Finite Group Databasebeta

← C131×#34388 #45720272

Subgroups — C131×#34388 #45720272

Order 33536.

2438 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#532×C131C211yesmaximal, characteristic
2#2381×C131C212yesmaximal, complemented
3#2380×C131C212yesmaximal, complemented
4#2364×C131C214yesmaximal, complemented
5#2353×C131C214yesmaximal, complemented
6#2339×C131C214yesmaximal, complemented
7#2337×C131C214yesmaximal, complemented
8#2325×C131C214yesmaximal, complemented
9#2318×C131C214yesmaximal, complemented
10#2283×C131C211yesmaximal, characteristic, complemented
11C2×#383×C131C211yesmaximal, characteristic, complemented
12C22×#41×C131C2212yes—
13#532×C131C4116yescomplemented
14C2×C4⋊Q8×C131C2211yescharacteristic
15C2×#127×C131C2211yescharacteristic
16C2×#124×C131C2212yes—
17C2×#121×C131C2212yes—
18C2×#121×C131C2212yes—
19C2×C4×D4×C131C2218yes—
20C2×C4×D4×C131C2212yes—
21C2×C4×D4×C131C2212yes—
22C22×C42×C131C2211yescharacteristic
23#391×C131C2218yes—
24#388×C131C2218yes—
25#388×C131C2218yes—
26#387×C131C2218yes—
27#387×C131C2218yes—
28#386×C131C2218yes—
29#386×C131C2218yes—
30#386×C131C2218yes—
31#386×C131C2218yes—
32#385×C131C2218yes—
33#383×C131C2212yes—
34#383×C131C2212yes—
35#383×C131C2212yes—
36#383×C131C2212yes—
37#383×C131C2212yes—
38#382×C131C2212yes—
39#381×C131C2212yes—
40#381×C131C2214yes—
41#380×C131C2212yes—
42#380×C131C2214yes—
43C4×C4⋊C4×C131C2212yes—
44C4×C22⋊C4×C131C2212yes—
45C4×C22⋊C4×C131C2214yes—
46C4×C22⋊C4×C131C2218yes—
47#376×C131C2212yes—
48C2×#95×C131C2212yes—
49C2×#95×C131C2212yes—
50C2×#95×C131C2212yes—
51C2×#95×C131C2214yes—
52C2×#41×C131—48no—
53C2×#41×C131C2×C4116yescomplemented
54C22×Q8×C131C2312yes—
55C22×D4×C131C2314yes—
56C22×D4×C131C2312yes—
57C22×D4×C131C2312yes—
58C22×D4×C131C2312yes—
59C23×C4×C131C2311yescharacteristic
60C23×C4×C131C2314yes—
61C23×C4×C131C2318yes—
62C23×C4×C131C2311yescharacteristic
63C23×C4×C131C2311yescharacteristic
64C23×C4×C131C2314yes—
65C4⋊Q8×C131C2×C418yescomplemented
66#127×C131C2×C418yescomplemented
67#124×C131C2×C4116yescomplemented
68#121×C131C2×C4116yescomplemented
69#121×C131C2×C4116yescomplemented
70C4×D4×C131—416no—
71C4×D4×C131C2×C4116yescomplemented
72C4×D4×C131C2×C4116yescomplemented
73C2×C4⋊C4×C131C2314yes—
74C2×C4⋊C4×C131C2318yes—
75C2×C4⋊C4×C131C2312yes—
76C2×C4⋊C4×C131C2312yes—
77C2×C22⋊C4×C131C2314yes—
78C2×C22⋊C4×C131C2318yes—
79C2×C22⋊C4×C131C2318yes—
80C2×C22⋊C4×C131C2318yes—
81C2×C22⋊C4×C131C2314yes—
82C2×C22⋊C4×C131C2314yes—
83C2×C42×C131C2×C418yescomplemented
84C2×C42×C131C2314yes—
85C2×C42×C131C2318yes—
86C2×C42×C131C2311yescharacteristic
87C2×C42×C131C2311yescharacteristic
88C2×C42×C131C2311yescharacteristic
89C2×C42×C131C2311yescharacteristic
90#95×C131C2318yes—
91#95×C131C2314yes—
92#95×C131C2314yes—
93#95×C131C2314yes—
94#95×C131C2314yes—
95#95×C131C2318yes—
96#95×C131C2314yes—
97#95×C131C2314yes—
98#95×C131C2314yes—
99#95×C131C2314yes—
100#95×C131C2314yes—
101#95×C131C2314yes—
102C24×C131C2414yes—
103C24×C131C2411yescharacteristic
104#41×C131—432no—
105C2×Q8×C131—44no—
106C2×Q8×C131C22×C418yes—
107C2×D4×C131—216no—
108C2×D4×C131—42no—
109C2×D4×C131—44no—
110C2×D4×C131—42no—
111C2×D4×C131—44no—
112C2×D4×C131C22×C4116yes—
113C2×D4×C131C22×C418yes—
114C2×D4×C131C22×C418yes—
115C2×D4×C131C22×C418yes—
116C22×C4×C131—24no—
117C22×C4×C131—28no—
118C22×C4×C131—42no—
119C22×C4×C131—42no—
120C22×C4×C131—48no—
121C22×C4×C131—48no—
122C22×C4×C131—48no—
123C22×C4×C131—48no—
124C22×C4×C131—44no—
125C22×C4×C131—44no—
126C22×C4×C131C22×C414yes—
127C22×C4×C131C22×C414yes—
128C22×C4×C131C22×C414yes—
129C22×C4×C131C22×C4116yes—
130C22×C4×C131C2418yes—
131C22×C4×C131C2418yes—
132C22×C4×C131C2411yescharacteristic
133C22×C4×C131C2411yescharacteristic
134C22×C4×C131C2411yescharacteristic
135C22×C4×C131C2411yescharacteristic
136C22×C4×C131C2411yescharacteristic
137C22×C4×C131C2411yescharacteristic
138C22×C4×C131C2414yes—
139C4⋊C4×C131—44no—
140C4⋊C4×C131—48no—
141C4⋊C4×C131C22×C418yes—
142C4⋊C4×C131C22×C418yes—
143C22⋊C4×C131—28no—
144C22⋊C4×C131—48no—
145C22⋊C4×C131—48no—
146C22⋊C4×C131—48no—
147C22⋊C4×C131C22×C4116yes—
148C22⋊C4×C131C22×C4116yes—
149C42×C131—44no—
150C42×C131—48no—
151C42×C131C22×C414yes—
152C42×C131C22×C414yes—
153C42×C131C22×C414yes—
154C42×C131C22×C414yes—
155C23×C131—22no—
156C23×C131—22no—
157C23×C131—28no—
158C23×C131—24no—
159C23×C131—28no—
160C23×C131—24no—
161C23×C131C23×C418yes—
162C23×C131C23×C412yes—
163C23×C131H(5,𝔽2)14yes—
164C23×C131C2511yescharacteristic
165Q8×C131—48no—
166D4×C131—232no—
167D4×C131—48no—
168D4×C131—48no—
169D4×C131—48no—
170C2×C4×C131—28no—
171C2×C4×C131—28no—
172C2×C4×C131—22no—
173C2×C4×C131—22no—
174C2×C4×C131—22no—
175C2×C4×C131—22no—
176C2×C4×C131—22no—
177C2×C4×C131—22no—
178C2×C4×C131—28no—
179C2×C4×C131—28no—
180C2×C4×C131—28no—
181C2×C4×C131—28no—
182C2×C4×C131—28no—
183C2×C4×C131—28no—
184C2×C4×C131—44no—
185C2×C4×C131—44no—
186C2×C4×C131—416no—
187C2×C4×C131—88no—
188C2×C4×C131—816no—
189C2×C4×C131C23×C412yes—
190C2×C4×C131C23×C412yes—
191C2×C4×C131C23×C412yes—
192C2×C4×C131C23×C412yes—
193C2×C4×C131C23×C412yes—
194C2×C4×C131C23×C412yes—
195C2×C4×C131C23×C418yes—
196C22×C131—22no—
197C22×C131—22no—
198C22×C131—28no—
199C22×C131—28no—
200C22×C131—28no—
201C22×C131—28no—
202C22×C131—48no—
203C22×C131—48no—
204C22×C131#51814yes—
205C22×C131C24×C411yescharacteristic
206C22×C131C2×H(5,𝔽2)12yes—
207C22×C131#58512yes—
208C22×C131#58512yes—
209C4×C131—22no—
210C4×C131—22no—
211C4×C131—28no—
212C4×C131—22no—
213C4×C131—22no—
214C4×C131—22no—
215C4×C131—22no—
216C4×C131—48no—
217C4×C131—48no—
218C2×C131—24no—
219C2×C131—48no—
220C2×C131C2×#51811yescharacteristic
221C2×C131#342811yescharacteristic
222C2×C131#342811yescharacteristic
223C2×C131#356014yes—
224#34388C13111yesmaximal, characteristic, direct product factor, Sylow
225C131#3438811yescharacteristic, direct product factor, Sylow
226C2×#532C2×C13111yescharacteristic
227#2381C2×C13112yescomplemented
228#2380C2×C13112yescomplemented
229#2364C2×C13114yescomplemented
230#2353C2×C13114yescomplemented
231#2339C2×C13114yescomplemented
232#2337C2×C13114yescomplemented
233#2325C2×C13114yescomplemented
234#2318C2×C13114yescomplemented
235#2283C2×C13111yescharacteristic, complemented
236C2×#383C2×C13111yescharacteristic, complemented
237C22×#41C22×C13112yes—
238#532C4×C131116yescomplemented
239C2×C4⋊Q8C22×C13111yescharacteristic
240C2×#127C22×C13111yescharacteristic
241C2×#124C22×C13112yes—
242C2×#121C22×C13112yes—
243C2×#121C22×C13112yes—
244C2×C4×D4C22×C13118yes—
245C2×C4×D4C22×C13112yes—
246C2×C4×D4C22×C13112yes—
247C22×C42C22×C13111yescharacteristic
248#391C22×C13118yes—
249#388C22×C13118yes—
250#388C22×C13118yes—
251#387C22×C13118yes—
252#387C22×C13118yes—
253#386C22×C13118yes—
254#386C22×C13118yes—
255#386C22×C13118yes—
256#386C22×C13118yes—
257#385C22×C13118yes—
258#383C22×C13112yes—
259#383C22×C13112yes—
260#383C22×C13112yes—
261#383C22×C13112yes—
262#383C22×C13112yes—
263#382C22×C13112yes—
264#381C22×C13114yes—
265#381C22×C13112yes—
266#380C22×C13114yes—
267#380C22×C13112yes—
268C4×C4⋊C4C22×C13112yes—
269C4×C22⋊C4C22×C13114yes—
270C4×C22⋊C4C22×C13118yes—
271C4×C22⋊C4C22×C13112yes—
272#376C22×C13112yes—
273C2×#95C22×C13114yes—
274C2×#95C22×C13112yes—
275C2×#95C22×C13112yes—
276C2×#95C22×C13112yes—
277C2×#41—48no—
278C2×#41C2×C4×C131116yescomplemented
279C22×Q8C23×C13112yes—
280C22×D4C23×C13114yes—
281C22×D4C23×C13112yes—
282C22×D4C23×C13112yes—
283C22×D4C23×C13112yes—
284C23×C4C23×C13111yescharacteristic
285C23×C4C23×C13114yes—
286C23×C4C23×C13118yes—
287C23×C4C23×C13114yes—
288C23×C4C23×C13111yescharacteristic
289C23×C4C23×C13111yescharacteristic
290C4⋊Q8C2×C4×C13118yescomplemented
291#127C2×C4×C13118yescomplemented
292#124C2×C4×C131116yescomplemented
293#121C2×C4×C131116yescomplemented
294#121C2×C4×C131116yescomplemented
295C4×D4—416no—
296C4×D4C2×C4×C131116yescomplemented
297C4×D4C2×C4×C131116yescomplemented
298C2×C4⋊C4C23×C13118yes—
299C2×C4⋊C4C23×C13114yes—
300C2×C4⋊C4C23×C13112yes—
301C2×C4⋊C4C23×C13112yes—
302C2×C22⋊C4C23×C13114yes—
303C2×C22⋊C4C23×C13118yes—
304C2×C22⋊C4C23×C13118yes—
305C2×C22⋊C4C23×C13118yes—
306C2×C22⋊C4C23×C13114yes—
307C2×C22⋊C4C23×C13114yes—
308C2×C42C2×C4×C13118yescomplemented
309C2×C42C23×C13118yes—
310C2×C42C23×C13114yes—
311C2×C42C23×C13111yescharacteristic
312C2×C42C23×C13111yescharacteristic
313C2×C42C23×C13111yescharacteristic
314C2×C42C23×C13111yescharacteristic
315#95C23×C13114yes—
316#95C23×C13114yes—
317#95C23×C13114yes—
318#95C23×C13114yes—
319#95C23×C13114yes—
320#95C23×C13114yes—
321#95C23×C13114yes—
322#95C23×C13118yes—
323#95C23×C13114yes—
324#95C23×C13118yes—
325#95C23×C13114yes—
326#95C23×C13114yes—
327C24C24×C13114yes—
328C24C24×C13111yescharacteristic
329#41—432no—
330C2×Q8—44no—
331C2×Q8C22×C4×C13118yes—
332C2×D4—216no—
333C2×D4—42no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—44no—
337C2×D4C22×C4×C131116yes—
338C2×D4C22×C4×C13118yes—
339C2×D4C22×C4×C13118yes—
340C2×D4C22×C4×C13118yes—
341C22×C4—24no—
342C22×C4—28no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4C22×C4×C13114yes—
352C22×C4C22×C4×C13114yes—
353C22×C4C22×C4×C13114yes—
354C22×C4C22×C4×C131116yes—
355C22×C4C24×C13118yes—
356C22×C4C24×C13118yes—
357C22×C4C24×C13111yescharacteristic
358C22×C4C24×C13111yescharacteristic
359C22×C4C24×C13111yescharacteristic
360C22×C4C24×C13111yescharacteristic
361C22×C4C24×C13111yescharacteristic
362C22×C4C24×C13111yescharacteristic
363C22×C4C24×C13114yes—
364C4⋊C4—44no—
365C4⋊C4—48no—
366C4⋊C4C22×C4×C13118yes—
367C4⋊C4C22×C4×C13118yes—
368C22⋊C4—28no—
369C22⋊C4—48no—
370C22⋊C4—48no—
371C22⋊C4—48no—
372C22⋊C4C22×C4×C131116yes—
373C22⋊C4C22×C4×C131116yes—
374C42—44no—
375C42—48no—
376C42C22×C4×C13114yes—
377C42C22×C4×C13114yes—
378C42C22×C4×C13114yes—
379C42C22×C4×C13114yes—
380C23—22no—
381C23—22no—
382C23—28no—
383C23—24no—
384C23—28no—
385C23—24no—
386C23C23×C4×C13118yes—
387C23C23×C4×C13112yes—
388C23H(5,𝔽2)×C13114yes—
389C23C25×C13111yescharacteristic
390Q8—48no—
391D4—232no—
392D4—48no—
393D4—48no—
394D4—48no—
395C2×C4—28no—
396C2×C4—28no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—28no—
409C2×C4—416no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—88no—
413C2×C4—816no—
414C2×C4C23×C4×C13112yes—
415C2×C4C23×C4×C13112yes—
416C2×C4C23×C4×C13112yes—
417C2×C4C23×C4×C13112yes—
418C2×C4C23×C4×C13112yes—
419C2×C4C23×C4×C13112yes—
420C2×C4C23×C4×C13118yes—
421C22—28no—
422C22—22no—
423C22—22no—
424C22—28no—
425C22—28no—
426C22—28no—
427C22—48no—
428C22—48no—
429C22#518×C13114yes—
430C22C24×C4×C13111yescharacteristic
431C22C2×H(5,𝔽2)×C13112yes—
432C22#585×C13112yes—
433C22#585×C13112yes—
434C4—22no—
435C4—22no—
436C4—28no—
437C4—22no—
438C4—22no—
439C4—22no—
440C4—22no—
441C4—48no—
442C4—48no—
443C2—24no—
444C2—48no—
445C2C2×#518×C13111yescharacteristic
446C2#3428×C13111yescharacteristic
447C2#3428×C13111yescharacteristic
448C2#3560×C13114yes—
subgroupG/H
center, socleC23×C131C25
commutatorC22C24×C4×C131
Frattini, cosocleC23C25×C131
prime radicalC2×#532×C131C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C131 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2381×C131C2semidirect
3. #2380×C131C2semidirect
4. #2364×C131C2semidirect
5. #2353×C131C2semidirect
6. #2339×C131C2semidirect
7. #2337×C131C2semidirect
8. #2325×C131C2semidirect
9. #2318×C131C2semidirect
10. #2283×C131C2semidirect
11. C2×#383×C131C2semidirect
13. #532×C131C4semidirect
53. C2×#41×C131C2×C4semidirect
65. C4⋊Q8×C131C2×C4semidirect
66. #127×C131C2×C4semidirect
67. #124×C131C2×C4semidirect
68. #121×C131C2×C4semidirect
69. #121×C131C2×C4semidirect
71. C4×D4×C131C2×C4semidirect
72. C4×D4×C131C2×C4semidirect
83. C2×C42×C131C2×C4semidirect
224. #34388C131direct (N × H)
225. C131#34388direct (N × H)
227. #2381C2×C131semidirect
228. #2380C2×C131semidirect
229. #2364C2×C131semidirect
230. #2353C2×C131semidirect
231. #2339C2×C131semidirect
232. #2337C2×C131semidirect
233. #2325C2×C131semidirect
234. #2318C2×C131semidirect
235. #2283C2×C131semidirect
236. C2×#383C2×C131semidirect
238. #532C4×C131semidirect
278. C2×#41C2×C4×C131semidirect
290. C4⋊Q8C2×C4×C131semidirect
291. #127C2×C4×C131semidirect
292. #124C2×C4×C131semidirect
293. #121C2×C4×C131semidirect
294. #121C2×C4×C131semidirect
296. C4×D4C2×C4×C131semidirect
297. C4×D4C2×C4×C131semidirect
308. C2×C42C2×C4×C131semidirect

Maximal subgroups

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

subgroupindex
C2×#532×C1312
#2381×C1312
#2380×C1312
#2364×C1312
#2353×C1312
#2339×C1312
#2337×C1312
#2325×C1312
#2318×C1312
#2283×C1312
C2×#383×C1312
#34388131