Finite Group Databasebeta

← C31×#13937 #21470696

Subgroups — C31×#13937 #21470696

Order 7936.

4470 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#41×C31C211yesmaximal, characteristic
2C2×C31×#391C212yesmaximal, complemented
3C2×C31×#390C212yesmaximal, complemented
4C2×C31×#380C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊C4×C31C211yesmaximal, characteristic, complemented
6C31×#1438C214yesmaximal, complemented
7C31×C22⋊#95C214yesmaximal, complemented
8C22×#41×C31—28no—
9C22×#41×C31—28no—
10C22×#41×C31—24no—
11C22×#41×C31—24no—
12C22×#41×C31C418yescomplemented
13C23×Q8×C31C2211yescharacteristic
14C23×D4×C31C2212yes—
15C23×D4×C31C2211yescharacteristic
16C24×C4×C31C2212yes—
17C24×C4×C31C2211yescharacteristic
18C22×C4⋊C4×C31C2212yes—
19C22×C22⋊C4×C31C2212yes—
20C22×C22⋊C4×C31C2218yes—
21C22×C22⋊C4×C31C2212yes—
22C22×C42×C31C2212yes—
23C31×#391—28no—
24C31×#390—28no—
25C31×#380—24no—
26C4×C22⋊C4×C31—24no—
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2218yes—
30C25×C31C2312yes—
31C25×C31C2311yescharacteristic
32C2×#41×C31—264no—
33C2×#41×C31—264no—
34C2×#41×C31—232no—
35C2×#41×C31—232no—
36C2×#41×C31—232no—
37C22×Q8×C31—22no—
38C22×Q8×C31—24no—
39C22×Q8×C31—22no—
40C22×Q8×C31C2×C414yes—
41C22×Q8×C31D418yes—
42C22×D4×C31—28no—
43C22×D4×C31—24no—
44C22×D4×C31—24no—
45C22×D4×C31—28no—
46C22×D4×C31—22no—
47C22×D4×C31—24no—
48C22×D4×C31—22no—
49C22×D4×C31C2×C418yes—
50C22×D4×C31C2×C414yes—
51C22×D4×C31D418yes—
52C23×C4×C31—28no—
53C23×C4×C31—28no—
54C23×C4×C31—24no—
55C23×C4×C31—24no—
56C23×C4×C31—24no—
57C23×C4×C31—24no—
58C23×C4×C31C2×C418yes—
59C23×C4×C31C2×C414yes—
60C23×C4×C31D414yes—
61C23×C4×C31D414yes—
62C23×C4×C31C2314yes—
63C23×C4×C31C2314yes—
64C23×C4×C31C2311yescharacteristic
65C23×C4×C31C2312yes—
66C23×C4×C31C2311yescharacteristic
67C2×C4⋊C4×C31—24no—
68C2×C4⋊C4×C31—24no—
69C2×C4⋊C4×C31—24no—
70C2×C22⋊C4×C31—24no—
71C2×C22⋊C4×C31—28no—
72C2×C22⋊C4×C31—24no—
73C2×C22⋊C4×C31—216no—
74C2×C22⋊C4×C31—216no—
75C2×C22⋊C4×C31—48no—
76C2×C22⋊C4×C31—44no—
77C2×C42×C31—24no—
78C2×C42×C31—24no—
79C2×C42×C31—24no—
80C31×#95—24no—
81C31×#95—24no—
82C31×#95—216no—
83C24×C31—22no—
84C24×C31—22no—
85C24×C31—22no—
86C24×C31—24no—
87C24×C31—22no—
88C24×C31—24no—
89C24×C31—28no—
90C24×C31—22no—
91C24×C31—24no—
92C24×C31—22no—
93C24×C31—24no—
94C24×C31C22×C414yes—
95C24×C31C22×C412yes—
96C24×C31C2×D418yes—
97C24×C31C2×D412yes—
98C24×C31C2×D412yes—
99C24×C31C2411yescharacteristic
100#41×C31—2256no—
101C2×Q8×C31—28no—
102C2×Q8×C31—28no—
103C2×Q8×C31—216no—
104C2×Q8×C31—216no—
105C2×Q8×C31C22⋊C4116yescomplemented
106C2×D4×C31—232no—
107C2×D4×C31—232no—
108C2×D4×C31—216no—
109C2×D4×C31—216no—
110C2×D4×C31—216no—
111C2×D4×C31—28no—
112C2×D4×C31—28no—
113C2×D4×C31—216no—
114C2×D4×C31—216no—
115C2×D4×C31C22⋊C4116yescomplemented
116C22×C4×C31—232no—
117C22×C4×C31—232no—
118C22×C4×C31—216no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—24no—
124C22×C4×C31—216no—
125C22×C4×C31—216no—
126C22×C4×C31—28no—
127C22×C4×C31—28no—
128C22×C4×C31—216no—
129C22×C4×C31—216no—
130C22×C4×C31—24no—
131C22×C4×C31—24no—
132C22×C4×C31—24no—
133C22×C4×C31—24no—
134C22×C4×C31—24no—
135C22×C4×C31—24no—
136C22×C4×C31—24no—
137C22×C4×C31—24no—
138C22×C4×C31—24no—
139C22×C4×C31—28no—
140C22×C4×C31—28no—
141C22×C4×C31C22⋊C4116yescomplemented
142C22×C4×C31C22×C412yes—
143C22×C4×C31C22×C414yes—
144C22×C4×C31C22×C412yes—
145C22×C4×C31C2×D412yes—
146C22×C4×C31C2×D412yes—
147C22×C4×C31C2×D418yes—
148C22×C4×C31C2×D412yes—
149C22×C4×C31C2×D414yes—
150C22×C4×C31C2×D414yes—
151C22×C4×C31C2×D412yes—
152C22×C4×C31#4118yescomplemented
153C4⋊C4×C31—48no—
154C22⋊C4×C31—216no—
155C22⋊C4×C31—432no—
156C22⋊C4×C31—48no—
157C42×C31—48no—
158#13937C3111yesmaximal, characteristic, direct product factor, Sylow
159C23×C31—24no—
160C23×C31—22no—
161C23×C31—24no—
162C23×C31—22no—
163C23×C31—28no—
164C23×C31—24no—
165C23×C31—24no—
166C23×C31—28no—
167C23×C31—28no—
168C23×C31—24no—
169C23×C31—28no—
170C23×C31—24no—
171C23×C31—24no—
172C23×C31—24no—
173C23×C31—22no—
174C23×C31—22no—
175C23×C31—24no—
176C23×C31—216no—
177C23×C31—216no—
178C23×C31—44no—
179C23×C31—44no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31—44no—
183C23×C31—44no—
184C23×C31—44no—
185C23×C31—44no—
186C23×C31—48no—
187C23×C31—44no—
188C23×C31—48no—
189C23×C31C2×C22⋊C418yes—
190C23×C31C2×C22⋊C414yes—
191C23×C31C23×C411yescharacteristic
192C23×C31C22×D411yescharacteristic
193C23×C31C22×D411yescharacteristic
194C23×C31C22×D412yes—
195C23×C31C22×D412yes—
196C23×C31C2×#4114yes—
197C23×C31H(5,𝔽2)12yes—
198C23×C312−1+412yes—
199Q8×C31—232no—
200D4×C31—264no—
201D4×C31—232no—
202C2×C4×C31—264no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—24no—
206C2×C4×C31—24no—
207C2×C4×C31—24no—
208C2×C4×C31—24no—
209C2×C4×C31—28no—
210C2×C4×C31—28no—
211C2×C4×C31—232no—
212C2×C4×C31—28no—
213C2×C4×C31—28no—
214C2×C4×C31—28no—
215C2×C4×C31—44no—
216C2×C4×C31—44no—
217C2×C4×C31—44no—
218C2×C4×C31—44no—
219C2×C4×C31—44no—
220C2×C4×C31—44no—
221C2×C4×C31—44no—
222C2×C4×C31—48no—
223C2×C4×C31—48no—
224C2×C4×C31C2×C22⋊C414yes—
225C2×C4×C31C2×C22⋊C418yes—
226C2×C4×C31C2×C22⋊C414yes—
227C2×C4×C31C2×C22⋊C414yes—
228C2×C4×C31C2×C22⋊C414yes—
229C2×C4×C31#12018yes—
230C2×C4×C31#12018yes—
231C23×#41C2×C3111yescharacteristic
232C2×#391C2×C3112yescomplemented
233C2×#390C2×C3112yescomplemented
234C2×#380C2×C3111yescharacteristic, complemented
235C2×C4×C22⋊C4C2×C3111yescharacteristic, complemented
236#1438C2×C3114yescomplemented
237C22⋊#95C2×C3114yescomplemented
238C22×C31—24no—
239C22×C31—216no—
240C22×C31—28no—
241C22×C31—24no—
242C22×C31—24no—
243C22×C31—44no—
244C22×C31—44no—
245C22×C31—44no—
246C22×C31—48no—
247C22×C31—48no—
248C22×C31—416no—
249C22×C31—44no—
250C22×C31—48no—
251C22×C31—44no—
252C22×C31—48no—
253C22×C31—44no—
254C22×C31C22×C22⋊C411yescharacteristic
255C22×C31C22×C22⋊C411yescharacteristic
256C22×C31C22×C22⋊C411yescharacteristic
257C22×C31C4×#4112yes—
258C22×C31#52012yes—
259C22×C31C2×#12012yes—
260C22×C31C2×#12012yes—
261C22×C31#52512yes—
262C22×C31#53212yes—
263C22×C31#53511yescharacteristic
264C22×C31#53611yescharacteristic
265C22×C31#53712yes—
266C22×C31#54614yes—
267C22×C31#54714yes—
268C22×C31#54814yes—
269C22×C31#55014yes—
270C4×C31—24no—
271C4×C31—28no—
272C4×C31—48no—
273C4×C31—84no—
274C4×C31#37918yes—
275C22×#41—28no—
276C22×#41—28no—
277C22×#41—24no—
278C22×#41—24no—
279C22×#41C4×C3118yescomplemented
280C23×Q8C22×C3111yescharacteristic
281C23×D4C22×C3112yes—
282C23×D4C22×C3111yescharacteristic
283C24×C4C22×C3112yes—
284C24×C4C22×C3111yescharacteristic
285C22×C4⋊C4C22×C3112yes—
286C22×C22⋊C4C22×C3112yes—
287C22×C22⋊C4C22×C3118yes—
288C22×C22⋊C4C22×C3112yes—
289C22×C42C22×C3112yes—
290#391—28no—
291#390—28no—
292#380—24no—
293C4×C22⋊C4—24no—
294C2×#95C22×C3112yes—
295C2×#95C22×C3112yes—
296C2×#95C22×C3118yes—
297C2×C31—44no—
298C2×C31—48no—
299C2×C31C2×#37911yescharacteristic
300C2×C31#229711yescharacteristic
301C2×C31#231111yescharacteristic
302C2×C31#233912yes—
303C2×C31#234112yes—
304C2×C31#240714yes—
305C2×C31#243112yes—
306C2×C31#243212yes—
307C25C23×C3112yes—
308C25C23×C3111yescharacteristic
309C2×#41—264no—
310C2×#41—264no—
311C2×#41—232no—
312C2×#41—232no—
313C2×#41—232no—
314C22×Q8—22no—
315C22×Q8—24no—
316C22×Q8—22no—
317C22×Q8C2×C4×C3114yes—
318C22×Q8D4×C3118yes—
319C22×D4—28no—
320C22×D4—24no—
321C22×D4—24no—
322C22×D4—28no—
323C22×D4—22no—
324C22×D4—24no—
325C22×D4—22no—
326C22×D4C2×C4×C3118yes—
327C22×D4C2×C4×C3114yes—
328C22×D4D4×C3118yes—
329C23×C4—28no—
330C23×C4—28no—
331C23×C4—24no—
332C23×C4—24no—
333C23×C4—24no—
334C23×C4—24no—
335C23×C4C2×C4×C3118yes—
336C23×C4C2×C4×C3114yes—
337C23×C4D4×C3114yes—
338C23×C4D4×C3114yes—
339C23×C4C23×C3114yes—
340C23×C4C23×C3114yes—
341C23×C4C23×C3111yescharacteristic
342C23×C4C23×C3112yes—
343C23×C4C23×C3111yescharacteristic
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—28no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—216no—
351C2×C22⋊C4—216no—
352C2×C22⋊C4—48no—
353C2×C22⋊C4—44no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—24no—
357#95—24no—
358#95—24no—
359#95—216no—
360C31#1393711yescharacteristic, direct product factor, Sylow
361C24—22no—
362C24—22no—
363C24—22no—
364C24—24no—
365C24—22no—
366C24—24no—
367C24—28no—
368C24—22no—
369C24—24no—
370C24—22no—
371C24—24no—
372C24C22×C4×C3114yes—
373C24C22×C4×C3112yes—
374C24C2×D4×C3118yes—
375C24C2×D4×C3112yes—
376C24C2×D4×C3112yes—
377C24C24×C3111yescharacteristic
378#41—2256no—
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—216no—
382C2×Q8—216no—
383C2×Q8C22⋊C4×C31116yescomplemented
384C2×D4—232no—
385C2×D4—232no—
386C2×D4—216no—
387C2×D4—216no—
388C2×D4—216no—
389C2×D4—28no—
390C2×D4—28no—
391C2×D4—216no—
392C2×D4—216no—
393C2×D4C22⋊C4×C31116yescomplemented
394C22×C4—232no—
395C22×C4—232no—
396C22×C4—216no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4C22⋊C4×C31116yescomplemented
420C22×C4C22×C4×C3112yes—
421C22×C4C22×C4×C3114yes—
422C22×C4C22×C4×C3112yes—
423C22×C4C2×D4×C3112yes—
424C22×C4C2×D4×C3112yes—
425C22×C4C2×D4×C3118yes—
426C22×C4C2×D4×C3112yes—
427C22×C4C2×D4×C3114yes—
428C22×C4C2×D4×C3114yes—
429C22×C4C2×D4×C3112yes—
430C22×C4#41×C3118yescomplemented
431C4⋊C4—48no—
432C22⋊C4—216no—
433C22⋊C4—432no—
434C22⋊C4—48no—
435C42—48no—
436C23—24no—
437C23—22no—
438C23—24no—
439C23—22no—
440C23—28no—
441C23—24no—
442C23—24no—
443C23—28no—
444C23—28no—
445C23—24no—
446C23—28no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—24no—
453C23—216no—
454C23—216no—
455C23—44no—
456C23—44no—
457C23—42no—
458C23—42no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—48no—
464C23—44no—
465C23—48no—
466C23C2×C22⋊C4×C3118yes—
467C23C2×C22⋊C4×C3114yes—
468C23C23×C4×C3111yescharacteristic
469C23C22×D4×C3111yescharacteristic
470C23C22×D4×C3111yescharacteristic
471C23C22×D4×C3112yes—
472C23C22×D4×C3112yes—
473C23C2×#41×C3114yes—
474C23C31×H(5,𝔽2)12yes—
475C23C31×2−1+412yes—
476Q8—232no—
477D4—264no—
478D4—232no—
479C2×C4—264no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—232no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—48no—
500C2×C4—48no—
501C2×C4C2×C22⋊C4×C3114yes—
502C2×C4C2×C22⋊C4×C3118yes—
503C2×C4C2×C22⋊C4×C3114yes—
504C2×C4C2×C22⋊C4×C3114yes—
505C2×C4C2×C22⋊C4×C3114yes—
506C2×C4C31×#12018yes—
507C2×C4C31×#12018yes—
508C22—24no—
509C22—216no—
510C22—28no—
511C22—24no—
512C22—24no—
513C22—44no—
514C22—44no—
515C22—44no—
516C22—48no—
517C22—48no—
518C22—416no—
519C22—44no—
520C22—48no—
521C22—44no—
522C22—48no—
523C22—44no—
524C22C22×C22⋊C4×C3111yescharacteristic
525C22C22×C22⋊C4×C3111yescharacteristic
526C22C22×C22⋊C4×C3111yescharacteristic
527C22C4×#41×C3112yes—
528C22C31×#52012yes—
529C22C2×C31×#12012yes—
530C22C2×C31×#12012yes—
531C22C31×#52512yes—
532C22C31×#53212yes—
533C22C31×#53511yescharacteristic
534C22C31×#53611yescharacteristic
535C22C31×#53712yes—
536C22C31×#54614yes—
537C22C31×#54714yes—
538C22C31×#54814yes—
539C22C31×#55014yes—
540C4—24no—
541C4—28no—
542C4—48no—
543C4—84no—
544C4C31×#37918yes—
545C2—44no—
546C2—48no—
547C2C2×C31×#37911yescharacteristic
548C2C31×#229711yescharacteristic
549C2C31×#231111yescharacteristic
550C2C31×#233912yes—
551C2C31×#234112yes—
552C2C31×#240714yes—
553C2C31×#243112yes—
554C2C31×#243212yes—
subgroupG/H
center, socleC24×C31C24
commutatorC23C23×C4×C31
Frattini, cosocleC24C24×C31
prime radicalC23×#41×C31C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#391C2semidirect
3. C2×C31×#390C2semidirect
4. C2×C31×#380C2semidirect
5. C2×C4×C22⋊C4×C31C2semidirect
6. C31×#1438C2semidirect
7. C31×C22⋊#95C2semidirect
12. C22×#41×C31C4semidirect
105. C2×Q8×C31C22⋊C4semidirect
115. C2×D4×C31C22⋊C4semidirect
141. C22×C4×C31C22⋊C4semidirect
152. C22×C4×C31#41semidirect
158. #13937C31direct (N × H)
232. C2×#391C2×C31semidirect
233. C2×#390C2×C31semidirect
234. C2×#380C2×C31semidirect
235. C2×C4×C22⋊C4C2×C31semidirect
236. #1438C2×C31semidirect
237. C22⋊#95C2×C31semidirect
279. C22×#41C4×C31semidirect
360. C31#13937direct (N × H)
383. C2×Q8C22⋊C4×C31semidirect
393. C2×D4C22⋊C4×C31semidirect
419. C22×C4C22⋊C4×C31semidirect
430. C22×C4#41×C31semidirect

Maximal subgroups

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

subgroupindex
C23×#41×C312
C2×C31×#3912
C2×C31×#3902
C2×C31×#3802
C2×C4×C22⋊C4×C312
C31×#14382
C31×C22⋊#952
#1393731