Finite Group Databasebeta

← C131×#29705 #45715589

Subgroups — C131×#29705 #45715589

Order 33536.

1018 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3460×C131C211yesmaximal, characteristic
2#3079×C131C212yesmaximal, complemented
3#2563×C131C211yesmaximal, characteristic, complemented
4#2061×C131C212yesmaximal, complemented
5#2040×C131C212yesmaximal, complemented
6#2037×C131C214yesmaximal, complemented
7#1806×C131C212yesmaximal, complemented
8#1800×C131C211yesmaximal, characteristic, complemented
9#544×C131C2211yescharacteristic
10#542×C131—22no—
11#540×C131—22no—
12#538×C131—22no—
13#537×C131C2211yescharacteristic
14#535×C131C2212yes—
15#532×C131—22no—
16C2×C4⋊Q8×C131C2211yescharacteristic
17#529×C131—22no—
18#525×C131—22no—
19C4×#41×C131C2212yes—
20C2×C4×D4×C131C2211yescharacteristic
21#477×C131—22no—
22#476×C131—22no—
23#475×C131—22no—
24#474×C131—22no—
25#428×C131C2214yes—
26#423×C131C2212yes—
27C2×#104×C131C2214yes—
28#417×C131C2214yes—
29C2×#100×C131C2212yes—
30#393×C131—22no—
31#390×C131C2212yes—
32#382×C131C2212yes—
33#380×C131—22no—
34C4×C4⋊C4×C131C2211yescharacteristic
35#343×C131C2212yes—
36#339×C131C2214yes—
37C2×#41×C131—22no—
38C2×#41×C131C2312yes—
39C22×Q8×C131—21no—
40C22×D4×C131C2311yescharacteristic
41C23×C4×C131D412yes—
42C2×M16×C131C2314yes—
43C4⋊Q8×C131—22no—
44C4⋊Q8×C131—22no—
45C4⋊Q8×C131—22no—
46#127×C131—22no—
47#126×C131—24no—
48#125×C131—22no—
49#124×C131—24no—
50#124×C131—22no—
51#123×C131—24no—
52#123×C131—24no—
53C22⋊Q8×C131—22no—
54C22⋊Q8×C131—22no—
55C22⋊Q8×C131—22no—
56#121×C131—22no—
57#121×C131—24no—
58#121×C131—22no—
59#121×C131—22no—
60#121×C131—22no—
61#121×C131—22no—
62#120×C131—24no—
63C4×Q8×C131D412yes—
64C4×Q8×C131D412yes—
65C4×D4×C131—24no—
66C4×D4×C131—22no—
67C4×D4×C131—22no—
68C4×D4×C131—22no—
69C4×D4×C131—22no—
70C4×D4×C131D412yes—
71C4×D4×C131D412yes—
72#117×C131—22no—
73#117×C131—22no—
74#117×C131C2312yes—
75C2×C4⋊C4×C131—21no—
76C2×C4⋊C4×C131—21no—
77C2×C4⋊C4×C131C2312yes—
78C2×C4⋊C4×C131C2311yescharacteristic
79C2×C22⋊C4×C131—24no—
80C2×C22⋊C4×C131D412yes—
81C2×C42×C131—22no—
82C2×C42×C131C2312yes—
83C2×C42×C131C2311yescharacteristic
84#107×C131—24no—
85#106×C131—24no—
86C4⋊C8×C131—22no—
87C4⋊C8×C131—22no—
88#104×C131—44no—
89#103×C131—24no—
90#102×C131—24no—
91#100×C131—42no—
92#95×C131—21no—
93#95×C131—24no—
94C24×C131C2×D412yes—
95#41×C131—22no—
96#41×C131—22no—
97#41×C131—22no—
98#41×C131—22no—
99#41×C131—84no—
100C2×Q8×C131—22no—
101C2×Q8×C131—42no—
102C2×Q8×C131—82no—
103C2×Q8×C131C2×D412yes—
104C2×D4×C131—22no—
105C2×D4×C131—24no—
106C2×D4×C131—24no—
107C2×D4×C131—22no—
108C2×D4×C131—22no—
109C2×D4×C131—21no—
110C2×D4×C131—21no—
111C2×D4×C131—22no—
112C2×D4×C131C2×D412yes—
113C22×C4×C131—22no—
114C22×C4×C131—22no—
115C22×C4×C131—22no—
116C22×C4×C131—22no—
117C22×C4×C131—22no—
118C22×C4×C131—22no—
119C22×C4×C131—22no—
120C22×C4×C131—44no—
121C22×C4×C131C2×D411yescharacteristic
122C22×C4×C131C2×D412yes—
123C22×C4×C131C2×D411yescharacteristic
124C22×C4×C131C2411yescharacteristic
125M16×C131—24no—
126M16×C131—24no—
127C2×C8×C131—24no—
128C4⋊C4×C131—22no—
129C4⋊C4×C131—22no—
130C4⋊C4×C131—24no—
131C4⋊C4×C131—22no—
132C4⋊C4×C131—22no—
133C4⋊C4×C131—22no—
134C4⋊C4×C131—24no—
135C4⋊C4×C131—42no—
136C4⋊C4×C131—41no—
137C4⋊C4×C131—41no—
138C4⋊C4×C131C2×D412yes—
139C4⋊C4×C131C2×D412yes—
140C22⋊C4×C131—24no—
141C22⋊C4×C131—24no—
142C22⋊C4×C131—24no—
143C22⋊C4×C131—22no—
144C22⋊C4×C131—24no—
145C22⋊C4×C131—22no—
146C22⋊C4×C131—22no—
147C22⋊C4×C131—84no—
148C42×C131—22no—
149C42×C131—22no—
150C42×C131—22no—
151C42×C131—22no—
152C42×C131—22no—
153C42×C131C2×D412yes—
154C42×C131C2×D412yes—
155C23×C131—22no—
156C23×C131—22no—
157C23×C131—22no—
158C23×C131—22no—
159C23×C131—42no—
160C23×C131—42no—
161C23×C131C22×D411yescharacteristic
162Q8×C131—22no—
163Q8×C131—22no—
164Q8×C131—84no—
165Q8×C131—82no—
166D4×C131—22no—
167D4×C131—22no—
168D4×C131—42no—
169D4×C131—42no—
170D4×C131—42no—
171D4×C131—42no—
172D4×C131—84no—
173D4×C131—82no—
174D4×C131—84no—
175C2×C4×C131—22no—
176C2×C4×C131—22no—
177C2×C4×C131—22no—
178C2×C4×C131—22no—
179C2×C4×C131—22no—
180C2×C4×C131—41no—
181C2×C4×C131—42no—
182C2×C4×C131—42no—
183C2×C4×C131—44no—
184C2×C4×C131—44no—
185C2×C4×C131—42no—
186C2×C4×C131—42no—
187C2×C4×C131—44no—
188C2×C4×C131—42no—
189C2×C4×C131—42no—
190C2×C4×C131—42no—
191C2×C4×C131—82no—
192C2×C4×C131—82no—
193C2×C4×C131#12012yes—
194C2×C4×C131#12011yescharacteristic
195C2×C4×C131#12011yescharacteristic
196C2×C4×C131C22×D412yes—
197C2×C4×C131H(5,𝔽2)11yescharacteristic
198C2×C4×C1312−1+411yescharacteristic
199C2×C4×C1312−1+412yes—
200C8×C131—44no—
201C22×C131—21no—
202C22×C131—21no—
203C22×C131—44no—
204C22×C131—42no—
205C22×C131—42no—
206C22×C131—42no—
207C22×C131—42no—
208C22×C131—42no—
209C22×C131—42no—
210C22×C131—82no—
211C22×C131C2×#12011yescharacteristic
212C22×C131#53611yescharacteristic
213C22×C131#53711yescharacteristic
214C4×C131—21no—
215C4×C131—22no—
216C4×C131—42no—
217C4×C131—42no—
218C4×C131—42no—
219C4×C131—42no—
220C4×C131—82no—
221C4×C131—82no—
222C4×C131#45512yes—
223C4×C131#53612yes—
224C4×C131#53712yes—
225C2×C131—21no—
226C2×C131—21no—
227C2×C131—82no—
228C2×C131—82no—
229C2×C131#261611yescharacteristic
230C2×C131C2×#45511yescharacteristic
231C2×C131#302111yescharacteristic
232#29705C13111yesmaximal, characteristic, direct product factor, Sylow
233C131#2970511yescharacteristic, direct product factor, Sylow
234#3460C2×C13111yescharacteristic
235#3079C2×C13112yescomplemented
236#2563C2×C13111yescharacteristic, complemented
237#2061C2×C13112yescomplemented
238#2040C2×C13112yescomplemented
239#2037C2×C13114yescomplemented
240#1806C2×C13112yescomplemented
241#1800C2×C13111yescharacteristic, complemented
242#544C22×C13111yescharacteristic
243#542—22no—
244#540—22no—
245#538—22no—
246#537C22×C13111yescharacteristic
247#535C22×C13112yes—
248#532—22no—
249C2×C4⋊Q8C22×C13111yescharacteristic
250#529—22no—
251#525—22no—
252C4×#41C22×C13112yes—
253C2×C4×D4C22×C13111yescharacteristic
254#477—22no—
255#476—22no—
256#475—22no—
257#474—22no—
258#428C22×C13114yes—
259#423C22×C13112yes—
260C2×#104C22×C13114yes—
261#417C22×C13114yes—
262C2×#100C22×C13112yes—
263#393—22no—
264#390C22×C13112yes—
265#382C22×C13112yes—
266#380—22no—
267C4×C4⋊C4C22×C13111yescharacteristic
268#343C22×C13112yes—
269#339C22×C13114yes—
270C2×#41—22no—
271C2×#41C23×C13112yes—
272C22×Q8—21no—
273C22×D4C23×C13111yescharacteristic
274C23×C4D4×C13112yes—
275C2×M16C23×C13114yes—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
279#127—22no—
280#126—24no—
281#125—22no—
282#124—24no—
283#124—22no—
284#123—24no—
285#123—24no—
286C22⋊Q8—22no—
287C22⋊Q8—22no—
288C22⋊Q8—22no—
289#121—22no—
290#121—24no—
291#121—22no—
292#121—22no—
293#121—22no—
294#121—22no—
295#120—24no—
296C4×Q8D4×C13112yes—
297C4×Q8D4×C13112yes—
298C4×D4—24no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4D4×C13112yes—
304C4×D4D4×C13112yes—
305#117—22no—
306#117—22no—
307#117C23×C13112yes—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4C23×C13112yes—
311C2×C4⋊C4C23×C13111yescharacteristic
312C2×C22⋊C4—24no—
313C2×C22⋊C4D4×C13112yes—
314C2×C42—22no—
315C2×C42C23×C13112yes—
316C2×C42C23×C13111yescharacteristic
317#107—24no—
318#106—24no—
319C4⋊C8—22no—
320C4⋊C8—22no—
321#104—44no—
322#103—24no—
323#102—24no—
324#100—42no—
325#95—21no—
326#95—24no—
327C24C2×D4×C13112yes—
328#41—22no—
329#41—22no—
330#41—22no—
331#41—22no—
332#41—84no—
333C2×Q8—22no—
334C2×Q8—42no—
335C2×Q8—82no—
336C2×Q8C2×D4×C13112yes—
337C2×D4—22no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—22no—
345C2×D4C2×D4×C13112yes—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—44no—
354C22×C4C2×D4×C13111yescharacteristic
355C22×C4C2×D4×C13112yes—
356C22×C4C2×D4×C13111yescharacteristic
357C22×C4C24×C13111yescharacteristic
358M16—24no—
359M16—24no—
360C2×C8—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—42no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4C2×D4×C13112yes—
372C4⋊C4C2×D4×C13112yes—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—22no—
377C22⋊C4—24no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—84no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42C2×D4×C13112yes—
387C42C2×D4×C13112yes—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—42no—
393C23—42no—
394C23C22×D4×C13111yescharacteristic
395Q8—22no—
396Q8—22no—
397Q8—84no—
398Q8—82no—
399D4—22no—
400D4—22no—
401D4—42no—
402D4—42no—
403D4—42no—
404D4—42no—
405D4—82no—
406D4—84no—
407D4—84no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—41no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4#120×C13112yes—
427C2×C4#120×C13111yescharacteristic
428C2×C4#120×C13111yescharacteristic
429C2×C4C22×D4×C13112yes—
430C2×C4H(5,𝔽2)×C13111yescharacteristic
431C2×C42−1+4×C13111yescharacteristic
432C2×C42−1+4×C13112yes—
433C8—44no—
434C22—21no—
435C22—21no—
436C22—44no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—82no—
444C22C2×#120×C13111yescharacteristic
445C22#536×C13111yescharacteristic
446C22#537×C13111yescharacteristic
447C4—21no—
448C4—22no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—42no—
453C4—82no—
454C4—82no—
455C4#455×C13112yes—
456C4#536×C13112yes—
457C4#537×C13112yes—
458C2—21no—
459C2—21no—
460C2—82no—
461C2—82no—
462C2#2616×C13111yescharacteristic
463C2C2×#455×C13111yescharacteristic
464C2#3021×C13111yescharacteristic
subgroupG/H
center, socleC22×C131C2×#120
commutator, Frattini, cosocleC22×C4C24×C131
prime radical#3460×C131C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C131 ◁ C2×C42×C131 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3079×C131C2semidirect
3. #2563×C131C2semidirect
4. #2061×C131C2semidirect
5. #2040×C131C2semidirect
6. #2037×C131C2semidirect
7. #1806×C131C2semidirect
8. #1800×C131C2semidirect
232. #29705C131direct (N × H)
233. C131#29705direct (N × H)
235. #3079C2×C131semidirect
236. #2563C2×C131semidirect
237. #2061C2×C131semidirect
238. #2040C2×C131semidirect
239. #2037C2×C131semidirect
240. #1806C2×C131semidirect
241. #1800C2×C131semidirect

Maximal subgroups

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

subgroupindex
#3460×C1312
#3079×C1312
#2563×C1312
#2061×C1312
#2040×C1312
#2037×C1312
#1806×C1312
#1800×C1312
#29705131