Finite Group Databasebeta

← C131×#36572 #45722456

Subgroups — C131×#36572 #45722456

Order 33536.

2446 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2848×C131C213yesmaximal, complemented
2#2564×C131C213yesmaximal, complemented
3#2549×C131C213yesmaximal, complemented
4#2505×C131C216yesmaximal, complemented
5#2450×C131C216yesmaximal, complemented
6#2368×C131C213yesmaximal, complemented
7#2349×C131C213yesmaximal, complemented
8C4×#120×C131C213yesmaximal
9C22×#95×C131C211yesmaximal, characteristic, complemented
10C24×C4×C131C2213yes—
11C2×C22⋊Q8×C131C2213yes—
12C2×C22⋊Q8×C131C2216yes—
13C2×#120×C131C2211yescharacteristic
14C2×C4×D4×C131C2213yes—
15C2×C4×D4×C131C2216yes—
16#400×C131C2213yescomplemented
17#400×C131C2216yescomplemented
18#399×C131C2211yescharacteristic, complemented
19#398×C131C2216yescomplemented
20#396×C131C2212yescomplemented
21#395×C131C2213yescomplemented
22#393×C131C2213yescomplemented
23#391×C131C2216yescomplemented
24#389×C131C2216yescomplemented
25#387×C131C2216yescomplemented
26#387×C131C2216yescomplemented
27#386×C131C2216yes—
28#386×C131C2213yes—
29#385×C131C2213yes—
30#385×C131C2216yes—
31#385×C131C2216yes—
32#385×C131C2216yes—
33#385×C131C2216yes—
34#385×C131C2216yescomplemented
35#382×C131C2213yescomplemented
36#380×C131C2216yes—
37#380×C131C2213yes—
38#380×C131C2213yescomplemented
39#379×C131C2213yes—
40C4×C4⋊C4×C131C2213yescomplemented
41C4×C22⋊C4×C131C2216yes—
42C4×C22⋊C4×C131C2213yes—
43C2×#95×C131C2216yes—
44C2×#95×C131C2213yes—
45C2×#95×C131C2213yes—
46C2×#95×C131C2211yescharacteristic, complemented
47C25×C131C2311yescharacteristic
48C22×Q8×C131C2313yes—
49C22×D4×C131C2313yes—
50C23×C4×C131—212no—
51C23×C4×C131—412no—
52C23×C4×C131C2316yes—
53C23×C4×C131C2316yes—
54C23×C4×C131C2313yes—
55C23×C4×C131C2313yes—
56C23×C4×C131C2313yes—
57C22⋊Q8×C131—212no—
58C22⋊Q8×C131—412no—
59#120×C131Q818yescomplemented
60C4×D4×C131—212no—
61C4×D4×C131—412no—
62C2×C4⋊C4×C131C2313yes—
63C2×C4⋊C4×C131C2313yes—
64C2×C4⋊C4×C131C2316yes—
65C2×C4⋊C4×C131C2316yes—
66C2×C4⋊C4×C131C2316yes—
67C2×C4⋊C4×C131C2316yes—
68C2×C4⋊C4×C131C2316yes—
69C2×C22⋊C4×C131C2313yes—
70C2×C22⋊C4×C131C2316yes—
71C2×C22⋊C4×C131C2313yes—
72C2×C22⋊C4×C131C2313yes—
73C2×C22⋊C4×C131C2316yes—
74C2×C22⋊C4×C131C2316yes—
75C2×C22⋊C4×C131C2313yes—
76C2×C42×C131C2313yes—
77C2×C42×C131C2316yes—
78#95×C131C2313yes—
79#95×C131C2316yes—
80#95×C131C2313yes—
81#95×C131C2316yes—
82#95×C131C2316yes—
83#95×C131C2313yes—
84#95×C131C2316yes—
85#95×C131C2311yescharacteristic
86#95×C131C2312yes—
87#95×C131C2316yes—
88#95×C131C2316yes—
89#95×C131C2311yescharacteristic
90#95×C131C2313yes—
91C24×C131—26no—
92C24×C131—26no—
93C24×C131C2×Q814yes—
94C24×C131C2411yescharacteristic
95C24×C131C2413yes—
96C2×Q8×C131—26no—
97C2×Q8×C131—46no—
98C2×D4×C131—26no—
99C2×D4×C131—26no—
100C2×D4×C131C2×Q8112yes—
101C22×C4×C131—26no—
102C22×C4×C131—212no—
103C22×C4×C131—212no—
104C22×C4×C131—212no—
105C22×C4×C131—212no—
106C22×C4×C131—26no—
107C22×C4×C131—46no—
108C22×C4×C131—46no—
109C22×C4×C131—46no—
110C22×C4×C131—46no—
111C22×C4×C131—46no—
112C22×C4×C131—448no—
113C22×C4×C131#41112yes—
114C22×C4×C131C2416yes—
115C22×C4×C131C2413yes—
116C22×C4×C131C2416yes—
117C22×C4×C131C2413yes—
118C22×C4×C131C2413yes—
119C22×C4×C131C2413yes—
120C22×C4×C131C2413yes—
121C4⋊C4×C131—26no—
122C4⋊C4×C131—26no—
123C4⋊C4×C131—212no—
124C4⋊C4×C131—46no—
125C4⋊C4×C131—46no—
126C4⋊C4×C131—46no—
127C4⋊C4×C131—46no—
128C22⋊C4×C131—26no—
129C22⋊C4×C131—212no—
130C22⋊C4×C131—212no—
131C22⋊C4×C131—26no—
132C22⋊C4×C131—26no—
133C22⋊C4×C131—46no—
134C22⋊C4×C131C2×Q8112yes—
135C42×C131—26no—
136C42×C131—46no—
137C23×C131—224no—
138C23×C131—28no—
139C23×C131—224no—
140C23×C131—26no—
141C23×C131—26no—
142C23×C131—23no—
143C23×C131—23no—
144C23×C131C22×Q812yes—
145C23×C131C22×Q816yes—
146C23×C131C2×#4116yes—
147C23×C131H(5,𝔽2)16yes—
148C23×C131C2511yescharacteristic
149Q8×C131—86no—
150D4×C131—224no—
151C2×C4×C131—23no—
152C2×C4×C131—23no—
153C2×C4×C131—26no—
154C2×C4×C131—26no—
155C2×C4×C131—23no—
156C2×C4×C131—26no—
157C2×C4×C131—23no—
158C2×C4×C131—26no—
159C2×C4×C131—26no—
160C2×C4×C131—26no—
161C2×C4×C131—26no—
162C2×C4×C131—26no—
163C2×C4×C131—26no—
164C2×C4×C131—26no—
165C2×C4×C131—412no—
166C2×C4×C131—424no—
167C2×C4×C131—424no—
168C2×C4×C131—412no—
169C2×C4×C131—86no—
170C2×C4×C131C22×Q816yes—
171C2×C4×C131C2×#4116yes—
172C2×C4×C131C2×#4116yes—
173C22×C131—232no—
174C22×C131—26no—
175C22×C131—212no—
176C22×C131—26no—
177C22×C131—212no—
178C22×C131—26no—
179C22×C131—41no—
180C22×C131—43no—
181C22×C131#554112yes—
182C22×C131C23×Q811yescharacteristic
183C22×C131C22×#4113yes—
184C22×C131C2×H(5,𝔽2)13yes—
185C4×C131—26no—
186C4×C131—26no—
187C4×C131—26no—
188C4×C131—43no—
189C4×C131—46no—
190C4×C131—43no—
191C4×C131—46no—
192C2×C131—212no—
193C2×C131—42no—
194C2×C131#344411yescharacteristic
195C2×C131C2×#55413yes—
196C2×C131#349013yes—
197#36572C13111yesmaximal, characteristic, direct product factor, Sylow
198C131#3657211yescharacteristic, direct product factor, Sylow
199#2848C2×C13113yescomplemented
200#2564C2×C13113yescomplemented
201#2549C2×C13113yescomplemented
202#2505C2×C13116yescomplemented
203#2450C2×C13116yescomplemented
204#2368C2×C13113yescomplemented
205#2349C2×C13113yescomplemented
206C4×#120C2×C13113yes—
207C22×#95C2×C13111yescharacteristic, complemented
208C24×C4C22×C13113yes—
209C2×C22⋊Q8C22×C13113yes—
210C2×C22⋊Q8C22×C13116yes—
211C2×#120C22×C13111yescharacteristic
212C2×C4×D4C22×C13113yes—
213C2×C4×D4C22×C13116yes—
214#400C22×C13113yescomplemented
215#400C22×C13116yescomplemented
216#399C22×C13111yescharacteristic, complemented
217#398C22×C13116yescomplemented
218#396C22×C13112yescomplemented
219#395C22×C13113yescomplemented
220#393C22×C13113yescomplemented
221#391C22×C13116yescomplemented
222#389C22×C13116yescomplemented
223#387C22×C13116yescomplemented
224#387C22×C13116yescomplemented
225#386C22×C13116yes—
226#386C22×C13113yes—
227#385C22×C13116yes—
228#385C22×C13116yes—
229#385C22×C13116yes—
230#385C22×C13113yes—
231#385C22×C13116yes—
232#385C22×C13116yescomplemented
233#382C22×C13113yescomplemented
234#380C22×C13116yes—
235#380C22×C13113yes—
236#380C22×C13113yescomplemented
237#379C22×C13113yes—
238C4×C4⋊C4C22×C13113yescomplemented
239C4×C22⋊C4C22×C13116yes—
240C4×C22⋊C4C22×C13113yes—
241C2×#95C22×C13113yes—
242C2×#95C22×C13113yes—
243C2×#95C22×C13116yes—
244C2×#95C22×C13111yescharacteristic, complemented
245C25C23×C13111yescharacteristic
246C22×Q8C23×C13113yes—
247C22×D4C23×C13113yes—
248C23×C4—212no—
249C23×C4—412no—
250C23×C4C23×C13116yes—
251C23×C4C23×C13116yes—
252C23×C4C23×C13113yes—
253C23×C4C23×C13113yes—
254C23×C4C23×C13113yes—
255C22⋊Q8—212no—
256C22⋊Q8—412no—
257#120Q8×C13118yescomplemented
258C4×D4—212no—
259C4×D4—412no—
260C2×C4⋊C4C23×C13113yes—
261C2×C4⋊C4C23×C13116yes—
262C2×C4⋊C4C23×C13113yes—
263C2×C4⋊C4C23×C13116yes—
264C2×C4⋊C4C23×C13116yes—
265C2×C4⋊C4C23×C13116yes—
266C2×C4⋊C4C23×C13116yes—
267C2×C22⋊C4C23×C13116yes—
268C2×C22⋊C4C23×C13113yes—
269C2×C22⋊C4C23×C13113yes—
270C2×C22⋊C4C23×C13113yes—
271C2×C22⋊C4C23×C13116yes—
272C2×C22⋊C4C23×C13116yes—
273C2×C22⋊C4C23×C13113yes—
274C2×C42C23×C13113yes—
275C2×C42C23×C13116yes—
276#95C23×C13111yescharacteristic
277#95C23×C13113yes—
278#95C23×C13116yes—
279#95C23×C13113yes—
280#95C23×C13111yescharacteristic
281#95C23×C13112yes—
282#95C23×C13113yes—
283#95C23×C13116yes—
284#95C23×C13116yes—
285#95C23×C13116yes—
286#95C23×C13113yes—
287#95C23×C13116yes—
288#95C23×C13116yes—
289C24—26no—
290C24—26no—
291C24C2×Q8×C13114yes—
292C24C24×C13111yescharacteristic
293C24C24×C13113yes—
294C2×Q8—26no—
295C2×Q8—46no—
296C2×D4—26no—
297C2×D4—26no—
298C2×D4C2×Q8×C131112yes—
299C22×C4—26no—
300C22×C4—212no—
301C22×C4—212no—
302C22×C4—212no—
303C22×C4—212no—
304C22×C4—26no—
305C22×C4—46no—
306C22×C4—46no—
307C22×C4—46no—
308C22×C4—46no—
309C22×C4—46no—
310C22×C4—448no—
311C22×C4#41×C131112yes—
312C22×C4C24×C13116yes—
313C22×C4C24×C13113yes—
314C22×C4C24×C13116yes—
315C22×C4C24×C13113yes—
316C22×C4C24×C13113yes—
317C22×C4C24×C13113yes—
318C22×C4C24×C13113yes—
319C4⋊C4—26no—
320C4⋊C4—26no—
321C4⋊C4—212no—
322C4⋊C4—46no—
323C4⋊C4—46no—
324C4⋊C4—46no—
325C4⋊C4—46no—
326C22⋊C4—26no—
327C22⋊C4—212no—
328C22⋊C4—212no—
329C22⋊C4—26no—
330C22⋊C4—26no—
331C22⋊C4—46no—
332C22⋊C4C2×Q8×C131112yes—
333C42—26no—
334C42—46no—
335C23—224no—
336C23—28no—
337C23—224no—
338C23—26no—
339C23—26no—
340C23—23no—
341C23—23no—
342C23C22×Q8×C13112yes—
343C23C22×Q8×C13116yes—
344C23C2×#41×C13116yes—
345C23H(5,𝔽2)×C13116yes—
346C23C25×C13111yescharacteristic
347Q8—86no—
348D4—224no—
349C2×C4—23no—
350C2×C4—23no—
351C2×C4—26no—
352C2×C4—26no—
353C2×C4—23no—
354C2×C4—26no—
355C2×C4—23no—
356C2×C4—26no—
357C2×C4—26no—
358C2×C4—26no—
359C2×C4—26no—
360C2×C4—26no—
361C2×C4—26no—
362C2×C4—26no—
363C2×C4—412no—
364C2×C4—424no—
365C2×C4—424no—
366C2×C4—412no—
367C2×C4—86no—
368C2×C4C22×Q8×C13116yes—
369C2×C4C2×#41×C13116yes—
370C2×C4C2×#41×C13116yes—
371C22—232no—
372C22—26no—
373C22—212no—
374C22—26no—
375C22—212no—
376C22—26no—
377C22—41no—
378C22—43no—
379C22#554×C131112yes—
380C22C23×Q8×C13111yescharacteristic
381C22C22×#41×C13113yes—
382C22C2×H(5,𝔽2)×C13113yes—
383C4—26no—
384C4—26no—
385C4—26no—
386C4—43no—
387C4—46no—
388C4—43no—
389C4—46no—
390C2—212no—
391C2—42no—
392C2#3444×C13111yescharacteristic
393C2C2×#554×C13113yes—
394C2#3490×C13113yes—
subgroupG/H
center, socleC23×C131C25
commutator, Frattini, cosocleC23C25×C131
prime radicalC2×#120×C131C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C131 ◁ G

Complements

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

NH ≅ G/Ntype
1. #2848×C131C2semidirect
2. #2564×C131C2semidirect
3. #2549×C131C2semidirect
4. #2505×C131C2semidirect
5. #2450×C131C2semidirect
6. #2368×C131C2semidirect
7. #2349×C131C2semidirect
9. C22×#95×C131C2semidirect
16. #400×C131C22semidirect
17. #400×C131C22semidirect
18. #399×C131C22semidirect
19. #398×C131C22semidirect
20. #396×C131C22semidirect
21. #395×C131C22semidirect
22. #393×C131C22semidirect
23. #391×C131C22semidirect
24. #389×C131C22semidirect
25. #387×C131C22semidirect
26. #387×C131C22semidirect
34. #385×C131C22semidirect
35. #382×C131C22semidirect
38. #380×C131C22semidirect
40. C4×C4⋊C4×C131C22semidirect
46. C2×#95×C131C22semidirect
59. #120×C131Q8semidirect
197. #36572C131direct (N × H)
198. C131#36572direct (N × H)
199. #2848C2×C131semidirect
200. #2564C2×C131semidirect
201. #2549C2×C131semidirect
202. #2505C2×C131semidirect
203. #2450C2×C131semidirect
204. #2368C2×C131semidirect
205. #2349C2×C131semidirect
207. C22×#95C2×C131semidirect
214. #400C22×C131semidirect
215. #400C22×C131semidirect
216. #399C22×C131semidirect
217. #398C22×C131semidirect
218. #396C22×C131semidirect
219. #395C22×C131semidirect
220. #393C22×C131semidirect
221. #391C22×C131semidirect
222. #389C22×C131semidirect
223. #387C22×C131semidirect
224. #387C22×C131semidirect
232. #385C22×C131semidirect
233. #382C22×C131semidirect
236. #380C22×C131semidirect
238. C4×C4⋊C4C22×C131semidirect
244. C2×#95C22×C131semidirect
257. #120Q8×C131semidirect

Maximal subgroups

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

subgroupindex
#2848×C1312
#2564×C1312
#2549×C1312
#2505×C1312
#2450×C1312
#2368×C1312
#2349×C1312
C4×#120×C1312
C22×#95×C1312
#36572131