Finite Group Databasebeta

← C31×#17106 #21473865

Subgroups — C31×#17106 #21473865

Order 7936.

1438 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#399C211yesmaximal, characteristic
2C2×C31×#397C212yesmaximal
3C2×C31×#395C212yesmaximal, complemented
4C2×C31×#393C211yesmaximal, characteristic
5C31×#1444C212yesmaximal, complemented
6C31×#1443C212yesmaximal, complemented
7C31×#1442C212yesmaximal, complemented
8C31×#1440C211yesmaximal, characteristic
9C31×#1439C212yesmaximal
10C23×Q8×C31C2211yescharacteristic
11C22×C4⋊C4×C31C2212yes—
12C22×C4⋊C4×C31C2212yes—
13C22×C4⋊C4×C31C2212yes—
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C22⋊C4×C31C2212yes—
16C22×C22⋊C4×C31C2211yescharacteristic
17C22×C22⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2211yescharacteristic
19C22×C22⋊C4×C31C2211yescharacteristic
20C31×#399—24no—
21C31×#397—28no—
22C31×#395—28no—
23C31×#393—24no—
24C2×C31×#95C2211yescharacteristic
25C2×C31×#95C2212yes—
26C2×C31×#95C2212yes—
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2212yes—
30C2×C31×#95C2211yescharacteristic
31C2×C31×#95C2212yes—
32C2×C31×#95C2212yes—
33C2×C31×#95C2212yes—
34C2×C31×#95C2212yes—
35C25×C31C2311yescharacteristic
36C22×Q8×C31—24no—
37C22×Q8×C31—22no—
38C22×Q8×C31—41no—
39C22×Q8×C31—42no—
40C22×Q8×C31—41no—
41C23×C4×C31C2312yes—
42C23×C4×C31C2312yes—
43C23×C4×C31C2312yes—
44C23×C4×C31C2311yescharacteristic
45C23×C4×C31C2312yes—
46C23×C4×C31C2312yes—
47C23×C4×C31C2311yescharacteristic
48C23×C4×C31C2312yes—
49C2×C4⋊C4×C31—24no—
50C2×C4⋊C4×C31—24no—
51C2×C4⋊C4×C31—24no—
52C2×C4⋊C4×C31—24no—
53C2×C4⋊C4×C31—24no—
54C2×C4⋊C4×C31—22no—
55C2×C4⋊C4×C31—42no—
56C2×C4⋊C4×C31—42no—
57C2×C4⋊C4×C31—42no—
58C2×C4⋊C4×C31—42no—
59C2×C4⋊C4×C31—41no—
60C2×C4⋊C4×C31—41no—
61C2×C22⋊C4×C31—24no—
62C2×C22⋊C4×C31—24no—
63C2×C22⋊C4×C31—24no—
64C2×C22⋊C4×C31—24no—
65C2×C22⋊C4×C31—24no—
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—41no—
68C2×C22⋊C4×C31—41no—
69C2×C22⋊C4×C31—41no—
70C2×C22⋊C4×C31—41no—
71C2×C22⋊C4×C31—42no—
72C2×C22⋊C4×C31—42no—
73C2×C22⋊C4×C31D414yes—
74C31×#95—24no—
75C31×#95—24no—
76C31×#95—24no—
77C31×#95—24no—
78C31×#95—22no—
79C31×#95—24no—
80C31×#95—22no—
81C31×#95—24no—
82C31×#95—24no—
83C31×#95—42no—
84C31×#95D418yes—
85C24×C31—22no—
86C24×C31—22no—
87C24×C31—21no—
88C24×C31—21no—
89C24×C31—22no—
90C24×C31—22no—
91C24×C31—22no—
92C24×C31—22no—
93C24×C31C2×D412yes—
94C24×C31C2411yescharacteristic
95C2×Q8×C31—48no—
96C2×Q8×C31—44no—
97C2×Q8×C31—44no—
98C2×Q8×C31—48no—
99C2×Q8×C31—44no—
100C22×C4×C31—22no—
101C22×C4×C31—22no—
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—22no—
105C22×C4×C31—22no—
106C22×C4×C31—22no—
107C22×C4×C31—22no—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—21no—
112C22×C4×C31—21no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31C2×D412yes—
147C22×C4×C31C2×D414yes—
148C22×C4×C31C2×D414yes—
149C22×C4×C31C2×D412yes—
150C22×C4×C31C2×D414yes—
151C22×C4×C31#4114yes—
152C4⋊C4×C31—48no—
153C4⋊C4×C31—48no—
154C4⋊C4×C31—44no—
155C4⋊C4×C31—48no—
156C22⋊C4×C31—44no—
157C22⋊C4×C31—44no—
158C22⋊C4×C31—44no—
159C22⋊C4×C31—48no—
160C22⋊C4×C31—48no—
161#17106C3111yesmaximal, characteristic, direct product factor, Sylow
162C23×C31—22no—
163C23×C31—22no—
164C23×C31—22no—
165C23×C31—24no—
166C23×C31—24no—
167C23×C31—42no—
168C23×C31—42no—
169C23×C31—42no—
170C23×C31—42no—
171C23×C31—42no—
172C23×C31—42no—
173C23×C31—41no—
174C23×C31—42no—
175C23×C31—42no—
176C23×C31—41no—
177C23×C31—41no—
178C23×C31—42no—
179C23×C31—42no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31—41no—
183C23×C31C22×D411yescharacteristic
184C23×C31C22×D412yes—
185C23×C31C2×#4112yes—
186C23×C31H(5,𝔽2)11yescharacteristic
187C23×C31H(5,𝔽2)11yescharacteristic
188C23×C31H(5,𝔽2)12yes—
189C23×C312−1+412yes—
190C23×C312−1+412yes—
191C23×C312−1+412yes—
192Q8×C31—416no—
193C2×C4×C31—22no—
194C2×C4×C31—22no—
195C2×C4×C31—24no—
196C2×C4×C31—24no—
197C2×C4×C31—24no—
198C2×C4×C31—24no—
199C2×C4×C31—24no—
200C2×C4×C31—24no—
201C2×C4×C31—24no—
202C2×C4×C31—24no—
203C2×C4×C31—24no—
204C2×C4×C31—24no—
205C2×C4×C31—24no—
206C2×C4×C31—24no—
207C2×C4×C31—24no—
208C2×C4×C31—24no—
209C2×C4×C31—42no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—42no—
213C2×C4×C31—42no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—41no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—41no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×#399C2×C3111yescharacteristic
246C2×#397C2×C3112yes—
247C2×#395C2×C3112yescomplemented
248C2×#393C2×C3111yescharacteristic
249#1444C2×C3112yescomplemented
250#1443C2×C3112yescomplemented
251#1442C2×C3112yescomplemented
252#1440C2×C3111yescharacteristic
253#1439C2×C3112yes—
254C22×C31—42no—
255C22×C31—42no—
256C22×C31—44no—
257C22×C31—44no—
258C22×C31—42no—
259C22×C31—82no—
260C22×C31—82no—
261C22×C31—82no—
262C22×C31—82no—
263C22×C31#53411yescharacteristic
264C22×C31#53512yes—
265C22×C31#53611yescharacteristic
266C22×C31#53611yescharacteristic
267C22×C31#53612yes—
268C22×C31#53612yes—
269C22×C31#53912yes—
270C22×C31#54212yes—
271C22×C31#54212yes—
272C22×C31D4211yescharacteristic
273C22×C31D4212yes—
274C22×C31#54712yes—
275C22×C31#54712yes—
276C22×C31#54812yes—
277C22×C31#55111yescharacteristic
278C22×C31#56011yescharacteristic
279C22×C31#56312yes—
280C22×C31#56312yes—
281C22×C31#56311yescharacteristic
282C22×C31#56312yes—
283C22×C31#56312yes—
284C4×C31—42no—
285C4×C31—44no—
286C4×C31—44no—
287C4×C31—44no—
288C4×C31—42no—
289C4×C31—44no—
290C4×C31—82no—
291C4×C31—82no—
292C23×Q8C22×C3111yescharacteristic
293C22×C4⋊C4C22×C3112yes—
294C22×C4⋊C4C22×C3112yes—
295C22×C4⋊C4C22×C3112yes—
296C22×C4⋊C4C22×C3111yescharacteristic
297C22×C22⋊C4C22×C3112yes—
298C22×C22⋊C4C22×C3111yescharacteristic
299C22×C22⋊C4C22×C3112yes—
300C22×C22⋊C4C22×C3111yescharacteristic
301C22×C22⋊C4C22×C3111yescharacteristic
302#399—24no—
303#397—28no—
304#395—28no—
305#393—24no—
306C2×#95C22×C3111yescharacteristic
307C2×#95C22×C3112yes—
308C2×#95C22×C3112yes—
309C2×#95C22×C3112yes—
310C2×#95C22×C3112yes—
311C2×#95C22×C3112yes—
312C2×#95C22×C3111yescharacteristic
313C2×#95C22×C3112yes—
314C2×#95C22×C3112yes—
315C2×#95C22×C3112yes—
316C2×#95C22×C3112yes—
317C2×C31—82no—
318C2×C31#266911yescharacteristic
319C2×C31#267412yes—
320C2×C31#267412yes—
321C2×C31#268012yes—
322C2×C31#271711yescharacteristic
323C2×C31#272312yes—
324C2×C31#281711yescharacteristic
325C2×C31#282112yes—
326C2×C31#283812yes—
327C25C23×C3111yescharacteristic
328C22×Q8—22no—
329C22×Q8—24no—
330C22×Q8—41no—
331C22×Q8—42no—
332C22×Q8—41no—
333C23×C4C23×C3112yes—
334C23×C4C23×C3112yes—
335C23×C4C23×C3112yes—
336C23×C4C23×C3111yescharacteristic
337C23×C4C23×C3112yes—
338C23×C4C23×C3112yes—
339C23×C4C23×C3111yescharacteristic
340C23×C4C23×C3112yes—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—42no—
348C2×C4⋊C4—42no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—41no—
352C2×C4⋊C4—41no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—41no—
360C2×C22⋊C4—41no—
361C2×C22⋊C4—41no—
362C2×C22⋊C4—41no—
363C2×C22⋊C4—42no—
364C2×C22⋊C4—42no—
365C2×C22⋊C4D4×C3114yes—
366#95—24no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—22no—
373#95—24no—
374#95—24no—
375#95—42no—
376#95D4×C3118yes—
377C31#1710611yescharacteristic, direct product factor, Sylow
378C24—22no—
379C24—22no—
380C24—21no—
381C24—21no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24C2×D4×C3112yes—
387C24C24×C3111yescharacteristic
388C2×Q8—48no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—48no—
392C2×Q8—44no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4C2×D4×C3112yes—
440C22×C4C2×D4×C3114yes—
441C22×C4C2×D4×C3114yes—
442C22×C4C2×D4×C3112yes—
443C22×C4C2×D4×C3114yes—
444C22×C4#41×C3114yes—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23C22×D4×C3111yescharacteristic
476C23C22×D4×C3112yes—
477C23C2×#41×C3112yes—
478C23C31×H(5,𝔽2)11yescharacteristic
479C23C31×H(5,𝔽2)11yescharacteristic
480C23C31×H(5,𝔽2)12yes—
481C23C31×2−1+412yes—
482C23C31×2−1+412yes—
483C23C31×2−1+412yes—
484Q8—416no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C22—42no—
538C22—44no—
539C22—44no—
540C22—42no—
541C22—42no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22C31×#53411yescharacteristic
547C22C31×#53512yes—
548C22C31×#53612yes—
549C22C31×#53612yes—
550C22C31×#53611yescharacteristic
551C22C31×#53611yescharacteristic
552C22C31×#53912yes—
553C22C31×#54212yes—
554C22C31×#54212yes—
555C22D42×C3111yescharacteristic
556C22D42×C3112yes—
557C22C31×#54712yes—
558C22C31×#54712yes—
559C22C31×#54812yes—
560C22C31×#55111yescharacteristic
561C22C31×#56011yescharacteristic
562C22C31×#56311yescharacteristic
563C22C31×#56312yes—
564C22C31×#56312yes—
565C22C31×#56312yes—
566C22C31×#56312yes—
567C4—42no—
568C4—44no—
569C4—44no—
570C4—44no—
571C4—42no—
572C4—44no—
573C4—82no—
574C4—82no—
575C2—82no—
576C2C31×#266911yescharacteristic
577C2C31×#267412yes—
578C2C31×#267412yes—
579C2C31×#268012yes—
580C2C31×#271711yescharacteristic
581C2C31×#272312yes—
582C2C31×#281711yescharacteristic
583C2C31×#282112yes—
584C2C31×#283812yes—
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC25×C31C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C31×#395C2semidirect
5. C31×#1444C2semidirect
6. C31×#1443C2semidirect
7. C31×#1442C2semidirect
161. #17106C31direct (N × H)
247. C2×#395C2×C31semidirect
249. #1444C2×C31semidirect
250. #1443C2×C31semidirect
251. #1442C2×C31semidirect
377. C31#17106direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3992
C2×C31×#3972
C2×C31×#3952
C2×C31×#3932
C31×#14442
C31×#14432
C31×#14422
C31×#14402
C31×#14392
#1710631