Finite Group Databasebeta

← C127×#8505 #44572178

Subgroups — C127×#8505 #44572178

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×#3531C211yesmaximal, characteristic
2C127×#1995C211yesmaximal, characteristic, complemented
3C127×#1544C211yesmaximal, characteristic, complemented
4C127×#1500C212yesmaximal, complemented
5C127×#1495C212yesmaximal, complemented
6#564×C127C411yescharacteristic, complemented
7#563×C127—22no—
8#563×C127—22no—
9#563×C127C412yescomplemented
10#562×C127—22no—
11#562×C127C411yescharacteristic, complemented
12#533×C127—22no—
13#533×C127C2211yescharacteristic
14#529×C127—22no—
15#529×C127—22no—
16#529×C127C2212yes—
17#431×C127C2212yes—
18#423×C127C2211yescharacteristic
19#376×C127C2211yescharacteristic
20#339×C127—22no—
21#332×C127—22no—
22#328×C127—22no—
23#327×C127—22no—
24Q8⋊C8×C127—22no—
25D4⋊C8×C127—22no—
26C2×M16×C127—21no—
27C4⋊Q8×C127—22no—
28C4⋊Q8×C127—22no—
29C4⋊Q8×C127D412yes—
30#126×C127—22no—
31#126×C127—22no—
32#126×C127—22no—
33#126×C127—22no—
34#126×C127—22no—
35#125×C127—22no—
36#125×C127—22no—
37#125×C127—22no—
38#125×C127—22no—
39#125×C127C2×C412yes—
40#125×C127C2×C411yescharacteristic
41#125×C127C2×C411yescharacteristic
42#124×C127—22no—
43#124×C127—22no—
44#124×C127C2×C412yes—
45#123×C127—22no—
46#123×C127—22no—
47#123×C127—22no—
48#123×C127—22no—
49#123×C127—22no—
50C22⋊Q8×C127—22no—
51C22⋊Q8×C127—22no—
52C22⋊Q8×C127—22no—
53C22⋊Q8×C127—22no—
54C22⋊Q8×C127—22no—
55C22⋊Q8×C127—22no—
56C22⋊Q8×C127—22no—
57C22⋊Q8×C127D412yes—
58#121×C127—22no—
59#121×C127—22no—
60#121×C127D412yes—
61C4×Q8×C127—22no—
62C4×Q8×C127—22no—
63C4×Q8×C127D412yes—
64C4×D4×C127—22no—
65C4×D4×C127—22no—
66C4×D4×C127D412yes—
67#117×C127—22no—
68#117×C127—22no—
69#117×C127D412yes—
70C2×C42×C127—22no—
71C2×C42×C127—21no—
72C2×C42×C127C2311yescharacteristic
73C4⋊C8×C127—22no—
74C4⋊C8×C127—22no—
75C22⋊C8×C127—22no—
76C4×C8×C127—22no—
77#95×C127—22no—
78C2×Q8×C127—22no—
79C2×Q8×C127—22no—
80C2×Q8×C127C22⋊C412yes—
81C2×D4×C127—22no—
82C2×D4×C127—22no—
83C2×D4×C127C22⋊C412yes—
84C22×C4×C127—22no—
85C22×C4×C127—22no—
86C22×C4×C127—22no—
87C22×C4×C127—41no—
88C22×C4×C127C2×D411yescharacteristic
89C22×C4×C127C2×D412yes—
90M16×C127—42no—
91C2×C8×C127—42no—
92C2×C8×C127—41no—
93C4⋊C4×C127—22no—
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×C127C22⋊C412yes—
107C4⋊C4×C127C22⋊C412yes—
108C4⋊C4×C127C22⋊C412yes—
109C4⋊C4×C127C22⋊C412yes—
110C22⋊C4×C127—22no—
111C22⋊C4×C127—22no—
112C22⋊C4×C127—22no—
113C22⋊C4×C127—22no—
114C22⋊C4×C127—22no—
115C22⋊C4×C127—22no—
116C22⋊C4×C127—22no—
117C22⋊C4×C127—22no—
118C22⋊C4×C127—22no—
119C22⋊C4×C127—22no—
120C42×C127—22no—
121C42×C127—22no—
122C42×C127—22no—
123C42×C127—22no—
124C42×C127—22no—
125C42×C127—41no—
126C42×C127C22×C411yescharacteristic
127C42×C127C2×D411yescharacteristic
128C42×C127C2×D412yes—
129C23×C127—22no—
130C23×C127#12011yescharacteristic
131Q8×C127—42no—
132Q8×C127—82no—
133Q8×C127—82no—
134D4×C127—42no—
135D4×C127—82no—
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—42no—
146C2×C4×C127—42no—
147C2×C4×C127—42no—
148C2×C4×C127—82no—
149C2×C4×C127—82no—
150C2×C4×C127—82no—
151C2×C4×C127C2×C22⋊C411yescharacteristic
152C2×C4×C127C2×C22⋊C412yes—
153C2×C4×C127#12012yes—
154C2×C4×C127#12011yescharacteristic
155C8×C127—82no—
156C8×C127—81no—
157C22×C127—21no—
158C22×C127—22no—
159C22×C127—42no—
160C22×C127—42no—
161C22×C127—42no—
162C22×C127#37911yescharacteristic
163C4×C127—21no—
164C4×C127—22no—
165C4×C127—22no—
166C4×C127—42no—
167C4×C127—42no—
168C4×C127—42no—
169C4×C127—42no—
170C4×C127—42no—
171C4×C127—82no—
172C4×C127#45411yescharacteristic
173C4×C127#45611yescharacteristic
174#8505C12711yesmaximal, characteristic, direct product factor, Sylow
175C2×C127—41no—
176C2×C127—82no—
177C2×C127#179311yescharacteristic
178C2×C127#179412yes—
179#3531C2×C12711yescharacteristic
180#1995C2×C12711yescharacteristic, complemented
181#1544C2×C12711yescharacteristic, complemented
182#1500C2×C12712yescomplemented
183#1495C2×C12712yescomplemented
184C127#850511yescharacteristic, direct product factor, Sylow
185#564C4×C12711yescharacteristic, complemented
186#563—22no—
187#563—22no—
188#563C4×C12712yescomplemented
189#562—22no—
190#562C4×C12711yescharacteristic, complemented
191#533—22no—
192#533C22×C12711yescharacteristic
193#529—22no—
194#529—22no—
195#529C22×C12712yes—
196#431C22×C12712yes—
197#423C22×C12711yescharacteristic
198#376C22×C12711yescharacteristic
199#339—22no—
200#332—22no—
201#328—22no—
202#327—22no—
203Q8⋊C8—22no—
204D4⋊C8—22no—
205C2×M16—21no—
206C4⋊Q8—22no—
207C4⋊Q8—22no—
208C4⋊Q8D4×C12712yes—
209#126—22no—
210#126—22no—
211#126—22no—
212#126—22no—
213#126—22no—
214#125—22no—
215#125—22no—
216#125—22no—
217#125—22no—
218#125C2×C4×C12712yes—
219#125C2×C4×C12711yescharacteristic
220#125C2×C4×C12711yescharacteristic
221#124—22no—
222#124—22no—
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⋊Q8—22no—
236C22⋊Q8D4×C12712yes—
237#121—22no—
238#121—22no—
239#121D4×C12712yes—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8D4×C12712yes—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4D4×C12712yes—
246#117—22no—
247#117—22no—
248#117D4×C12712yes—
249C2×C42—22no—
250C2×C42—21no—
251C2×C42C23×C12711yescharacteristic
252C4⋊C8—22no—
253C4⋊C8—22no—
254C22⋊C8—22no—
255C4×C8—22no—
256#95—22no—
257C2×Q8—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—41no—
267C22×C4C2×D4×C12711yescharacteristic
268C22×C4C2×D4×C12712yes—
269M16—42no—
270C2×C8—42no—
271C2×C8—41no—
272C4⋊C4—22no—
273C4⋊C4—22no—
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⋊C4C22⋊C4×C12712yes—
286C4⋊C4C22⋊C4×C12712yes—
287C4⋊C4C22⋊C4×C12712yes—
288C4⋊C4C22⋊C4×C12712yes—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—41no—
305C42C22×C4×C12711yescharacteristic
306C42C2×D4×C12712yes—
307C42C2×D4×C12711yescharacteristic
308C23—22no—
309C23#120×C12711yescharacteristic
310Q8—42no—
311Q8—82no—
312Q8—82no—
313D4—42no—
314D4—82no—
315D4—82no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4C2×C22⋊C4×C12711yescharacteristic
331C2×C4C2×C22⋊C4×C12712yes—
332C2×C4#120×C12712yes—
333C2×C4#120×C12711yescharacteristic
334C8—82no—
335C8—81no—
336C22—21no—
337C22—22no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22#379×C12711yescharacteristic
342C4—21no—
343C4—22no—
344C4—22no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—42no—
350C4—82no—
351C4#454×C12711yescharacteristic
352C4#456×C12711yescharacteristic
353C2—41no—
354C2—82no—
355C2C127×#179311yescharacteristic
356C2C127×#179412yes—
subgroupG/H
centerC2×C4×C127#120
commutatorC42C22×C4×C127
Frattini, cosocleC2×C42C23×C127
socleC22×C127#379
prime radicalC127×#3531C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1995C2semidirect
3. C127×#1544C2semidirect
4. C127×#1500C2semidirect
5. C127×#1495C2semidirect
6. #564×C127C4semidirect
9. #563×C127C4semidirect
11. #562×C127C4semidirect
174. #8505C127direct (N × H)
180. #1995C2×C127semidirect
181. #1544C2×C127semidirect
182. #1500C2×C127semidirect
183. #1495C2×C127semidirect
184. C127#8505direct (N × H)
185. #564C4×C127semidirect
188. #563C4×C127semidirect
190. #562C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#35312
C127×#19952
C127×#15442
C127×#15002
C127×#14952
#8505127