Finite Group Databasebeta

← C127×#11726 #44575399

Subgroups — C127×#11726 #44575399

Order 32512.

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

normal subgroups only

This group has 606 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×#2397C211yesmaximal, characteristic
2C4×#99×C127C211yesmaximal, characteristic, complemented
3C127×#1459C211yesmaximal, characteristic, complemented
4C127×#1352C212yesmaximal, complemented
5C127×#1343C212yesmaximal, complemented
6C2×C4×D4×C127—21no—
7C2×C4×D4×C127—21no—
8C2×C4×D4×C127C411yescharacteristic, complemented
9C2×C4×D4×C127C2211yescharacteristic
10C2×#99×C127—21no—
11C2×C22⋊C8×C127—21no—
12#399×C127—21no—
13#399×C127C411yescharacteristic, complemented
14#399×C127C2211yescharacteristic
15#392×C127—21no—
16#392×C127C411yescharacteristic, complemented
17#392×C127C2211yescharacteristic
18C4×C22⋊C4×C127—21no—
19#376×C127C411yescharacteristic, complemented
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
30C22×C8×C127—41no—
31C4×D4×C127—41no—
32C4×D4×C127—41no—
33C4×D4×C127—44no—
34C4×D4×C127—44no—
35C4×D4×C127—41no—
36C4×D4×C127—41no—
37C2×C4⋊C4×C127—21no—
38C2×C4⋊C4×C127—21no—
39C2×C4⋊C4×C127C2×C411yescharacteristic
40C2×C4⋊C4×C127D411yescharacteristic
41C2×C22⋊C4×C127—22no—
42C2×C22⋊C4×C127—21no—
43C2×C22⋊C4×C127—21no—
44C2×C22⋊C4×C127—21no—
45C2×C22⋊C4×C127—21no—
46C2×C22⋊C4×C127—21no—
47C2×C22⋊C4×C127C2×C411yescharacteristic
48C2×C22⋊C4×C127C2311yescharacteristic
49C2×C42×C127—21no—
50C2×C42×C127—41no—
51C2×C42×C127C2×C411yescharacteristic
52#99×C127—24no—
53C22⋊C8×C127—22no—
54C22⋊C8×C127—41no—
55#95×C127—21no—
56#95×C127—41no—
57#95×C127C2×C411yescharacteristic
58#95×C127C2×C411yescharacteristic
59C24×C127—21no—
60C24×C127C22⋊C411yescharacteristic
61C24×C127C2×D411yescharacteristic
62C2×D4×C127—22no—
63C2×D4×C127—22no—
64C2×D4×C127—41no—
65C2×D4×C127—41no—
66C2×D4×C127—42no—
67C2×D4×C127—42no—
68C2×D4×C127—42no—
69C2×D4×C127—42no—
70C2×D4×C127—42no—
71C2×D4×C127—42no—
72C2×D4×C127—42no—
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—21no—
80C22×C4×C127—42no—
81C22×C4×C127—42no—
82C22×C4×C127—42no—
83C22×C4×C127—42no—
84C22×C4×C127—42no—
85C22×C4×C127—41no—
86C22×C4×C127—41no—
87C22×C4×C127C22⋊C411yescharacteristic
88C22×C4×C127C22⋊C411yescharacteristic
89C22×C4×C127C22⋊C411yescharacteristic
90C22×C4×C127C22×C411yescharacteristic
91C22×C4×C127C2×D411yescharacteristic
92C2×C8×C127—42no—
93C2×C8×C127—82no—
94C4⋊C4×C127—41no—
95C4⋊C4×C127—41no—
96C4⋊C4×C127—42no—
97C4⋊C4×C127—42no—
98C22⋊C4×C127—24no—
99C22⋊C4×C127—41no—
100C22⋊C4×C127—41no—
101C22⋊C4×C127—42no—
102C22⋊C4×C127—42no—
103C22⋊C4×C127—42no—
104C22⋊C4×C127—42no—
105C22⋊C4×C127—42no—
106C22⋊C4×C127—42no—
107C42×C127—24no—
108C42×C127—21no—
109C42×C127—21no—
110C42×C127—82no—
111C23×C127—21no—
112C23×C127—21no—
113C23×C127—21no—
114C23×C127—21no—
115C23×C127—22no—
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×C127—41no—
125C23×C127—41no—
126C23×C127#10412yes—
127C23×C127C2×C22⋊C411yescharacteristic
128C23×C127#13512yes—
129D4×C127—44no—
130D4×C127—42no—
131D4×C127—48no—
132D4×C127—48no—
133D4×C127—42no—
134C2×C4×C127—22no—
135C2×C4×C127—22no—
136C2×C4×C127—22no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
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—42no—
148C2×C4×C127—42no—
149C2×C4×C127—41no—
150C2×C4×C127—41no—
151C2×C4×C127—41no—
152C2×C4×C127—44no—
153C2×C4×C127—41no—
154C2×C4×C127—41no—
155C2×C4×C127—41no—
156C2×C4×C127—81no—
157C2×C4×C127—81no—
158C2×C4×C127—82no—
159C2×C4×C127—82no—
160C2×C4×C127—82no—
161C2×C4×C127—82no—
162C2×C4×C127#9911yescharacteristic
163C2×C4×C127#9911yescharacteristic
164C8×C127—82no—
165C22×C127—21no—
166C22×C127—21no—
167C22×C127—22no—
168C22×C127—22no—
169C22×C127—42no—
170C22×C127—42no—
171C22×C127—42no—
172C22×C127—42no—
173C22×C127—41no—
174C22×C127—41no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127—42no—
178C22×C127—81no—
179C22×C127—81no—
180C22×C127—81no—
181C22×C127—81no—
182C22×C127—81no—
183C22×C127—81no—
184C22×C127C2×#9911yescharacteristic
185C22×C127#41611yescharacteristic
186C22×C127C2×#10411yescharacteristic
187C4×C127—21no—
188C4×C127—42no—
189C4×C127—42no—
190C4×C127—41no—
191C4×C127—41no—
192C4×C127—41no—
193C4×C127—41no—
194C4×C127—84no—
195C4×C127—81no—
196C4×C127#35312yes—
197#11726C12711yesmaximal, characteristic, direct product factor, Sylow
198C2×C127—21no—
199C2×C127—21no—
200C2×C127—42no—
201C2×C127—42no—
202C2×C127—82no—
203C2×C127#150111yescharacteristic
204C2×C127#212011yescharacteristic
205C2×C127C2×#35311yescharacteristic
206#2397C2×C12711yescharacteristic
207C4×#99C2×C12711yescharacteristic, complemented
208#1459C2×C12711yescharacteristic, complemented
209#1352C2×C12712yescomplemented
210#1343C2×C12712yescomplemented
211C127#1172611yescharacteristic, direct product factor, Sylow
212C2×C4×D4—21no—
213C2×C4×D4—21no—
214C2×C4×D4C4×C12711yescharacteristic, complemented
215C2×C4×D4C22×C12711yescharacteristic
216C2×#99—21no—
217C2×C22⋊C8—21no—
218#399—21no—
219#399C4×C12711yescharacteristic, complemented
220#399C22×C12711yescharacteristic
221#392—21no—
222#392C4×C12711yescharacteristic, complemented
223#392C22×C12711yescharacteristic
224C4×C22⋊C4—21no—
225#376C4×C12711yescharacteristic, complemented
226#342C22×C12712yes—
227#324—21no—
228#323C22×C12712yes—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2×C4×C12711yescharacteristic
232C22×D4D4×C12711yescharacteristic
233C23×C4—21no—
234C23×C4D4×C12711yescharacteristic
235C23×C4D4×C12711yescharacteristic
236C22×C8—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—44no—
240C4×D4—44no—
241C4×D4—41no—
242C4×D4—41no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4C2×C4×C12711yescharacteristic
246C2×C4⋊C4D4×C12711yescharacteristic
247C2×C22⋊C4—22no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4C2×C4×C12711yescharacteristic
254C2×C22⋊C4C23×C12711yescharacteristic
255C2×C42—21no—
256C2×C42—41no—
257C2×C42C2×C4×C12711yescharacteristic
258#99—24no—
259C22⋊C8—22no—
260C22⋊C8—41no—
261#95—21no—
262#95—41no—
263#95C2×C4×C12711yescharacteristic
264#95C2×C4×C12711yescharacteristic
265C24—21no—
266C24C22⋊C4×C12711yescharacteristic
267C24C2×D4×C12711yescharacteristic
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4C22⋊C4×C12711yescharacteristic
294C22×C4C22⋊C4×C12711yescharacteristic
295C22×C4C22⋊C4×C12711yescharacteristic
296C22×C4C22×C4×C12711yescharacteristic
297C22×C4C2×D4×C12711yescharacteristic
298C2×C8—42no—
299C2×C8—82no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C22⋊C4—24no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C42—24no—
314C42—21no—
315C42—21no—
316C42—82no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23#104×C12712yes—
333C23C2×C22⋊C4×C12711yescharacteristic
334C23#135×C12712yes—
335D4—44no—
336D4—42no—
337D4—48no—
338D4—48no—
339D4—42no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—44no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4#99×C12711yescharacteristic
369C2×C4#99×C12711yescharacteristic
370C8—82no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—22no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—42no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22C2×#99×C12711yescharacteristic
391C22#416×C12711yescharacteristic
392C22C2×#104×C12711yescharacteristic
393C4—21no—
394C4—42no—
395C4—42no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—84no—
401C4—81no—
402C4#353×C12712yes—
403C2—21no—
404C2—21no—
405C2—42no—
406C2—42no—
407C2—82no—
408C2C127×#150111yescharacteristic
409C2C127×#212011yescharacteristic
410C2C2×#353×C12711yescharacteristic
subgroupG/H
centerC2×C4×C127#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C22⋊C4C23×C127
socleC22×C127C2×#99
prime radicalC127×#2397C2
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
2. C4×#99×C127C2semidirect
3. C127×#1459C2semidirect
4. C127×#1352C2semidirect
5. C127×#1343C2semidirect
8. C2×C4×D4×C127C4semidirect
13. #399×C127C4semidirect
16. #392×C127C4semidirect
19. #376×C127C4semidirect
197. #11726C127direct (N × H)
207. C4×#99C2×C127semidirect
208. #1459C2×C127semidirect
209. #1352C2×C127semidirect
210. #1343C2×C127semidirect
211. C127#11726direct (N × H)
214. C2×C4×D4C4×C127semidirect
219. #399C4×C127semidirect
222. #392C4×C127semidirect
225. #376C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#23972
C4×#99×C1272
C127×#14592
C127×#13522
C127×#13432
#11726127