Finite Group Databasebeta

← C31×#13347 #21470106

Subgroups — C31×#13347 #21470106

Order 7936.

830 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C2×C31×#458C211yesmaximal, characteristic, complemented
2C2×C31×#453C211yesmaximal, characteristic
3C31×#2709C211yesmaximal, characteristic, complemented
4C2×C31×#353C211yesmaximal, characteristic, complemented
5C2×C31×#352C211yesmaximal, characteristic, complemented
6C31×#1790C211yesmaximal, characteristic, complemented
7C31×#1390C211yesmaximal, characteristic, complemented
8C2×C31×H(5,𝔽2)C2211yescharacteristic
9C2×C31×AGL(1,ℤ/8)—21no—
10C2×C31×#127C2211yescharacteristic
11C2×C31×#123—21no—
12C2×C31×#123C2211yescharacteristic
13C31×#458—22no—
14C31×#458—22no—
15C31×#453—22no—
16C31×#453—22no—
17C2×C31×#104—21no—
18C2×C31×#100C2211yescharacteristic
19C2×C31×#99—21no—
20C2×C31×#99—21no—
21C2×C31×#99C2211yescharacteristic
22C2×C31×#99C2211yescharacteristic
23C31×#394—21no—
24C31×#394—21no—
25C31×#394—21no—
26C31×#387—21no—
27C31×#386—21no—
28C31×#386—21no—
29C31×#386—21no—
30C31×#386C2211yescharacteristic
31C31×#353—22no—
32C31×#352—22no—
33C31×H(5,𝔽2)—21no—
34C31×H(5,𝔽2)—21no—
35C31×H(5,𝔽2)—21no—
36C31×H(5,𝔽2)—42no—
37C31×H(5,𝔽2)D412yes—
38C2×#41×C31—21no—
39C2×#41×C31—41no—
40C22×D4×C31—21no—
41C22×D4×C31—21no—
42C22×D4×C31—21no—
43C22×D4×C31—21no—
44C22×D4×C31—41no—
45C22×D4×C31D411yescharacteristic
46C22×D4×C31C2311yescharacteristic
47C23×C4×C31D411yescharacteristic
48C23×C4×C31D411yescharacteristic
49C31×AGL(1,ℤ/8)—24no—
50C31×AGL(1,ℤ/8)—42no—
51C2×SD16×C31—41no—
52C2×D8×C31—41no—
53C2×M16×C31—21no—
54C31×#127—21no—
55C31×#127—21no—
56C31×#127—21no—
57C31×#127—21no—
58C31×#123—24no—
59C31×#123—21no—
60C31×#123—21no—
61C31×#123—21no—
62C31×#123—21no—
63C31×#123—42no—
64C2×C4⋊C4×C31—21no—
65C2×C4⋊C4×C31—21no—
66C2×C4⋊C4×C31—41no—
67C2×C22⋊C4×C31—21no—
68C2×C22⋊C4×C31—21no—
69C2×C22⋊C4×C31—21no—
70C2×C22⋊C4×C31—21no—
71C2×C22⋊C4×C31—21no—
72C2×C22⋊C4×C31—21no—
73C2×C22⋊C4×C31—21no—
74C2×C22⋊C4×C31—21no—
75C2×C22⋊C4×C31—41no—
76C2×C22⋊C4×C31—41no—
77C2×C22⋊C4×C31D411yescharacteristic
78C2×C22⋊C4×C31D411yescharacteristic
79C2×C42×C31D411yescharacteristic
80C31×#104—24no—
81C31×#100—21no—
82C31×#100D412yes—
83C31×#99—24no—
84C31×#99—21no—
85C31×#99—21no—
86C31×#99—21no—
87C31×#99—42no—
88C31×#99D412yes—
89C31×#95—21no—
90C31×#95—21no—
91C31×#95—21no—
92C31×#95—21no—
93C31×#95—41no—
94C24×C31—21no—
95C24×C31—21no—
96C24×C31—21no—
97C24×C31C2×D411yescharacteristic
98C24×C31C2×D411yescharacteristic
99#41×C31—22no—
100#41×C31—22no—
101#41×C31—42no—
102#41×C31—41no—
103#41×C31—42no—
104#41×C31—42no—
105#41×C31—42no—
106#41×C31—41no—
107C2×Q8×C31—21no—
108C2×D4×C31—21no—
109C2×D4×C31—22no—
110C2×D4×C31—22no—
111C2×D4×C31—22no—
112C2×D4×C31—22no—
113C2×D4×C31—21no—
114C2×D4×C31—21no—
115C2×D4×C31—41no—
116C2×D4×C31—41no—
117C2×D4×C31—41no—
118C2×D4×C31—41no—
119C2×D4×C31—42no—
120C2×D4×C31—42no—
121C2×D4×C31—42no—
122C2×D4×C31—42no—
123C2×D4×C31—41no—
124C2×D4×C31—41no—
125C2×D4×C31—41no—
126C2×D4×C31—41no—
127C2×D4×C31—42no—
128C2×D4×C31—42no—
129C2×D4×C31—41no—
130C2×D4×C31—41no—
131C2×D4×C31—41no—
132C2×D4×C31—41no—
133C2×D4×C31—41no—
134C2×D4×C31—41no—
135C2×D4×C31—41no—
136C2×D4×C31—81no—
137C2×D4×C31—81no—
138C2×D4×C31C2×D411yescharacteristic
139C2×D4×C31C2×D411yescharacteristic
140C2×D4×C31C2×D411yescharacteristic
141C2×D4×C31#4111yescharacteristic
142C22×C4×C31—21no—
143C22×C4×C31—21no—
144C22×C4×C31—21no—
145C22×C4×C31—21no—
146C22×C4×C31—21no—
147C22×C4×C31—21no—
148C22×C4×C31—21no—
149C22×C4×C31—21no—
150C22×C4×C31—21no—
151C22×C4×C31—41no—
152C22×C4×C31—41no—
153C22×C4×C31—41no—
154C22×C4×C31—41no—
155C22×C4×C31—41no—
156C22×C4×C31—41no—
157C22×C4×C31—81no—
158C22×C4×C31C2×D411yescharacteristic
159SD16×C31—44no—
160D8×C31—44no—
161M16×C31—22no—
162M16×C31—41no—
163C2×C8×C31—41no—
164C4⋊C4×C31—42no—
165C4⋊C4×C31—44no—
166C4⋊C4×C31—41no—
167C4⋊C4×C31—41no—
168C22⋊C4×C31—21no—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—21no—
171C22⋊C4×C31—41no—
172C22⋊C4×C31—44no—
173C22⋊C4×C31—41no—
174C22⋊C4×C31—41no—
175C22⋊C4×C31—41no—
176C22⋊C4×C31—41no—
177C22⋊C4×C31—41no—
178C22⋊C4×C31—81no—
179C22⋊C4×C31—81no—
180C22⋊C4×C31—81no—
181C22⋊C4×C31—81no—
182C22⋊C4×C31—81no—
183C22⋊C4×C31—82no—
184C42×C31—21no—
185C42×C31—21no—
186#13347C3111yesmaximal, characteristic, direct product factor, Sylow
187C23×C31—21no—
188C23×C31—21no—
189C23×C31—21no—
190C23×C31—21no—
191C23×C31—22no—
192C23×C31—22no—
193C23×C31—41no—
194C23×C31—41no—
195C23×C31—41no—
196C23×C31—41no—
197C23×C31—41no—
198C23×C31—41no—
199C23×C31—41no—
200C23×C31—41no—
201C23×C31—41no—
202C23×C31—41no—
203C23×C31—41no—
204C23×C31—41no—
205C23×C31—41no—
206C23×C31—41no—
207C23×C31—81no—
208C23×C31—81no—
209C23×C31—81no—
210C23×C31—81no—
211C23×C31#12011yescharacteristic
212C23×C31#12011yescharacteristic
213C23×C31#12111yescharacteristic
214C23×C31#12111yescharacteristic
215C23×C31#12711yescharacteristic
216Q8×C31—22no—
217Q8×C31—41no—
218D4×C31—22no—
219D4×C31—41no—
220D4×C31—41no—
221D4×C31—41no—
222D4×C31—41no—
223D4×C31—41no—
224D4×C31—42no—
225D4×C31—42no—
226D4×C31—42no—
227D4×C31—42no—
228D4×C31—42no—
229D4×C31—42no—
230D4×C31—42no—
231D4×C31—81no—
232D4×C31—82no—
233D4×C31—82no—
234D4×C31—82no—
235D4×C31—82no—
236C2×C4×C31—21no—
237C2×C4×C31—21no—
238C2×C4×C31—22no—
239C2×C4×C31—22no—
240C2×C4×C31—22no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—41no—
244C2×C4×C31—41no—
245C2×C4×C31—41no—
246C2×C4×C31—41no—
247C2×C4×C31—41no—
248C2×C4×C31—42no—
249C2×C4×C31—42no—
250C2×C4×C31—41no—
251C2×C4×C31—41no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—42no—
257C2×C4×C31—41no—
258C2×C4×C31—41no—
259C2×C4×C31—41no—
260C2×C4×C31—41no—
261C2×C4×C31—41no—
262C2×C4×C31—41no—
263C2×C4×C31—81no—
264C2×C4×C31—81no—
265C2×C4×C31—81no—
266C2×C4×C31—81no—
267C2×C4×C31—81no—
268C2×C4×C31—81no—
269C2×C4×C31—81no—
270C2×C4×C31#12011yescharacteristic
271C2×C4×C31#12111yescharacteristic
272C8×C31—42no—
273C2×#458C2×C3111yescharacteristic, complemented
274C2×#453C2×C3111yescharacteristic
275#2709C2×C3111yescharacteristic, complemented
276C2×#353C2×C3111yescharacteristic, complemented
277C2×#352C2×C3111yescharacteristic, complemented
278#1790C2×C3111yescharacteristic, complemented
279#1390C2×C3111yescharacteristic, complemented
280C22×C31—21no—
281C22×C31—21no—
282C22×C31—21no—
283C22×C31—21no—
284C22×C31—21no—
285C22×C31—21no—
286C22×C31—41no—
287C22×C31—41no—
288C22×C31—41no—
289C22×C31—41no—
290C22×C31—41no—
291C22×C31—41no—
292C22×C31—41no—
293C22×C31—41no—
294C22×C31—42no—
295C22×C31—81no—
296C22×C31—81no—
297C22×C31—81no—
298C22×C31—81no—
299C22×C31—81no—
300C22×C31—81no—
301C22×C31—82no—
302C22×C31—82no—
303C22×C31—81no—
304C22×C31—81no—
305C22×C31—81no—
306C22×C31—81no—
307C22×C31—81no—
308C22×C31—161no—
309C22×C31—161no—
310C22×C31#39211yescharacteristic
311C22×C31B(4,𝔽2)11yescharacteristic
312C22×C31B(4,𝔽2)11yescharacteristic
313C4×C31—21no—
314C4×C31—21no—
315C4×C31—42no—
316C4×C31—42no—
317C4×C31—81no—
318C4×C31—81no—
319C4×C31—82no—
320C4×C31—81no—
321C4×C31—81no—
322C4×C31—161no—
323C2×H(5,𝔽2)C22×C3111yescharacteristic
324C2×AGL(1,ℤ/8)—21no—
325C2×#127C22×C3111yescharacteristic
326C2×#123—21no—
327C2×#123C22×C3111yescharacteristic
328#458—22no—
329#458—22no—
330#453—22no—
331#453—22no—
332C2×#104—21no—
333C2×#100C22×C3111yescharacteristic
334C2×#99—21no—
335C2×#99—21no—
336C2×#99C22×C3111yescharacteristic
337C2×#99C22×C3111yescharacteristic
338#394—21no—
339#394—21no—
340#394—21no—
341#387—21no—
342#386—21no—
343#386—21no—
344#386—21no—
345#386C22×C3111yescharacteristic
346#353—22no—
347#352—22no—
348C2×C31—21no—
349C2×C31—21no—
350C2×C31—41no—
351C2×C31—41no—
352C2×C31—41no—
353C2×C31—41no—
354C2×C31—82no—
355C2×C31—161no—
356C2×C31#202111yescharacteristic
357C2×C31#219811yescharacteristic
358C2×C31#220011yescharacteristic
359H(5,𝔽2)—21no—
360H(5,𝔽2)—21no—
361H(5,𝔽2)—21no—
362H(5,𝔽2)—42no—
363H(5,𝔽2)D4×C3112yes—
364C2×#41—21no—
365C2×#41—41no—
366C22×D4—21no—
367C22×D4—21no—
368C22×D4—21no—
369C22×D4—21no—
370C22×D4—41no—
371C22×D4D4×C3111yescharacteristic
372C22×D4C23×C3111yescharacteristic
373C23×C4D4×C3111yescharacteristic
374C23×C4D4×C3111yescharacteristic
375AGL(1,ℤ/8)—24no—
376AGL(1,ℤ/8)—42no—
377C2×SD16—41no—
378C2×D8—41no—
379C2×M16—21no—
380#127—21no—
381#127—21no—
382#127—21no—
383#127—21no—
384#123—24no—
385#123—21no—
386#123—21no—
387#123—21no—
388#123—21no—
389#123—42no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—41no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—41no—
402C2×C22⋊C4—41no—
403C2×C22⋊C4D4×C3111yescharacteristic
404C2×C22⋊C4D4×C3111yescharacteristic
405C2×C42D4×C3111yescharacteristic
406#104—24no—
407#100—21no—
408#100D4×C3112yes—
409#99—24no—
410#99—21no—
411#99—21no—
412#99—21no—
413#99—42no—
414#99D4×C3112yes—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—41no—
420C31#1334711yescharacteristic, direct product factor, Sylow
421C24—21no—
422C24—21no—
423C24—21no—
424C24C2×D4×C3111yescharacteristic
425C24C2×D4×C3111yescharacteristic
426#41—22no—
427#41—22no—
428#41—42no—
429#41—41no—
430#41—42no—
431#41—42no—
432#41—42no—
433#41—41no—
434C2×Q8—21no—
435C2×D4—21no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—41no—
443C2×D4—41no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—41no—
451C2×D4—41no—
452C2×D4—41no—
453C2×D4—41no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—41no—
457C2×D4—41no—
458C2×D4—41no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—81no—
464C2×D4—81no—
465C2×D4C2×D4×C3111yescharacteristic
466C2×D4C2×D4×C3111yescharacteristic
467C2×D4C2×D4×C3111yescharacteristic
468C2×D4#41×C3111yescharacteristic
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—81no—
485C22×C4C2×D4×C3111yescharacteristic
486SD16—44no—
487D8—44no—
488M16—22no—
489M16—41no—
490C2×C8—41no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C22⋊C4—21no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—41no—
499C22⋊C4—44no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C22⋊C4—81no—
508C22⋊C4—81no—
509C22⋊C4—81no—
510C22⋊C4—82no—
511C42—21no—
512C42—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—81no—
534C23—81no—
535C23—81no—
536C23—81no—
537C23C31×#12011yescharacteristic
538C23C31×#12011yescharacteristic
539C23C31×#12111yescharacteristic
540C23C31×#12111yescharacteristic
541C23C31×#12711yescharacteristic
542Q8—22no—
543Q8—41no—
544D4—22no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—81no—
558D4—82no—
559D4—82no—
560D4—82no—
561D4—82no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4C31×#12011yescharacteristic
597C2×C4C31×#12111yescharacteristic
598C8—42no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—82no—
621C22—82no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—161no—
628C22—161no—
629C22C31×#39211yescharacteristic
630C22C31×B(4,𝔽2)11yescharacteristic
631C22C31×B(4,𝔽2)11yescharacteristic
632C4—21no—
633C4—21no—
634C4—42no—
635C4—42no—
636C4—81no—
637C4—81no—
638C4—82no—
639C4—81no—
640C4—81no—
641C4—161no—
642C2—21no—
643C2—21no—
644C2—41no—
645C2—41no—
646C2—41no—
647C2—41no—
648C2—82no—
649C2—161no—
650C2C31×#202111yescharacteristic
651C2C31×#219811yescharacteristic
652C2C31×#220011yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C31
prime radicalC2×C31×#453C2
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C22×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#458C2semidirect
3. C31×#2709C2semidirect
4. C2×C31×#353C2semidirect
5. C2×C31×#352C2semidirect
6. C31×#1790C2semidirect
7. C31×#1390C2semidirect
186. #13347C31direct (N × H)
273. C2×#458C2×C31semidirect
275. #2709C2×C31semidirect
276. C2×#353C2×C31semidirect
277. C2×#352C2×C31semidirect
278. #1790C2×C31semidirect
279. #1390C2×C31semidirect
420. C31#13347direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4582
C2×C31×#4532
C31×#27092
C2×C31×#3532
C2×C31×#3522
C31×#17902
C31×#13902
#1334731