Finite Group Databasebeta

← C31×#20597 #21477356

Subgroups — C31×#20597 #21477356

Order 7936.

3998 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#41×C31C211yesmaximal, characteristic
2C2×C31×#416C214yesmaximal, complemented
3C22×C31×C22⋊C8C211yesmaximal, characteristic, complemented
4C31×#2280C211yesmaximal, characteristic, complemented
5C31×#1788C214yesmaximal, complemented
6C31×#1787C212yesmaximal, complemented
7C31×#1786C212yesmaximal, complemented
8C22×#41×C31—216no—
9C22×#41×C31—24no—
10C22×#41×C31—21no—
11C22×#41×C31—21no—
12C22×#41×C31C418yescomplemented
13C22×#41×C31C2212yes—
14C22×#41×C31C2212yes—
15C23×Q8×C31C2211yescharacteristic
16C23×D4×C31—21no—
17C23×D4×C31C2211yescharacteristic
18C24×C4×C31—21no—
19C24×C4×C31C2211yescharacteristic
20C23×C8×C31C2212yescomplemented
21C2×C31×#117C2212yescomplemented
22C31×#416—216no—
23C2×C31×#103C2218yescomplemented
24C2×C31×#102C2218yescomplemented
25C2×C31×C22⋊C8—24no—
26C2×C31×C22⋊C8C2212yes—
27C2×C31×C22⋊C8C2212yescomplemented
28C31×#381—21no—
29C31×#380C2212yes—
30C31×#380C2212yescomplemented
31C31×#379—21no—
32C4×C22⋊C4×C31—21no—
33C4×C22⋊C4×C31—21no—
34C25×C31—21no—
35C25×C31D411yescharacteristic
36C2×#41×C31—216no—
37C2×#41×C31—264no—
38C2×#41×C31—216no—
39C2×#41×C31—264no—
40C2×#41×C31—28no—
41C2×#41×C31—28no—
42C2×#41×C31—28no—
43C2×#41×C31—28no—
44C2×#41×C31—24no—
45C2×#41×C31—24no—
46C2×#41×C31—28no—
47C2×#41×C31C2×C418yes—
48C2×#41×C31C2×C418yes—
49C2×#41×C31D4116yes—
50C22×Q8×C31—28no—
51C22×Q8×C31—22no—
52C22×Q8×C31—21no—
53C22×Q8×C31C2×C414yes—
54C22×Q8×C31C2312yes—
55C22×D4×C31—216no—
56C22×D4×C31—28no—
57C22×D4×C31—22no—
58C22×D4×C31—22no—
59C22×D4×C31—28no—
60C22×D4×C31—21no—
61C22×D4×C31—22no—
62C22×D4×C31C2×C414yes—
63C22×D4×C31C2312yes—
64C23×C4×C31—216no—
65C23×C4×C31—21no—
66C23×C4×C31—28no—
67C23×C4×C31—22no—
68C23×C4×C31—28no—
69C23×C4×C31—22no—
70C23×C4×C31C2×C414yes—
71C23×C4×C31D411yescharacteristic
72C23×C4×C31D411yescharacteristic
73C23×C4×C31D411yescharacteristic
74C23×C4×C31D414yes—
75C23×C4×C31C2311yescharacteristic
76C23×C4×C31C2311yescharacteristic
77C23×C4×C31C2311yescharacteristic
78C22×C8×C31—24no—
79C22×C8×C31—28no—
80C22×C8×C31C2314yes—
81C31×#117—28no—
82C2×C4⋊C4×C31C2314yes—
83C2×C22⋊C4×C31—22no—
84C2×C22⋊C4×C31—22no—
85C2×C22⋊C4×C31—22no—
86C2×C42×C31—22no—
87C31×#103—216no—
88C31×#102—216no—
89C31×C22⋊C8—24no—
90C31×C22⋊C8—24no—
91C31×#95—22no—
92C31×#95—22no—
93C24×C31—28no—
94C24×C31—24no—
95C24×C31—24no—
96C24×C31—24no—
97C24×C31—22no—
98C24×C31—42no—
99C24×C31—44no—
100C24×C31—44no—
101C24×C31C22⋊C414yes—
102C24×C31C2×D411yescharacteristic
103C24×C31C2×D411yescharacteristic
104C24×C31C2×D411yescharacteristic
105#41×C31—264no—
106#41×C31—232no—
107#41×C31—264no—
108#41×C31—216no—
109#41×C31—216no—
110#41×C31—216no—
111#41×C31—232no—
112#41×C31C22⋊C4132yescomplemented
113C2×Q8×C31—216no—
114C2×Q8×C31—28no—
115C2×Q8×C31—216no—
116C2×Q8×C31—24no—
117C2×Q8×C31—22no—
118C2×Q8×C31—24no—
119C2×Q8×C31C22×C414yes—
120C2×Q8×C31C2×D418yes—
121C2×D4×C31—264no—
122C2×D4×C31—216no—
123C2×D4×C31—28no—
124C2×D4×C31—216no—
125C2×D4×C31—28no—
126C2×D4×C31—28no—
127C2×D4×C31—216no—
128C2×D4×C31—216no—
129C2×D4×C31—24no—
130C2×D4×C31—24no—
131C2×D4×C31—22no—
132C2×D4×C31C22×C414yes—
133C2×D4×C31C2×D418yes—
134C22×C4×C31—28no—
135C22×C4×C31—264no—
136C22×C4×C31—216no—
137C22×C4×C31—24no—
138C22×C4×C31—24no—
139C22×C4×C31—22no—
140C22×C4×C31—28no—
141C22×C4×C31—28no—
142C22×C4×C31—216no—
143C22×C4×C31—24no—
144C22×C4×C31—216no—
145C22×C4×C31—24no—
146C22×C4×C31—216no—
147C22×C4×C31—24no—
148C22×C4×C31—24no—
149C22×C4×C31—24no—
150C22×C4×C31C22⋊C414yes—
151C22×C4×C31C22⋊C414yes—
152C22×C4×C31C22⋊C414yes—
153C22×C4×C31C22×C412yes—
154C22×C4×C31C22×C412yes—
155C22×C4×C31C22×C412yes—
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D412yes—
159C22×C4×C31C2×D412yes—
160C22×C4×C31C2×D418yes—
161C22×C4×C31C2×D412yes—
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2411yescharacteristic
164C2×C8×C31—24no—
165C2×C8×C31—28no—
166C2×C8×C31—48no—
167C4⋊C4×C31—28no—
168C22⋊C4×C31—44no—
169C22⋊C4×C31—44no—
170C22⋊C4×C31—44no—
171C42×C31—44no—
172#20597C3111yesmaximal, characteristic, direct product factor, Sylow
173C23×C31—22no—
174C23×C31—22no—
175C23×C31—22no—
176C23×C31—22no—
177C23×C31—216no—
178C23×C31—24no—
179C23×C31—22no—
180C23×C31—28no—
181C23×C31—22no—
182C23×C31—22no—
183C23×C31—28no—
184C23×C31—28no—
185C23×C31—44no—
186C23×C31—416no—
187C23×C31—48no—
188C23×C31—432no—
189C23×C31—44no—
190C23×C31—44no—
191C23×C31—48no—
192C23×C31C2×C22⋊C412yes—
193C23×C31C2×C22⋊C412yes—
194C23×C31C2×C22⋊C412yes—
195C23×C31#12014yes—
196C23×C31#13514yes—
197C23×C31C22×D411yescharacteristic
198Q8×C31—216no—
199Q8×C31—24no—
200Q8×C31—28no—
201Q8×C31C2×C22⋊C418yes—
202D4×C31—232no—
203D4×C31—216no—
204D4×C31—232no—
205D4×C31—24no—
206D4×C31—28no—
207D4×C31C2×C22⋊C418yes—
208C2×C4×C31—216no—
209C2×C4×C31—232no—
210C2×C4×C31—24no—
211C2×C4×C31—22no—
212C2×C4×C31—28no—
213C2×C4×C31—22no—
214C2×C4×C31—28no—
215C2×C4×C31—24no—
216C2×C4×C31—24no—
217C2×C4×C31—24no—
218C2×C4×C31—24no—
219C2×C4×C31—28no—
220C2×C4×C31—232no—
221C2×C4×C31—28no—
222C2×C4×C31—44no—
223C2×C4×C31—44no—
224C2×C4×C31—44no—
225C2×C4×C31C2×C22⋊C412yes—
226C2×C4×C31C2×C22⋊C412yes—
227C2×C4×C31C2×C22⋊C418yes—
228C2×C4×C31C2×C22⋊C412yes—
229C2×C4×C31#12018yes—
230C2×C4×C31#12014yes—
231C2×C4×C31C23×C411yescharacteristic
232C2×C4×C31C22×D414yes—
233C2×C4×C31C22×D411yescharacteristic
234C8×C31—44no—
235C23×#41C2×C3111yescharacteristic
236C2×#416C2×C3114yescomplemented
237C22×C22⋊C8C2×C3111yescharacteristic, complemented
238#2280C2×C3111yescharacteristic, complemented
239#1788C2×C3114yescomplemented
240#1787C2×C3112yescomplemented
241#1786C2×C3112yescomplemented
242C22×C31—22no—
243C22×C31—22no—
244C22×C31—22no—
245C22×C31—28no—
246C22×C31—28no—
247C22×C31—22no—
248C22×C31—22no—
249C22×C31—22no—
250C22×C31—22no—
251C22×C31—28no—
252C22×C31—416no—
253C22×C31—432no—
254C22×C31—44no—
255C22×C31—44no—
256C22×C31—44no—
257C22×C31—416no—
258C22×C31#37914yes—
259C22×C31#41614yescomplemented
260C22×C31C22×C22⋊C411yescharacteristic
261C22×C31C2×#12012yes—
262C22×C31C2×#13512yes—
263C22×C31#57512yes—
264C4×C31—22no—
265C4×C31—28no—
266C4×C31—22no—
267C4×C31—84no—
268C4×C31#37914yes—
269C4×C31C22×C22⋊C412yes—
270C4×C31C2×#12012yes—
271C22×#41—216no—
272C22×#41—24no—
273C22×#41—21no—
274C22×#41—21no—
275C22×#41C4×C3118yescomplemented
276C22×#41C22×C3112yes—
277C22×#41C22×C3112yes—
278C23×Q8C22×C3111yescharacteristic
279C23×D4—21no—
280C23×D4C22×C3111yescharacteristic
281C24×C4—21no—
282C24×C4C22×C3111yescharacteristic
283C23×C8C22×C3112yescomplemented
284C2×#117C22×C3112yescomplemented
285#416—216no—
286C2×#103C22×C3118yescomplemented
287C2×#102C22×C3118yescomplemented
288C2×C22⋊C8—24no—
289C2×C22⋊C8C22×C3112yes—
290C2×C22⋊C8C22×C3112yescomplemented
291#381—21no—
292#380C22×C3112yes—
293#380C22×C3112yescomplemented
294#379—21no—
295C4×C22⋊C4—21no—
296C4×C22⋊C4—21no—
297C2×C31—24no—
298C2×C31—24no—
299C2×C31—42no—
300C2×C31—48no—
301C2×C31C2×#37911yescharacteristic
302C2×C31C2×#41611yescharacteristic
303C2×C31#289611yescharacteristic
304C2×C31#300214yes—
305C25—21no—
306C25D4×C3111yescharacteristic
307C2×#41—216no—
308C2×#41—264no—
309C2×#41—216no—
310C2×#41—264no—
311C2×#41—28no—
312C2×#41—28no—
313C2×#41—28no—
314C2×#41—28no—
315C2×#41—24no—
316C2×#41—24no—
317C2×#41—28no—
318C2×#41C2×C4×C3118yes—
319C2×#41C2×C4×C3118yes—
320C2×#41D4×C31116yes—
321C22×Q8—28no—
322C22×Q8—22no—
323C22×Q8—21no—
324C22×Q8C2×C4×C3114yes—
325C22×Q8C23×C3112yes—
326C22×D4—216no—
327C22×D4—28no—
328C22×D4—22no—
329C22×D4—22no—
330C22×D4—28no—
331C22×D4—21no—
332C22×D4—22no—
333C22×D4C2×C4×C3114yes—
334C22×D4C23×C3112yes—
335C23×C4—216no—
336C23×C4—21no—
337C23×C4—28no—
338C23×C4—22no—
339C23×C4—22no—
340C23×C4—28no—
341C23×C4C2×C4×C3114yes—
342C23×C4D4×C3111yescharacteristic
343C23×C4D4×C3111yescharacteristic
344C23×C4D4×C3111yescharacteristic
345C23×C4D4×C3114yes—
346C23×C4C23×C3111yescharacteristic
347C23×C4C23×C3111yescharacteristic
348C23×C4C23×C3111yescharacteristic
349C22×C8—24no—
350C22×C8—28no—
351C22×C8C23×C3114yes—
352#117—28no—
353C2×C4⋊C4C23×C3114yes—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C42—22no—
358#103—216no—
359#102—216no—
360C22⋊C8—24no—
361C22⋊C8—24no—
362#95—22no—
363#95—22no—
364C31#2059711yescharacteristic, direct product factor, Sylow
365C24—28no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—22no—
370C24—42no—
371C24—44no—
372C24—44no—
373C24C22⋊C4×C3114yes—
374C24C2×D4×C3111yescharacteristic
375C24C2×D4×C3111yescharacteristic
376C24C2×D4×C3111yescharacteristic
377#41—264no—
378#41—232no—
379#41—264no—
380#41—216no—
381#41—216no—
382#41—216no—
383#41—232no—
384#41C22⋊C4×C31132yescomplemented
385C2×Q8—216no—
386C2×Q8—28no—
387C2×Q8—216no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8C22×C4×C3114yes—
392C2×Q8C2×D4×C3118yes—
393C2×D4—264no—
394C2×D4—216no—
395C2×D4—28no—
396C2×D4—216no—
397C2×D4—28no—
398C2×D4—28no—
399C2×D4—216no—
400C2×D4—216no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4C22×C4×C3114yes—
405C2×D4C2×D4×C3118yes—
406C22×C4—28no—
407C22×C4—264no—
408C22×C4—216no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—216no—
415C22×C4—24no—
416C22×C4—216no—
417C22×C4—24no—
418C22×C4—216no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4C22⋊C4×C3114yes—
423C22×C4C22⋊C4×C3114yes—
424C22×C4C22⋊C4×C3114yes—
425C22×C4C22×C4×C3112yes—
426C22×C4C22×C4×C3112yes—
427C22×C4C22×C4×C3112yes—
428C22×C4C2×D4×C3111yescharacteristic
429C22×C4C2×D4×C3111yescharacteristic
430C22×C4C2×D4×C3112yes—
431C22×C4C2×D4×C3112yes—
432C22×C4C2×D4×C3111yescharacteristic
433C22×C4C2×D4×C3118yes—
434C22×C4C2×D4×C3112yes—
435C22×C4C24×C3111yescharacteristic
436C2×C8—24no—
437C2×C8—28no—
438C2×C8—48no—
439C4⋊C4—28no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C42—44no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—216no—
449C23—28no—
450C23—24no—
451C23—22no—
452C23—28no—
453C23—22no—
454C23—22no—
455C23—28no—
456C23—44no—
457C23—416no—
458C23—48no—
459C23—432no—
460C23—48no—
461C23—44no—
462C23—44no—
463C23C2×C22⋊C4×C3112yes—
464C23C2×C22⋊C4×C3112yes—
465C23C2×C22⋊C4×C3112yes—
466C23C31×#12014yes—
467C23C31×#13514yes—
468C23C22×D4×C3111yescharacteristic
469Q8—216no—
470Q8—24no—
471Q8—28no—
472Q8C2×C22⋊C4×C3118yes—
473D4—232no—
474D4—216no—
475D4—232no—
476D4—24no—
477D4—28no—
478D4C2×C22⋊C4×C3118yes—
479C2×C4—216no—
480C2×C4—232no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—28no—
484C2×C4—22no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—232no—
492C2×C4—28no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4C2×C22⋊C4×C3112yes—
497C2×C4C2×C22⋊C4×C3112yes—
498C2×C4C2×C22⋊C4×C3118yes—
499C2×C4C2×C22⋊C4×C3112yes—
500C2×C4C31×#12018yes—
501C2×C4C31×#12014yes—
502C2×C4C23×C4×C3111yescharacteristic
503C2×C4C22×D4×C3114yes—
504C2×C4C22×D4×C3111yescharacteristic
505C8—44no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—28no—
510C22—28no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—28no—
516C22—416no—
517C22—432no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—416no—
522C22C31×#37914yes—
523C22C31×#41614yescomplemented
524C22C22×C22⋊C4×C3111yescharacteristic
525C22C2×C31×#12012yes—
526C22C2×C31×#13512yes—
527C22C31×#57512yes—
528C4—22no—
529C4—28no—
530C4—22no—
531C4—84no—
532C4C31×#37914yes—
533C4C22×C22⋊C4×C3112yes—
534C4C2×C31×#12012yes—
535C2—24no—
536C2—24no—
537C2—42no—
538C2—48no—
539C2C2×C31×#37911yescharacteristic
540C2C2×C31×#41611yescharacteristic
541C2C31×#289611yescharacteristic
542C2C31×#300214yes—
subgroupG/H
centerC22×C4×C31C2×D4
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
socleC23×C31C22×D4
prime radicalC23×#41×C31C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#416C2semidirect
3. C22×C31×C22⋊C8C2semidirect
4. C31×#2280C2semidirect
5. C31×#1788C2semidirect
6. C31×#1787C2semidirect
7. C31×#1786C2semidirect
12. C22×#41×C31C4semidirect
20. C23×C8×C31C22semidirect
21. C2×C31×#117C22semidirect
23. C2×C31×#103C22semidirect
24. C2×C31×#102C22semidirect
27. C2×C31×C22⋊C8C22semidirect
30. C31×#380C22semidirect
112. #41×C31C22⋊C4semidirect
172. #20597C31direct (N × H)
236. C2×#416C2×C31semidirect
237. C22×C22⋊C8C2×C31semidirect
238. #2280C2×C31semidirect
239. #1788C2×C31semidirect
240. #1787C2×C31semidirect
241. #1786C2×C31semidirect
259. C22×C31#416semidirect
275. C22×#41C4×C31semidirect
283. C23×C8C22×C31semidirect
284. C2×#117C22×C31semidirect
286. C2×#103C22×C31semidirect
287. C2×#102C22×C31semidirect
290. C2×C22⋊C8C22×C31semidirect
293. #380C22×C31semidirect
364. C31#20597direct (N × H)
384. #41C22⋊C4×C31semidirect
523. C22C31×#416semidirect

Maximal subgroups

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

subgroupindex
C23×#41×C312
C2×C31×#4162
C22×C31×C22⋊C82
C31×#22802
C31×#17882
C31×#17872
C31×#17862
#2059731