Finite Group Databasebeta

← C127×#8529 #44572202

Subgroups — C127×#8529 #44572202

Order 32512.

648 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 142 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×#3553C211yesmaximal, characteristic
2C127×#1991C211yesmaximal, characteristic, complemented
3C127×#1534C211yesmaximal, characteristic, complemented
4C127×#1512C212yesmaximal, complemented
5C127×#1494C212yesmaximal, complemented
6#564×C127C411yescharacteristic, complemented
7#563×C127C412yescomplemented
8#562×C127C411yescharacteristic, complemented
9#554×C127—22no—
10#553×C127—22no—
11#552×C127—22no—
12#541×C127—22no—
13#539×C127—22no—
14#539×C127—22no—
15#529×C127C2212yes—
16C2×#125×C127C2211yescharacteristic
17#432×C127C2212yes—
18C2×C4×C8×C127C2211yescharacteristic
19#382×C127C2211yescharacteristic
20#340×C127—22no—
21#330×C127—22no—
22#328×C127—22no—
23#327×C127—22no—
24Q8⋊C8×C127—22no—
25D4⋊C8×C127—22no—
26C22×C8×C127—21no—
27C4⋊Q8×C127—22no—
28#126×C127—22no—
29#126×C127—22no—
30#126×C127—22no—
31#126×C127—22no—
32#126×C127—22no—
33#125×C127—22no—
34#125×C127—22no—
35#125×C127—22no—
36#125×C127—22no—
37#125×C127—22no—
38#125×C127—22no—
39#125×C127C2×C411yescharacteristic
40#125×C127C2×C411yescharacteristic
41#125×C127C2×C412yes—
42#125×C127D412yes—
43#124×C127C2×C412yes—
44#123×C127—22no—
45#123×C127—22no—
46#123×C127—22no—
47#123×C127—22no—
48#123×C127—22no—
49C22⋊Q8×C127—22no—
50C22⋊Q8×C127—22no—
51C22⋊Q8×C127—22no—
52C22⋊Q8×C127—22no—
53C22⋊Q8×C127—22no—
54C22⋊Q8×C127—22no—
55C22⋊Q8×C127D412yes—
56#121×C127—22no—
57#121×C127—22no—
58#121×C127D412yes—
59C4×Q8×C127—22no—
60C4×Q8×C127D412yes—
61C4×D4×C127—22no—
62C4×D4×C127—22no—
63C4×D4×C127—22no—
64C4×D4×C127—22no—
65C4×D4×C127D412yes—
66#117×C127—22no—
67#117×C127—22no—
68C2×C4⋊C4×C127—22no—
69C2×C4⋊C4×C127—22no—
70C2×C4⋊C4×C127—21no—
71C2×C4⋊C4×C127D412yes—
72C2×C42×C127C2311yescharacteristic
73C4⋊C8×C127—22no—
74C22⋊C8×C127—22no—
75#97×C127—22no—
76C4×C8×C127—22no—
77#95×C127—22no—
78C2×Q8×C127—22no—
79C2×Q8×C127C22⋊C412yes—
80C2×D4×C127—22no—
81C2×D4×C127—22no—
82C2×D4×C127C22⋊C412yes—
83C22×C4×C127—22no—
84C22×C4×C127—22no—
85C22×C4×C127—22no—
86C22×C4×C127—22no—
87C22×C4×C127—41no—
88C22×C4×C127C2×D411yescharacteristic
89C22×C4×C127C2×D412yes—
90C2×C8×C127—22no—
91C2×C8×C127—41no—
92C2×C8×C127—41no—
93C2×C8×C127—42no—
94C4⋊C4×C127—22no—
95C4⋊C4×C127—22no—
96C4⋊C4×C127—22no—
97C4⋊C4×C127—22no—
98C4⋊C4×C127—22no—
99C4⋊C4×C127—22no—
100C4⋊C4×C127—22no—
101C4⋊C4×C127—22no—
102C4⋊C4×C127—22no—
103C4⋊C4×C127—22no—
104C4⋊C4×C127—22no—
105C4⋊C4×C127—22no—
106C4⋊C4×C127—22no—
107C4⋊C4×C127—22no—
108C4⋊C4×C127—22no—
109C4⋊C4×C127—42no—
110C4⋊C4×C127C22⋊C412yes—
111C4⋊C4×C127C22⋊C412yes—
112C4⋊C4×C127C22⋊C412yes—
113C4⋊C4×C127C22⋊C412yes—
114C22⋊C4×C127—22no—
115C22⋊C4×C127—22no—
116C22⋊C4×C127—22no—
117C22⋊C4×C127—22no—
118C22⋊C4×C127—22no—
119C22⋊C4×C127—22no—
120C22⋊C4×C127—22no—
121C22⋊C4×C127—22no—
122C22⋊C4×C127—22no—
123C42×C127—22no—
124C42×C127—22no—
125C42×C127—22no—
126C42×C127C22×C411yescharacteristic
127C42×C127C2×D412yes—
128C42×C127C2×D411yescharacteristic
129C23×C127—22no—
130C23×C127#12011yescharacteristic
131Q8×C127—42no—
132Q8×C127—82no—
133D4×C127—42no—
134D4×C127—42no—
135D4×C127—42no—
136D4×C127—82no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—22no—
140C2×C4×C127—22no—
141C2×C4×C127—22no—
142C2×C4×C127—22no—
143C2×C4×C127—22no—
144C2×C4×C127—42no—
145C2×C4×C127—82no—
146C2×C4×C127—82no—
147C2×C4×C127—82no—
148C2×C4×C127—82no—
149C2×C4×C127—82no—
150C2×C4×C127—81no—
151C2×C4×C127C2×C22⋊C411yescharacteristic
152C2×C4×C127C2×C22⋊C412yes—
153C2×C4×C127#12012yes—
154C2×C4×C127#12011yescharacteristic
155C8×C127—42no—
156C8×C127—82no—
157C22×C127—22no—
158C22×C127—42no—
159C22×C127—42no—
160C22×C127—42no—
161C22×C127#37911yescharacteristic
162C22×C127#45411yescharacteristic
163C22×C127#45611yescharacteristic
164C4×C127—21no—
165C4×C127—21no—
166C4×C127—22no—
167C4×C127—22no—
168C4×C127—42no—
169C4×C127—42no—
170C4×C127—42no—
171C4×C127—42no—
172C4×C127—42no—
173C4×C127—161no—
174#8529C12711yesmaximal, characteristic, direct product factor, Sylow
175C2×C127—21no—
176C2×C127—21no—
177C2×C127—82no—
178C2×C127#179311yescharacteristic
179C2×C127#179412yes—
180#3553C2×C12711yescharacteristic
181#1991C2×C12711yescharacteristic, complemented
182#1534C2×C12711yescharacteristic, complemented
183#1512C2×C12712yescomplemented
184#1494C2×C12712yescomplemented
185C127#852911yescharacteristic, direct product factor, Sylow
186#564C4×C12711yescharacteristic, complemented
187#563C4×C12712yescomplemented
188#562C4×C12711yescharacteristic, complemented
189#554—22no—
190#553—22no—
191#552—22no—
192#541—22no—
193#539—22no—
194#539—22no—
195#529C22×C12712yes—
196C2×#125C22×C12711yescharacteristic
197#432C22×C12712yes—
198C2×C4×C8C22×C12711yescharacteristic
199#382C22×C12711yescharacteristic
200#340—22no—
201#330—22no—
202#328—22no—
203#327—22no—
204Q8⋊C8—22no—
205D4⋊C8—22no—
206C22×C8—21no—
207C4⋊Q8—22no—
208#126—22no—
209#126—22no—
210#126—22no—
211#126—22no—
212#126—22no—
213#125—22no—
214#125—22no—
215#125—22no—
216#125—22no—
217#125—22no—
218#125—22no—
219#125C2×C4×C12711yescharacteristic
220#125C2×C4×C12711yescharacteristic
221#125C2×C4×C12712yes—
222#125D4×C12712yes—
223#124C2×C4×C12712yes—
224#123—22no—
225#123—22no—
226#123—22no—
227#123—22no—
228#123—22no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C22⋊Q8D4×C12712yes—
236#121—22no—
237#121—22no—
238#121D4×C12712yes—
239C4×Q8—22no—
240C4×Q8D4×C12712yes—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4D4×C12712yes—
246#117—22no—
247#117—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4D4×C12712yes—
252C2×C42C23×C12711yescharacteristic
253C4⋊C8—22no—
254C22⋊C8—22no—
255#97—22no—
256C4×C8—22no—
257#95—22no—
258C2×Q8—22no—
259C2×Q8C22⋊C4×C12712yes—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4C22⋊C4×C12712yes—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4C2×D4×C12711yescharacteristic
269C22×C4C2×D4×C12712yes—
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—42no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—42no—
290C4⋊C4C22⋊C4×C12712yes—
291C4⋊C4C22⋊C4×C12712yes—
292C4⋊C4C22⋊C4×C12712yes—
293C4⋊C4C22⋊C4×C12712yes—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42C22×C4×C12711yescharacteristic
307C42C2×D4×C12712yes—
308C42C2×D4×C12711yescharacteristic
309C23—22no—
310C23#120×C12711yescharacteristic
311Q8—42no—
312Q8—82no—
313D4—42no—
314D4—42no—
315D4—42no—
316D4—82no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—42no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—81no—
331C2×C4C2×C22⋊C4×C12711yescharacteristic
332C2×C4C2×C22⋊C4×C12712yes—
333C2×C4#120×C12712yes—
334C2×C4#120×C12711yescharacteristic
335C8—42no—
336C8—82no—
337C22—22no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22#379×C12711yescharacteristic
342C22#454×C12711yescharacteristic
343C22#456×C12711yescharacteristic
344C4—21no—
345C4—21no—
346C4—22no—
347C4—22no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4—42no—
353C4—161no—
354C2—21no—
355C2—21no—
356C2—82no—
357C2C127×#179311yescharacteristic
358C2C127×#179412yes—
subgroupG/H
center, socleC22×C127#379
commutatorC42C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC127×#3553C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1991C2semidirect
3. C127×#1534C2semidirect
4. C127×#1512C2semidirect
5. C127×#1494C2semidirect
6. #564×C127C4semidirect
7. #563×C127C4semidirect
8. #562×C127C4semidirect
174. #8529C127direct (N × H)
181. #1991C2×C127semidirect
182. #1534C2×C127semidirect
183. #1512C2×C127semidirect
184. #1494C2×C127semidirect
185. C127#8529direct (N × H)
186. #564C4×C127semidirect
187. #563C4×C127semidirect
188. #562C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#35532
C127×#19912
C127×#15342
C127×#15122
C127×#14942
#8529127