Finite Group Databasebeta

← C127×#8523 #44572196

Subgroups — C127×#8523 #44572196

Order 32512.

720 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#542×C127C211yesmaximal, characteristic
2C127×#1556C212yesmaximal, complemented
3C127×#1533C212yesmaximal, complemented
4C127×#1464C211yesmaximal, characteristic, complemented
5C127×#1464C211yesmaximal, characteristic, complemented
6#542×C127—24no—
7#542×C127—24no—
8C2×C4⋊Q8×C127C2211yescharacteristic
9C2×#126×C127—21no—
10C2×#124×C127—21no—
11C2×#124×C127C2211yescharacteristic
12C2×#123×C127—21no—
13C2×C22⋊Q8×C127—21no—
14C2×C22⋊Q8×C127C411yescharacteristic
15C2×#121×C127C411yescharacteristic
16C2×C4×Q8×C127C411yescharacteristic
17C2×C4×D4×C127C411yescharacteristic
18C2×#117×C127C2211yescharacteristic
19#432×C127C2212yes—
20#432×C127C2212yes—
21#332×C127—24no—
22#329×C127—24no—
23#324×C127—21no—
24#324×C127—21no—
25#323×C127—21no—
26#323×C127—21no—
27C22×Q8×C127—21no—
28C22×Q8×C127C2×C411yescharacteristic
29C22×D4×C127—21no—
30C22×D4×C127C2×C411yescharacteristic
31C23×C4×C127—21no—
32C23×C4×C127C2×C411yescharacteristic
33C4⋊Q8×C127—22no—
34C4⋊Q8×C127D414yes—
35#126×C127—28no—
36#124×C127—28no—
37#124×C127—22no—
38#124×C127D414yes—
39#123×C127—28no—
40C22⋊Q8×C127—28no—
41C22⋊Q8×C127—22no—
42C22⋊Q8×C127—22no—
43#121×C127—22no—
44#121×C127—22no—
45C4×Q8×C127—22no—
46C4×Q8×C127—22no—
47C4×D4×C127—22no—
48C4×D4×C127—22no—
49#117×C127—22no—
50#117×C127—22no—
51C2×C4⋊C4×C127—21no—
52C2×C4⋊C4×C127—21no—
53C2×C4⋊C4×C127—21no—
54C2×C4⋊C4×C127C2×C411yescharacteristic
55C2×C4⋊C4×C127C2×C411yescharacteristic
56C2×C4⋊C4×C127D411yescharacteristic
57C2×C4⋊C4×C127D411yescharacteristic
58C2×C22⋊C4×C127—21no—
59C2×C22⋊C4×C127—21no—
60C2×C22⋊C4×C127—21no—
61C2×C22⋊C4×C127—21no—
62C2×C22⋊C4×C127—21no—
63C2×C22⋊C4×C127D411yescharacteristic
64C2×C22⋊C4×C127D411yescharacteristic
65C2×C42×C127—21no—
66C2×C42×C127C2×C411yescharacteristic
67C2×C42×C127C2311yescharacteristic
68C4⋊C8×C127—22no—
69C4⋊C8×C127—22no—
70C22⋊C8×C127—22no—
71C22⋊C8×C127—22no—
72#97×C127—22no—
73#97×C127—22no—
74C24×C127—21no—
75C24×C127C22⋊C411yescharacteristic
76C2×Q8×C127—21no—
77C2×Q8×C127—24no—
78C2×Q8×C127—42no—
79C2×Q8×C127—82no—
80C2×Q8×C127C22⋊C412yes—
81C2×D4×C127—24no—
82C2×D4×C127—21no—
83C2×D4×C127—42no—
84C2×D4×C127—82no—
85C2×D4×C127C22⋊C412yes—
86C22×C4×C127—21no—
87C22×C4×C127—21no—
88C22×C4×C127—21no—
89C22×C4×C127—24no—
90C22×C4×C127—21no—
91C22×C4×C127—21no—
92C22×C4×C127—42no—
93C22×C4×C127—82no—
94C22×C4×C127C22⋊C411yescharacteristic
95C22×C4×C127C22⋊C411yescharacteristic
96C22×C4×C127C22⋊C411yescharacteristic
97C22×C4×C127C22×C411yescharacteristic
98C22×C4×C127C2×D411yescharacteristic
99C22×C4×C127C2×D411yescharacteristic
100C2×C8×C127—42no—
101C2×C8×C127—42no—
102C4⋊C4×C127—24no—
103C4⋊C4×C127—24no—
104C4⋊C4×C127—21no—
105C4⋊C4×C127—21no—
106C4⋊C4×C127—24no—
107C4⋊C4×C127—22no—
108C4⋊C4×C127—22no—
109C4⋊C4×C127C22⋊C412yes—
110C4⋊C4×C127C22⋊C412yes—
111C22⋊C4×C127—24no—
112C22⋊C4×C127—24no—
113C22⋊C4×C127—24no—
114C22⋊C4×C127—24no—
115C22⋊C4×C127—24no—
116C22⋊C4×C127—22no—
117C22⋊C4×C127—22no—
118C42×C127—24no—
119C42×C127—21no—
120C42×C127—21no—
121C42×C127C2×D412yes—
122C42×C127C2×D412yes—
123C23×C127—21no—
124C23×C127—22no—
125C23×C127—22no—
126C23×C127—42no—
127C23×C127—42no—
128C23×C127—41no—
129C23×C127—41no—
130C23×C127—42no—
131C23×C127C2×C22⋊C411yescharacteristic
132Q8×C127—42no—
133Q8×C127—42no—
134Q8×C127—84no—
135D4×C127—42no—
136D4×C127—42no—
137D4×C127—84no—
138C2×C4×C127—22no—
139C2×C4×C127—21no—
140C2×C4×C127—22no—
141C2×C4×C127—21no—
142C2×C4×C127—22no—
143C2×C4×C127—21no—
144C2×C4×C127—22no—
145C2×C4×C127—22no—
146C2×C4×C127—41no—
147C2×C4×C127—41no—
148C2×C4×C127—42no—
149C2×C4×C127—42no—
150C2×C4×C127—42no—
151C2×C4×C127—42no—
152C2×C4×C127—42no—
153C2×C4×C127—41no—
154C2×C4×C127—41no—
155C2×C4×C127—84no—
156C2×C4×C127C2×C22⋊C411yescharacteristic
157C2×C4×C127C2×C22⋊C411yescharacteristic
158C2×C4×C127#12012yes—
159C2×C4×C127#12012yes—
160C8×C127—82no—
161C8×C127—82no—
162C22×C127—21no—
163C22×C127—21no—
164C22×C127—42no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—42no—
168C22×C127—41no—
169C22×C127—42no—
170C22×C127—82no—
171C22×C127—84no—
172C22×C127#37911yescharacteristic
173C22×C127#41312yes—
174C4×C127—22no—
175C4×C127—22no—
176C4×C127—21no—
177C4×C127—21no—
178C4×C127—42no—
179C4×C127—42no—
180C4×C127—41no—
181C4×C127—41no—
182C4×C127—41no—
183C4×C127—42no—
184#8523C12711yesmaximal, characteristic, direct product factor, Sylow
185C2×C127—22no—
186C2×C127—81no—
187C2×C127—82no—
188C2×C127#178511yescharacteristic
189C2×C127#179511yescharacteristic
190C2×C127#179611yescharacteristic
191C2×#542C2×C12711yescharacteristic
192#1556C2×C12712yescomplemented
193#1533C2×C12712yescomplemented
194#1464C2×C12711yescharacteristic, complemented
195#1464C2×C12711yescharacteristic, complemented
196C127#852311yescharacteristic, direct product factor, Sylow
197#542—24no—
198#542—24no—
199C2×C4⋊Q8C22×C12711yescharacteristic
200C2×#126—21no—
201C2×#124—21no—
202C2×#124C22×C12711yescharacteristic
203C2×#123—21no—
204C2×C22⋊Q8—21no—
205C2×C22⋊Q8C4×C12711yescharacteristic
206C2×#121C4×C12711yescharacteristic
207C2×C4×Q8C4×C12711yescharacteristic
208C2×C4×D4C4×C12711yescharacteristic
209C2×#117C22×C12711yescharacteristic
210#432C22×C12712yes—
211#432C22×C12712yes—
212#332—24no—
213#329—24no—
214#324—21no—
215#324—21no—
216#323—21no—
217#323—21no—
218C22×Q8—21no—
219C22×Q8C2×C4×C12711yescharacteristic
220C22×D4—21no—
221C22×D4C2×C4×C12711yescharacteristic
222C23×C4—21no—
223C23×C4C2×C4×C12711yescharacteristic
224C4⋊Q8—22no—
225C4⋊Q8D4×C12714yes—
226#126—28no—
227#124—28no—
228#124—22no—
229#124D4×C12714yes—
230#123—28no—
231C22⋊Q8—28no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234#121—22no—
235#121—22no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×D4—22no—
239C4×D4—22no—
240#117—22no—
241#117—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4C2×C4×C12711yescharacteristic
246C2×C4⋊C4C2×C4×C12711yescharacteristic
247C2×C4⋊C4D4×C12711yescharacteristic
248C2×C4⋊C4D4×C12711yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4D4×C12711yescharacteristic
255C2×C22⋊C4D4×C12711yescharacteristic
256C2×C42—21no—
257C2×C42C2×C4×C12711yescharacteristic
258C2×C42C23×C12711yescharacteristic
259C4⋊C8—22no—
260C4⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263#97—22no—
264#97—22no—
265C24—21no—
266C24C22⋊C4×C12711yescharacteristic
267C2×Q8—21no—
268C2×Q8—24no—
269C2×Q8—42no—
270C2×Q8—82no—
271C2×Q8C22⋊C4×C12712yes—
272C2×D4—24no—
273C2×D4—21no—
274C2×D4—42no—
275C2×D4—82no—
276C2×D4C22⋊C4×C12712yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—82no—
285C22×C4C22⋊C4×C12711yescharacteristic
286C22×C4C22⋊C4×C12711yescharacteristic
287C22×C4C22⋊C4×C12711yescharacteristic
288C22×C4C22×C4×C12711yescharacteristic
289C22×C4C2×D4×C12711yescharacteristic
290C22×C4C2×D4×C12711yescharacteristic
291C2×C8—42no—
292C2×C8—42no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4C22⋊C4×C12712yes—
301C4⋊C4C22⋊C4×C12712yes—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—24no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C42—24no—
310C42—21no—
311C42—21no—
312C42C2×D4×C12712yes—
313C42C2×D4×C12712yes—
314C23—21no—
315C23—22no—
316C23—22no—
317C23—42no—
318C23—42no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23C2×C22⋊C4×C12711yescharacteristic
323Q8—42no—
324Q8—42no—
325Q8—84no—
326D4—42no—
327D4—42no—
328D4—84no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—84no—
347C2×C4C2×C22⋊C4×C12711yescharacteristic
348C2×C4C2×C22⋊C4×C12711yescharacteristic
349C2×C4#120×C12712yes—
350C2×C4#120×C12712yes—
351C8—82no—
352C8—82no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—42no—
357C22—42no—
358C22—41no—
359C22—42no—
360C22—41no—
361C22—82no—
362C22—84no—
363C22#379×C12711yescharacteristic
364C22#413×C12712yes—
365C4—21no—
366C4—21no—
367C4—22no—
368C4—22no—
369C4—41no—
370C4—42no—
371C4—42no—
372C4—42no—
373C4—41no—
374C4—41no—
375C2—22no—
376C2—82no—
377C2—81no—
378C2C127×#178511yescharacteristic
379C2C127×#179511yescharacteristic
380C2C127×#179611yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC2×#121×C127C4
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×#1556C2semidirect
3. C127×#1533C2semidirect
4. C127×#1464C2semidirect
5. C127×#1464C2semidirect
184. #8523C127direct (N × H)
192. #1556C2×C127semidirect
193. #1533C2×C127semidirect
194. #1464C2×C127semidirect
195. #1464C2×C127semidirect
196. C127#8523direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#542×C1272
C127×#15562
C127×#15332
C127×#14642
#8523127