Finite Group Databasebeta

← C127×#9735 #44573408

Subgroups — C127×#9735 #44573408

Order 32512.

522 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#486×C127C211yesmaximal, characteristic
2C127×#1987C211yesmaximal, characteristic, complemented
3C127×#1982C211yesmaximal, characteristic
4C8×C4⋊C4×C127C211yesmaximal, characteristic, complemented
5C2×#334×C127C211yesmaximal, characteristic, complemented
6C2×#333×C127C211yesmaximal, characteristic, complemented
7C2×#331×C127C211yesmaximal, characteristic
8C2×C4⋊Q8×C127C2211yescharacteristic
9C2×#127×C127C2211yescharacteristic
10#486×C127—22no—
11#486×C127C414yescomplemented
12C2×C4⋊C8×C127C2211yescharacteristic
13C2×C4⋊C8×C127C2211yescharacteristic
14C2×C4⋊C8×C127C2211yescharacteristic
15C2×#102×C127—21no—
16C2×#102×C127—21no—
17C2×#102×C127—22no—
18C2×C4×C8×C127—21no—
19C2×C4×C8×C127C2211yescharacteristic
20C4×C4⋊C4×C127C2211yescharacteristic
21#340×C127—22no—
22#336×C127—21no—
23#334×C127—22no—
24#333×C127—22no—
25#331×C127—22no—
26C22×Q8×C127—21no—
27C22×D4×C127—21no—
28C22×D4×C127—21no—
29C22×D4×C127—21no—
30C22×C8×C127—21no—
31C22×C8×C127—21no—
32C22×C8×C127—21no—
33C22×C8×C127D411yescharacteristic
34C22×C8×C127D411yescharacteristic
35C4⋊Q8×C127—21no—
36C4⋊Q8×C127—21no—
37C4⋊Q8×C127—21no—
38C4⋊Q8×C127C2×C412yes—
39#127×C127—21no—
40#127×C127—21no—
41#127×C127—21no—
42#127×C127C2×C412yes—
43C2×C4⋊C4×C127—21no—
44C2×C4⋊C4×C127—21no—
45C2×C4⋊C4×C127D412yes—
46C2×C42×C127—21no—
47C2×C42×C127C2311yescharacteristic
48C4⋊C8×C127—21no—
49C4⋊C8×C127—21no—
50C4⋊C8×C127—21no—
51C4⋊C8×C127—21no—
52C4⋊C8×C127—21no—
53C4⋊C8×C127D412yes—
54#102×C127—24no—
55#102×C127—24no—
56#102×C127—44no—
57C4×C8×C127—42no—
58C4×C8×C127C2×C412yes—
59C4×C8×C127D412yes—
60#95×C127—21no—
61C24×C127—41no—
62C2×Q8×C127—22no—
63C2×Q8×C127—41no—
64C2×Q8×C127—82no—
65C2×D4×C127—22no—
66C2×D4×C127—22no—
67C2×D4×C127—22no—
68C2×D4×C127—41no—
69C2×D4×C127—41no—
70C2×D4×C127—41no—
71C2×D4×C127—42no—
72C2×D4×C127—41no—
73C2×D4×C127—41no—
74C2×D4×C127—42no—
75C2×D4×C127—41no—
76C2×D4×C127—41no—
77C2×D4×C127—82no—
78C22×C4×C127—22no—
79C22×C4×C127—41no—
80C22×C4×C127C2×D411yescharacteristic
81C22×C4×C127C2×D411yescharacteristic
82C22×C4×C127#4111yescharacteristic
83C2×C8×C127—21no—
84C2×C8×C127—21no—
85C2×C8×C127—21no—
86C2×C8×C127—21no—
87C2×C8×C127—22no—
88C2×C8×C127—41no—
89C2×C8×C127—41no—
90C2×C8×C127—41no—
91C2×C8×C127—41no—
92C2×C8×C127—42no—
93C2×C8×C127—42no—
94C2×C8×C127C22⋊C412yes—
95C2×C8×C127C22⋊C412yes—
96C4⋊C4×C127—22no—
97C4⋊C4×C127—22no—
98C4⋊C4×C127—22no—
99C4⋊C4×C127—41no—
100C4⋊C4×C127—41no—
101C4⋊C4×C127SD1614yes—
102C42×C127—41no—
103C42×C127—41no—
104C42×C127C22×C411yescharacteristic
105C42×C127C2×D411yescharacteristic
106C42×C127C2×D411yescharacteristic
107C42×C127#4111yescharacteristic
108C23×C127—42no—
109C23×C127—81no—
110C23×C127—81no—
111C23×C127—82no—
112C23×C127—82no—
113C23×C127#12111yescharacteristic
114Q8×C127—82no—
115Q8×C127—82no—
116D4×C127—42no—
117D4×C127—42no—
118D4×C127—44no—
119D4×C127—82no—
120D4×C127—82no—
121D4×C127—81no—
122D4×C127—81no—
123D4×C127—81no—
124D4×C127—81no—
125C2×C4×C127—22no—
126C2×C4×C127—24no—
127C2×C4×C127—42no—
128C2×C4×C127—42no—
129C2×C4×C127—42no—
130C2×C4×C127—82no—
131C2×C4×C127C2×C22⋊C411yescharacteristic
132C2×C4×C127C4×D411yescharacteristic
133C2×C4×C127C4×D411yescharacteristic
134C2×C4×C127#12111yescharacteristic
135C2×C4×C127#12411yescharacteristic
136C2×C4×C127#12711yescharacteristic
137C2×C4×C127C2×SD1612yes—
138C2×C4×C127#13511yescharacteristic
139C2×C4×C127#13511yescharacteristic
140C2×C4×C127#13512yes—
141C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C127#13711yescharacteristic
143C8×C127—22no—
144C8×C127—22no—
145C8×C127—42no—
146C8×C127—81no—
147C8×C127—81no—
148C22×C127—82no—
149C22×C127—82no—
150C22×C127—82no—
151C22×C127—162no—
152C22×C127—162no—
153C22×C127#39011yescharacteristic
154C22×C127#46311yescharacteristic
155C22×C127#46411yescharacteristic
156C22×C127#46512yes—
157C22×C127#47212yes—
158C4×C127—21no—
159C4×C127—21no—
160C4×C127—44no—
161C4×C127—82no—
162C4×C127#41611yescharacteristic
163C4×C127#41711yescharacteristic
164C4×C127C4×SD1612yes—
165C4×C127#46311yescharacteristic
166C4×C127#46411yescharacteristic
167C4×C127#49211yescharacteristic
168C4×C127#49511yescharacteristic
169#9735C12711yesmaximal, characteristic, direct product factor, Sylow
170C2×C127—162no—
171C2×C127#167712yes—
172C2×C127#167812yes—
173C2×C127#196811yescharacteristic
174C2×C127#197811yescharacteristic
175C2×C127#198311yescharacteristic
176C2×#486C2×C12711yescharacteristic
177#1987C2×C12711yescharacteristic, complemented
178#1982C2×C12711yescharacteristic
179C8×C4⋊C4C2×C12711yescharacteristic, complemented
180C2×#334C2×C12711yescharacteristic, complemented
181C2×#333C2×C12711yescharacteristic, complemented
182C2×#331C2×C12711yescharacteristic
183C127#973511yescharacteristic, direct product factor, Sylow
184C2×C4⋊Q8C22×C12711yescharacteristic
185C2×#127C22×C12711yescharacteristic
186#486—22no—
187#486C4×C12714yescomplemented
188C2×C4⋊C8C22×C12711yescharacteristic
189C2×C4⋊C8C22×C12711yescharacteristic
190C2×C4⋊C8C22×C12711yescharacteristic
191C2×#102—21no—
192C2×#102—21no—
193C2×#102—22no—
194C2×C4×C8—21no—
195C2×C4×C8C22×C12711yescharacteristic
196C4×C4⋊C4C22×C12711yescharacteristic
197#340—22no—
198#336—21no—
199#334—22no—
200#333—22no—
201#331—22no—
202C22×Q8—21no—
203C22×D4—21no—
204C22×D4—21no—
205C22×D4—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8D4×C12711yescharacteristic
210C22×C8D4×C12711yescharacteristic
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8C2×C4×C12712yes—
215#127—21no—
216#127—21no—
217#127—21no—
218#127C2×C4×C12712yes—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4D4×C12712yes—
222C2×C42—21no—
223C2×C42C23×C12711yescharacteristic
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8D4×C12712yes—
230#102—24no—
231#102—24no—
232#102—44no—
233C4×C8—42no—
234C4×C8C2×C4×C12712yes—
235C4×C8D4×C12712yes—
236#95—21no—
237C24—41no—
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—82no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—42no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—82no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4C2×D4×C12711yescharacteristic
257C22×C4C2×D4×C12711yescharacteristic
258C22×C4#41×C12711yescharacteristic
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8C22⋊C4×C12712yes—
271C2×C8C22⋊C4×C12712yes—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4SD16×C12714yes—
278C42—41no—
279C42—41no—
280C42C22×C4×C12711yescharacteristic
281C42C2×D4×C12711yescharacteristic
282C42C2×D4×C12711yescharacteristic
283C42#41×C12711yescharacteristic
284C23—42no—
285C23—81no—
286C23—81no—
287C23—82no—
288C23—82no—
289C23#121×C12711yescharacteristic
290Q8—82no—
291Q8—82no—
292D4—42no—
293D4—42no—
294D4—44no—
295D4—82no—
296D4—82no—
297D4—81no—
298D4—81no—
299D4—81no—
300D4—81no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—82no—
307C2×C4C2×C22⋊C4×C12711yescharacteristic
308C2×C4C4×D4×C12711yescharacteristic
309C2×C4C4×D4×C12711yescharacteristic
310C2×C4#121×C12711yescharacteristic
311C2×C4#124×C12711yescharacteristic
312C2×C4#127×C12711yescharacteristic
313C2×C4C2×SD16×C12712yes—
314C2×C4#135×C12711yescharacteristic
315C2×C4#135×C12711yescharacteristic
316C2×C4#135×C12712yes—
317C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
318C2×C4#137×C12711yescharacteristic
319C8—22no—
320C8—22no—
321C8—42no—
322C8—81no—
323C8—81no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—162no—
328C22—162no—
329C22#390×C12711yescharacteristic
330C22#463×C12711yescharacteristic
331C22#464×C12711yescharacteristic
332C22#465×C12712yes—
333C22#472×C12712yes—
334C4—21no—
335C4—21no—
336C4—44no—
337C4—82no—
338C4#416×C12711yescharacteristic
339C4#417×C12711yescharacteristic
340C4C4×SD16×C12712yes—
341C4#463×C12711yescharacteristic
342C4#464×C12711yescharacteristic
343C4#492×C12711yescharacteristic
344C4#495×C12711yescharacteristic
345C2—162no—
346C2C127×#167712yes—
347C2C127×#167812yes—
348C2C127×#196811yescharacteristic
349C2C127×#197811yescharacteristic
350C2C127×#198311yescharacteristic
subgroupG/H
center, socleC23×C127#121
commutatorC42C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC2×#127×C127C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1987C2semidirect
4. C8×C4⋊C4×C127C2semidirect
5. C2×#334×C127C2semidirect
6. C2×#333×C127C2semidirect
11. #486×C127C4semidirect
169. #9735C127direct (N × H)
177. #1987C2×C127semidirect
179. C8×C4⋊C4C2×C127semidirect
180. C2×#334C2×C127semidirect
181. C2×#333C2×C127semidirect
183. C127#9735direct (N × H)
187. #486C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#486×C1272
C127×#19872
C127×#19822
C8×C4⋊C4×C1272
C2×#334×C1272
C2×#333×C1272
C2×#331×C1272
#9735127