Finite Group Databasebeta

← C31×#14327 #21471086

Subgroups — C31×#14327 #21471086

Order 7936.

1346 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#395C211yesmaximal, characteristic
2C2×C31×#391C211yesmaximal, characteristic
3C2×C31×#387C211yesmaximal, characteristic
4C31×#1446C211yesmaximal, characteristic
5C31×#1444C211yesmaximal, characteristic
6C31×#1442C212yesmaximal
7C31×#1441C211yesmaximal, characteristic
8C31×#1441C212yesmaximal
9C31×C4⋊#95C212yesmaximal
10C31×#1434C212yesmaximal
11C4×C31×#95C211yesmaximal, characteristic
12C23×Q8×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2211yescharacteristic
14C22×C4⋊C4×C31C2212yes—
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C4⋊C4×C31C2211yescharacteristic
17C22×C4⋊C4×C31C2212yes—
18C22×C42×C31C2211yescharacteristic
19C22×C42×C31C2212yes—
20C31×#395C418yescomplemented
21C31×#391—24no—
22C31×#387—24no—
23C2×C31×#95C2212yes—
24C2×C31×#95C2211yescharacteristic
25C2×C31×#95C2211yescharacteristic
26C2×C31×#95C2212yes—
27C2×C31×#95C2212yes—
28C2×C31×#95C2211yescharacteristic
29C2×C31×#95C2212yes—
30C2×C31×#95C2211yescharacteristic
31C2×C31×#95C2211yescharacteristic
32C2×C31×#95C2212yes—
33C2×C31×#95C2212yes—
34C2×C31×#95C2212yes—
35C2×C31×#95C2212yes—
36C2×C31×#95C2211yescharacteristic
37C2×C31×#95C2212yes—
38C22×Q8×C31—22no—
39C22×Q8×C31—22no—
40C22×Q8×C31—42no—
41C22×Q8×C31—42no—
42C22×Q8×C31C2×C414yes—
43C23×C4×C31C2311yescharacteristic
44C23×C4×C31C2312yes—
45C23×C4×C31C2311yescharacteristic
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2312yes—
48C23×C4×C31C2311yescharacteristic
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2312yes—
51C23×C4×C31C2311yescharacteristic
52C23×C4×C31C2312yes—
53C23×C4×C31C2311yescharacteristic
54C2×C4⋊C4×C31—22no—
55C2×C4⋊C4×C31—22no—
56C2×C4⋊C4×C31—22no—
57C2×C4⋊C4×C31—22no—
58C2×C4⋊C4×C31—24no—
59C2×C4⋊C4×C31—42no—
60C2×C4⋊C4×C31—42no—
61C2×C4⋊C4×C31—42no—
62C2×C4⋊C4×C31—42no—
63C2×C4⋊C4×C31—42no—
64C2×C4⋊C4×C31—42no—
65C2×C4⋊C4×C31C2×C414yes—
66C2×C4⋊C4×C31C2×C418yes—
67C2×C42×C31—42no—
68C2×C42×C31—42no—
69C2×C42×C31—42no—
70C2×C42×C31D414yes—
71C2×C42×C31Q818yescomplemented
72C31×#95—22no—
73C31×#95—22no—
74C31×#95—22no—
75C31×#95—22no—
76C31×#95—24no—
77C31×#95—24no—
78C31×#95—22no—
79C31×#95—24no—
80C31×#95—24no—
81C31×#95—42no—
82C31×#95—42no—
83C31×#95—42no—
84C31×#95—42no—
85C31×#95C2×C418yes—
86C31×#95C2×C414yes—
87C24×C31C2411yescharacteristic
88C2×Q8×C31—28no—
89C2×Q8×C31—48no—
90C2×Q8×C31—44no—
91C2×Q8×C31—44no—
92C2×Q8×C31—48no—
93C22×C4×C31—22no—
94C22×C4×C31—21no—
95C22×C4×C31—22no—
96C22×C4×C31—22no—
97C22×C4×C31—22no—
98C22×C4×C31—22no—
99C22×C4×C31—22no—
100C22×C4×C31—22no—
101C22×C4×C31—21no—
102C22×C4×C31—21no—
103C22×C4×C31—22no—
104C22×C4×C31—21no—
105C22×C4×C31—22no—
106C22×C4×C31—22no—
107C22×C4×C31—22no—
108C22×C4×C31—21no—
109C22×C4×C31—21no—
110C22×C4×C31—22no—
111C22×C4×C31—21no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—22no—
128C22×C4×C31—21no—
129C22×C4×C31—21no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—21no—
144C22×C4×C31—22no—
145C22×C4×C31C22×C412yes—
146C22×C4×C31C22×C414yes—
147C22×C4×C31C22×C412yes—
148C22×C4×C31C22×C412yes—
149C22×C4×C31C22×C414yes—
150C22×C4×C31C2×D412yes—
151C22×C4×C31C2×D412yes—
152C22×C4×C31C2×D412yes—
153C22×C4×C31C2×Q814yes—
154C22×C4×C31C2×Q814yes—
155C22×C4×C31C2×Q814yes—
156C22×C4×C31#4112yes—
157C22×C4×C31#4114yes—
158C4⋊C4×C31—28no—
159C4⋊C4×C31—48no—
160C4⋊C4×C31—44no—
161C4⋊C4×C31—48no—
162C4⋊C4×C31—44no—
163C42×C31—44no—
164C42×C31—48no—
165#14327C3111yesmaximal, characteristic, direct product factor, Sylow
166C23×C31C23×C411yescharacteristic
167C23×C31C22×D411yescharacteristic
168C23×C31C22×Q812yes—
169C23×C31C2×#4111yescharacteristic
170C23×C31C2×#4112yes—
171C23×C31H(5,𝔽2)12yes—
172C23×C31H(5,𝔽2)12yes—
173C23×C31H(5,𝔽2)11yescharacteristic
174C23×C312−1+411yescharacteristic
175C23×C312−1+411yescharacteristic
176C23×C312−1+411yescharacteristic
177Q8×C31—416no—
178C2×C4×C31—24no—
179C2×C4×C31—24no—
180C2×C4×C31—24no—
181C2×C4×C31—22no—
182C2×C4×C31—22no—
183C2×C4×C31—22no—
184C2×C4×C31—22no—
185C2×C4×C31—24no—
186C2×C4×C31—24no—
187C2×C4×C31—24no—
188C2×C4×C31—24no—
189C2×C4×C31—24no—
190C2×C4×C31—24no—
191C2×C4×C31—24no—
192C2×C4×C31—22no—
193C2×C4×C31—22no—
194C2×C4×C31—24no—
195C2×C4×C31—24no—
196C2×C4×C31—24no—
197C2×C4×C31—24no—
198C2×C4×C31—22no—
199C2×C4×C31—22no—
200C2×C4×C31—22no—
201C2×C4×C31—24no—
202C2×C4×C31—22no—
203C2×C4×C31—22no—
204C2×C4×C31—24no—
205C2×C4×C31—24no—
206C2×C4×C31—41no—
207C2×C4×C31—42no—
208C2×C4×C31—41no—
209C2×C4×C31—42no—
210C2×C4×C31—41no—
211C2×C4×C31—41no—
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—41no—
219C2×C4×C31—41no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—41no—
224C2×C4×C31—42no—
225C2×C4×C31—41no—
226C2×C4×C31—41no—
227C2×C4×C31—42no—
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—42no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31C4×D414yes—
241C2×C4×C31C4×Q818yes—
242C2×C4×C31C4⋊Q814yes—
243C2×#395C2×C3111yescharacteristic
244C2×#391C2×C3111yescharacteristic
245C2×#387C2×C3111yescharacteristic
246#1446C2×C3111yescharacteristic
247#1444C2×C3111yescharacteristic
248#1442C2×C3112yes—
249#1441C2×C3111yescharacteristic
250#1441C2×C3112yes—
251C4⋊#95C2×C3112yes—
252#1434C2×C3112yes—
253C4×#95C2×C3111yescharacteristic
254C22×C31C2×C4×D411yescharacteristic
255C22×C31C2×C4×Q812yes—
256C22×C31#51812yes—
257C22×C31#51911yescharacteristic
258C22×C31#52011yescharacteristic
259C22×C31C2×C4⋊Q811yescharacteristic
260C22×C31#53211yescharacteristic
261C22×C31#53312yes—
262C22×C31#53411yescharacteristic
263C22×C31#53511yescharacteristic
264C22×C31#53611yescharacteristic
265C22×C31#53711yescharacteristic
266C22×C31#53812yes—
267C22×C31#53812yes—
268C22×C31#53912yes—
269C22×C31#54111yescharacteristic
270C22×C31#54211yescharacteristic
271C22×C31#54212yes—
272C22×C31#54212yes—
273C22×C31#54312yes—
274C22×C31#54412yes—
275C22×C31#54412yes—
276C22×C31#54412yes—
277C4×C31—24no—
278C4×C31—28no—
279C4×C31—24no—
280C4×C31—42no—
281C4×C31—44no—
282C4×C31—42no—
283C4×C31—42no—
284C4×C31—44no—
285C4×C31—82no—
286C4×C31—81no—
287C4×C31—81no—
288C23×Q8C22×C3111yescharacteristic
289C22×C4⋊C4C22×C3111yescharacteristic
290C22×C4⋊C4C22×C3112yes—
291C22×C4⋊C4C22×C3111yescharacteristic
292C22×C4⋊C4C22×C3111yescharacteristic
293C22×C4⋊C4C22×C3112yes—
294C22×C42C22×C3111yescharacteristic
295C22×C42C22×C3112yes—
296#395C4×C3118yescomplemented
297#391—24no—
298#387—24no—
299C2×#95C22×C3112yes—
300C2×#95C22×C3111yescharacteristic
301C2×#95C22×C3111yescharacteristic
302C2×#95C22×C3112yes—
303C2×#95C22×C3112yes—
304C2×#95C22×C3111yescharacteristic
305C2×#95C22×C3112yes—
306C2×#95C22×C3111yescharacteristic
307C2×#95C22×C3111yescharacteristic
308C2×#95C22×C3112yes—
309C2×#95C22×C3112yes—
310C2×#95C22×C3112yes—
311C2×#95C22×C3112yes—
312C2×#95C22×C3111yescharacteristic
313C2×#95C22×C3112yes—
314C2×C31C4×C4⋊Q811yescharacteristic
315C2×C31#232111yescharacteristic
316C2×C31#232311yescharacteristic
317C2×C31#232912yes—
318C2×C31#233112yes—
319C2×C31#265911yescharacteristic
320C2×C31#266011yescharacteristic
321C2×C31#266211yescharacteristic
322C2×C31#266311yescharacteristic
323C2×C31#266512yes—
324C2×C31#266612yes—
325C22×Q8—22no—
326C22×Q8—22no—
327C22×Q8—42no—
328C22×Q8—42no—
329C22×Q8C2×C4×C3114yes—
330C23×C4C23×C3111yescharacteristic
331C23×C4C23×C3112yes—
332C23×C4C23×C3111yescharacteristic
333C23×C4C23×C3111yescharacteristic
334C23×C4C23×C3112yes—
335C23×C4C23×C3111yescharacteristic
336C23×C4C23×C3111yescharacteristic
337C23×C4C23×C3112yes—
338C23×C4C23×C3111yescharacteristic
339C23×C4C23×C3112yes—
340C23×C4C23×C3111yescharacteristic
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—42no—
347C2×C4⋊C4—42no—
348C2×C4⋊C4—42no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4C2×C4×C3114yes—
353C2×C4⋊C4C2×C4×C3118yes—
354C2×C42—42no—
355C2×C42—42no—
356C2×C42—42no—
357C2×C42D4×C3114yes—
358C2×C42Q8×C3118yescomplemented
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—42no—
369#95—42no—
370#95—42no—
371#95—42no—
372#95C2×C4×C3118yes—
373#95C2×C4×C3114yes—
374C31#1432711yescharacteristic, direct product factor, Sylow
375C24C24×C3111yescharacteristic
376C2×Q8—28no—
377C2×Q8—48no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—48no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4C22×C4×C3112yes—
434C22×C4C22×C4×C3114yes—
435C22×C4C22×C4×C3112yes—
436C22×C4C22×C4×C3112yes—
437C22×C4C22×C4×C3114yes—
438C22×C4C2×D4×C3112yes—
439C22×C4C2×D4×C3112yes—
440C22×C4C2×D4×C3112yes—
441C22×C4C2×Q8×C3114yes—
442C22×C4C2×Q8×C3114yes—
443C22×C4C2×Q8×C3114yes—
444C22×C4#41×C3112yes—
445C22×C4#41×C3114yes—
446C4⋊C4—28no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C4⋊C4—44no—
451C42—44no—
452C42—48no—
453C23C23×C4×C3111yescharacteristic
454C23C22×D4×C3111yescharacteristic
455C23C22×Q8×C3112yes—
456C23C2×#41×C3111yescharacteristic
457C23C2×#41×C3112yes—
458C23C31×H(5,𝔽2)12yes—
459C23C31×H(5,𝔽2)12yes—
460C23C31×H(5,𝔽2)11yescharacteristic
461C23C31×2−1+411yescharacteristic
462C23C31×2−1+411yescharacteristic
463C23C31×2−1+411yescharacteristic
464Q8—416no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4C4×D4×C3114yes—
528C2×C4C4×Q8×C3118yes—
529C2×C4C31×C4⋊Q814yes—
530C22C2×C4×D4×C3111yescharacteristic
531C22C2×C4×Q8×C3112yes—
532C22C31×#51812yes—
533C22C31×#51911yescharacteristic
534C22C31×#52011yescharacteristic
535C22C2×C31×C4⋊Q811yescharacteristic
536C22C31×#53211yescharacteristic
537C22C31×#53312yes—
538C22C31×#53411yescharacteristic
539C22C31×#53511yescharacteristic
540C22C31×#53611yescharacteristic
541C22C31×#53711yescharacteristic
542C22C31×#53812yes—
543C22C31×#53812yes—
544C22C31×#53912yes—
545C22C31×#54111yescharacteristic
546C22C31×#54211yescharacteristic
547C22C31×#54212yes—
548C22C31×#54212yes—
549C22C31×#54312yes—
550C22C31×#54412yes—
551C22C31×#54412yes—
552C22C31×#54412yes—
553C4—24no—
554C4—28no—
555C4—24no—
556C4—44no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—44no—
561C4—82no—
562C4—81no—
563C4—81no—
564C2C4×C31×C4⋊Q811yescharacteristic
565C2C31×#232111yescharacteristic
566C2C31×#232311yescharacteristic
567C2C31×#232912yes—
568C2C31×#233112yes—
569C2C31×#265911yescharacteristic
570C2C31×#266011yescharacteristic
571C2C31×#266211yescharacteristic
572C2C31×#266311yescharacteristic
573C2C31×#266512yes—
574C2C31×#266612yes—
subgroupG/H
center, socle, prime radicalC24×C31C24
commutatorC23C23×C4×C31
Frattini, cosocleC24C24×C31
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
20. C31×#395C4semidirect
71. C2×C42×C31Q8semidirect
165. #14327C31direct (N × H)
296. #395C4×C31semidirect
358. C2×C42Q8×C31semidirect
374. C31#14327direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3952
C2×C31×#3912
C2×C31×#3872
C31×#14462
C31×#14442
C31×#14422
C31×#14412
C31×C4⋊#952
C31×#14342
C4×C31×#952
#1432731