Finite Group Databasebeta

← C31×#13349 #21470108

Subgroups — C31×#13349 #21470108

Order 7936.

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

normal subgroups only

This group has 780 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
2C2×C31×#455C211yesmaximal, characteristic
3C31×#2717C211yesmaximal, characteristic, complemented
4C2×C31×#354C211yesmaximal, characteristic, complemented
5C2×C31×#352C211yesmaximal, characteristic, complemented
6C31×#1790C211yesmaximal, characteristic
7C31×#1390C211yesmaximal, characteristic, complemented
8C2×C31×H(5,𝔽2)C2211yescharacteristic
9C2×C31×#137—21no—
10C2×C31×#124C2211yescharacteristic
11C2×C31×#123—21no—
12C2×C31×#123C2211yescharacteristic
13C31×#458—22no—
14C31×#458—22no—
15C31×#455—22no—
16C31×#455—22no—
17C2×C31×#104—21no—
18C2×C31×#100C2211yescharacteristic
19C2×C31×#99—21no—
20C2×C31×#99—21no—
21C2×C31×#99C2211yescharacteristic
22C2×C31×#99C2211yescharacteristic
23C31×#397—21no—
24C31×#397—21no—
25C31×#395—21no—
26C31×#387—21no—
27C31×#386—21no—
28C31×#386C2211yescharacteristic
29C31×#385—21no—
30C31×#380—21no—
31C31×#354—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×Q8×C31D411yescharacteristic
41C22×D4×C31—21no—
42C22×D4×C31—21no—
43C22×D4×C31—41no—
44C22×D4×C31C2311yescharacteristic
45C23×C4×C31D411yescharacteristic
46C23×C4×C31D411yescharacteristic
47C31×#137—24no—
48C31×#137—42no—
49C2×Q16×C31—41no—
50C2×SD16×C31—41no—
51C2×M16×C31—21no—
52C31×#124—21no—
53C31×#124—21no—
54C31×#124—21no—
55C31×#124—21no—
56C31×#123—24no—
57C31×#123—21no—
58C31×#123—21no—
59C31×#123—21no—
60C31×#123—21no—
61C31×#123—42no—
62C2×C4⋊C4×C31—21no—
63C2×C4⋊C4×C31—21no—
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—41no—
74C2×C22⋊C4×C31—41no—
75C2×C22⋊C4×C31D411yescharacteristic
76C2×C22⋊C4×C31D411yescharacteristic
77C2×C42×C31D411yescharacteristic
78C31×#104—24no—
79C31×#100—21no—
80C31×#100D412yes—
81C31×#99—24no—
82C31×#99—21no—
83C31×#99—21no—
84C31×#99—21no—
85C31×#99—42no—
86C31×#99D412yes—
87C31×#95—21no—
88C31×#95—21no—
89C31×#95—21no—
90C31×#95—21no—
91C31×#95—21no—
92C31×#95—21no—
93C31×#95—41no—
94C24×C31—21no—
95C24×C31—21no—
96C24×C31C2×D411yescharacteristic
97C24×C31C2×D411yescharacteristic
98#41×C31—22no—
99#41×C31—22no—
100#41×C31—42no—
101#41×C31—41no—
102#41×C31—42no—
103#41×C31—42no—
104#41×C31—42no—
105#41×C31—41no—
106C2×Q8×C31—21no—
107C2×Q8×C31—21no—
108C2×Q8×C31—21no—
109C2×Q8×C31—41no—
110C2×Q8×C31—41no—
111C2×D4×C31—21no—
112C2×D4×C31—22no—
113C2×D4×C31—22no—
114C2×D4×C31—22no—
115C2×D4×C31—22no—
116C2×D4×C31—41no—
117C2×D4×C31—41no—
118C2×D4×C31—42no—
119C2×D4×C31—42no—
120C2×D4×C31—41no—
121C2×D4×C31—41no—
122C2×D4×C31—41no—
123C2×D4×C31—41no—
124C2×D4×C31—42no—
125C2×D4×C31—42no—
126C2×D4×C31—41no—
127C2×D4×C31—41no—
128C2×D4×C31—41no—
129C2×D4×C31C2×D411yescharacteristic
130C2×D4×C31C2×D411yescharacteristic
131C2×D4×C31C2×D411yescharacteristic
132C2×D4×C31#4111yescharacteristic
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—21no—
137C22×C4×C31—21no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—41no—
144C22×C4×C31—42no—
145C22×C4×C31—41no—
146C22×C4×C31—41no—
147C22×C4×C31—41no—
148C22×C4×C31—41no—
149C22×C4×C31—41no—
150C22×C4×C31C2×D411yescharacteristic
151Q16×C31—44no—
152SD16×C31—44no—
153M16×C31—22no—
154M16×C31—41no—
155C2×C8×C31—41no—
156C4⋊C4×C31—42no—
157C4⋊C4×C31—44no—
158C4⋊C4×C31—41no—
159C4⋊C4×C31—41no—
160C4⋊C4×C31—81no—
161C4⋊C4×C31—81no—
162C22⋊C4×C31—21no—
163C22⋊C4×C31—22no—
164C22⋊C4×C31—21no—
165C22⋊C4×C31—42no—
166C22⋊C4×C31—41no—
167C22⋊C4×C31—44no—
168C22⋊C4×C31—41no—
169C22⋊C4×C31—41no—
170C22⋊C4×C31—41no—
171C22⋊C4×C31—41no—
172C22⋊C4×C31—41no—
173C22⋊C4×C31—41no—
174C22⋊C4×C31—41no—
175C22⋊C4×C31—41no—
176C22⋊C4×C31—41no—
177C22⋊C4×C31—82no—
178C42×C31—21no—
179C42×C31—21no—
180#13349C3111yesmaximal, characteristic, direct product factor, Sylow
181C23×C31—21no—
182C23×C31—21no—
183C23×C31—21no—
184C23×C31—21no—
185C23×C31—22no—
186C23×C31—22no—
187C23×C31—41no—
188C23×C31—41no—
189C23×C31—41no—
190C23×C31—41no—
191C23×C31—41no—
192C23×C31—41no—
193C23×C31—41no—
194C23×C31—81no—
195C23×C31—81no—
196C23×C31—81no—
197C23×C31—81no—
198C23×C31#12011yescharacteristic
199C23×C31#12011yescharacteristic
200C23×C31#12111yescharacteristic
201C23×C31#12111yescharacteristic
202C23×C31#12711yescharacteristic
203Q8×C31—22no—
204Q8×C31—41no—
205Q8×C31—42no—
206Q8×C31—81no—
207D4×C31—22no—
208D4×C31—41no—
209D4×C31—41no—
210D4×C31—41no—
211D4×C31—41no—
212D4×C31—41no—
213D4×C31—42no—
214D4×C31—42no—
215D4×C31—42no—
216D4×C31—42no—
217D4×C31—42no—
218D4×C31—42no—
219C2×C4×C31—21no—
220C2×C4×C31—21no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—22no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—41no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—41no—
230C2×C4×C31—41no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—41no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—41no—
245C2×C4×C31—41no—
246C2×C4×C31—41no—
247C2×C4×C31—41no—
248C2×C4×C31—41no—
249C2×C4×C31—81no—
250C2×C4×C31—81no—
251C2×C4×C31—81no—
252C2×C4×C31—81no—
253C2×C4×C31—81no—
254C2×C4×C31—81no—
255C2×C4×C31—81no—
256C2×C4×C31#12011yescharacteristic
257C2×C4×C31#12111yescharacteristic
258C8×C31—42no—
259C2×#458C2×C3111yescharacteristic
260C2×#455C2×C3111yescharacteristic
261#2717C2×C3111yescharacteristic, complemented
262C2×#354C2×C3111yescharacteristic, complemented
263C2×#352C2×C3111yescharacteristic, complemented
264#1790C2×C3111yescharacteristic
265#1390C2×C3111yescharacteristic, complemented
266C22×C31—21no—
267C22×C31—21no—
268C22×C31—21no—
269C22×C31—21no—
270C22×C31—21no—
271C22×C31—21no—
272C22×C31—41no—
273C22×C31—41no—
274C22×C31—41no—
275C22×C31—41no—
276C22×C31—41no—
277C22×C31—41no—
278C22×C31—41no—
279C22×C31—41no—
280C22×C31—42no—
281C22×C31—81no—
282C22×C31—81no—
283C22×C31—82no—
284C22×C31—82no—
285C22×C31—81no—
286C22×C31—81no—
287C22×C31—161no—
288C22×C31—161no—
289C22×C31#39211yescharacteristic
290C22×C31B(4,𝔽2)11yescharacteristic
291C22×C31B(4,𝔽2)11yescharacteristic
292C4×C31—21no—
293C4×C31—21no—
294C4×C31—42no—
295C4×C31—42no—
296C4×C31—81no—
297C4×C31—81no—
298C4×C31—81no—
299C4×C31—82no—
300C4×C31—81no—
301C4×C31—81no—
302C4×C31—161no—
303C2×H(5,𝔽2)C22×C3111yescharacteristic
304C2×#137—21no—
305C2×#124C22×C3111yescharacteristic
306C2×#123—21no—
307C2×#123C22×C3111yescharacteristic
308#458—22no—
309#458—22no—
310#455—22no—
311#455—22no—
312C2×#104—21no—
313C2×#100C22×C3111yescharacteristic
314C2×#99—21no—
315C2×#99—21no—
316C2×#99C22×C3111yescharacteristic
317C2×#99C22×C3111yescharacteristic
318#397—21no—
319#397—21no—
320#395—21no—
321#387—21no—
322#386—21no—
323#386C22×C3111yescharacteristic
324#385—21no—
325#380—21no—
326#354—22no—
327#352—22no—
328C2×C31—21no—
329C2×C31—21no—
330C2×C31—41no—
331C2×C31—41no—
332C2×C31—41no—
333C2×C31—41no—
334C2×C31—82no—
335C2×C31#202111yescharacteristic
336C2×C31#219711yescharacteristic
337C2×C31#220111yescharacteristic
338H(5,𝔽2)—21no—
339H(5,𝔽2)—21no—
340H(5,𝔽2)—21no—
341H(5,𝔽2)—42no—
342H(5,𝔽2)D4×C3112yes—
343C2×#41—21no—
344C2×#41—41no—
345C22×Q8D4×C3111yescharacteristic
346C22×D4—21no—
347C22×D4—21no—
348C22×D4—41no—
349C22×D4C23×C3111yescharacteristic
350C23×C4D4×C3111yescharacteristic
351C23×C4D4×C3111yescharacteristic
352#137—24no—
353#137—42no—
354C2×Q16—41no—
355C2×SD16—41no—
356C2×M16—21no—
357#124—21no—
358#124—21no—
359#124—21no—
360#124—21no—
361#123—24no—
362#123—21no—
363#123—21no—
364#123—21no—
365#123—21no—
366#123—42no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—41no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—41no—
379C2×C22⋊C4—41no—
380C2×C22⋊C4D4×C3111yescharacteristic
381C2×C22⋊C4D4×C3111yescharacteristic
382C2×C42D4×C3111yescharacteristic
383#104—24no—
384#100—21no—
385#100D4×C3112yes—
386#99—24no—
387#99—21no—
388#99—21no—
389#99—21no—
390#99—42no—
391#99D4×C3112yes—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—41no—
399C31#1334911yescharacteristic, direct product factor, Sylow
400C24—21no—
401C24—21no—
402C24C2×D4×C3111yescharacteristic
403C24C2×D4×C3111yescharacteristic
404#41—22no—
405#41—22no—
406#41—42no—
407#41—41no—
408#41—42no—
409#41—42no—
410#41—42no—
411#41—41no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—41no—
416C2×Q8—41no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—41no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4C2×D4×C3111yescharacteristic
436C2×D4C2×D4×C3111yescharacteristic
437C2×D4C2×D4×C3111yescharacteristic
438C2×D4#41×C3111yescharacteristic
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4C2×D4×C3111yescharacteristic
457Q16—44no—
458SD16—44no—
459M16—22no—
460M16—41no—
461C2×C8—41no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C22⋊C4—21no—
469C22⋊C4—22no—
470C22⋊C4—21no—
471C22⋊C4—42no—
472C22⋊C4—41no—
473C22⋊C4—44no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—82no—
484C42—21no—
485C42—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—81no—
500C23—81no—
501C23—81no—
502C23—81no—
503C23C31×#12011yescharacteristic
504C23C31×#12011yescharacteristic
505C23C31×#12111yescharacteristic
506C23C31×#12111yescharacteristic
507C23C31×#12711yescharacteristic
508Q8—22no—
509Q8—41no—
510Q8—42no—
511Q8—81no—
512D4—22no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—41no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—42no—
522D4—42no—
523D4—42no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4C31×#12011yescharacteristic
562C2×C4C31×#12111yescharacteristic
563C8—42no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—81no—
580C22—81no—
581C22—82no—
582C22—82no—
583C22—81no—
584C22—81no—
585C22—161no—
586C22—161no—
587C22C31×#39211yescharacteristic
588C22C31×B(4,𝔽2)11yescharacteristic
589C22C31×B(4,𝔽2)11yescharacteristic
590C4—21no—
591C4—21no—
592C4—42no—
593C4—42no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—82no—
598C4—81no—
599C4—81no—
600C4—161no—
601C2—21no—
602C2—21no—
603C2—41no—
604C2—41no—
605C2—41no—
606C2—41no—
607C2—82no—
608C2C31×#202111yescharacteristic
609C2C31×#219711yescharacteristic
610C2C31×#220111yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C31
prime radicalC2×C31×H(5,𝔽2)C22
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
3. C31×#2717C2semidirect
4. C2×C31×#354C2semidirect
5. C2×C31×#352C2semidirect
7. C31×#1390C2semidirect
180. #13349C31direct (N × H)
261. #2717C2×C31semidirect
262. C2×#354C2×C31semidirect
263. C2×#352C2×C31semidirect
265. #1390C2×C31semidirect
399. C31#13349direct (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×#4552
C31×#27172
C2×C31×#3542
C2×C31×#3522
C31×#17902
C31×#13902
#1334931