Finite Group Databasebeta

← C127×#8461 #44572134

Subgroups — C127×#8461 #44572134

Order 32512.

682 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3507C211yesmaximal, characteristic
2C127×#1985C211yesmaximal, characteristic, complemented
3C127×#1552C211yesmaximal, characteristic, complemented
4C127×#1551C211yesmaximal, characteristic
5C127×#1531C211yesmaximal, characteristic, complemented
6C127×#1530C211yesmaximal, characteristic
7C127×#1467C211yesmaximal, characteristic, complemented
8#559×C127—21no—
9#557×C127—21no—
10#556×C127—21no—
11#556×C127—22no—
12#554×C127—21no—
13#552×C127—22no—
14#539×C127—22no—
15#536×C127C412yes—
16#533×C127C2211yescharacteristic
17#532×C127C2211yescharacteristic
18#529×C127—22no—
19#520×C127C412yes—
20C2×C4×Q8×C127C2211yescharacteristic
21#432×C127C2211yescharacteristic
22#432×C127C2211yescharacteristic
23C2×C4⋊C8×C127C2211yescharacteristic
24C2×#97×C127C2211yescharacteristic
25#336×C127—22no—
26#333×C127—21no—
27#332×C127—21no—
28#332×C127—21no—
29#331×C127—21no—
30#330×C127—22no—
31#329×C127—22no—
32#324×C127—22no—
33C2×#41×C127C2×C412yes—
34C22×Q8×C127C2×C411yescharacteristic
35C22×C8×C127—21no—
36C22×C8×C127—21no—
37C4⋊Q8×C127—22no—
38C4⋊Q8×C127D411yescharacteristic
39C4⋊Q8×C127D411yescharacteristic
40C4⋊Q8×C127D411yescharacteristic
41#127×C127D411yescharacteristic
42#126×C127—24no—
43#126×C127—24no—
44#125×C127—22no—
45#125×C127—24no—
46#125×C127D412yes—
47#124×C127—22no—
48#124×C127—22no—
49#124×C127D412yes—
50#123×C127—24no—
51#123×C127—24no—
52C22⋊Q8×C127—24no—
53C22⋊Q8×C127—21no—
54C22⋊Q8×C127—21no—
55C22⋊Q8×C127—22no—
56C22⋊Q8×C127—22no—
57#121×C127—21no—
58#121×C127—21no—
59C4×Q8×C127—22no—
60C4×Q8×C127—22no—
61C4×Q8×C127—21no—
62C4×Q8×C127—21no—
63C4×Q8×C127—21no—
64C4×Q8×C127—21no—
65C4×Q8×C127—21no—
66C4×Q8×C127—21no—
67C4×D4×C127—22no—
68C4×D4×C127—22no—
69C4×D4×C127—24no—
70C4×D4×C127—21no—
71C4×D4×C127—21no—
72#117×C127—24no—
73#117×C127—22no—
74#117×C127C2×C412yes—
75C2×C4⋊C4×C127—22no—
76C2×C4⋊C4×C127C2×C411yescharacteristic
77C2×C4⋊C4×C127D412yes—
78C2×C42×C127D412yes—
79C2×C42×C127C2311yescharacteristic
80C4⋊C8×C127—21no—
81C4⋊C8×C127—21no—
82C4⋊C8×C127—21no—
83C4⋊C8×C127—21no—
84C22⋊C8×C127—21no—
85C22⋊C8×C127—21no—
86#97×C127—21no—
87#97×C127—21no—
88#97×C127—21no—
89#97×C127—21no—
90#41×C127—44no—
91C2×Q8×C127—21no—
92C2×Q8×C127—21no—
93C2×Q8×C127—21no—
94C2×Q8×C127—21no—
95C2×Q8×C127—42no—
96C2×Q8×C127C22⋊C412yes—
97C2×D4×C127—22no—
98C2×D4×C127—21no—
99C2×D4×C127—21no—
100C2×D4×C127C22⋊C412yes—
101C22×C4×C127—24no—
102C22×C4×C127—22no—
103C22×C4×C127C22⋊C412yes—
104C22×C4×C127C22⋊C412yes—
105C22×C4×C127C22×C411yescharacteristic
106C22×C4×C127C2×D411yescharacteristic
107C22×C4×C127C2×D411yescharacteristic
108C2×C8×C127—22no—
109C2×C8×C127—41no—
110C2×C8×C127—41no—
111C2×C8×C127—42no—
112C2×C8×C127—41no—
113C2×C8×C127—41no—
114C2×C8×C127—41no—
115C4⋊C4×C127—24no—
116C4⋊C4×C127—24no—
117C4⋊C4×C127—24no—
118C4⋊C4×C127—21no—
119C4⋊C4×C127—21no—
120C4⋊C4×C127—22no—
121C4⋊C4×C127—22no—
122C4⋊C4×C127—22no—
123C4⋊C4×C127—21no—
124C4⋊C4×C127—21no—
125C4⋊C4×C127—21no—
126C4⋊C4×C127—22no—
127C4⋊C4×C127—21no—
128C4⋊C4×C127C22⋊C412yes—
129C4⋊C4×C127C22⋊C412yes—
130C22⋊C4×C127—24no—
131C22⋊C4×C127—24no—
132C22⋊C4×C127—24no—
133C22⋊C4×C127—22no—
134C22⋊C4×C127—22no—
135C22⋊C4×C127—22no—
136C42×C127—24no—
137C42×C127—22no—
138C42×C127—22no—
139C42×C127—22no—
140C42×C127C2×D411yescharacteristic
141C42×C127C2×D411yescharacteristic
142C42×C127C2×D411yescharacteristic
143C42×C127C2×D411yescharacteristic
144C23×C127—22no—
145C23×C127C2×C22⋊C411yescharacteristic
146Q8×C127—42no—
147Q8×C127—42no—
148Q8×C127—42no—
149Q8×C127—41no—
150Q8×C127—41no—
151Q8×C127—81no—
152D4×C127—42no—
153D4×C127—44no—
154D4×C127—41no—
155D4×C127—41no—
156D4×C127—81no—
157C2×C4×C127—22no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127—22no—
161C2×C4×C127—22no—
162C2×C4×C127—22no—
163C2×C4×C127—22no—
164C2×C4×C127—22no—
165C2×C4×C127—24no—
166C2×C4×C127—41no—
167C2×C4×C127—44no—
168C2×C4×C127—42no—
169C2×C4×C127—84no—
170C2×C4×C127C2×C22⋊C411yescharacteristic
171C2×C4×C127C2×C22⋊C411yescharacteristic
172C2×C4×C127#12011yescharacteristic
173C2×C4×C127#12011yescharacteristic
174C2×C4×C127#12011yescharacteristic
175C2×C4×C127#12011yescharacteristic
176C8×C127—42no—
177C8×C127—81no—
178C8×C127—81no—
179C8×C127—81no—
180C22×C127—21no—
181C22×C127—21no—
182C22×C127—42no—
183C22×C127—42no—
184C22×C127—42no—
185C22×C127#37911yescharacteristic
186C22×C127#41312yes—
187C4×C127—21no—
188C4×C127—21no—
189C4×C127—21no—
190C4×C127—21no—
191C4×C127—21no—
192C4×C127—42no—
193C4×C127—42no—
194C4×C127—42no—
195C4×C127—42no—
196C4×C127—42no—
197C4×C127#45411yescharacteristic
198C4×C127#45611yescharacteristic
199#8461C12711yesmaximal, characteristic, direct product factor, Sylow
200C2×C127—22no—
201C2×C127—82no—
202C2×C127#178511yescharacteristic
203C2×C127#179311yescharacteristic
204C2×C127#179411yescharacteristic
205#3507C2×C12711yescharacteristic
206#1985C2×C12711yescharacteristic, complemented
207#1552C2×C12711yescharacteristic, complemented
208#1551C2×C12711yescharacteristic
209#1531C2×C12711yescharacteristic, complemented
210#1530C2×C12711yescharacteristic
211#1467C2×C12711yescharacteristic, complemented
212C127#846111yescharacteristic, direct product factor, Sylow
213#559—21no—
214#557—21no—
215#556—21no—
216#556—22no—
217#554—21no—
218#552—22no—
219#539—22no—
220#536C4×C12712yes—
221#533C22×C12711yescharacteristic
222#532C22×C12711yescharacteristic
223#529—22no—
224#520C4×C12712yes—
225C2×C4×Q8C22×C12711yescharacteristic
226#432C22×C12711yescharacteristic
227#432C22×C12711yescharacteristic
228C2×C4⋊C8C22×C12711yescharacteristic
229C2×#97C22×C12711yescharacteristic
230#336—22no—
231#333—21no—
232#332—21no—
233#332—21no—
234#331—21no—
235#330—22no—
236#329—22no—
237#324—22no—
238C2×#41C2×C4×C12712yes—
239C22×Q8C2×C4×C12711yescharacteristic
240C22×C8—21no—
241C22×C8—21no—
242C4⋊Q8—22no—
243C4⋊Q8D4×C12711yescharacteristic
244C4⋊Q8D4×C12711yescharacteristic
245C4⋊Q8D4×C12711yescharacteristic
246#127D4×C12711yescharacteristic
247#126—24no—
248#126—24no—
249#125—22no—
250#125—24no—
251#125D4×C12712yes—
252#124—22no—
253#124—22no—
254#124D4×C12712yes—
255#123—24no—
256#123—24no—
257C22⋊Q8—24no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260C22⋊Q8—22no—
261C22⋊Q8—22no—
262#121—21no—
263#121—21no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—24no—
275C4×D4—21no—
276C4×D4—21no—
277#117—24no—
278#117—22no—
279#117C2×C4×C12712yes—
280C2×C4⋊C4—22no—
281C2×C4⋊C4C2×C4×C12711yescharacteristic
282C2×C4⋊C4D4×C12712yes—
283C2×C42D4×C12712yes—
284C2×C42C23×C12711yescharacteristic
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291#97—21no—
292#97—21no—
293#97—21no—
294#97—21no—
295#41—44no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—42no—
301C2×Q8C22⋊C4×C12712yes—
302C2×D4—22no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4C22⋊C4×C12712yes—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4C22⋊C4×C12712yes—
309C22×C4C22⋊C4×C12712yes—
310C22×C4C22×C4×C12711yescharacteristic
311C22×C4C2×D4×C12711yescharacteristic
312C22×C4C2×D4×C12711yescharacteristic
313C2×C8—22no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4C22⋊C4×C12712yes—
334C4⋊C4C22⋊C4×C12712yes—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C42—24no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42C2×D4×C12711yescharacteristic
346C42C2×D4×C12711yescharacteristic
347C42C2×D4×C12711yescharacteristic
348C42C2×D4×C12711yescharacteristic
349C23—22no—
350C23C2×C22⋊C4×C12711yescharacteristic
351Q8—42no—
352Q8—42no—
353Q8—42no—
354Q8—41no—
355Q8—41no—
356Q8—81no—
357D4—42no—
358D4—44no—
359D4—41no—
360D4—41no—
361D4—81no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—41no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—84no—
375C2×C4C2×C22⋊C4×C12711yescharacteristic
376C2×C4C2×C22⋊C4×C12711yescharacteristic
377C2×C4#120×C12711yescharacteristic
378C2×C4#120×C12711yescharacteristic
379C2×C4#120×C12711yescharacteristic
380C2×C4#120×C12711yescharacteristic
381C8—42no—
382C8—81no—
383C8—81no—
384C8—81no—
385C22—21no—
386C22—21no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22#379×C12711yescharacteristic
391C22#413×C12712yes—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—42no—
402C4#454×C12711yescharacteristic
403C4#456×C12711yescharacteristic
404C2—22no—
405C2—82no—
406C2C127×#178511yescharacteristic
407C2C127×#179311yescharacteristic
408C2C127×#179411yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radical#532×C127C22
LCSG ▷ C22×C4 ▷ 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×#1985C2semidirect
3. C127×#1552C2semidirect
5. C127×#1531C2semidirect
7. C127×#1467C2semidirect
199. #8461C127direct (N × H)
206. #1985C2×C127semidirect
207. #1552C2×C127semidirect
209. #1531C2×C127semidirect
211. #1467C2×C127semidirect
212. C127#8461direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#35072
C127×#19852
C127×#15522
C127×#15512
C127×#15312
C127×#15302
C127×#14672
#8461127