Finite Group Databasebeta

← C31×#13361 #21470120

Subgroups — C31×#13361 #21470120

Order 7936.

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
1C31×#3200C211yesmaximal, characteristic, complemented
2C31×#3022C211yesmaximal, characteristic, complemented
3C31×#3021C211yesmaximal, characteristic
4C31×#2123C211yesmaximal, characteristic, complemented
5C31×#2122C211yesmaximal, characteristic, complemented
6C31×#1794C211yesmaximal, characteristic, complemented
7C31×#1392C211yesmaximal, characteristic, complemented
8C31×#585C2211yescharacteristic
9C2×C31×#137—21no—
10C2×C31×AGL(1,ℤ/8)—21no—
11C31×#536C2211yescharacteristic
12C31×#535C2211yescharacteristic
13C31×#485—21no—
14C31×#482—21no—
15C31×#468—21no—
16C31×#465—21no—
17C31×#456—22no—
18C31×#455—22no—
19C31×#454—22no—
20C31×#453—22no—
21C31×#450—21no—
22C31×#448—21no—
23C31×#421—21no—
24C31×#421—21no—
25C31×#421—21no—
26C31×#416—21no—
27C31×#413C2211yescharacteristic
28C31×#413C2211yescharacteristic
29C31×#410C2211yescharacteristic
30C31×#408C2211yescharacteristic
31C31×#352—22no—
32C31×AΣL(1,𝔽16)—22no—
33C31×2−1+4—41no—
34C31×2−1+4D412yes—
35C31×H(5,𝔽2)—21no—
36C31×H(5,𝔽2)—21no—
37C31×H(5,𝔽2)—21no—
38C31×H(5,𝔽2)—41no—
39C2×#41×C31—21no—
40C2×#41×C31—21no—
41C2×#41×C31—21no—
42C2×#41×C31—42no—
43C2×#41×C31C2311yescharacteristic
44C22×Q8×C31D411yescharacteristic
45C22×D4×C31D411yescharacteristic
46C31×#137—24no—
47C31×#137—42no—
48C31×AGL(1,ℤ/8)—24no—
49C31×AGL(1,ℤ/8)—42no—
50C2×Q16×C31—41no—
51C2×SD16×C31—21no—
52C2×SD16×C31—21no—
53C2×SD16×C31—41no—
54C2×SD16×C31—41no—
55C2×D8×C31—41no—
56C2×M16×C31—21no—
57C2×M16×C31—21no—
58C2×M16×C31D411yescharacteristic
59C22×C8×C31D411yescharacteristic
60C31×C4⋊Q8—21no—
61C31×#127—21no—
62C31×#124—21no—
63C31×#124—21no—
64C31×#123—21no—
65C31×#123—21no—
66C31×C22⋊Q8—21no—
67C31×C22⋊Q8—21no—
68C31×#121—21no—
69C31×#121—21no—
70C31×#120—21no—
71C31×#120—21no—
72C31×#117—21no—
73C31×#117D411yescharacteristic
74C31×#117D411yescharacteristic
75C31×#107—21no—
76C31×#106—21no—
77C31×#104—24no—
78C31×#104—24no—
79C31×#104—42no—
80C31×#103—21no—
81C31×#103—21no—
82C31×#103—41no—
83C31×#102—21no—
84C31×#102—21no—
85C31×#102—41no—
86C31×#101—21no—
87C31×#101—21no—
88C31×#100D412yes—
89C31×#100D412yes—
90C31×#99—21no—
91C31×#99—21no—
92C31×C22⋊C8—21no—
93C31×C22⋊C8—21no—
94C24×C31—21no—
95#41×C31—22no—
96#41×C31—22no—
97#41×C31—22no—
98#41×C31—22no—
99#41×C31—22no—
100#41×C31—22no—
101#41×C31—41no—
102#41×C31—41no—
103#41×C31—41no—
104#41×C31—41no—
105#41×C31—41no—
106#41×C31—42no—
107#41×C31—42no—
108#41×C31—42no—
109#41×C31—42no—
110#41×C31—41no—
111C2×Q8×C31—21no—
112C2×Q8×C31—21no—
113C2×Q8×C31—21no—
114C2×Q8×C31—21no—
115C2×Q8×C31—21no—
116C2×Q8×C31—41no—
117C2×Q8×C31—41no—
118C2×Q8×C31—42no—
119C2×Q8×C31C2×D411yescharacteristic
120C2×D4×C31—21no—
121C2×D4×C31—21no—
122C2×D4×C31—21no—
123C2×D4×C31—21no—
124C2×D4×C31—21no—
125C2×D4×C31—41no—
126C2×D4×C31—41no—
127C2×D4×C31—42no—
128C2×D4×C31—42no—
129C2×D4×C31—42no—
130C2×D4×C31—41no—
131C2×D4×C31—41no—
132C2×D4×C31—41no—
133C2×D4×C31—41no—
134C2×D4×C31—41no—
135C2×D4×C31C2×D411yescharacteristic
136C2×D4×C31C2×D411yescharacteristic
137C2×D4×C31#4111yescharacteristic
138C22×C4×C31—21no—
139C22×C4×C31—41no—
140C22×C4×C31—41no—
141C22×C4×C31—41no—
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D411yescharacteristic
145Q16×C31—44no—
146SD16×C31—44no—
147SD16×C31—44no—
148SD16×C31—81no—
149SD16×C31—81no—
150D8×C31—44no—
151M16×C31—22no—
152M16×C31—22no—
153M16×C31—41no—
154M16×C31—41no—
155M16×C31—41no—
156C2×C8×C31—21no—
157C2×C8×C31—21no—
158C2×C8×C31—41no—
159C2×C8×C31—41no—
160C2×C8×C31—41no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—41no—
164C4⋊C4×C31—41no—
165C4⋊C4×C31—41no—
166C22⋊C4×C31—21no—
167C22⋊C4×C31—21no—
168C22⋊C4×C31—41no—
169C22⋊C4×C31—41no—
170C22⋊C4×C31—41no—
171C22⋊C4×C31—41no—
172C22⋊C4×C31—41no—
173C42×C31—21no—
174C42×C31—21no—
175C42×C31—41no—
176#13361C3111yesmaximal, characteristic, direct product factor, Sylow
177C23×C31—41no—
178C23×C31—41no—
179C23×C31—41no—
180C23×C31—41no—
181C23×C31—81no—
182C23×C31—161no—
183C23×C31#12111yescharacteristic
184C23×C31#12111yescharacteristic
185C23×C31#12711yescharacteristic
186Q8×C31—22no—
187Q8×C31—22no—
188Q8×C31—22no—
189Q8×C31—41no—
190Q8×C31—42no—
191Q8×C31—41no—
192Q8×C31—41no—
193Q8×C31—41no—
194D4×C31—22no—
195D4×C31—22no—
196D4×C31—22no—
197D4×C31—41no—
198D4×C31—41no—
199D4×C31—41no—
200D4×C31—42no—
201D4×C31—42no—
202D4×C31—42no—
203D4×C31—42no—
204D4×C31—42no—
205D4×C31—41no—
206D4×C31—41no—
207D4×C31—41no—
208D4×C31—161no—
209D4×C31—161no—
210C2×C4×C31—21no—
211C2×C4×C31—21no—
212C2×C4×C31—21no—
213C2×C4×C31—21no—
214C2×C4×C31—21no—
215C2×C4×C31—21no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—41no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
222C2×C4×C31—42no—
223C2×C4×C31—41no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—81no—
227C2×C4×C31—81no—
228C2×C4×C31#12011yescharacteristic
229C2×C4×C31#12011yescharacteristic
230C2×C4×C31#12011yescharacteristic
231C2×C4×C31#12111yescharacteristic
232C8×C31—42no—
233C8×C31—42no—
234C8×C31—41no—
235C8×C31—41no—
236#3200C2×C3111yescharacteristic, complemented
237#3022C2×C3111yescharacteristic, complemented
238#3021C2×C3111yescharacteristic
239#2123C2×C3111yescharacteristic, complemented
240#2122C2×C3111yescharacteristic, complemented
241#1794C2×C3111yescharacteristic, complemented
242#1392C2×C3111yescharacteristic, complemented
243C22×C31—21no—
244C22×C31—21no—
245C22×C31—21no—
246C22×C31—41no—
247C22×C31—41no—
248C22×C31—41no—
249C22×C31—42no—
250C22×C31—81no—
251C22×C31—81no—
252C22×C31—81no—
253C22×C31—161no—
254C22×C31—161no—
255C22×C31—161no—
256C22×C31—162no—
257C22×C31#39211yescharacteristic
258C4×C31—21no—
259C4×C31—21no—
260C4×C31—21no—
261C4×C31—42no—
262C4×C31—81no—
263C4×C31—81no—
264C4×C31—81no—
265C4×C31—161no—
266C4×C31B(4,𝔽2)11yescharacteristic
267C4×C31B(4,𝔽2)11yescharacteristic
268#585C22×C3111yescharacteristic
269C2×#137—21no—
270C2×AGL(1,ℤ/8)—21no—
271#536C22×C3111yescharacteristic
272#535C22×C3111yescharacteristic
273#485—21no—
274#482—21no—
275#468—21no—
276#465—21no—
277#456—22no—
278#455—22no—
279#454—22no—
280#453—22no—
281#450—21no—
282#448—21no—
283#421—21no—
284#421—21no—
285#421—21no—
286#416—21no—
287#413C22×C3111yescharacteristic
288#413C22×C3111yescharacteristic
289#410C22×C3111yescharacteristic
290#408C22×C3111yescharacteristic
291#352—22no—
292AΣL(1,𝔽16)—22no—
293C2×C31—21no—
294C2×C31—41no—
295C2×C31—41no—
296C2×C31—41no—
297C2×C31—82no—
298C2×C31—161no—
299C2×C31#202111yescharacteristic
3002−1+4—41no—
3012−1+4D4×C3112yes—
302H(5,𝔽2)—21no—
303H(5,𝔽2)—21no—
304H(5,𝔽2)—21no—
305H(5,𝔽2)—41no—
306C2×#41—21no—
307C2×#41—21no—
308C2×#41—21no—
309C2×#41—42no—
310C2×#41C23×C3111yescharacteristic
311C22×Q8D4×C3111yescharacteristic
312C22×D4D4×C3111yescharacteristic
313#137—24no—
314#137—42no—
315AGL(1,ℤ/8)—24no—
316AGL(1,ℤ/8)—42no—
317C2×Q16—41no—
318C2×SD16—21no—
319C2×SD16—21no—
320C2×SD16—41no—
321C2×SD16—41no—
322C2×D8—41no—
323C2×M16—21no—
324C2×M16—21no—
325C2×M16D4×C3111yescharacteristic
326C22×C8D4×C3111yescharacteristic
327C4⋊Q8—21no—
328#127—21no—
329#124—21no—
330#124—21no—
331#123—21no—
332#123—21no—
333C22⋊Q8—21no—
334C22⋊Q8—21no—
335#121—21no—
336#121—21no—
337#120—21no—
338#120—21no—
339#117—21no—
340#117D4×C3111yescharacteristic
341#117D4×C3111yescharacteristic
342#107—21no—
343#106—21no—
344#104—24no—
345#104—24no—
346#104—42no—
347#103—21no—
348#103—21no—
349#103—41no—
350#102—21no—
351#102—21no—
352#102—41no—
353#101—21no—
354#101—21no—
355#100D4×C3112yes—
356#100D4×C3112yes—
357#99—21no—
358#99—21no—
359C22⋊C8—21no—
360C22⋊C8—21no—
361C31#1336111yescharacteristic, direct product factor, Sylow
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×C3111yescharacteristic
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×C3111yescharacteristic
404C2×D4C2×D4×C3111yescharacteristic
405C2×D4#41×C3111yescharacteristic
406C22×C4—21no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4C2×D4×C3111yescharacteristic
411C22×C4C2×D4×C3111yescharacteristic
412C22×C4C2×D4×C3111yescharacteristic
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—
450C23C31×#12111yescharacteristic
451C23C31×#12111yescharacteristic
452C23C31×#12711yescharacteristic
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×C4C31×#12011yescharacteristic
496C2×C4C31×#12011yescharacteristic
497C2×C4C31×#12011yescharacteristic
498C2×C4C31×#12111yescharacteristic
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—
517C22C31×#39211yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—42no—
522C4—81no—
523C4—81no—
524C4—81no—
525C4—161no—
526C4C31×B(4,𝔽2)11yescharacteristic
527C4C31×B(4,𝔽2)11yescharacteristic
528C2—21no—
529C2—41no—
530C2—41no—
531C2—41no—
532C2—82no—
533C2—161no—
534C2C31×#202111yescharacteristic
subgroupG/H
center, socleC2×C31#2021
commutator, Frattini, cosocleC2×#41C23×C31
prime radicalC31×#3021C2
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3200C2semidirect
2. C31×#3022C2semidirect
4. C31×#2123C2semidirect
5. C31×#2122C2semidirect
6. C31×#1794C2semidirect
7. C31×#1392C2semidirect
176. #13361C31direct (N × H)
236. #3200C2×C31semidirect
237. #3022C2×C31semidirect
239. #2123C2×C31semidirect
240. #2122C2×C31semidirect
241. #1794C2×C31semidirect
242. #1392C2×C31semidirect
361. C31#13361direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32002
C31×#30222
C31×#30212
C31×#21232
C31×#21222
C31×#17942
C31×#13922
#1336131