Finite Group Databasebeta

← C31×#19017 #21475776

Subgroups — C31×#19017 #21475776

Order 7936.

944 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3559C211yesmaximal, characteristic
2C31×#3238C211yesmaximal, characteristic, complemented
3C2×C31×#501C211yesmaximal, characteristic, complemented
4C31×#3152C211yesmaximal, characteristic, complemented
5C31×#1661C214yesmaximal, complemented
6C31×#1660C214yesmaximal, complemented
7C31×#1530C211yesmaximal, characteristic, complemented
8C31×#1528C211yesmaximal, characteristic, complemented
9C31×#1524C211yesmaximal, characteristic, complemented
10C2×C31×#137—22no—
11C31×#563C2212yes—
12C31×#562C2212yes—
13C31×#554—21no—
14C31×#553—21no—
15C31×#553—21no—
16C31×#547—22no—
17C31×#547—21no—
18C31×#544—21no—
19C31×#540—21no—
20C31×#539—22no—
21C31×#539—21no—
22C31×#537—21no—
23C31×#532C2211yescharacteristic
24C2×C31×C4⋊Q8C2211yescharacteristic
25C31×#529C2211yescharacteristic
26C2×C31×#125C2211yescharacteristic
27C31×#501—22no—
28C31×#501C2214yes—
29C31×#501C2212yes—
30C31×#497C2214yes—
31C31×#494—21no—
32C31×#492—21no—
33C31×#489C2212yes—
34C31×#483—21no—
35C31×#480—21no—
36C31×#477—21no—
37C31×#476—21no—
38C31×#432C2211yescharacteristic
39C2×C31×#107—21no—
40C2×C31×#106—21no—
41C4×M16×C31C2211yescharacteristic
42C2×C31×#97C2211yescharacteristic
43C31×#330C2212yes—
44C31×#330C2214yes—
45C31×#329C2214yes—
46C31×#329C2212yes—
47C2×#41×C31—22no—
48C2×#41×C31D412yes—
49C22×Q8×C31—21no—
50C31×#137—48no—
51C2×Q16×C31—24no—
52C2×SD16×C31—24no—
53C2×M16×C31—21no—
54C22×C8×C31—21no—
55C31×C4⋊Q8—21no—
56C31×C4⋊Q8—22no—
57C31×C4⋊Q8—21no—
58C31×C4⋊Q8—21no—
59C31×C4⋊Q8—21no—
60C31×C4⋊Q8D411yescharacteristic
61C31×C4⋊Q8C2312yes—
62C31×#127D411yescharacteristic
63C31×#126—22no—
64C31×#126—22no—
65C31×#126—21no—
66C31×#125—22no—
67C31×#125—21no—
68C31×#125—21no—
69C31×#125—21no—
70C31×#125D412yes—
71C31×#125C2312yes—
72C31×#125C2311yescharacteristic
73C31×#124C2311yescharacteristic
74C31×#124C2312yes—
75C31×#123—22no—
76C31×#123—22no—
77C31×#123—22no—
78C31×#123—22no—
79C31×#123—21no—
80C31×#123—22no—
81C31×#123—22no—
82C31×C22⋊Q8—22no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8—22no—
85C31×C22⋊Q8—22no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—22no—
88C31×C22⋊Q8D411yescharacteristic
89C31×#121—22no—
90C31×#121—22no—
91C31×#121—21no—
92C31×#121—21no—
93C31×#121—22no—
94C31×#121—22no—
95C31×#121—21no—
96C31×#121—21no—
97C31×#121D411yescharacteristic
98C4×Q8×C31D411yescharacteristic
99C4×D4×C31—22no—
100C4×D4×C31—22no—
101C4×D4×C31—21no—
102C4×D4×C31—21no—
103C4×D4×C31—21no—
104C4×D4×C31—21no—
105C4×D4×C31—21no—
106C4×D4×C31D411yescharacteristic
107C31×#117—21no—
108C31×#117—21no—
109C2×C4⋊C4×C31—22no—
110C2×C4⋊C4×C31—21no—
111C2×C4⋊C4×C31—21no—
112C2×C4⋊C4×C31—21no—
113C2×C4⋊C4×C31—21no—
114C2×C4⋊C4×C31—21no—
115C2×C4⋊C4×C31D411yescharacteristic
116C2×C4⋊C4×C31D411yescharacteristic
117C2×C42×C31C2311yescharacteristic
118C31×#107—24no—
119C31×#107—22no—
120C31×#106—24no—
121C31×#106—22no—
122C31×C4⋊C8—21no—
123C31×#103—22no—
124C31×#102—22no—
125C31×C22⋊C8—21no—
126C31×#97—21no—
127C31×#97C2312yes—
128C31×#97C2312yes—
129C31×#97C2312yes—
130C4×C8×C31—21no—
131#41×C31—44no—
132#41×C31—84no—
133C2×Q8×C31—22no—
134C2×Q8×C31—22no—
135C2×Q8×C31—44no—
136C2×Q8×C31—41no—
137C2×Q8×C31C2×D411yescharacteristic
138C2×Q8×C31C2×D412yes—
139C2×D4×C31—22no—
140C2×D4×C31—21no—
141C2×D4×C31—22no—
142C2×D4×C31—22no—
143C2×D4×C31—21no—
144C2×D4×C31—21no—
145C2×D4×C31C2×D411yescharacteristic
146C2×D4×C31C2×D412yes—
147C22×C4×C31—21no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—21no—
152C22×C4×C31—21no—
153C22×C4×C31—22no—
154C22×C4×C31—41no—
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158Q16×C31—48no—
159SD16×C31—48no—
160M16×C31—24no—
161C2×C8×C31—24no—
162C2×C8×C31—22no—
163C2×C8×C31—22no—
164C2×C8×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—41no—
184C4⋊C4×C31—41no—
185C4⋊C4×C31C2×D412yes—
186C4⋊C4×C31C2×D411yescharacteristic
187C4⋊C4×C31C2×D412yes—
188C4⋊C4×C31C2×D411yescharacteristic
189C22⋊C4×C31—22no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—21no—
195C22⋊C4×C31—21no—
196C22⋊C4×C31—21no—
197C22⋊C4×C31—21no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—21no—
201C42×C31—22no—
202C42×C31—21no—
203C42×C31C2×D411yescharacteristic
204C42×C31C2×D411yescharacteristic
205C42×C31C2×D411yescharacteristic
206C42×C31C2411yescharacteristic
207#19017C3111yesmaximal, characteristic, direct product factor, Sylow
208C23×C31—22no—
209C23×C31—21no—
210C23×C31#12011yescharacteristic
211Q8×C31—42no—
212Q8×C31—41no—
213Q8×C31—44no—
214Q8×C31—82no—
215Q8×C31—82no—
216D4×C31—42no—
217D4×C31—41no—
218D4×C31—44no—
219D4×C31—42no—
220D4×C31—42no—
221D4×C31—81no—
222D4×C31—82no—
223D4×C31—82no—
224C2×C4×C31—22no—
225C2×C4×C31—22no—
226C2×C4×C31—21no—
227C2×C4×C31—21no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—21no—
231C2×C4×C31—22no—
232C2×C4×C31—41no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—82no—
237C2×C4×C31—81no—
238C2×C4×C31—82no—
239C2×C4×C31—82no—
240C2×C4×C31—82no—
241C2×C4×C31—82no—
242C2×C4×C31#12011yescharacteristic
243C2×C4×C31#12011yescharacteristic
244C2×C4×C31#12011yescharacteristic
245C2×C4×C31C22×D411yescharacteristic
246C2×C4×C31C22×D411yescharacteristic
247C2×C4×C31C22×D411yescharacteristic
248C8×C31—24no—
249C8×C31—24no—
250C8×C31—42no—
251#3559C2×C3111yescharacteristic
252#3238C2×C3111yescharacteristic, complemented
253C2×#501C2×C3111yescharacteristic, complemented
254#3152C2×C3111yescharacteristic, complemented
255#1661C2×C3114yescomplemented
256#1660C2×C3114yescomplemented
257#1530C2×C3111yescharacteristic, complemented
258#1528C2×C3111yescharacteristic, complemented
259#1524C2×C3111yescharacteristic, complemented
260C22×C31—21no—
261C22×C31—21no—
262C22×C31—42no—
263C22×C31—41no—
264C22×C31—41no—
265C22×C31—42no—
266C22×C31—42no—
267C22×C31—41no—
268C22×C31#45312yes—
269C22×C31C2×#12011yescharacteristic
270C4×C31—21no—
271C4×C31—21no—
272C4×C31—21no—
273C4×C31—21no—
274C4×C31—21no—
275C4×C31—42no—
276C4×C31—42no—
277C4×C31—41no—
278C4×C31—41no—
279C4×C31—42no—
280C4×C31—41no—
281C4×C31—82no—
282C4×C31#45612yes—
283C2×#137—22no—
284#563C22×C3112yes—
285#562C22×C3112yes—
286#554—21no—
287#553—21no—
288#553—21no—
289#547—22no—
290#547—21no—
291#544—21no—
292#540—21no—
293#539—22no—
294#539—21no—
295#537—21no—
296#532C22×C3111yescharacteristic
297C2×C4⋊Q8C22×C3111yescharacteristic
298#529C22×C3111yescharacteristic
299C2×#125C22×C3111yescharacteristic
300#501—22no—
301#501C22×C3114yes—
302#501C22×C3112yes—
303#497C22×C3114yes—
304#494—21no—
305#492—21no—
306#489C22×C3112yes—
307#483—21no—
308#480—21no—
309#477—21no—
310#476—21no—
311#432C22×C3111yescharacteristic
312C2×#107—21no—
313C2×#106—21no—
314C4×M16C22×C3111yescharacteristic
315C2×#97C22×C3111yescharacteristic
316#330C22×C3112yes—
317#330C22×C3114yes—
318#329C22×C3114yes—
319#329C22×C3112yes—
320C2×C31—22no—
321C2×C31—82no—
322C2×C31—81no—
323C2×C31C2×#45311yescharacteristic
324C2×C31C2×#45611yescharacteristic
325C2×C31#302211yescharacteristic
326C2×#41—22no—
327C2×#41D4×C3112yes—
328C22×Q8—21no—
329#137—48no—
330C2×Q16—24no—
331C2×SD16—24no—
332C2×M16—21no—
333C22×C8—21no—
334C4⋊Q8—21no—
335C4⋊Q8—22no—
336C4⋊Q8—21no—
337C4⋊Q8—21no—
338C4⋊Q8—21no—
339C4⋊Q8D4×C3111yescharacteristic
340C4⋊Q8C23×C3112yes—
341#127D4×C3111yescharacteristic
342#126—22no—
343#126—22no—
344#126—21no—
345#125—22no—
346#125—21no—
347#125—21no—
348#125—21no—
349#125D4×C3112yes—
350#125C23×C3112yes—
351#125C23×C3111yescharacteristic
352#124C23×C3111yescharacteristic
353#124C23×C3112yes—
354#123—22no—
355#123—22no—
356#123—22no—
357#123—22no—
358#123—21no—
359#123—22no—
360#123—22no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8D4×C3111yescharacteristic
368#121—22no—
369#121—22no—
370#121—21no—
371#121—21no—
372#121—22no—
373#121—22no—
374#121—21no—
375#121—21no—
376#121D4×C3111yescharacteristic
377C4×Q8D4×C3111yescharacteristic
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4D4×C3111yescharacteristic
386#117—21no—
387#117—21no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4D4×C3111yescharacteristic
395C2×C4⋊C4D4×C3111yescharacteristic
396C2×C42C23×C3111yescharacteristic
397#107—24no—
398#107—22no—
399#106—24no—
400#106—22no—
401C4⋊C8—21no—
402#103—22no—
403#102—22no—
404C22⋊C8—21no—
405#97—21no—
406#97C23×C3112yes—
407#97C23×C3112yes—
408#97C23×C3112yes—
409C4×C8—21no—
410C31#1901711yescharacteristic, direct product factor, Sylow
411#41—44no—
412#41—84no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—44no—
416C2×Q8—41no—
417C2×Q8C2×D4×C3111yescharacteristic
418C2×Q8C2×D4×C3112yes—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4C2×D4×C3111yescharacteristic
426C2×D4C2×D4×C3112yes—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—41no—
435C22×C4C2×D4×C3111yescharacteristic
436C22×C4C2×D4×C3111yescharacteristic
437C22×C4C2×D4×C3111yescharacteristic
438Q16—48no—
439SD16—48no—
440M16—24no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4C2×D4×C3112yes—
466C4⋊C4C2×D4×C3111yescharacteristic
467C4⋊C4C2×D4×C3112yes—
468C4⋊C4C2×D4×C3111yescharacteristic
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—21no—
481C42—22no—
482C42—21no—
483C42C2×D4×C3111yescharacteristic
484C42C2×D4×C3111yescharacteristic
485C42C2×D4×C3111yescharacteristic
486C42C24×C3111yescharacteristic
487C23—22no—
488C23—21no—
489C23C31×#12011yescharacteristic
490Q8—42no—
491Q8—41no—
492Q8—44no—
493Q8—82no—
494Q8—82no—
495D4—42no—
496D4—41no—
497D4—44no—
498D4—42no—
499D4—42no—
500D4—81no—
501D4—82no—
502D4—82no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4C31×#12011yescharacteristic
522C2×C4C31×#12011yescharacteristic
523C2×C4C31×#12011yescharacteristic
524C2×C4C22×D4×C3111yescharacteristic
525C2×C4C22×D4×C3111yescharacteristic
526C2×C4C22×D4×C3111yescharacteristic
527C8—24no—
528C8—24no—
529C8—42no—
530C22—21no—
531C22—21no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—42no—
536C22—42no—
537C22—41no—
538C22C31×#45312yes—
539C22C2×C31×#12011yescharacteristic
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—42no—
546C4—42no—
547C4—41no—
548C4—41no—
549C4—42no—
550C4—41no—
551C4—82no—
552C4C31×#45612yes—
553C2—22no—
554C2—82no—
555C2—81no—
556C2C2×C31×#45311yescharacteristic
557C2C2×C31×#45611yescharacteristic
558C2C31×#302211yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3559C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3238C2semidirect
3. C2×C31×#501C2semidirect
4. C31×#3152C2semidirect
5. C31×#1661C2semidirect
6. C31×#1660C2semidirect
7. C31×#1530C2semidirect
8. C31×#1528C2semidirect
9. C31×#1524C2semidirect
207. #19017C31direct (N × H)
252. #3238C2×C31semidirect
253. C2×#501C2×C31semidirect
254. #3152C2×C31semidirect
255. #1661C2×C31semidirect
256. #1660C2×C31semidirect
257. #1530C2×C31semidirect
258. #1528C2×C31semidirect
259. #1524C2×C31semidirect
410. C31#19017direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35592
C31×#32382
C2×C31×#5012
C31×#31522
C31×#16612
C31×#16602
C31×#15302
C31×#15282
C31×#15242
#1901731