Finite Group Databasebeta

← C31×#14538 #21471297

Subgroups — C31×#14538 #21471297

Order 7936.

1402 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#395C211yesmaximal, characteristic
2C2×C31×#391C212yesmaximal
3C2×C31×#387C212yesmaximal
4C31×#1443C212yesmaximal
5C31×#1441C212yesmaximal
6C31×#1441C211yesmaximal, characteristic
7C31×#1436C212yesmaximal
8C31×#1436C212yesmaximal
9C31×C4⋊#95C211yesmaximal, characteristic
10C23×Q8×C31C2211yescharacteristic
11C22×C4⋊C4×C31C2212yes—
12C22×C4⋊C4×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2212yes—
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C42×C31C2212yes—
17C22×C42×C31C2211yescharacteristic
18C31×#395C418yescomplemented
19C31×#391—28no—
20C31×#387—28no—
21C2×C31×#95C2212yes—
22C2×C31×#95C2212yes—
23C2×C31×#95C2212yes—
24C2×C31×#95C2212yes—
25C2×C31×#95C2212yes—
26C2×C31×#95C2212yes—
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2212yes—
30C2×C31×#95C2211yescharacteristic
31C2×C31×#95C2212yes—
32C2×C31×#95C2211yescharacteristic
33C2×C31×#95C2212yes—
34C22×Q8×C31—24no—
35C22×Q8×C31—24no—
36C22×Q8×C31—41no—
37C22×Q8×C31—41no—
38C22×Q8×C31C2×C414yes—
39C23×C4×C31C2312yes—
40C23×C4×C31C2311yescharacteristic
41C23×C4×C31C2311yescharacteristic
42C23×C4×C31C2312yes—
43C23×C4×C31C2312yes—
44C23×C4×C31C2312yes—
45C23×C4×C31C2312yes—
46C23×C4×C31C2312yes—
47C23×C4×C31C2311yescharacteristic
48C2×C4⋊C4×C31—24no—
49C2×C4⋊C4×C31—24no—
50C2×C4⋊C4×C31—24no—
51C2×C4⋊C4×C31—24no—
52C2×C4⋊C4×C31—24no—
53C2×C4⋊C4×C31—41no—
54C2×C4⋊C4×C31—41no—
55C2×C4⋊C4×C31—41no—
56C2×C4⋊C4×C31—41no—
57C2×C4⋊C4×C31—42no—
58C2×C4⋊C4×C31—41no—
59C2×C4⋊C4×C31C2×C418yes—
60C2×C4⋊C4×C31C2×C414yes—
61C2×C4⋊C4×C31Q814yescomplemented
62C2×C42×C31—24no—
63C2×C42×C31—24no—
64C2×C42×C31—22no—
65C2×C42×C31—42no—
66C2×C42×C31—41no—
67C2×C42×C31—41no—
68C31×#95—24no—
69C31×#95—24no—
70C31×#95—24no—
71C31×#95—24no—
72C31×#95—24no—
73C31×#95—24no—
74C31×#95—24no—
75C31×#95—42no—
76C31×#95—42no—
77C31×#95—41no—
78C31×#95C2×C414yes—
79C31×#95C2×C418yes—
80C31×#95D418yes—
81C24×C31C2411yescharacteristic
82C2×Q8×C31—216no—
83C2×Q8×C31—44no—
84C2×Q8×C31—44no—
85C2×Q8×C31—48no—
86C2×Q8×C31—44no—
87C22×C4×C31—22no—
88C22×C4×C31—22no—
89C22×C4×C31—21no—
90C22×C4×C31—21no—
91C22×C4×C31—22no—
92C22×C4×C31—21no—
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—22no—
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—22no—
105C22×C4×C31—22no—
106C22×C4×C31—22no—
107C22×C4×C31—22no—
108C22×C4×C31—22no—
109C22×C4×C31—21no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—21no—
115C22×C4×C31—21no—
116C22×C4×C31—21no—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31C22×C414yes—
135C22×C4×C31C22×C412yes—
136C22×C4×C31C22×C412yes—
137C22×C4×C31C22×C414yes—
138C22×C4×C31C22×C412yes—
139C22×C4×C31C2×D414yes—
140C22×C4×C31C2×D414yes—
141C22×C4×C31C2×D414yes—
142C22×C4×C31C2×Q814yes—
143C22×C4×C31C2×Q812yes—
144C22×C4×C31#4114yes—
145C22×C4×C31#4112yes—
146C4⋊C4×C31—216no—
147C4⋊C4×C31—44no—
148C4⋊C4×C31—44no—
149C4⋊C4×C31—48no—
150C4⋊C4×C31—44no—
151C42×C31—48no—
152C42×C31—44no—
153#14538C3111yesmaximal, characteristic, direct product factor, Sylow
154C23×C31C23×C411yescharacteristic
155C23×C31C22×D412yes—
156C23×C31C22×Q811yescharacteristic
157C23×C31C2×#4112yes—
158C23×C31C2×#4111yescharacteristic
159C23×C31H(5,𝔽2)11yescharacteristic
160C23×C31H(5,𝔽2)11yescharacteristic
161C23×C31H(5,𝔽2)11yescharacteristic
162C23×C312−1+412yes—
163C23×C312−1+412yes—
164C23×C312−1+411yescharacteristic
165Q8×C31—416no—
166C2×C4×C31—24no—
167C2×C4×C31—22no—
168C2×C4×C31—24no—
169C2×C4×C31—22no—
170C2×C4×C31—22no—
171C2×C4×C31—22no—
172C2×C4×C31—22no—
173C2×C4×C31—22no—
174C2×C4×C31—24no—
175C2×C4×C31—24no—
176C2×C4×C31—24no—
177C2×C4×C31—24no—
178C2×C4×C31—24no—
179C2×C4×C31—24no—
180C2×C4×C31—24no—
181C2×C4×C31—22no—
182C2×C4×C31—22no—
183C2×C4×C31—24no—
184C2×C4×C31—24no—
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—24no—
193C2×C4×C31—24no—
194C2×C4×C31—42no—
195C2×C4×C31—42no—
196C2×C4×C31—42no—
197C2×C4×C31—42no—
198C2×C4×C31—42no—
199C2×C4×C31—42no—
200C2×C4×C31—42no—
201C2×C4×C31—42no—
202C2×C4×C31—42no—
203C2×C4×C31—42no—
204C2×C4×C31—42no—
205C2×C4×C31—41no—
206C2×C4×C31—41no—
207C2×C4×C31—42no—
208C2×C4×C31—42no—
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—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31C4×D418yes—
224C2×C4×C31C4×Q814yes—
225C2×#395C2×C3111yescharacteristic
226C2×#391C2×C3112yes—
227C2×#387C2×C3112yes—
228#1443C2×C3112yes—
229#1441C2×C3112yes—
230#1441C2×C3111yescharacteristic
231#1436C2×C3112yes—
232#1436C2×C3112yes—
233C4⋊#95C2×C3111yescharacteristic
234C22×C31C2×C4×D412yes—
235C22×C31C2×C4×Q811yescharacteristic
236C22×C31#51811yescharacteristic
237C22×C31#51912yes—
238C22×C31#52011yescharacteristic
239C22×C31#53612yes—
240C22×C31#53712yes—
241C22×C31#53811yescharacteristic
242C22×C31#53911yescharacteristic
243C22×C31#54112yes—
244C22×C31#54212yes—
245C22×C31#54311yescharacteristic
246C22×C31#54411yescharacteristic
247C22×C31D4211yescharacteristic
248C22×C31#54612yes—
249C22×C31#54712yes—
250C22×C31D4×Q812yes—
251C22×C31#55111yescharacteristic
252C22×C31#55612yes—
253C22×C31#55712yes—
254C22×C31#56311yescharacteristic
255C22×C31#56311yescharacteristic
256C22×C31#56312yes—
257C4×C31—24no—
258C4×C31—28no—
259C4×C31—44no—
260C4×C31—42no—
261C4×C31—42no—
262C4×C31—44no—
263C4×C31—44no—
264C4×C31—44no—
265C4×C31—82no—
266C23×Q8C22×C3111yescharacteristic
267C22×C4⋊C4C22×C3112yes—
268C22×C4⋊C4C22×C3111yescharacteristic
269C22×C4⋊C4C22×C3112yes—
270C22×C4⋊C4C22×C3111yescharacteristic
271C22×C4⋊C4C22×C3111yescharacteristic
272C22×C42C22×C3112yes—
273C22×C42C22×C3111yescharacteristic
274#395C4×C3118yescomplemented
275#391—28no—
276#387—28no—
277C2×#95C22×C3112yes—
278C2×#95C22×C3112yes—
279C2×#95C22×C3112yes—
280C2×#95C22×C3112yes—
281C2×#95C22×C3112yes—
282C2×#95C22×C3112yes—
283C2×#95C22×C3112yes—
284C2×#95C22×C3112yes—
285C2×#95C22×C3112yes—
286C2×#95C22×C3111yescharacteristic
287C2×#95C22×C3112yes—
288C2×#95C22×C3111yescharacteristic
289C2×#95C22×C3112yes—
290C2×C31#232312yes—
291C2×C31#232911yescharacteristic
292C2×C31#235811yescharacteristic
293C2×C31#236512yes—
294C2×C31#238111yescharacteristic
295C2×C31#268011yescharacteristic
296C2×C31#268311yescharacteristic
297C2×C31#271312yes—
298C2×C31#271512yes—
299C2×C31#278911yescharacteristic
300C2×C31#279911yescharacteristic
301C22×Q8—24no—
302C22×Q8—24no—
303C22×Q8—41no—
304C22×Q8—41no—
305C22×Q8C2×C4×C3114yes—
306C23×C4C23×C3112yes—
307C23×C4C23×C3111yescharacteristic
308C23×C4C23×C3111yescharacteristic
309C23×C4C23×C3112yes—
310C23×C4C23×C3112yes—
311C23×C4C23×C3112yes—
312C23×C4C23×C3112yes—
313C23×C4C23×C3112yes—
314C23×C4C23×C3111yescharacteristic
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—41no—
321C2×C4⋊C4—41no—
322C2×C4⋊C4—41no—
323C2×C4⋊C4—41no—
324C2×C4⋊C4—42no—
325C2×C4⋊C4—41no—
326C2×C4⋊C4C2×C4×C3118yes—
327C2×C4⋊C4C2×C4×C3114yes—
328C2×C4⋊C4Q8×C3114yescomplemented
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—42no—
333C2×C42—41no—
334C2×C42—41no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—42no—
343#95—42no—
344#95—41no—
345#95C2×C4×C3114yes—
346#95C2×C4×C3118yes—
347#95D4×C3118yes—
348C31#1453811yescharacteristic, direct product factor, Sylow
349C24C24×C3111yescharacteristic
350C2×Q8—216no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—48no—
354C2×Q8—44no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4C22×C4×C3114yes—
403C22×C4C22×C4×C3112yes—
404C22×C4C22×C4×C3112yes—
405C22×C4C22×C4×C3114yes—
406C22×C4C22×C4×C3112yes—
407C22×C4C2×D4×C3114yes—
408C22×C4C2×D4×C3114yes—
409C22×C4C2×D4×C3114yes—
410C22×C4C2×Q8×C3114yes—
411C22×C4C2×Q8×C3112yes—
412C22×C4#41×C3114yes—
413C22×C4#41×C3112yes—
414C4⋊C4—216no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4—44no—
419C42—48no—
420C42—44no—
421C23C23×C4×C3111yescharacteristic
422C23C22×D4×C3112yes—
423C23C22×Q8×C3111yescharacteristic
424C23C2×#41×C3112yes—
425C23C2×#41×C3111yescharacteristic
426C23C31×H(5,𝔽2)11yescharacteristic
427C23C31×H(5,𝔽2)11yescharacteristic
428C23C31×H(5,𝔽2)11yescharacteristic
429C23C31×2−1+412yes—
430C23C31×2−1+412yes—
431C23C31×2−1+411yescharacteristic
432Q8—416no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
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—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4C4×D4×C3118yes—
491C2×C4C4×Q8×C3114yes—
492C22C2×C4×D4×C3112yes—
493C22C2×C4×Q8×C3111yescharacteristic
494C22C31×#51811yescharacteristic
495C22C31×#51912yes—
496C22C31×#52011yescharacteristic
497C22C31×#53612yes—
498C22C31×#53712yes—
499C22C31×#53811yescharacteristic
500C22C31×#53911yescharacteristic
501C22C31×#54112yes—
502C22C31×#54212yes—
503C22C31×#54311yescharacteristic
504C22C31×#54411yescharacteristic
505C22D42×C3111yescharacteristic
506C22C31×#54612yes—
507C22C31×#54712yes—
508C22D4×Q8×C3112yes—
509C22C31×#55111yescharacteristic
510C22C31×#55612yes—
511C22C31×#55712yes—
512C22C31×#56311yescharacteristic
513C22C31×#56311yescharacteristic
514C22C31×#56312yes—
515C4—24no—
516C4—28no—
517C4—44no—
518C4—42no—
519C4—42no—
520C4—44no—
521C4—44no—
522C4—44no—
523C4—82no—
524C2C31×#232312yes—
525C2C31×#232911yescharacteristic
526C2C31×#235811yescharacteristic
527C2C31×#236512yes—
528C2C31×#238111yescharacteristic
529C2C31×#268011yescharacteristic
530C2C31×#268311yescharacteristic
531C2C31×#271312yes—
532C2C31×#271512yes—
533C2C31×#278911yescharacteristic
534C2C31×#279911yescharacteristic
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
18. C31×#395C4semidirect
61. C2×C4⋊C4×C31Q8semidirect
153. #14538C31direct (N × H)
274. #395C4×C31semidirect
328. C2×C4⋊C4Q8×C31semidirect
348. C31#14538direct (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×#14432
C31×#14412
C31×#14362
C31×C4⋊#952
#1453831