Finite Group Databasebeta

← C2×C31×#2010 #21480172

Subgroups — C2×C31×#2010 #21480172

Order 7936.

2190 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#575C211yesmaximal, characteristic, complemented
2C2×C31×#525C212yesmaximal, complemented
3C2×C31×#421C212yesmaximal, complemented
4C2×C31×#407C211yesmaximal, characteristic, complemented
5C31×#2010C218yesmaximal, direct product factor
6C2×C31×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
7C22×#41×C31C2212yes—
8C24×C4×C31C2211yescharacteristic
9C31×#575—24no—
10C31×#575—22no—
11C31×#575C2214yescomplemented
12C2×C31×#137—22no—
13C2×C31×AGL(1,ℤ/8)—22no—
14C2×C31×#135—22no—
15C22×M16×C31C2211yescharacteristic
16C22×M16×C31C2211yescharacteristic
17C31×#525—24no—
18C31×#525—24no—
19C31×#525—24no—
20C31×#525C2218yescomplemented
21C2×C31×#123—22no—
22C2×C31×C22⋊Q8—22no—
23C2×C31×#121—22no—
24C2×C31×#120—22no—
25C2×C4×D4×C31—24no—
26C2×C31×#117C2212yes—
27C31×#421—24no—
28C31×#421C2218yescomplemented
29C2×C31×#104—24no—
30C31×#407—22no—
31C31×#407C2214yescomplemented
32C2×C31×C22⋊C8—22no—
33C31×H(3,ℤ/4)C2214yescomplemented
34C25×C31D411yescharacteristic
35C2×#41×C31—22no—
36C2×#41×C31—22no—
37C2×#41×C31—22no—
38C2×#41×C31—22no—
39C2×#41×C31—22no—
40C2×#41×C31—22no—
41C2×#41×C31—22no—
42C2×#41×C31—22no—
43C2×#41×C31—22no—
44C2×#41×C31—22no—
45C2×#41×C31D414yescomplemented
46C2×#41×C31C2314yes—
47C22×Q8×C31D412yescomplemented
48C22×D4×C31—24no—
49C22×D4×C31—22no—
50C22×D4×C31D412yescomplemented
51C23×C4×C31—21no—
52C23×C4×C31—21no—
53C23×C4×C31—22no—
54C23×C4×C31—22no—
55C23×C4×C31—22no—
56C23×C4×C31—22no—
57C23×C4×C31—22no—
58C23×C4×C31—21no—
59C23×C4×C31—21no—
60C23×C4×C31D411yescharacteristic
61C23×C4×C31D412yes—
62C23×C4×C31C2312yes—
63C23×C4×C31C2311yescharacteristic
64C31×#137—28no—
65C31×#137—28no—
66C31×AGL(1,ℤ/8)—28no—
67C31×AGL(1,ℤ/8)—28no—
68C31×#135—28no—
69C31×#135—28no—
70C2×Q16×C31—22no—
71C2×SD16×C31—24no—
72C2×D8×C31—22no—
73C2×M16×C31—22no—
74C2×M16×C31—22no—
75C2×M16×C31—21no—
76C2×M16×C31—22no—
77C2×M16×C31—22no—
78C2×M16×C31D412yescomplemented
79C2×M16×C31C2312yes—
80C2×M16×C31C2312yes—
81C22×C8×C31—21no—
82C22×C8×C31D412yescomplemented
83C31×#123—28no—
84C31×#123—28no—
85C31×C22⋊Q8—28no—
86C31×C22⋊Q8—28no—
87C31×#121—28no—
88C31×#121—28no—
89C31×#120—28no—
90C31×#120—28no—
91C4×D4×C31—216no—
92C4×D4×C31—216no—
93C31×#117—22no—
94C31×#117—22no—
95C31×#117D414yescomplemented
96C31×#117C2314yes—
97C2×C4⋊C4×C31—24no—
98C2×C4⋊C4×C31—22no—
99C2×C22⋊C4×C31—24no—
100C2×C22⋊C4×C31—24no—
101C2×C22⋊C4×C31—22no—
102C2×C42×C31D414yescomplemented
103C31×#104—216no—
104C31×C22⋊C8—28no—
105C24×C31—22no—
106C24×C31—21no—
107C24×C31—22no—
108C24×C31—22no—
109C24×C31—21no—
110C24×C31—44no—
111C24×C31C2×D412yes—
112C24×C31C2×D411yescharacteristic
113#41×C31—24no—
114#41×C31—24no—
115#41×C31—24no—
116#41×C31—24no—
117#41×C31—24no—
118#41×C31—24no—
119#41×C31—24no—
120#41×C31—24no—
121#41×C31—42no—
122#41×C31—42no—
123#41×C31—42no—
124#41×C31—42no—
125#41×C31—42no—
126#41×C31—42no—
127#41×C31—42no—
128#41×C31—42no—
129C2×Q8×C31—22no—
130C2×Q8×C31—22no—
131C2×Q8×C31—22no—
132C2×Q8×C31—22no—
133C2×Q8×C31—22no—
134C2×Q8×C31C2×D414yescomplemented
135C2×D4×C31—216no—
136C2×D4×C31—24no—
137C2×D4×C31—24no—
138C2×D4×C31—22no—
139C2×D4×C31—22no—
140C2×D4×C31—22no—
141C2×D4×C31—22no—
142C2×D4×C31—22no—
143C2×D4×C31—416no—
144C2×D4×C31—42no—
145C2×D4×C31—42no—
146C2×D4×C31—42no—
147C2×D4×C31—42no—
148C2×D4×C31C2×D414yescomplemented
149C22×C4×C31—24no—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—24no—
156C22×C4×C31—24no—
157C22×C4×C31—24no—
158C22×C4×C31—22no—
159C22×C4×C31—22no—
160C22×C4×C31—22no—
161C22×C4×C31—24no—
162C22×C4×C31—24no—
163C22×C4×C31—24no—
164C22×C4×C31—22no—
165C22×C4×C31—41no—
166C22×C4×C31—41no—
167C22×C4×C31—42no—
168C22×C4×C31—42no—
169C22×C4×C31—42no—
170C22×C4×C31—42no—
171C22×C4×C31—42no—
172C22×C4×C31—42no—
173C22×C4×C31—42no—
174C22×C4×C31—41no—
175C22×C4×C31—42no—
176C22×C4×C31—42no—
177C22×C4×C31—41no—
178C22×C4×C31—42no—
179C22×C4×C31C2×D412yes—
180C22×C4×C31C2×D411yescharacteristic
181C22×C4×C31C2×D412yes—
182C22×C4×C31C2×D412yes—
183C22×C4×C31C2×D411yescharacteristic
184C22×C4×C31C2×D411yescharacteristic
185C22×C4×C31C2×D412yes—
186C22×C4×C31C2×D412yes—
187C22×C4×C31#4111yescharacteristic
188C22×C4×C31#4111yescharacteristic
189C22×C4×C31C2411yescharacteristic
190Q16×C31—28no—
191SD16×C31—216no—
192D8×C31—28no—
193M16×C31—24no—
194M16×C31—24no—
195M16×C31—24no—
196M16×C31—42no—
197C2×C8×C31—22no—
198C2×C8×C31—22no—
199C2×C8×C31—22no—
200C2×C8×C31—41no—
201C2×C8×C31—41no—
202C2×C8×C31C2×D414yescomplemented
203C4⋊C4×C31—216no—
204C4⋊C4×C31—24no—
205C4⋊C4×C31—24no—
206C22⋊C4×C31—216no—
207C22⋊C4×C31—216no—
208C22⋊C4×C31—24no—
209C22⋊C4×C31—24no—
210C42×C31—24no—
211C42×C31C2×D418yescomplemented
212C2×#2010C3111yesmaximal, characteristic, direct product factor, Sylow
213C23×C31—28no—
214C23×C31—24no—
215C23×C31—24no—
216C23×C31—22no—
217C23×C31—24no—
218C23×C31—42no—
219C23×C31—44no—
220C23×C31—44no—
221C23×C31—42no—
222C23×C31—48no—
223C23×C31—44no—
224C23×C31—48no—
225C23×C31—42no—
226C23×C31—41no—
227C23×C31—41no—
228C23×C31—42no—
229C23×C31—42no—
230C23×C31#12012yes—
231C23×C31#12012yes—
232C23×C31#12111yescharacteristic
233C23×C31#12111yescharacteristic
234C23×C31C22×D411yescharacteristic
235Q8×C31—24no—
236Q8×C31—24no—
237Q8×C31—42no—
238Q8×C31—42no—
239D4×C31—24no—
240D4×C31—24no—
241D4×C31—432no—
242D4×C31—44no—
243D4×C31—44no—
244D4×C31—44no—
245D4×C31—44no—
246D4×C31—42no—
247D4×C31—42no—
248C2×C4×C31—28no—
249C2×C4×C31—24no—
250C2×C4×C31—24no—
251C2×C4×C31—22no—
252C2×C4×C31—48no—
253C2×C4×C31—42no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—44no—
260C2×C4×C31—44no—
261C2×C4×C31—42no—
262C2×C4×C31—44no—
263C2×C4×C31—42no—
264C2×C4×C31—44no—
265C2×C4×C31—42no—
266C2×C4×C31—44no—
267C2×C4×C31—44no—
268C2×C4×C31#12011yescharacteristic
269C2×C4×C31#12011yescharacteristic
270C2×C4×C31#12012yes—
271C2×C4×C31#12012yes—
272C2×C4×C31#12012yes—
273C2×C4×C31#12112yes—
274C2×C4×C31#12112yes—
275C2×C4×C31#12112yes—
276C2×C4×C31#12112yes—
277C2×C4×C31#12111yescharacteristic
278C2×C4×C31#12111yescharacteristic
279C2×C4×C31#12711yescharacteristic
280C2×C4×C31#12712yes—
281C2×C4×C31#12711yescharacteristic
282C2×C4×C31C22×D412yes—
283C2×C4×C31C22×D411yescharacteristic
284C2×C4×C31C22×D412yes—
285C2×C4×C31C2×#4111yescharacteristic
286C8×C31—24no—
287C8×C31—42no—
288C2×#575C2×C3111yescharacteristic, complemented
289C2×#525C2×C3112yescomplemented
290C2×#421C2×C3112yescomplemented
291C2×#407C2×C3111yescharacteristic, complemented
292#2010C2×C3118yesdirect product factor
293C2×H(3,ℤ/4)C2×C3111yescharacteristic, complemented
294C22×C31—24no—
295C22×C31—22no—
296C22×C31—28no—
297C22×C31—24no—
298C22×C31—42no—
299C22×C31—41no—
300C22×C31—41no—
301C22×C31—44no—
302C22×C31—42no—
303C22×C31—48no—
304C22×C31—44no—
305C22×C31—48no—
306C22×C31—42no—
307C22×C31—44no—
308C22×C31—44no—
309C22×C31—84no—
310C22×C31—84no—
311C22×C31#39212yes—
312C22×C31#39211yescharacteristic
313C22×C31#39211yescharacteristic
314C22×C31C2×#12012yes—
315C22×C31C2×#12111yescharacteristic
316C4×C31—48no—
317C4×C31—42no—
318C4×C31—44no—
319C4×C31#39211yescharacteristic
320C4×C31#39212yes—
321C4×C31#39211yescharacteristic
322C4×C31C2×#12011yescharacteristic
323C4×C31C2×#12112yes—
324C4×C31C2×#12711yescharacteristic
325C22×#41C22×C3112yes—
326C24×C4C22×C3111yescharacteristic
327#575—24no—
328#575—22no—
329#575C22×C3114yescomplemented
330C2×#137—22no—
331C2×AGL(1,ℤ/8)—22no—
332C2×#135—22no—
333C22×M16C22×C3111yescharacteristic
334C22×M16C22×C3111yescharacteristic
335#525—24no—
336#525—24no—
337#525—24no—
338#525C22×C3118yescomplemented
339C2×#123—22no—
340C2×C22⋊Q8—22no—
341C2×#121—22no—
342C2×#120—22no—
343C2×C4×D4—24no—
344C2×#117C22×C3112yes—
345#421—24no—
346#421C22×C3118yescomplemented
347C2×#104—24no—
348#407—22no—
349#407C22×C3114yescomplemented
350C2×C22⋊C8—22no—
351H(3,ℤ/4)C22×C3114yescomplemented
352C2×C31—22no—
353C2×C31—21no—
354C2×C31—22no—
355C2×C31—21no—
356C2×C31—44no—
357C2×C31—84no—
358C2×C31#201012yesdirect product factor
359C2×C31C2×#39211yescharacteristic
360C25D4×C3111yescharacteristic
361C2×#41—22no—
362C2×#41—22no—
363C2×#41—22no—
364C2×#41—22no—
365C2×#41—22no—
366C2×#41—22no—
367C2×#41—22no—
368C2×#41—22no—
369C2×#41—22no—
370C2×#41—22no—
371C2×#41D4×C3114yescomplemented
372C2×#41C23×C3114yes—
373C22×Q8D4×C3112yescomplemented
374C22×D4—24no—
375C22×D4—22no—
376C22×D4D4×C3112yescomplemented
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—22no—
380C23×C4—22no—
381C23×C4—22no—
382C23×C4—21no—
383C23×C4—21no—
384C23×C4—21no—
385C23×C4—21no—
386C23×C4D4×C3111yescharacteristic
387C23×C4D4×C3112yes—
388C23×C4C23×C3112yes—
389C23×C4C23×C3111yescharacteristic
390#137—28no—
391#137—28no—
392AGL(1,ℤ/8)—28no—
393AGL(1,ℤ/8)—28no—
394#135—28no—
395#135—28no—
396C2×Q16—22no—
397C2×SD16—24no—
398C2×D8—22no—
399C2×M16—22no—
400C2×M16—22no—
401C2×M16—21no—
402C2×M16—22no—
403C2×M16—22no—
404C2×M16D4×C3112yescomplemented
405C2×M16C23×C3112yes—
406C2×M16C23×C3112yes—
407C22×C8—21no—
408C22×C8D4×C3112yescomplemented
409#123—28no—
410#123—28no—
411C22⋊Q8—28no—
412C22⋊Q8—28no—
413#121—28no—
414#121—28no—
415#120—28no—
416#120—28no—
417C4×D4—216no—
418C4×D4—216no—
419#117—22no—
420#117—22no—
421#117D4×C3114yescomplemented
422#117C23×C3114yes—
423C2×C4⋊C4—24no—
424C2×C4⋊C4—22no—
425C2×C22⋊C4—24no—
426C2×C22⋊C4—24no—
427C2×C22⋊C4—22no—
428C2×C42D4×C3114yescomplemented
429#104—216no—
430C22⋊C8—28no—
431C31C2×#201011yescharacteristic, direct product factor, Sylow
432C24—22no—
433C24—21no—
434C24—22no—
435C24—22no—
436C24—21no—
437C24—44no—
438C24C2×D4×C3112yes—
439C24C2×D4×C3111yescharacteristic
440#41—24no—
441#41—24no—
442#41—24no—
443#41—24no—
444#41—24no—
445#41—24no—
446#41—24no—
447#41—24no—
448#41—42no—
449#41—42no—
450#41—42no—
451#41—42no—
452#41—42no—
453#41—42no—
454#41—42no—
455#41—42no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8C2×D4×C3114yescomplemented
462C2×D4—216no—
463C2×D4—24no—
464C2×D4—24no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—416no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4C2×D4×C3114yescomplemented
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—41no—
502C22×C4—42no—
503C22×C4—41no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4C2×D4×C3112yes—
507C22×C4C2×D4×C3111yescharacteristic
508C22×C4C2×D4×C3112yes—
509C22×C4C2×D4×C3112yes—
510C22×C4C2×D4×C3111yescharacteristic
511C22×C4C2×D4×C3111yescharacteristic
512C22×C4C2×D4×C3112yes—
513C22×C4C2×D4×C3112yes—
514C22×C4#41×C3111yescharacteristic
515C22×C4#41×C3111yescharacteristic
516C22×C4C24×C3111yescharacteristic
517Q16—28no—
518SD16—216no—
519D8—28no—
520M16—24no—
521M16—24no—
522M16—24no—
523M16—42no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—41no—
528C2×C8—41no—
529C2×C8C2×D4×C3114yescomplemented
530C4⋊C4—216no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C22⋊C4—216no—
534C22⋊C4—216no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C42—24no—
538C42C2×D4×C3118yescomplemented
539C23—28no—
540C23—24no—
541C23—24no—
542C23—22no—
543C23—24no—
544C23—42no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—48no—
549C23—44no—
550C23—48no—
551C23—42no—
552C23—41no—
553C23—41no—
554C23—42no—
555C23—42no—
556C23C31×#12012yes—
557C23C31×#12012yes—
558C23C31×#12111yescharacteristic
559C23C31×#12111yescharacteristic
560C23C22×D4×C3111yescharacteristic
561Q8—24no—
562Q8—24no—
563Q8—42no—
564Q8—42no—
565D4—24no—
566D4—24no—
567D4—432no—
568D4—44no—
569D4—44no—
570D4—44no—
571D4—44no—
572D4—42no—
573D4—42no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4C31×#12011yescharacteristic
595C2×C4C31×#12011yescharacteristic
596C2×C4C31×#12012yes—
597C2×C4C31×#12012yes—
598C2×C4C31×#12012yes—
599C2×C4C31×#12112yes—
600C2×C4C31×#12112yes—
601C2×C4C31×#12112yes—
602C2×C4C31×#12112yes—
603C2×C4C31×#12111yescharacteristic
604C2×C4C31×#12111yescharacteristic
605C2×C4C31×#12711yescharacteristic
606C2×C4C31×#12712yes—
607C2×C4C31×#12711yescharacteristic
608C2×C4C22×D4×C3112yes—
609C2×C4C22×D4×C3111yescharacteristic
610C2×C4C22×D4×C3112yes—
611C2×C4C2×#41×C3111yescharacteristic
612C8—24no—
613C8—42no—
614C22—24no—
615C22—22no—
616C22—28no—
617C22—24no—
618C22—42no—
619C22—41no—
620C22—41no—
621C22—44no—
622C22—42no—
623C22—48no—
624C22—44no—
625C22—48no—
626C22—42no—
627C22—44no—
628C22—44no—
629C22—84no—
630C22—84no—
631C22C31×#39212yes—
632C22C31×#39211yescharacteristic
633C22C31×#39211yescharacteristic
634C22C2×C31×#12012yes—
635C22C2×C31×#12111yescharacteristic
636C4—48no—
637C4—42no—
638C4—44no—
639C4C31×#39211yescharacteristic
640C4C31×#39212yes—
641C4C31×#39211yescharacteristic
642C4C2×C31×#12011yescharacteristic
643C4C2×C31×#12112yes—
644C4C2×C31×#12711yescharacteristic
645C2—22no—
646C2—21no—
647C2—22no—
648C2—21no—
649C2—44no—
650C2—84no—
651C2C31×#201012yesdirect product factor
652C2C2×C31×#39211yescharacteristic
subgroupG/H
centerC2×C4×C31#127
commutator, Frattini, cosocleC22×C4C24×C31
socleC22×C31#392
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#575C2semidirect
2. C2×C31×#525C2semidirect
3. C2×C31×#421C2semidirect
4. C2×C31×#407C2semidirect
5. C31×#2010C2direct (N × H)
6. C2×C31×H(3,ℤ/4)C2semidirect
11. C31×#575C22semidirect
20. C31×#525C22semidirect
28. C31×#421C22semidirect
31. C31×#407C22semidirect
33. C31×H(3,ℤ/4)C22semidirect
45. C2×#41×C31D4semidirect
47. C22×Q8×C31D4semidirect
50. C22×D4×C31D4semidirect
78. C2×M16×C31D4semidirect
82. C22×C8×C31D4semidirect
95. C31×#117D4semidirect
102. C2×C42×C31D4semidirect
134. C2×Q8×C31C2×D4semidirect
148. C2×D4×C31C2×D4semidirect
202. C2×C8×C31C2×D4semidirect
211. C42×C31C2×D4semidirect
212. C2×#2010C31direct (N × H)
288. C2×#575C2×C31semidirect
289. C2×#525C2×C31semidirect
290. C2×#421C2×C31semidirect
291. C2×#407C2×C31semidirect
292. #2010C2×C31direct (N × H)
293. C2×H(3,ℤ/4)C2×C31semidirect
329. #575C22×C31semidirect
338. #525C22×C31semidirect
346. #421C22×C31semidirect
349. #407C22×C31semidirect
351. H(3,ℤ/4)C22×C31semidirect
358. C2×C31#2010direct (N × H)
371. C2×#41D4×C31semidirect
373. C22×Q8D4×C31semidirect
376. C22×D4D4×C31semidirect
404. C2×M16D4×C31semidirect
408. C22×C8D4×C31semidirect
421. #117D4×C31semidirect
428. C2×C42D4×C31semidirect
431. C31C2×#2010direct (N × H)
461. C2×Q8C2×D4×C31semidirect
475. C2×D4C2×D4×C31semidirect
529. C2×C8C2×D4×C31semidirect
538. C42C2×D4×C31semidirect
651. C2C31×#2010direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5752
C2×C31×#5252
C2×C31×#4212
C2×C31×#4072
C31×#20102
C2×C31×H(3,ℤ/4)2
C2×#201031