yescharacteristic, complemented
204#1465C2×C12711yescharacteristic, complemented
205#1344C2×C12712yescomplemented
206#1343C2×C12712yescomplemented
207C127#1177411yescharacteristic, direct product factor, Sylow
208C2×#121—21no—
209C2×C4×D4C4×C12711yescharacteristic, complemented
210C2×C4×D4C22×C12711yescharacteristic
211C2×#99—21no—
212#407—21no—
213#397C4×C12711yescharacteristic, complemented
214#397C22×C12711yescharacteristic
215#392C4×C12711yescharacteristic, complemented
216#392C22×C12711yescharacteristic
217#386—21no—
218#386—21no—
219#382C4×C12711yescharacteristic, complemented
220#380—21no—
221C4×C22⋊C4—21no—
222#342C22×C12712yes—
223#324—21no—
224#323C22×C12712yes—
225C22×D4—21no—
226C22×D4—21no—
227C22×D4C2×C4×C12711yescharacteristic
228C22×D4D4×C12711yescharacteristic
229C23×C4—21no—
230C23×C4D4×C12711yescharacteristic
231C23×C4D4×C12711yescharacteristic
232C2×M16—41no—
233#121—44no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—41no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C2×C4×C12711yescharacteristic
240C2×C4⋊C4D4×C12711yescharacteristic
241C2×C22⋊C4—22no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4C2×C4×C12711yescharacteristic
248C2×C22⋊C4C23×C12711yescharacteristic
249C2×C42—41no—
250C2×C42C2×C4×C12711yescharacteristic
251#99—24no—
252C22⋊C8—22no—
253C22⋊C8—41no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—41no—
258#95C2×C4×C12711yescharacteristic
259#95C2×C4×C12711yescharacteristic
260C24—21no—
261C24C22⋊C4×C12711yescharacteristic
262C24C2×D4×C12711yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—81no—
273C2×D4—81no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4C22⋊C4×C12711yescharacteristic
288C22×C4C22⋊C4×C12711yescharacteristic
289C22×C4C22⋊C4×C12711yescharacteristic
290C22×C4C22×C4×C12711yescharacteristic
291C22×C4C2×D4×C12711yescharacteristic
292M16—82no—
293C2×C8—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C22⋊C4—24no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—81no—
306C22⋊C4—81no—
307C42—21no—
308C42—21no—
309C42—82no—
310C23—21no—
311C23—21no—
312C23—22no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23C2×C22⋊C4×C12711yescharacteristic
327C23AGL(1,ℤ/8)×C12712yes—
328D4—44no—
329D4—42no—
330D4—42no—
331D4—84no—
332D4—84no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—44no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—84no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4#99×C12711yescharacteristic
360C2×C4#99×C12711yescharacteristic
361C2×C4#104×C12712yes—
362C8—82no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22C2×#99×C12711yescharacteristic
383C22#418×C12711yescharacteristic
384C22C2×#104×C12711yescharacteristic
385C4—21no—
386C4—21no—
387C4—22no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—81no—
392C4—81no—
393C4—84no—
394C4—81no—
395C2—21no—
396C2—21no—
397C2—42no—
398C2—81no—
399C2—82no—
400C2C127×#150711yescharacteristic
401C2C127×#212011yescharacteristic
402C2C127×#212711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC127×#2535C2
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×#1344C2semidirect
5. C127×#1343C2semidirect
7. C2×C4×D4×C127C4semidirect
11. #397×C127C4semidirect
13. #392×C127C4semidirect
17. #382×C127C4semidirect
193. #11774C127direct (N × H)
203. C4×#99C2×C127semidirect
204. #1465C2×C127semidirect
205. #1344C2×C127semidirect
206. #1343C2×C127semidirect
207. C127#11774direct (N × H)
209. C2×C4×D4C4×C127semidirect
213. #397C4×C127semidirect
215. #392C4×C127semidirect
219. #382C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#25352
C4×#99×C1272
C127×#14652
C127×#13442
C127×#13432
#11774127