Finite Group Databasebeta

← C127×#8498 #44572171

Subgroups — C127×#8498 #44572171

Order 32512.

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

normal subgroups only

This group has 692 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×#3544C211yesmaximal, characteristic
2C127×#1960C211yesmaximal, characteristic, complemented
3C127×#1558C211yesmaximal, characteristic, complemented
4C127×#1532C212yesmaximal, complemented
5C127×#1512C212yesmaximal, complemented
6#563×C127—22no—
7Q82×C127C411yescharacteristic, complemented
8#555×C127C411yescharacteristic, complemented
9#554×C127C412yescomplemented
10#552×C127—22no—
11#547×C127—22no—
12#544×C127—22no—
13#542×C127—22no—
14#536×C127—22no—
15C2×C4⋊Q8×C127C2211yescharacteristic
16#529×C127C2212yes—
17#432×C127C2212yes—
18C2×#103×C127—22no—
19C2×#97×C127C2211yescharacteristic
20#382×C127C2211yescharacteristic
21#333×C127—22no—
22#330×C127—22no—
23#329×C127—22no—
24#328×C127—22no—
25#327×C127—22no—
26C2×#41×C127—22no—
27C22×Q8×C127D412yes—
28C22×C8×C127—21no—
29C4⋊Q8×C127—22no—
30C4⋊Q8×C127—22no—
31C4⋊Q8×C127—22no—
32C4⋊Q8×C127—22no—
33C4⋊Q8×C127C2×C411yescharacteristic
34C4⋊Q8×C127C2×C411yescharacteristic
35C4⋊Q8×C127D412yes—
36#126×C127—24no—
37#126×C127—22no—
38#125×C127—24no—
39#125×C127—22no—
40#125×C127D412yes—
41#124×C127—22no—
42#124×C127—22no—
43#124×C127D412yes—
44#123×C127—24no—
45#123×C127—24no—
46#123×C127—22no—
47C22⋊Q8×C127—24no—
48C22⋊Q8×C127—24no—
49C22⋊Q8×C127—24no—
50C22⋊Q8×C127—22no—
51C22⋊Q8×C127D412yes—
52#121×C127—22no—
53#121×C127D412yes—
54C4×Q8×C127—22no—
55C4×Q8×C127C2×C412yes—
56C4×D4×C127—22no—
57C4×D4×C127—22no—
58C4×D4×C127C2×C412yes—
59#117×C127—22no—
60#117×C127—22no—
61#117×C127—22no—
62C2×C4⋊C4×C127—22no—
63C2×C4⋊C4×C127—22no—
64C2×C4⋊C4×C127—21no—
65C2×C42×C127C2311yescharacteristic
66C4⋊C8×C127—22no—
67#103×C127—44no—
68C22⋊C8×C127—22no—
69#97×C127—22no—
70#97×C127—22no—
71#95×C127—22no—
72#41×C127—82no—
73#41×C127—82no—
74C2×Q8×C127—22no—
75C2×Q8×C127—22no—
76C2×Q8×C127—42no—
77C2×Q8×C127—42no—
78C2×Q8×C127C22⋊C412yes—
79C2×Q8×C127C22⋊C412yes—
80C2×Q8×C127C22⋊C412yes—
81C2×D4×C127—22no—
82C2×D4×C127—22no—
83C2×D4×C127C22⋊C412yes—
84C22×C4×C127—22no—
85C22×C4×C127—24no—
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—24no—
95C4⋊C4×C127—24no—
96C4⋊C4×C127—24no—
97C4⋊C4×C127—22no—
98C4⋊C4×C127—22no—
99C4⋊C4×C127—22no—
100C4⋊C4×C127—24no—
101C4⋊C4×C127—22no—
102C4⋊C4×C127—22no—
103C4⋊C4×C127—22no—
104C4⋊C4×C127—42no—
105C4⋊C4×C127C22⋊C412yes—
106C4⋊C4×C127C22⋊C412yes—
107C22⋊C4×C127—24no—
108C22⋊C4×C127—24no—
109C22⋊C4×C127—22no—
110C22⋊C4×C127—22no—
111C22⋊C4×C127—22no—
112C22⋊C4×C127—22no—
113C22⋊C4×C127—22no—
114C42×C127—22no—
115C42×C127—22no—
116C42×C127—22no—
117C42×C127C22×C411yescharacteristic
118C42×C127C2×D412yes—
119C42×C127C2×D411yescharacteristic
120C23×C127—22no—
121C23×C127#12011yescharacteristic
122Q8×C127—42no—
123Q8×C127—42no—
124Q8×C127—42no—
125Q8×C127—82no—
126Q8×C127—82no—
127D4×C127—42no—
128D4×C127—42no—
129D4×C127—42no—
130D4×C127—82no—
131C2×C4×C127—21no—
132C2×C4×C127—21no—
133C2×C4×C127—22no—
134C2×C4×C127—22no—
135C2×C4×C127—22no—
136C2×C4×C127—22no—
137C2×C4×C127—22no—
138C2×C4×C127—42no—
139C2×C4×C127—44no—
140C2×C4×C127—42no—
141C2×C4×C127—42no—
142C2×C4×C127—84no—
143C2×C4×C127—82no—
144C2×C4×C127—81no—
145C2×C4×C127C2×C22⋊C412yes—
146C2×C4×C127C2×C22⋊C411yescharacteristic
147C2×C4×C127#12012yes—
148C2×C4×C127#12011yescharacteristic
149C8×C127—42no—
150C8×C127—82no—
151C22×C127—22no—
152C22×C127—42no—
153C22×C127—42no—
154C22×C127—42no—
155C22×C127#37911yescharacteristic
156C22×C127#45311yescharacteristic
157C22×C127#45511yescharacteristic
158C4×C127—21no—
159C4×C127—21no—
160C4×C127—22no—
161C4×C127—22no—
162C4×C127—42no—
163C4×C127—42no—
164C4×C127—42no—
165C4×C127—42no—
166C4×C127—42no—
167C4×C127—161no—
168#8498C12711yesmaximal, characteristic, direct product factor, Sylow
169C2×C127—21no—
170C2×C127—21no—
171C2×C127—82no—
172C2×C127#179211yescharacteristic
173C2×C127#179612yes—
174#3544C2×C12711yescharacteristic
175#1960C2×C12711yescharacteristic, complemented
176#1558C2×C12711yescharacteristic, complemented
177#1532C2×C12712yescomplemented
178#1512C2×C12712yescomplemented
179C127#849811yescharacteristic, direct product factor, Sylow
180#563—22no—
181Q82C4×C12711yescharacteristic, complemented
182#555C4×C12711yescharacteristic, complemented
183#554C4×C12712yescomplemented
184#552—22no—
185#547—22no—
186#544—22no—
187#542—22no—
188#536—22no—
189C2×C4⋊Q8C22×C12711yescharacteristic
190#529C22×C12712yes—
191#432C22×C12712yes—
192C2×#103—22no—
193C2×#97C22×C12711yescharacteristic
194#382C22×C12711yescharacteristic
195#333—22no—
196#330—22no—
197#329—22no—
198#328—22no—
199#327—22no—
200C2×#41—22no—
201C22×Q8D4×C12712yes—
202C22×C8—21no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8C2×C4×C12711yescharacteristic
208C4⋊Q8C2×C4×C12711yescharacteristic
209C4⋊Q8D4×C12712yes—
210#126—24no—
211#126—22no—
212#125—24no—
213#125—22no—
214#125D4×C12712yes—
215#124—22no—
216#124—22no—
217#124D4×C12712yes—
218#123—24no—
219#123—24no—
220#123—22no—
221C22⋊Q8—24no—
222C22⋊Q8—24no—
223C22⋊Q8—24no—
224C22⋊Q8—22no—
225C22⋊Q8D4×C12712yes—
226#121—22no—
227#121D4×C12712yes—
228C4×Q8—22no—
229C4×Q8C2×C4×C12712yes—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4C2×C4×C12712yes—
233#117—22no—
234#117—22no—
235#117—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C42C23×C12711yescharacteristic
240C4⋊C8—22no—
241#103—44no—
242C22⋊C8—22no—
243#97—22no—
244#97—22no—
245#95—22no—
246#41—82no—
247#41—82no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8C22⋊C4×C12712yes—
253C2×Q8C22⋊C4×C12712yes—
254C2×Q8C22⋊C4×C12712yes—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4C22⋊C4×C12712yes—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4C2×D4×C12711yescharacteristic
263C22×C4C2×D4×C12712yes—
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—42no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—42no—
279C4⋊C4C22⋊C4×C12712yes—
280C4⋊C4C22⋊C4×C12712yes—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42C22×C4×C12711yescharacteristic
292C42C2×D4×C12712yes—
293C42C2×D4×C12711yescharacteristic
294C23—22no—
295C23#120×C12711yescharacteristic
296Q8—42no—
297Q8—42no—
298Q8—42no—
299Q8—82no—
300Q8—82no—
301D4—42no—
302D4—42no—
303D4—42no—
304D4—82no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—84no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4C2×C22⋊C4×C12712yes—
320C2×C4C2×C22⋊C4×C12711yescharacteristic
321C2×C4#120×C12712yes—
322C2×C4#120×C12711yescharacteristic
323C8—42no—
324C8—82no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22#379×C12711yescharacteristic
330C22#453×C12711yescharacteristic
331C22#455×C12711yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—22no—
336C4—42no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—161no—
342C2—21no—
343C2—21no—
344C2—82no—
345C2C127×#179211yescharacteristic
346C2C127×#179612yes—
subgroupG/H
center, socleC22×C127#379
commutatorC42C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC127×#3544C2
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×#1960C2semidirect
3. C127×#1558C2semidirect
4. C127×#1532C2semidirect
5. C127×#1512C2semidirect
7. Q82×C127C4semidirect
8. #555×C127C4semidirect
9. #554×C127C4semidirect
168. #8498C127direct (N × H)
175. #1960C2×C127semidirect
176. #1558C2×C127semidirect
177. #1532C2×C127semidirect
178. #1512C2×C127semidirect
179. C127#8498direct (N × H)
181. Q82C4×C127semidirect
182. #555C4×C127semidirect
183. #554C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#35442
C127×#19602
C127×#15582
C127×#15322
C127×#15122
#8498127