Finite Group Databasebeta

← C131×#13361 #45699245

Subgroups — C131×#13361 #45699245

Order 33536.

686 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3200×C131C211yesmaximal, characteristic, complemented
2#3022×C131C211yesmaximal, characteristic, complemented
3#3021×C131C211yesmaximal, characteristic
4#2123×C131C211yesmaximal, characteristic, complemented
5#2122×C131C211yesmaximal, characteristic, complemented
6#1794×C131C211yesmaximal, characteristic, complemented
7#1392×C131C211yesmaximal, characteristic, complemented
8#585×C131C2211yescharacteristic
9C2×#137×C131—21no—
10C2×AGL(1,ℤ/8)×C131—21no—
11#536×C131C2211yescharacteristic
12#535×C131C2211yescharacteristic
13#485×C131—21no—
14#482×C131—21no—
15#468×C131—21no—
16#465×C131—21no—
17#456×C131—22no—
18#455×C131—22no—
19#454×C131—22no—
20#453×C131—22no—
21#450×C131—21no—
22#448×C131—21no—
23#421×C131—21no—
24#421×C131—21no—
25#421×C131—21no—
26#416×C131—21no—
27#413×C131C2211yescharacteristic
28#413×C131C2211yescharacteristic
29#410×C131C2211yescharacteristic
30#408×C131C2211yescharacteristic
31#352×C131—22no—
32AΣL(1,𝔽16)×C131—22no—
332−1+4×C131—41no—
342−1+4×C131D412yes—
35H(5,𝔽2)×C131—21no—
36H(5,𝔽2)×C131—21no—
37H(5,𝔽2)×C131—21no—
38H(5,𝔽2)×C131—41no—
39C2×#41×C131—21no—
40C2×#41×C131—21no—
41C2×#41×C131—21no—
42C2×#41×C131—42no—
43C2×#41×C131C2311yescharacteristic
44C22×Q8×C131D411yescharacteristic
45C22×D4×C131D411yescharacteristic
46#137×C131—24no—
47#137×C131—42no—
48AGL(1,ℤ/8)×C131—24no—
49AGL(1,ℤ/8)×C131—42no—
50C2×Q16×C131—41no—
51C2×SD16×C131—21no—
52C2×SD16×C131—21no—
53C2×SD16×C131—41no—
54C2×SD16×C131—41no—
55C2×D8×C131—41no—
56C2×M16×C131—21no—
57C2×M16×C131—21no—
58C2×M16×C131D411yescharacteristic
59C22×C8×C131D411yescharacteristic
60C4⋊Q8×C131—21no—
61#127×C131—21no—
62#124×C131—21no—
63#124×C131—21no—
64#123×C131—21no—
65#123×C131—21no—
66C22⋊Q8×C131—21no—
67C22⋊Q8×C131—21no—
68#121×C131—21no—
69#121×C131—21no—
70#120×C131—21no—
71#120×C131—21no—
72#117×C131—21no—
73#117×C131D411yescharacteristic
74#117×C131D411yescharacteristic
75#107×C131—21no—
76#106×C131—21no—
77#104×C131—24no—
78#104×C131—24no—
79#104×C131—42no—
80#103×C131—21no—
81#103×C131—21no—
82#103×C131—41no—
83#102×C131—21no—
84#102×C131—21no—
85#102×C131—41no—
86#101×C131—21no—
87#101×C131—21no—
88#100×C131D412yes—
89#100×C131D412yes—
90#99×C131—21no—
91#99×C131—21no—
92C22⋊C8×C131—21no—
93C22⋊C8×C131—21no—
94C24×C131—21no—
95#41×C131—22no—
96#41×C131—22no—
97#41×C131—22no—
98#41×C131—22no—
99#41×C131—22no—
100#41×C131—22no—
101#41×C131—41no—
102#41×C131—41no—
103#41×C131—41no—
104#41×C131—41no—
105#41×C131—41no—
106#41×C131—42no—
107#41×C131—42no—
108#41×C131—42no—
109#41×C131—42no—
110#41×C131—41no—
111C2×Q8×C131—21no—
112C2×Q8×C131—21no—
113C2×Q8×C131—21no—
114C2×Q8×C131—21no—
115C2×Q8×C131—21no—
116C2×Q8×C131—41no—
117C2×Q8×C131—41no—
118C2×Q8×C131—42no—
119C2×Q8×C131C2×D411yescharacteristic
120C2×D4×C131—21no—
121C2×D4×C131—21no—
122C2×D4×C131—21no—
123C2×D4×C131—21no—
124C2×D4×C131—21no—
125C2×D4×C131—41no—
126C2×D4×C131—41no—
127C2×D4×C131—42no—
128C2×D4×C131—42no—
129C2×D4×C131—42no—
130C2×D4×C131—41no—
131C2×D4×C131—41no—
132C2×D4×C131—41no—
133C2×D4×C131—41no—
134C2×D4×C131—41no—
135C2×D4×C131C2×D411yescharacteristic
136C2×D4×C131C2×D411yescharacteristic
137C2×D4×C131#4111yescharacteristic
138C22×C4×C131—21no—
139C22×C4×C131—41no—
140C22×C4×C131—41no—
141C22×C4×C131—41no—
142C22×C4×C131C2×D411yescharacteristic
143C22×C4×C131C2×D411yescharacteristic
144C22×C4×C131C2×D411yescharacteristic
145Q16×C131—44no—
146SD16×C131—44no—
147SD16×C131—44no—
148SD16×C131—81no—
149SD16×C131—81no—
150D8×C131—44no—
151M16×C131—22no—
152M16×C131—22no—
153M16×C131—41no—
154M16×C131—41no—
155M16×C131—41no—
156C2×C8×C131—21no—
157C2×C8×C131—21no—
158C2×C8×C131—41no—
159C2×C8×C131—41no—
160C2×C8×C131—41no—
161C4⋊C4×C131—21no—
162C4⋊C4×C131—21no—
163C4⋊C4×C131—41no—
164C4⋊C4×C131—41no—
165C4⋊C4×C131—41no—
166C22⋊C4×C131—21no—
167C22⋊C4×C131—21no—
168C22⋊C4×C131—41no—
169C22⋊C4×C131—41no—
170C22⋊C4×C131—41no—
171C22⋊C4×C131—41no—
172C22⋊C4×C131—41no—
173C42×C131—21no—
174C42×C131—21no—
175C42×C131—41no—
176C23×C131—41no—
177C23×C131—41no—
178C23×C131—41no—
179C23×C131—41no—
180C23×C131—81no—
181C23×C131—161no—
182C23×C131#12111yescharacteristic
183C23×C131#12111yescharacteristic
184C23×C131#12711yescharacteristic
185Q8×C131—22no—
186Q8×C131—22no—
187Q8×C131—22no—
188Q8×C131—41no—
189Q8×C131—42no—
190Q8×C131—41no—
191Q8×C131—41no—
192Q8×C131—41no—
193D4×C131—22no—
194D4×C131—22no—
195D4×C131—22no—
196D4×C131—41no—
197D4×C131—41no—
198D4×C131—41no—
199D4×C131—42no—
200D4×C131—42no—
201D4×C131—42no—
202D4×C131—42no—
203D4×C131—42no—
204D4×C131—41no—
205D4×C131—41no—
206D4×C131—41no—
207D4×C131—161no—
208D4×C131—161no—
209C2×C4×C131—21no—
210C2×C4×C131—21no—
211C2×C4×C131—21no—
212C2×C4×C131—21no—
213C2×C4×C131—21no—
214C2×C4×C131—21no—
215C2×C4×C131—42no—
216C2×C4×C131—42no—
217C2×C4×C131—42no—
218C2×C4×C131—41no—
219C2×C4×C131—41no—
220C2×C4×C131—41no—
221C2×C4×C131—42no—
222C2×C4×C131—41no—
223C2×C4×C131—41no—
224C2×C4×C131—41no—
225C2×C4×C131—81no—
226C2×C4×C131—81no—
227C2×C4×C131#12011yescharacteristic
228C2×C4×C131#12011yescharacteristic
229C2×C4×C131#12011yescharacteristic
230C2×C4×C131#12111yescharacteristic
231C8×C131—42no—
232C8×C131—42no—
233C8×C131—41no—
234C8×C131—41no—
235C22×C131—21no—
236C22×C131—21no—
237C22×C131—21no—
238C22×C131—41no—
239C22×C131—41no—
240C22×C131—41no—
241C22×C131—42no—
242C22×C131—81no—
243C22×C131—81no—
244C22×C131—81no—
245C22×C131—161no—
246C22×C131—161no—
247C22×C131—161no—
248C22×C131—162no—
249C22×C131#39211yescharacteristic
250C4×C131—21no—
251C4×C131—21no—
252C4×C131—21no—
253C4×C131—42no—
254C4×C131—81no—
255C4×C131—81no—
256C4×C131—81no—
257C4×C131—161no—
258C4×C131B(4,𝔽2)11yescharacteristic
259C4×C131B(4,𝔽2)11yescharacteristic
260C2×C131—21no—
261C2×C131—41no—
262C2×C131—41no—
263C2×C131—41no—
264C2×C131—82no—
265C2×C131—161no—
266C2×C131#202111yescharacteristic
267#13361C13111yesmaximal, characteristic, direct product factor, Sylow
268C131#1336111yescharacteristic, direct product factor, Sylow
269#3200C2×C13111yescharacteristic, complemented
270#3022C2×C13111yescharacteristic, complemented
271#3021C2×C13111yescharacteristic
272#2123C2×C13111yescharacteristic, complemented
273#2122C2×C13111yescharacteristic, complemented
274#1794C2×C13111yescharacteristic, complemented
275#1392C2×C13111yescharacteristic, complemented
276#585C22×C13111yescharacteristic
277C2×#137—21no—
278C2×AGL(1,ℤ/8)—21no—
279#536C22×C13111yescharacteristic
280#535C22×C13111yescharacteristic
281#485—21no—
282#482—21no—
283#468—21no—
284#465—21no—
285#456—22no—
286#455—22no—
287#454—22no—
288#453—22no—
289#450—21no—
290#448—21no—
291#421—21no—
292#421—21no—
293#421—21no—
294#416—21no—
295#413C22×C13111yescharacteristic
296#413C22×C13111yescharacteristic
297#410C22×C13111yescharacteristic
298#408C22×C13111yescharacteristic
299#352—22no—
300AΣL(1,𝔽16)—22no—
3012−1+4—41no—
3022−1+4D4×C13112yes—
303H(5,𝔽2)—21no—
304H(5,𝔽2)—21no—
305H(5,𝔽2)—21no—
306H(5,𝔽2)—41no—
307C2×#41—21no—
308C2×#41—21no—
309C2×#41—21no—
310C2×#41—42no—
311C2×#41C23×C13111yescharacteristic
312C22×Q8D4×C13111yescharacteristic
313C22×D4D4×C13111yescharacteristic
314#137—24no—
315#137—42no—
316AGL(1,ℤ/8)—24no—
317AGL(1,ℤ/8)—42no—
318C2×Q16—41no—
319C2×SD16—21no—
320C2×SD16—21no—
321C2×SD16—41no—
322C2×SD16—41no—
323C2×D8—41no—
324C2×M16—21no—
325C2×M16—21no—
326C2×M16D4×C13111yescharacteristic
327C22×C8D4×C13111yescharacteristic
328C4⋊Q8—21no—
329#127—21no—
330#124—21no—
331#124—21no—
332#123—21no—
333#123—21no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336#121—21no—
337#121—21no—
338#120—21no—
339#120—21no—
340#117—21no—
341#117D4×C13111yescharacteristic
342#117D4×C13111yescharacteristic
343#107—21no—
344#106—21no—
345#104—24no—
346#104—24no—
347#104—42no—
348#103—21no—
349#103—21no—
350#103—41no—
351#102—21no—
352#102—21no—
353#102—41no—
354#101—21no—
355#101—21no—
356#100D4×C13112yes—
357#100D4×C13112yes—
358#99—21no—
359#99—21no—
360C22⋊C8—21no—
361C22⋊C8—21no—
362C24—21no—
363#41—22no—
364#41—22no—
365#41—22no—
366#41—22no—
367#41—22no—
368#41—22no—
369#41—41no—
370#41—41no—
371#41—41no—
372#41—41no—
373#41—41no—
374#41—42no—
375#41—42no—
376#41—42no—
377#41—42no—
378#41—41no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—42no—
387C2×Q8C2×D4×C13111yescharacteristic
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4C2×D4×C13111yescharacteristic
404C2×D4C2×D4×C13111yescharacteristic
405C2×D4#41×C13111yescharacteristic
406C22×C4—21no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4C2×D4×C13111yescharacteristic
411C22×C4C2×D4×C13111yescharacteristic
412C22×C4C2×D4×C13111yescharacteristic
413Q16—44no—
414SD16—44no—
415SD16—44no—
416SD16—81no—
417SD16—81no—
418D8—44no—
419M16—22no—
420M16—22no—
421M16—41no—
422M16—41no—
423M16—41no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C42—21no—
442C42—21no—
443C42—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—81no—
449C23—161no—
450C23#121×C13111yescharacteristic
451C23#121×C13111yescharacteristic
452C23#127×C13111yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—22no—
456Q8—41no—
457Q8—42no—
458Q8—41no—
459Q8—41no—
460Q8—41no—
461D4—22no—
462D4—22no—
463D4—22no—
464D4—41no—
465D4—41no—
466D4—41no—
467D4—42no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—42no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—161no—
476D4—161no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4#120×C13111yescharacteristic
496C2×C4#120×C13111yescharacteristic
497C2×C4#120×C13111yescharacteristic
498C2×C4#121×C13111yescharacteristic
499C8—42no—
500C8—42no—
501C8—41no—
502C8—41no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—42no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—161no—
514C22—161no—
515C22—161no—
516C22—162no—
517C22#392×C13111yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—42no—
522C4—81no—
523C4—81no—
524C4—81no—
525C4—161no—
526C4B(4,𝔽2)×C13111yescharacteristic
527C4B(4,𝔽2)×C13111yescharacteristic
528C2—21no—
529C2—41no—
530C2—41no—
531C2—41no—
532C2—82no—
533C2—161no—
534C2#2021×C13111yescharacteristic
subgroupG/H
center, socleC2×C131#2021
commutator, Frattini, cosocleC2×#41C23×C131
prime radical#3021×C131C2
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C131 ◁ C2×C4×C131 ◁ C2×#41×C131 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3200×C131C2semidirect
2. #3022×C131C2semidirect
4. #2123×C131C2semidirect
5. #2122×C131C2semidirect
6. #1794×C131C2semidirect
7. #1392×C131C2semidirect
267. #13361C131direct (N × H)
268. C131#13361direct (N × H)
269. #3200C2×C131semidirect
270. #3022C2×C131semidirect
272. #2123C2×C131semidirect
273. #2122C2×C131semidirect
274. #1794C2×C131semidirect
275. #1392C2×C131semidirect

Maximal subgroups

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

subgroupindex
#3200×C1312
#3022×C1312
#3021×C1312
#2123×C1312
#2122×C1312
#1794×C1312
#1392×C1312
#13361131