Finite Group Databasebeta

← C2×C127×#1720 #44583285

Subgroups — C2×C127×#1720 #44583285

Order 32512.

850 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#498×C127C211yesmaximal, characteristic, complemented
2C2×#494×C127C211yesmaximal, characteristic, complemented
3C2×#487×C127C211yesmaximal, characteristic, complemented
4C2×#486×C127C211yesmaximal, characteristic
5C2×#486×C127C211yesmaximal, characteristic
6C2×#486×C127C211yesmaximal, characteristic
7C127×#1720C218yesmaximal, direct product factor
8C2×#322×C127C211yesmaximal, characteristic, complemented
9C22×Q16×C127—22no—
10C2×C4⋊Q8×C127C2211yescharacteristic
11C2×C4⋊Q8×C127C2211yescharacteristic
12C2×C4⋊Q8×C127C2211yescharacteristic
13C2×#127×C127C2211yescharacteristic
14#498×C127—22no—
15#498×C127C2214yescomplemented
16#494×C127—22no—
17#494×C127C2214yescomplemented
18#487×C127—22no—
19#487×C127C2214yescomplemented
20#486×C127—22no—
21#486×C127—22no—
22#486×C127—22no—
23#486×C127C2214yes—
24#486×C127C2214yes—
25#486×C127C2214yes—
26C2×#106×C127—22no—
27C2×#103×C127—22no—
28C2×#102×C127—22no—
29C2×#102×C127—22no—
30C2×#102×C127—22no—
31C2×C4×C8×C127C2211yescharacteristic
32C2×C4×C8×C127C2211yescharacteristic
33C2×C4×C8×C127C2211yescharacteristic
34#322×C127C2214yescomplemented
35C22×Q8×C127—21no—
36C22×Q8×C127—21no—
37C22×Q8×C127—21no—
38C22×D4×C127—21no—
39C22×D4×C127—21no—
40C22×D4×C127—21no—
41C2×Q16×C127—28no—
42C2×Q16×C127—48no—
43C22×C8×C127D412yes—
44C22×C8×C127D412yes—
45C22×C8×C127D412yes—
46C4⋊Q8×C127—21no—
47C4⋊Q8×C127—21no—
48C4⋊Q8×C127—21no—
49C4⋊Q8×C127—21no—
50C4⋊Q8×C127—21no—
51C4⋊Q8×C127—21no—
52C4⋊Q8×C127—21no—
53C4⋊Q8×C127—21no—
54C4⋊Q8×C127—21no—
55C4⋊Q8×C127C2312yes—
56C4⋊Q8×C127C2312yes—
57C4⋊Q8×C127C2312yes—
58#127×C127—21no—
59#127×C127—21no—
60#127×C127—21no—
61#127×C127C2312yes—
62C2×C4⋊C4×C127—21no—
63C2×C4⋊C4×C127—21no—
64C2×C4⋊C4×C127—21no—
65C2×C4⋊C4×C127—21no—
66C2×C4⋊C4×C127—21no—
67C2×C4⋊C4×C127—21no—
68C2×C42×C127C2311yescharacteristic
69#106×C127—28no—
70#103×C127—28no—
71#102×C127—28no—
72#102×C127—28no—
73#102×C127—28no—
74C4×C8×C127D412yes—
75C4×C8×C127D412yes—
76C4×C8×C127D412yes—
77C4×C8×C127C2312yes—
78C4×C8×C127C2312yes—
79C4×C8×C127C2312yes—
80C24×C127—41no—
81C2×Q8×C127—22no—
82C2×Q8×C127—22no—
83C2×Q8×C127—22no—
84C2×Q8×C127—41no—
85C2×Q8×C127—41no—
86C2×Q8×C127—41no—
87C2×Q8×C127—44no—
88C2×Q8×C127—44no—
89C2×Q8×C127—44no—
90C2×D4×C127—22no—
91C2×D4×C127—22no—
92C2×D4×C127—22no—
93C2×D4×C127—41no—
94C2×D4×C127—41no—
95C2×D4×C127—41no—
96C2×D4×C127—44no—
97C2×D4×C127—44no—
98C2×D4×C127—44no—
99C22×C4×C127—41no—
100C22×C4×C127—41no—
101C22×C4×C127—41no—
102C22×C4×C127C2×D411yescharacteristic
103C22×C4×C127C2×D411yescharacteristic
104C22×C4×C127C2×D411yescharacteristic
105Q16×C127—416no—
106C2×C8×C127—24no—
107C2×C8×C127—24no—
108C2×C8×C127—24no—
109C2×C8×C127C2×D414yes—
110C2×C8×C127C2×D414yes—
111C2×C8×C127C2×D414yes—
112C4⋊C4×C127—22no—
113C4⋊C4×C127—22no—
114C4⋊C4×C127—22no—
115C4⋊C4×C127—22no—
116C4⋊C4×C127—22no—
117C4⋊C4×C127—22no—
118C4⋊C4×C127—41no—
119C4⋊C4×C127—41no—
120C4⋊C4×C127—41no—
121C4⋊C4×C127—41no—
122C4⋊C4×C127—41no—
123C4⋊C4×C127—41no—
124C42×C127C2×D411yescharacteristic
125C42×C127C2×D411yescharacteristic
126C42×C127C2×D411yescharacteristic
127C42×C127C2411yescharacteristic
128C23×C127—42no—
129C23×C127—82no—
130C23×C127—82no—
131C23×C127—82no—
132C23×C127#12711yescharacteristic
133Q8×C127—44no—
134Q8×C127—44no—
135Q8×C127—44no—
136Q8×C127—82no—
137Q8×C127—82no—
138Q8×C127—82no—
139D4×C127—44no—
140D4×C127—44no—
141D4×C127—44no—
142D4×C127—82no—
143D4×C127—82no—
144D4×C127—82no—
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×C127#12711yescharacteristic
152C2×C4×C127#12711yescharacteristic
153C2×C4×C127#12711yescharacteristic
154C2×C4×C127#13514yes—
155C2×C4×C127AGL(1,ℤ/8)14yes—
156C2×C4×C127#13714yes—
157C2×C4×C127C22×D411yescharacteristic
158C2×C4×C127C22×D411yescharacteristic
159C2×C4×C127C22×D411yescharacteristic
160C8×C127—24no—
161C8×C127—24no—
162C8×C127—24no—
163C22×C127—82no—
164C22×C127—82no—
165C22×C127—82no—
166C22×C127—164no—
167C22×C127#49512yes—
168C22×C127#49612yes—
169C22×C127#49712yes—
170C22×C127C2×#12711yescharacteristic
171C4×C127—82no—
172C4×C127—82no—
173C4×C127—82no—
174C4×C127#49512yes—
175C4×C127#49612yes—
176C4×C127#49712yes—
177C4×C127C2×#13512yes—
178C4×C127C2×AGL(1,ℤ/8)12yes—
179C4×C127C2×#13712yes—
180C2×#1720C12711yesmaximal, characteristic, direct product factor, Sylow
181C2×C127—162no—
182C2×C127#172014yesdirect product factor
183C2×C127C2×#49511yescharacteristic
184C2×C127C2×#49611yescharacteristic
185C2×C127C2×#49711yescharacteristic
186C2×#498C2×C12711yescharacteristic, complemented
187C2×#494C2×C12711yescharacteristic, complemented
188C2×#487C2×C12711yescharacteristic, complemented
189C2×#486C2×C12711yescharacteristic
190C2×#486C2×C12711yescharacteristic
191C2×#486C2×C12711yescharacteristic
192#1720C2×C12718yesdirect product factor
193C2×#322C2×C12711yescharacteristic, complemented
194C127C2×#172011yescharacteristic, direct product factor, Sylow
195C22×Q16—22no—
196C2×C4⋊Q8C22×C12711yescharacteristic
197C2×C4⋊Q8C22×C12711yescharacteristic
198C2×C4⋊Q8C22×C12711yescharacteristic
199C2×#127C22×C12711yescharacteristic
200#498—22no—
201#498C22×C12714yescomplemented
202#494—22no—
203#494C22×C12714yescomplemented
204#487—22no—
205#487C22×C12714yescomplemented
206#486—22no—
207#486—22no—
208#486—22no—
209#486C22×C12714yes—
210#486C22×C12714yes—
211#486C22×C12714yes—
212C2×#106—22no—
213C2×#103—22no—
214C2×#102—22no—
215C2×#102—22no—
216C2×#102—22no—
217C2×C4×C8C22×C12711yescharacteristic
218C2×C4×C8C22×C12711yescharacteristic
219C2×C4×C8C22×C12711yescharacteristic
220#322C22×C12714yescomplemented
221C22×Q8—21no—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—21no—
227C2×Q16—28no—
228C2×Q16—48no—
229C22×C8D4×C12712yes—
230C22×C8D4×C12712yes—
231C22×C8D4×C12712yes—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8C23×C12712yes—
242C4⋊Q8C23×C12712yes—
243C4⋊Q8C23×C12712yes—
244#127—21no—
245#127—21no—
246#127—21no—
247#127C23×C12712yes—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C42C23×C12711yescharacteristic
255#106—28no—
256#103—28no—
257#102—28no—
258#102—28no—
259#102—28no—
260C4×C8D4×C12712yes—
261C4×C8D4×C12712yes—
262C4×C8D4×C12712yes—
263C4×C8C23×C12712yes—
264C4×C8C23×C12712yes—
265C4×C8C23×C12712yes—
266C24—41no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—44no—
274C2×Q8—44no—
275C2×Q8—44no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4—44no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4C2×D4×C12711yescharacteristic
289C22×C4C2×D4×C12711yescharacteristic
290C22×C4C2×D4×C12711yescharacteristic
291Q16—416no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8C2×D4×C12714yes—
296C2×C8C2×D4×C12714yes—
297C2×C8C2×D4×C12714yes—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C42C2×D4×C12711yescharacteristic
311C42C2×D4×C12711yescharacteristic
312C42C2×D4×C12711yescharacteristic
313C42C24×C12711yescharacteristic
314C23—42no—
315C23—82no—
316C23—82no—
317C23—82no—
318C23#127×C12711yescharacteristic
319Q8—44no—
320Q8—44no—
321Q8—44no—
322Q8—82no—
323Q8—82no—
324Q8—82no—
325D4—44no—
326D4—44no—
327D4—44no—
328D4—82no—
329D4—82no—
330D4—82no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4#127×C12711yescharacteristic
338C2×C4#127×C12711yescharacteristic
339C2×C4#127×C12711yescharacteristic
340C2×C4#135×C12714yes—
341C2×C4AGL(1,ℤ/8)×C12714yes—
342C2×C4#137×C12714yes—
343C2×C4C22×D4×C12711yescharacteristic
344C2×C4C22×D4×C12711yescharacteristic
345C2×C4C22×D4×C12711yescharacteristic
346C8—24no—
347C8—24no—
348C8—24no—
349C22—82no—
350C22—82no—
351C22—82no—
352C22—164no—
353C22#495×C12712yes—
354C22#496×C12712yes—
355C22#497×C12712yes—
356C22C2×#127×C12711yescharacteristic
357C4—82no—
358C4—82no—
359C4—82no—
360C4#495×C12712yes—
361C4#496×C12712yes—
362C4#497×C12712yes—
363C4C2×#135×C12712yes—
364C4C2×AGL(1,ℤ/8)×C12712yes—
365C4C2×#137×C12712yes—
366C2—162no—
367C2C127×#172014yesdirect product factor
368C2C2×#495×C12711yescharacteristic
369C2C2×#496×C12711yescharacteristic
370C2C2×#497×C12711yescharacteristic
subgroupG/H
center, socleC23×C127#127
commutator, Frattini, cosocleC42C24×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
1. C2×#498×C127C2semidirect
2. C2×#494×C127C2semidirect
3. C2×#487×C127C2semidirect
7. C127×#1720C2direct (N × H)
8. C2×#322×C127C2semidirect
15. #498×C127C22semidirect
17. #494×C127C22semidirect
19. #487×C127C22semidirect
34. #322×C127C22semidirect
180. C2×#1720C127direct (N × H)
182. C2×C127#1720direct (N × H)
186. C2×#498C2×C127semidirect
187. C2×#494C2×C127semidirect
188. C2×#487C2×C127semidirect
192. #1720C2×C127direct (N × H)
193. C2×#322C2×C127semidirect
194. C127C2×#1720direct (N × H)
201. #498C22×C127semidirect
203. #494C22×C127semidirect
205. #487C22×C127semidirect
220. #322C22×C127semidirect
367. C2C127×#1720direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#498×C1272
C2×#494×C1272
C2×#487×C1272
C2×#486×C1272
C127×#17202
C2×#322×C1272
C2×#1720127