Finite Group Databasebeta

← C31×#13102 #21469861

Subgroups — C31×#13102 #21469861

Order 7936.

688 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3008C211yesmaximal, characteristic
2C31×#2883C211yesmaximal, characteristic, complemented
3C31×#2128C211yesmaximal, characteristic, complemented
4C31×#2122C211yesmaximal, characteristic, complemented
5C31×#1795C211yesmaximal, characteristic, complemented
6C31×#1395C211yesmaximal, characteristic, complemented
7C31×#1391C211yesmaximal, characteristic, complemented
8C2×C31×H(5,𝔽2)C2211yescharacteristic
9C2×C31×AGL(1,ℤ/8)C411yescharacteristic, complemented
10C22×D8×C31C411yescharacteristic, complemented
11C31×#535C2211yescharacteristic
12C31×#518—21no—
13C31×#449—21no—
14C31×#449—21no—
15C31×#447—21no—
16C31×#447—21no—
17C31×#421—21no—
18C31×#418—21no—
19C31×#418C411yescharacteristic, complemented
20C31×#416C411yescharacteristic, complemented
21C31×#413C2211yescharacteristic
22C31×#410—21no—
23C31×#410C2211yescharacteristic
24C31×#410C2211yescharacteristic
25C31×#410C2211yescharacteristic
26C2×C31×#99—22no—
27C2×C31×#99—22no—
28C31×#408C2211yescharacteristic
29C31×#353—22no—
30C31×AΣL(1,𝔽16)—22no—
31C31×H(5,𝔽2)—42no—
32C31×H(5,𝔽2)—42no—
33C2×#41×C31—21no—
34C2×#41×C31C2×C411yescharacteristic
35C2×#41×C31D411yescharacteristic
36C2×#41×C31D411yescharacteristic
37C2×#41×C31C2311yescharacteristic
38C22×D4×C31—21no—
39C22×D4×C31—21no—
40C22×D4×C31—21no—
41C22×D4×C31C2×C411yescharacteristic
42C22×D4×C31C2×C411yescharacteristic, complemented
43C22×D4×C31D411yescharacteristic
44C22×D4×C31D411yescharacteristic
45C31×AGL(1,ℤ/8)—42no—
46C2×SD16×C31—21no—
47C2×D8×C31—21no—
48C2×D8×C31—21no—
49C2×D8×C31—21no—
50C2×D8×C31—42no—
51C2×M16×C31—21no—
52C2×M16×C31C2×C411yescharacteristic, complemented
53C22×C8×C31C2×C411yescharacteristic, complemented
54C31×#127—21no—
55C31×#124—21no—
56C31×#121—21no—
57C31×#121—21no—
58C31×#120—21no—
59C31×#120—21no—
60C4×D4×C31—42no—
61C4×D4×C31—42no—
62C31×#117—21no—
63C31×#117—21no—
64C31×#117—21no—
65C31×#117C2×C411yescharacteristic, complemented
66C2×C22⋊C4×C31—24no—
67C31×#104—42no—
68C31×#103—21no—
69C31×#102—21no—
70C31×#102—21no—
71C31×#102—21no—
72C31×#102—42no—
73C31×#101—21no—
74C31×#100—21no—
75C31×#99—24no—
76C31×#99—21no—
77C31×#99—21no—
78C31×#99—21no—
79C31×#99—21no—
80C31×#99D412yes—
81C31×#99D412yes—
82C31×C22⋊C8—21no—
83C31×C22⋊C8—21no—
84C24×C31—21no—
85C24×C31—21no—
86C24×C31—21no—
87C24×C31—21no—
88#41×C31—22no—
89#41×C31—22no—
90#41×C31—41no—
91#41×C31—41no—
92#41×C31—41no—
93#41×C31—41no—
94#41×C31—41no—
95#41×C31—41no—
96#41×C31—44no—
97C2×Q8×C31C22⋊C411yescharacteristic
98C2×Q8×C31#4111yescharacteristic
99C2×D4×C31—22no—
100C2×D4×C31—22no—
101C2×D4×C31—22no—
102C2×D4×C31—22no—
103C2×D4×C31—21no—
104C2×D4×C31—21no—
105C2×D4×C31—21no—
106C2×D4×C31—21no—
107C2×D4×C31—21no—
108C2×D4×C31—21no—
109C2×D4×C31—21no—
110C2×D4×C31—21no—
111C2×D4×C31—41no—
112C2×D4×C31—41no—
113C2×D4×C31—41no—
114C2×D4×C31—41no—
115C2×D4×C31—44no—
116C2×D4×C31—44no—
117C2×D4×C31—44no—
118C2×D4×C31—41no—
119C2×D4×C31—41no—
120C2×D4×C31C22⋊C411yescharacteristic
121C2×D4×C31C22⋊C411yescharacteristic
122C2×D4×C31C22⋊C411yescharacteristic
123C2×D4×C31C2×D411yescharacteristic
124C2×D4×C31C2×D411yescharacteristic
125C2×D4×C31#4111yescharacteristic
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—44no—
130C22×C4×C31C22×C411yescharacteristic
131C22×C4×C31C2×D411yescharacteristic
132C22×C4×C31C2×D411yescharacteristic
133SD16×C31—42no—
134D8×C31—42no—
135D8×C31—42no—
136D8×C31—42no—
137M16×C31—41no—
138M16×C31—41no—
139M16×C31—41no—
140C2×C8×C31—21no—
141C2×C8×C31—21no—
142C2×C8×C31—41no—
143C2×C8×C31—41no—
144C4⋊C4×C31—21no—
145C4⋊C4×C31—41no—
146C4⋊C4×C31—41no—
147C4⋊C4×C31—41no—
148C22⋊C4×C31—21no—
149C22⋊C4×C31—22no—
150C22⋊C4×C31—22no—
151C22⋊C4×C31—44no—
152C22⋊C4×C31—41no—
153C22⋊C4×C31—41no—
154C22⋊C4×C31—41no—
155C42×C31—21no—
156C42×C31—41no—
157C42×C31—41no—
158C42×C31—41no—
159#13102C3111yesmaximal, characteristic, direct product factor, Sylow
160C23×C31—24no—
161C23×C31—21no—
162C23×C31—21no—
163C23×C31—21no—
164C23×C31—42no—
165C23×C31—42no—
166C23×C31—41no—
167C23×C31—81no—
168C23×C31—161no—
169C23×C31—161no—
170C23×C31—161no—
171C23×C31—161no—
172C23×C31C4×D411yescharacteristic
173C23×C31#12011yescharacteristic
174C23×C31#12111yescharacteristic
175Q8×C31—21no—
176Q8×C31—21no—
177Q8×C31—41no—
178D4×C31—24no—
179D4×C31—21no—
180D4×C31—21no—
181D4×C31—21no—
182D4×C31—21no—
183D4×C31—41no—
184D4×C31—41no—
185D4×C31—41no—
186D4×C31—42no—
187D4×C31—42no—
188D4×C31—42no—
189D4×C31—42no—
190D4×C31—42no—
191D4×C31—41no—
192D4×C31—41no—
193D4×C31—81no—
194D4×C31—161no—
195D4×C31—161no—
196D4×C31#9912yes—
197D4×C31#9912yes—
198C2×C4×C31—21no—
199C2×C4×C31—21no—
200C2×C4×C31—21no—
201C2×C4×C31—21no—
202C2×C4×C31—21no—
203C2×C4×C31—21no—
204C2×C4×C31—21no—
205C2×C4×C31—42no—
206C2×C4×C31—42no—
207C2×C4×C31—42no—
208C2×C4×C31—42no—
209C2×C4×C31—42no—
210C2×C4×C31—41no—
211C2×C4×C31—81no—
212C2×C4×C31—164no—
213C2×C4×C31C2×C22⋊C411yescharacteristic
214C2×C4×C31C4×D411yescharacteristic
215C2×C4×C31#12111yescharacteristic
216C2×C4×C31#12311yescharacteristic
217C8×C31—41no—
218C8×C31—41no—
219C8×C31—81no—
220#3008C2×C3111yescharacteristic
221#2883C2×C3111yescharacteristic, complemented
222#2128C2×C3111yescharacteristic, complemented
223#2122C2×C3111yescharacteristic, complemented
224#1795C2×C3111yescharacteristic, complemented
225#1395C2×C3111yescharacteristic, complemented
226#1391C2×C3111yescharacteristic, complemented
227C22×C31—21no—
228C22×C31—21no—
229C22×C31—21no—
230C22×C31—22no—
231C22×C31—22no—
232C22×C31—41no—
233C22×C31—41no—
234C22×C31—41no—
235C22×C31—41no—
236C22×C31—81no—
237C22×C31—81no—
238C22×C31—81no—
239C22×C31—81no—
240C22×C31—81no—
241C22×C31—84no—
242C22×C31—81no—
243C22×C31—162no—
244C22×C31—162no—
245C22×C31—162no—
246C22×C31#38611yescharacteristic
247C4×C31—21no—
248C4×C31—21no—
249C4×C31—21no—
250C4×C31—41no—
251C4×C31—81no—
252C4×C31—82no—
253C4×C31—82no—
254C4×C31—161no—
255C4×C31C2×#9911yescharacteristic
256C4×C31C2×#9911yescharacteristic
257C2×H(5,𝔽2)C22×C3111yescharacteristic
258C2×AGL(1,ℤ/8)C4×C3111yescharacteristic, complemented
259C22×D8C4×C3111yescharacteristic, complemented
260#535C22×C3111yescharacteristic
261#518—21no—
262#449—21no—
263#449—21no—
264#447—21no—
265#447—21no—
266#421—21no—
267#418—21no—
268#418C4×C3111yescharacteristic, complemented
269#416C4×C3111yescharacteristic, complemented
270#413C22×C3111yescharacteristic
271#410—21no—
272#410C22×C3111yescharacteristic
273#410C22×C3111yescharacteristic
274#410C22×C3111yescharacteristic
275C2×#99—22no—
276C2×#99—22no—
277#408C22×C3111yescharacteristic
278#353—22no—
279AΣL(1,𝔽16)—22no—
280C2×C31—21no—
281C2×C31—41no—
282C2×C31—41no—
283C2×C31—41no—
284C2×C31—42no—
285C2×C31—42no—
286C2×C31—81no—
287C2×C31—161no—
288C2×C31#189811yescharacteristic
289H(5,𝔽2)—42no—
290H(5,𝔽2)—42no—
291C2×#41—21no—
292C2×#41C2×C4×C3111yescharacteristic
293C2×#41D4×C3111yescharacteristic
294C2×#41D4×C3111yescharacteristic
295C2×#41C23×C3111yescharacteristic
296C22×D4—21no—
297C22×D4—21no—
298C22×D4—21no—
299C22×D4C2×C4×C3111yescharacteristic
300C22×D4C2×C4×C3111yescharacteristic, complemented
301C22×D4D4×C3111yescharacteristic
302C22×D4D4×C3111yescharacteristic
303AGL(1,ℤ/8)—42no—
304C2×SD16—21no—
305C2×D8—21no—
306C2×D8—21no—
307C2×D8—21no—
308C2×D8—42no—
309C2×M16—21no—
310C2×M16C2×C4×C3111yescharacteristic, complemented
311C22×C8C2×C4×C3111yescharacteristic, complemented
312#127—21no—
313#124—21no—
314#121—21no—
315#121—21no—
316#120—21no—
317#120—21no—
318C4×D4—42no—
319C4×D4—42no—
320#117—21no—
321#117—21no—
322#117—21no—
323#117C2×C4×C3111yescharacteristic, complemented
324C2×C22⋊C4—24no—
325#104—42no—
326#103—21no—
327#102—21no—
328#102—21no—
329#102—21no—
330#102—42no—
331#101—21no—
332#100—21no—
333#99—24no—
334#99—21no—
335#99—21no—
336#99—21no—
337#99—21no—
338#99D4×C3112yes—
339#99D4×C3112yes—
340C22⋊C8—21no—
341C22⋊C8—21no—
342C31#1310211yescharacteristic, direct product factor, Sylow
343C24—21no—
344C24—21no—
345C24—21no—
346C24—21no—
347#41—22no—
348#41—22no—
349#41—41no—
350#41—41no—
351#41—41no—
352#41—41no—
353#41—41no—
354#41—41no—
355#41—44no—
356C2×Q8C22⋊C4×C3111yescharacteristic
357C2×Q8#41×C3111yescharacteristic
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4C22⋊C4×C3111yescharacteristic
380C2×D4C22⋊C4×C3111yescharacteristic
381C2×D4C22⋊C4×C3111yescharacteristic
382C2×D4C2×D4×C3111yescharacteristic
383C2×D4C2×D4×C3111yescharacteristic
384C2×D4#41×C3111yescharacteristic
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—44no—
389C22×C4C22×C4×C3111yescharacteristic
390C22×C4C2×D4×C3111yescharacteristic
391C22×C4C2×D4×C3111yescharacteristic
392SD16—42no—
393D8—42no—
394D8—42no—
395D8—42no—
396M16—41no—
397M16—41no—
398M16—41no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—41no—
402C2×C8—41no—
403C4⋊C4—21no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C22⋊C4—21no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—44no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C42—21no—
415C42—41no—
416C42—41no—
417C42—41no—
418C23—24no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—42no—
423C23—42no—
424C23—41no—
425C23—81no—
426C23—161no—
427C23—161no—
428C23—161no—
429C23—161no—
430C23C4×D4×C3111yescharacteristic
431C23C31×#12011yescharacteristic
432C23C31×#12111yescharacteristic
433Q8—21no—
434Q8—21no—
435Q8—41no—
436D4—24no—
437D4—21no—
438D4—21no—
439D4—21no—
440D4—21no—
441D4—41no—
442D4—41no—
443D4—41no—
444D4—42no—
445D4—42no—
446D4—42no—
447D4—42no—
448D4—42no—
449D4—41no—
450D4—41no—
451D4—81no—
452D4—161no—
453D4—161no—
454D4C31×#9912yes—
455D4C31×#9912yes—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—81no—
470C2×C4—164no—
471C2×C4C2×C22⋊C4×C3111yescharacteristic
472C2×C4C4×D4×C3111yescharacteristic
473C2×C4C31×#12111yescharacteristic
474C2×C4C31×#12311yescharacteristic
475C8—41no—
476C8—41no—
477C8—81no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—22no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—84no—
493C22—81no—
494C22—162no—
495C22—162no—
496C22—162no—
497C22C31×#38611yescharacteristic
498C4—21no—
499C4—21no—
500C4—21no—
501C4—41no—
502C4—81no—
503C4—82no—
504C4—82no—
505C4—161no—
506C4C2×C31×#9911yescharacteristic
507C4C2×C31×#9911yescharacteristic
508C2—21no—
509C2—41no—
510C2—41no—
511C2—41no—
512C2—42no—
513C2—42no—
514C2—81no—
515C2—161no—
516C2C31×#189811yescharacteristic
subgroupG/H
center, socleC2×C31#1898
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×#41C23×C31
prime radicalC31×#3008C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×C31×H(5,𝔽2) ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2883C2semidirect
3. C31×#2128C2semidirect
4. C31×#2122C2semidirect
5. C31×#1795C2semidirect
6. C31×#1395C2semidirect
7. C31×#1391C2semidirect
9. C2×C31×AGL(1,ℤ/8)C4semidirect
10. C22×D8×C31C4semidirect
19. C31×#418C4semidirect
20. C31×#416C4semidirect
42. C22×D4×C31C2×C4semidirect
52. C2×M16×C31C2×C4semidirect
53. C22×C8×C31C2×C4semidirect
65. C31×#117C2×C4semidirect
159. #13102C31direct (N × H)
221. #2883C2×C31semidirect
222. #2128C2×C31semidirect
223. #2122C2×C31semidirect
224. #1795C2×C31semidirect
225. #1395C2×C31semidirect
226. #1391C2×C31semidirect
258. C2×AGL(1,ℤ/8)C4×C31semidirect
259. C22×D8C4×C31semidirect
268. #418C4×C31semidirect
269. #416C4×C31semidirect
300. C22×D4C2×C4×C31semidirect
310. C2×M16C2×C4×C31semidirect
311. C22×C8C2×C4×C31semidirect
323. #117C2×C4×C31semidirect
342. C31#13102direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#30082
C31×#28832
C31×#21282
C31×#21222
C31×#17952
C31×#13952
C31×#13912
#1310231