Finite Group Databasebeta

← C127×#8497 #44572170

Subgroups — C127×#8497 #44572170

Order 32512.

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

normal subgroups only

This group has 820 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×#3543C211yesmaximal, characteristic
2C127×#1961C211yesmaximal, characteristic, complemented
3C127×#1555C211yesmaximal, characteristic, complemented
4C127×#1532C212yesmaximal, complemented
5C127×#1512C212yesmaximal, complemented
6#560×C127—22no—
7#559×C127C411yescharacteristic, complemented
8#550×C127C412yescomplemented
9#546×C127—22no—
10D42×C127—22no—
11D42×C127C411yescharacteristic, complemented
12#540×C127—22no—
13#535×C127—22no—
14#534×C127—22no—
15C2×#127×C127C2211yescharacteristic
16#529×C127C2212yes—
17#432×C127C2212yescomplemented
18C2×#102×C127—22no—
19C2×#97×C127C2211yescharacteristic, complemented
20#382×C127C2211yescharacteristic, complemented
21#331×C127—22no—
22#330×C127—22no—
23#329×C127—22no—
24#328×C127—22no—
25#327×C127—22no—
26C2×#41×C127—22no—
27C22×D4×C127—22no—
28C22×D4×C127—22no—
29C22×D4×C127—22no—
30C22×D4×C127D412yes—
31C22×C8×C127—21no—
32#127×C127—22no—
33#127×C127—22no—
34#127×C127C2×C411yescharacteristic
35#127×C127C2×C411yescharacteristic
36#127×C127D412yes—
37#126×C127—22no—
38#125×C127D412yes—
39#124×C127—22no—
40#124×C127D412yes—
41#123×C127—22no—
42#123×C127—24no—
43#123×C127—22no—
44#123×C127—22no—
45C22⋊Q8×C127D412yes—
46#121×C127—24no—
47#121×C127—24no—
48#121×C127—24no—
49#121×C127—24no—
50#121×C127—22no—
51#121×C127—22no—
52#121×C127D412yes—
53#120×C127—22no—
54#120×C127—24no—
55#120×C127—24no—
56#120×C127—22no—
57C4×Q8×C127C2×C412yes—
58C4×D4×C127—22no—
59C4×D4×C127—22no—
60C4×D4×C127—22no—
61C4×D4×C127C2×C412yes—
62#117×C127—22no—
63C2×C4⋊C4×C127—21no—
64C2×C22⋊C4×C127—22no—
65C2×C42×C127C2311yescharacteristic
66C4⋊C8×C127—22no—
67#102×C127—44no—
68C22⋊C8×C127—22no—
69#97×C127—22no—
70#97×C127—22no—
71#95×C127—22no—
72C24×C127—22no—
73C24×C127—22no—
74#41×C127—84no—
75C2×Q8×C127C22⋊C412yes—
76C2×D4×C127—22no—
77C2×D4×C127—24no—
78C2×D4×C127—24no—
79C2×D4×C127—24no—
80C2×D4×C127—22no—
81C2×D4×C127—22no—
82C2×D4×C127—22no—
83C2×D4×C127—24no—
84C2×D4×C127—42no—
85C2×D4×C127—42no—
86C2×D4×C127—84no—
87C2×D4×C127—84no—
88C2×D4×C127—82no—
89C2×D4×C127—82no—
90C2×D4×C127C22⋊C412yes—
91C2×D4×C127C22⋊C412yes—
92C2×D4×C127C22⋊C412yes—
93C22×C4×C127—24no—
94C22×C4×C127—22no—
95C22×C4×C127—22no—
96C22×C4×C127—41no—
97C22×C4×C127C2×D411yescharacteristic
98C22×C4×C127C2×D412yes—
99C2×C8×C127—22no—
100C2×C8×C127—41no—
101C2×C8×C127—41no—
102C2×C8×C127—42no—
103C4⋊C4×C127—22no—
104C4⋊C4×C127—22no—
105C4⋊C4×C127—22no—
106C4⋊C4×C127—22no—
107C4⋊C4×C127—42no—
108C4⋊C4×C127C22⋊C412yes—
109C4⋊C4×C127C22⋊C412yes—
110C22⋊C4×C127—22no—
111C22⋊C4×C127—22no—
112C22⋊C4×C127—24no—
113C22⋊C4×C127—24no—
114C22⋊C4×C127—22no—
115C22⋊C4×C127—22no—
116C22⋊C4×C127—22no—
117C22⋊C4×C127—22no—
118C22⋊C4×C127—22no—
119C42×C127—22no—
120C42×C127C22×C411yescharacteristic
121C42×C127C2×D412yes—
122C42×C127C2×D411yescharacteristic
123C23×C127—22no—
124C23×C127—22no—
125C23×C127—22no—
126C23×C127—42no—
127C23×C127—42no—
128C23×C127—84no—
129C23×C127—82no—
130C23×C127—84no—
131C23×C127#12011yescharacteristic
132Q8×C127—42no—
133D4×C127—42no—
134D4×C127—42no—
135D4×C127—42no—
136D4×C127—44no—
137D4×C127—44no—
138D4×C127—42no—
139D4×C127—42no—
140D4×C127—82no—
141D4×C127—84no—
142D4×C127—84no—
143D4×C127—84no—
144D4×C127—84no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—22no—
148C2×C4×C127—22no—
149C2×C4×C127—22no—
150C2×C4×C127—42no—
151C2×C4×C127—44no—
152C2×C4×C127—82no—
153C2×C4×C127—84no—
154C2×C4×C127—82no—
155C2×C4×C127—81no—
156C2×C4×C127C2×C22⋊C412yes—
157C2×C4×C127C2×C22⋊C411yescharacteristic
158C2×C4×C127#12011yescharacteristic
159C2×C4×C127#12012yes—
160C8×C127—42no—
161C8×C127—82no—
162C22×C127—22no—
163C22×C127—42no—
164C22×C127—42no—
165C22×C127—42no—
166C22×C127—42no—
167C22×C127—44no—
168C22×C127—44no—
169C22×C127—42no—
170C22×C127—162no—
171C22×C127—162no—
172C22×C127—164no—
173C22×C127#37911yescharacteristic
174C22×C127#45311yescharacteristic
175C22×C127#45511yescharacteristic
176C4×C127—21no—
177C4×C127—21no—
178C4×C127—22no—
179C4×C127—22no—
180C4×C127—42no—
181C4×C127—42no—
182C4×C127—42no—
183C4×C127—161no—
184#8497C12711yesmaximal, characteristic, direct product factor, Sylow
185C2×C127—21no—
186C2×C127—21no—
187C2×C127—82no—
188C2×C127—82no—
189C2×C127—82no—
190C2×C127#179211yescharacteristic
191C2×C127#179512yes—
192#3543C2×C12711yescharacteristic
193#1961C2×C12711yescharacteristic, complemented
194#1555C2×C12711yescharacteristic, complemented
195#1532C2×C12712yescomplemented
196#1512C2×C12712yescomplemented
197C127#849711yescharacteristic, direct product factor, Sylow
198#560—22no—
199#559C4×C12711yescharacteristic, complemented
200#550C4×C12712yescomplemented
201#546—22no—
202D42—22no—
203D42C4×C12711yescharacteristic, complemented
204#540—22no—
205#535—22no—
206#534—22no—
207C2×#127C22×C12711yescharacteristic
208#529C22×C12712yes—
209#432C22×C12712yescomplemented
210C2×#102—22no—
211C2×#97C22×C12711yescharacteristic, complemented
212#382C22×C12711yescharacteristic, complemented
213#331—22no—
214#330—22no—
215#329—22no—
216#328—22no—
217#327—22no—
218C2×#41—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4D4×C12712yes—
223C22×C8—21no—
224#127—22no—
225#127—22no—
226#127C2×C4×C12711yescharacteristic
227#127C2×C4×C12711yescharacteristic
228#127D4×C12712yes—
229#126—22no—
230#125D4×C12712yes—
231#124—22no—
232#124D4×C12712yes—
233#123—22no—
234#123—24no—
235#123—22no—
236#123—22no—
237C22⋊Q8D4×C12712yes—
238#121—24no—
239#121—24no—
240#121—24no—
241#121—24no—
242#121—22no—
243#121—22no—
244#121D4×C12712yes—
245#120—22no—
246#120—24no—
247#120—24no—
248#120—22no—
249C4×Q8C2×C4×C12712yes—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4C2×C4×C12712yes—
254#117—22no—
255C2×C4⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C42C23×C12711yescharacteristic
258C4⋊C8—22no—
259#102—44no—
260C22⋊C8—22no—
261#97—22no—
262#97—22no—
263#95—22no—
264C24—22no—
265C24—22no—
266#41—84no—
267C2×Q8C22⋊C4×C12712yes—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—84no—
279C2×D4—84no—
280C2×D4—82no—
281C2×D4—82no—
282C2×D4C22⋊C4×C12712yes—
283C2×D4C22⋊C4×C12712yes—
284C2×D4C22⋊C4×C12712yes—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4C2×D4×C12711yescharacteristic
290C22×C4C2×D4×C12712yes—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—42no—
300C4⋊C4C22⋊C4×C12712yes—
301C4⋊C4C22⋊C4×C12712yes—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C42—22no—
312C42C22×C4×C12711yescharacteristic
313C42C2×D4×C12712yes—
314C42C2×D4×C12711yescharacteristic
315C23—22no—
316C23—22no—
317C23—22no—
318C23—42no—
319C23—42no—
320C23—84no—
321C23—82no—
322C23—84no—
323C23#120×C12711yescharacteristic
324Q8—42no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—44no—
329D4—44no—
330D4—42no—
331D4—42no—
332D4—82no—
333D4—84no—
334D4—84no—
335D4—84no—
336D4—84no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—82no—
345C2×C4—84no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4C2×C22⋊C4×C12712yes—
349C2×C4C2×C22⋊C4×C12711yescharacteristic
350C2×C4#120×C12711yescharacteristic
351C2×C4#120×C12712yes—
352C8—42no—
353C8—82no—
354C22—22no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—44no—
360C22—44no—
361C22—42no—
362C22—162no—
363C22—162no—
364C22—164no—
365C22#379×C12711yescharacteristic
366C22#453×C12711yescharacteristic
367C22#455×C12711yescharacteristic
368C4—21no—
369C4—21no—
370C4—22no—
371C4—22no—
372C4—42no—
373C4—42no—
374C4—42no—
375C4—161no—
376C2—21no—
377C2—21no—
378C2—82no—
379C2—82no—
380C2—82no—
381C2C127×#179211yescharacteristic
382C2C127×#179512yes—
subgroupG/H
center, socleC22×C127#379
commutatorC42C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC127×#3543C2
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×#1961C2semidirect
3. C127×#1555C2semidirect
4. C127×#1532C2semidirect
5. C127×#1512C2semidirect
7. #559×C127C4semidirect
8. #550×C127C4semidirect
11. D42×C127C4semidirect
17. #432×C127C22semidirect
19. C2×#97×C127C22semidirect
20. #382×C127C22semidirect
184. #8497C127direct (N × H)
193. #1961C2×C127semidirect
194. #1555C2×C127semidirect
195. #1532C2×C127semidirect
196. #1512C2×C127semidirect
197. C127#8497direct (N × H)
199. #559C4×C127semidirect
200. #550C4×C127semidirect
203. D42C4×C127semidirect
209. #432C22×C127semidirect
211. C2×#97C22×C127semidirect
212. #382C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#35432
C127×#19612
C127×#15552
C127×#15322
C127×#15122
#8497127