Finite Group Databasebeta

← C2×C31×#1615 #21474865

Subgroups — C2×C31×#1615 #21474865

Order 7936.

1016 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#539C211yesmaximal, characteristic
2C2×C31×#484C211yesmaximal, characteristic, complemented
3C2×C31×#481C211yesmaximal, characteristic
4C2×C31×#400C211yesmaximal, characteristic, complemented
5C31×#1615C218yesmaximal, direct product factor
6C2×C31×#328C211yesmaximal, characteristic, complemented
7C2×C31×#327C211yesmaximal, characteristic
8C2×C31×#324C211yesmaximal, characteristic, complemented
9C31×#539—22no—
10C31×#539—22no—
11C31×#539—22no—
12C31×#539C2214yes—
13C2×C31×#126—21no—
14C2×C31×#125—21no—
15C2×C31×#123—21no—
16C2×C31×#123—21no—
17C2×C31×C22⋊Q8—21no—
18C2×C31×C22⋊Q8C2211yescharacteristic
19C2×C31×#121C2211yescharacteristic
20C2×C4×D4×C31—21no—
21C22×C4⋊C4×C31C2211yescharacteristic
22C22×C4⋊C4×C31C2211yescharacteristic
23C31×#484—22no—
24C31×#484C2214yescomplemented
25C31×#481—22no—
26C31×#481C2214yes—
27C2×C31×#107—21no—
28C2×C31×#106—21no—
29C2×C31×#103—21no—
30C2×C31×#102—21no—
31C2×C31×C22⋊C8C2211yescharacteristic
32C2×C31×C22⋊C8C2211yescharacteristic
33C31×#400—22no—
34C31×#400C2214yescomplemented
35C2×C31×#95—21no—
36C2×C31×#95—21no—
37C2×C31×#95C2211yescharacteristic
38C31×#328C2214yescomplemented
39C31×#327C2214yes—
40C31×#324C2214yescomplemented
41C22×Q8×C31D411yescharacteristic
42C22×D4×C31—21no—
43C22×D4×C31D411yescharacteristic
44C23×C4×C31—21no—
45C23×C4×C31—21no—
46C23×C4×C31—21no—
47C23×C4×C31D411yescharacteristic
48C23×C4×C31D411yescharacteristic
49C23×C4×C31C2311yescharacteristic
50C22×C8×C31—21no—
51C22×C8×C31—21no—
52C31×#126—24no—
53C31×#126—24no—
54C31×#125—24no—
55C31×#125—24no—
56C31×#123—24no—
57C31×#123—24no—
58C31×#123—24no—
59C31×#123—24no—
60C31×C22⋊Q8—24no—
61C31×C22⋊Q8—24no—
62C31×C22⋊Q8—21no—
63C31×C22⋊Q8—21no—
64C31×C22⋊Q8D412yes—
65C31×C22⋊Q8C2312yes—
66C31×#121—21no—
67C31×#121—21no—
68C31×#121D412yes—
69C31×#121C2312yes—
70C4×D4×C31—24no—
71C4×D4×C31—24no—
72C2×C4⋊C4×C31—21no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31—21no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31D412yes—
86C2×C4⋊C4×C31D411yescharacteristic
87C2×C4⋊C4×C31D411yescharacteristic
88C2×C4⋊C4×C31C2312yes—
89C2×C4⋊C4×C31C2312yes—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C42×C31—21no—
96C31×#107—24no—
97C31×#106—24no—
98C31×#103—24no—
99C31×#102—24no—
100C31×C22⋊C8—21no—
101C31×C22⋊C8—21no—
102C31×C22⋊C8C2312yes—
103C31×C22⋊C8C2312yes—
104C31×#95—21no—
105C31×#95—24no—
106C31×#95—24no—
107C31×#95C2312yes—
108C24×C31—21no—
109C24×C31C2×D411yescharacteristic
110C2×Q8×C31—21no—
111C2×Q8×C31—42no—
112C2×Q8×C31C2×D412yes—
113C2×D4×C31—22no—
114C2×D4×C31—22no—
115C2×D4×C31—21no—
116C2×D4×C31—42no—
117C2×D4×C31—42no—
118C2×D4×C31—42no—
119C2×D4×C31C2×D412yes—
120C22×C4×C31—21no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—24no—
129C22×C4×C31—21no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—42no—
133C22×C4×C31—42no—
134C22×C4×C31—42no—
135C22×C4×C31—42no—
136C22×C4×C31—44no—
137C22×C4×C31—41no—
138C22×C4×C31—41no—
139C22×C4×C31—42no—
140C22×C4×C31—82no—
141C22×C4×C31C2×D412yes—
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D412yes—
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2411yescharacteristic
149C2×C8×C31—22no—
150C2×C8×C31—22no—
151C2×C8×C31—42no—
152C2×C8×C31—42no—
153C4⋊C4×C31—24no—
154C4⋊C4×C31—24no—
155C4⋊C4×C31—24no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—41no—
167C4⋊C4×C31—41no—
168C4⋊C4×C31C2×D412yes—
169C4⋊C4×C31C2×D412yes—
170C22⋊C4×C31—24no—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—24no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—22no—
177C42×C31—24no—
178C2×#1615C3111yesmaximal, characteristic, direct product factor, Sylow
179C23×C31—22no—
180C23×C31—22no—
181C23×C31—22no—
182C23×C31—42no—
183C23×C31—42no—
184C23×C31—42no—
185C23×C31#12011yescharacteristic
186C23×C31#12011yescharacteristic
187C23×C31#13512yes—
188C23×C31#13712yes—
189C23×C31C22×D411yescharacteristic
190Q8×C31—42no—
191Q8×C31—81no—
192D4×C31—42no—
193D4×C31—42no—
194D4×C31—42no—
195D4×C31—44no—
196D4×C31—81no—
197C2×C4×C31—22no—
198C2×C4×C31—22no—
199C2×C4×C31—22no—
200C2×C4×C31—22no—
201C2×C4×C31—22no—
202C2×C4×C31—22no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—41no—
207C2×C4×C31—41no—
208C2×C4×C31—41no—
209C2×C4×C31—41no—
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—44no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—81no—
223C2×C4×C31—84no—
224C2×C4×C31—84no—
225C2×C4×C31—82no—
226C2×C4×C31—82no—
227C2×C4×C31#12011yescharacteristic
228C2×C4×C31#12011yescharacteristic
229C2×C4×C31C22×D411yescharacteristic
230C2×C4×C31C22×D411yescharacteristic
231C8×C31—42no—
232C8×C31—42no—
233C2×#539C2×C3111yescharacteristic
234C2×#484C2×C3111yescharacteristic, complemented
235C2×#481C2×C3111yescharacteristic
236C2×#400C2×C3111yescharacteristic, complemented
237#1615C2×C3118yesdirect product factor
238C2×#328C2×C3111yescharacteristic, complemented
239C2×#327C2×C3111yescharacteristic
240C2×#324C2×C3111yescharacteristic, complemented
241C22×C31—24no—
242C22×C31—24no—
243C22×C31—21no—
244C22×C31—21no—
245C22×C31—21no—
246C22×C31—21no—
247C22×C31—42no—
248C22×C31—42no—
249C22×C31—42no—
250C22×C31—84no—
251C22×C31#45212yes—
252C22×C31#45212yes—
253C22×C31#45412yes—
254C22×C31#45812yes—
255C22×C31C2×#12011yescharacteristic
256C22×C31C2×#13511yescharacteristic
257C22×C31C2×#13711yescharacteristic
258C4×C31—21no—
259C4×C31—21no—
260C4×C31—21no—
261C4×C31—21no—
262C4×C31—42no—
263C4×C31—42no—
264C4×C31—42no—
265C4×C31—42no—
266C4×C31—42no—
267C4×C31—82no—
268C4×C31—82no—
269#539—22no—
270#539—22no—
271#539—22no—
272#539C22×C3114yes—
273C2×#126—21no—
274C2×#125—21no—
275C2×#123—21no—
276C2×#123—21no—
277C2×C22⋊Q8—21no—
278C2×C22⋊Q8C22×C3111yescharacteristic
279C2×#121C22×C3111yescharacteristic
280C2×C4×D4—21no—
281C22×C4⋊C4C22×C3111yescharacteristic
282C22×C4⋊C4C22×C3111yescharacteristic
283#484—22no—
284#484C22×C3114yescomplemented
285#481—22no—
286#481C22×C3114yes—
287C2×#107—21no—
288C2×#106—21no—
289C2×#103—21no—
290C2×#102—21no—
291C2×C22⋊C8C22×C3111yescharacteristic
292C2×C22⋊C8C22×C3111yescharacteristic
293#400—22no—
294#400C22×C3114yescomplemented
295C2×#95—21no—
296C2×#95—21no—
297C2×#95C22×C3111yescharacteristic
298#328C22×C3114yescomplemented
299#327C22×C3114yes—
300#324C22×C3114yescomplemented
301C2×C31—21no—
302C2×C31—21no—
303C2×C31—22no—
304C2×C31—82no—
305C2×C31#161514yesdirect product factor
306C2×C31C2×#45211yescharacteristic
307C2×C31C2×#45411yescharacteristic
308C2×C31C2×#45811yescharacteristic
309C22×Q8D4×C3111yescharacteristic
310C22×D4—21no—
311C22×D4D4×C3111yescharacteristic
312C23×C4—21no—
313C23×C4—21no—
314C23×C4—21no—
315C23×C4D4×C3111yescharacteristic
316C23×C4D4×C3111yescharacteristic
317C23×C4C23×C3111yescharacteristic
318C22×C8—21no—
319C22×C8—21no—
320#126—24no—
321#126—24no—
322#125—24no—
323#125—24no—
324#123—24no—
325#123—24no—
326#123—24no—
327#123—24no—
328C22⋊Q8—24no—
329C22⋊Q8—24no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8D4×C3112yes—
333C22⋊Q8C23×C3112yes—
334#121—21no—
335#121—21no—
336#121D4×C3112yes—
337#121C23×C3112yes—
338C4×D4—24no—
339C4×D4—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4D4×C3112yes—
354C2×C4⋊C4D4×C3111yescharacteristic
355C2×C4⋊C4D4×C3111yescharacteristic
356C2×C4⋊C4C23×C3112yes—
357C2×C4⋊C4C23×C3112yes—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C42—21no—
364#107—24no—
365#106—24no—
366#103—24no—
367#102—24no—
368C22⋊C8—21no—
369C22⋊C8—21no—
370C22⋊C8C23×C3112yes—
371C22⋊C8C23×C3112yes—
372#95—24no—
373#95—24no—
374#95—21no—
375#95C23×C3112yes—
376C31C2×#161511yescharacteristic, direct product factor, Sylow
377C24—21no—
378C24C2×D4×C3111yescharacteristic
379C2×Q8—21no—
380C2×Q8—42no—
381C2×Q8C2×D4×C3112yes—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4C2×D4×C3112yes—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—82no—
410C22×C4C2×D4×C3112yes—
411C22×C4C2×D4×C3111yescharacteristic
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4C2×D4×C3112yes—
414C22×C4C2×D4×C3111yescharacteristic
415C22×C4C2×D4×C3111yescharacteristic
416C22×C4C2×D4×C3111yescharacteristic
417C22×C4C24×C3111yescharacteristic
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—42no—
421C2×C8—42no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4C2×D4×C3112yes—
438C4⋊C4C2×D4×C3112yes—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C42—24no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23C31×#12011yescharacteristic
454C23C31×#12011yescharacteristic
455C23C31×#13512yes—
456C23C31×#13712yes—
457C23C22×D4×C3111yescharacteristic
458Q8—42no—
459Q8—81no—
460D4—42no—
461D4—44no—
462D4—42no—
463D4—42no—
464D4—81no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—81no—
491C2×C4—84no—
492C2×C4—84no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4C31×#12011yescharacteristic
496C2×C4C31×#12011yescharacteristic
497C2×C4C22×D4×C3111yescharacteristic
498C2×C4C22×D4×C3111yescharacteristic
499C8—42no—
500C8—42no—
501C22—24no—
502C22—24no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—84no—
511C22C31×#45212yes—
512C22C31×#45212yes—
513C22C31×#45412yes—
514C22C31×#45812yes—
515C22C2×C31×#12011yescharacteristic
516C22C2×C31×#13511yescharacteristic
517C22C2×C31×#13711yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—82no—
528C4—82no—
529C2—21no—
530C2—22no—
531C2—21no—
532C2—82no—
533C2C31×#161514yesdirect product factor
534C2C2×C31×#45211yescharacteristic
535C2C2×C31×#45411yescharacteristic
536C2C2×C31×#45811yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#484C2semidirect
4. C2×C31×#400C2semidirect
5. C31×#1615C2direct (N × H)
6. C2×C31×#328C2semidirect
8. C2×C31×#324C2semidirect
24. C31×#484C22semidirect
34. C31×#400C22semidirect
38. C31×#328C22semidirect
40. C31×#324C22semidirect
178. C2×#1615C31direct (N × H)
234. C2×#484C2×C31semidirect
236. C2×#400C2×C31semidirect
237. #1615C2×C31direct (N × H)
238. C2×#328C2×C31semidirect
240. C2×#324C2×C31semidirect
284. #484C22×C31semidirect
294. #400C22×C31semidirect
298. #328C22×C31semidirect
300. #324C22×C31semidirect
305. C2×C31#1615direct (N × H)
376. C31C2×#1615direct (N × H)
533. C2C31×#1615direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5392
C2×C31×#4842
C2×C31×#4812
C2×C31×#4002
C31×#16152
C2×C31×#3282
C2×C31×#3272
C2×C31×#3242
C2×#161531