Finite Group Databasebeta

← C31×#9037 #21465796

Subgroups — C31×#9037 #21465796

Order 7936.

676 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3002C211yesmaximal, characteristic
2C31×#2976C211yesmaximal, characteristic, complemented
3C31×#1789C211yesmaximal, characteristic, complemented
4C31×#1780C211yesmaximal, characteristic, complemented
5C31×#1501C211yesmaximal, characteristic, complemented
6C31×#1298C211yesmaximal, characteristic, complemented
7C31×#1282C211yesmaximal, characteristic, complemented
8C22×#41×C31C2211yescharacteristic
9C2×C31×#135—21no—
10C31×#525C2211yescharacteristic
11C31×#452—21no—
12C31×#451C411yescharacteristic, complemented
13C31×#450C411yescharacteristic, complemented
14C31×#449—21no—
15C31×#448C411yescharacteristic, complemented
16C31×#447C411yescharacteristic, complemented
17C31×#436—21no—
18C31×#435—21no—
19C31×#435—21no—
20C8×D4×C31—21no—
21C31×#433—21no—
22C2×C31×#104—21no—
23C2×C31×#104—21no—
24C31×#416—21no—
25C31×#408—21no—
26C31×#407C2211yescharacteristic
27C2×C31×C22⋊C8—21no—
28C2×C31×C22⋊C8—21no—
29C2×C31×C22⋊C8C2211yescharacteristic
30C2×C31×C22⋊C8C2211yescharacteristic
31C2×C31×C22⋊C8C2211yescharacteristic
32C4×C22⋊C4×C31C2211yescharacteristic
33C31×#328—21no—
34C31×#327—21no—
35C2×#41×C31—21no—
36C2×#41×C31—21no—
37C2×#41×C31—21no—
38C2×#41×C31—21no—
39C2×#41×C31—42no—
40C2×#41×C31—42no—
41C2×#41×C31D411yescharacteristic
42C22×Q8×C31C2×C411yescharacteristic
43C22×D4×C31—21no—
44C22×D4×C31C2×C411yescharacteristic
45C23×C4×C31—21no—
46C23×C4×C31C2311yescharacteristic
47C31×#135—42no—
48C31×#135—42no—
49C2×Q16×C31—21no—
50C2×SD16×C31—21no—
51C2×SD16×C31—21no—
52C2×D8×C31—21no—
53C2×M16×C31—21no—
54C2×M16×C31D411yescharacteristic
55C2×M16×C31D411yescharacteristic
56C22×C8×C31—21no—
57C22×C8×C31—21no—
58C22×C8×C31—21no—
59C22×C8×C31D411yescharacteristic
60C22×C8×C31D411yescharacteristic
61C31×#123—21no—
62C31×C22⋊Q8C2×C411yescharacteristic
63C31×C22⋊Q8D411yescharacteristic
64C31×#121C2×C411yescharacteristic
65C31×#121D411yescharacteristic
66C31×#120—21no—
67C4×D4×C31—21no—
68C4×D4×C31—21no—
69C31×#117D411yescharacteristic
70C2×C22⋊C4×C31—21no—
71C2×C42×C31—21no—
72C31×C4⋊C8—21no—
73C31×C4⋊C8—21no—
74C31×#104—42no—
75C31×#104—42no—
76C31×#103—21no—
77C31×#103—21no—
78C31×#102—21no—
79C31×#102—21no—
80C31×C22⋊C8—21no—
81C31×C22⋊C8—21no—
82C31×C22⋊C8—21no—
83C31×C22⋊C8—21no—
84C31×C22⋊C8—21no—
85C31×C22⋊C8—21no—
86C31×C22⋊C8—21no—
87C31×C22⋊C8—21no—
88C31×C22⋊C8—21no—
89C31×C22⋊C8—42no—
90C31×C22⋊C8—42no—
91C31×C22⋊C8C2×C411yescharacteristic
92C31×C22⋊C8C2×C411yescharacteristic
93C31×#97—21no—
94C4×C8×C31—21no—
95C31×#95—21no—
96C24×C31—21no—
97C24×C31#4111yescharacteristic
98#41×C31—22no—
99#41×C31—22no—
100#41×C31—22no—
101#41×C31—22no—
102#41×C31—41no—
103#41×C31—41no—
104#41×C31—41no—
105#41×C31—41no—
106#41×C31—41no—
107#41×C31—41no—
108#41×C31—42no—
109#41×C31—42no—
110#41×C31—42no—
111#41×C31—42no—
112C2×Q8×C31—21no—
113C2×Q8×C31—21no—
114C2×Q8×C31—42no—
115C2×Q8×C31C22⋊C411yescharacteristic
116C2×D4×C31—21no—
117C2×D4×C31—21no—
118C2×D4×C31—21no—
119C2×D4×C31—21no—
120C2×D4×C31—21no—
121C2×D4×C31—42no—
122C2×D4×C31—42no—
123C2×D4×C31—42no—
124C2×D4×C31—41no—
125C2×D4×C31—41no—
126C2×D4×C31C22⋊C411yescharacteristic
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—41no—
132C22×C4×C31—41no—
133C22×C4×C31—41no—
134C22×C4×C31—41no—
135C22×C4×C31—41no—
136C22×C4×C31—41no—
137C22×C4×C31—41no—
138C22×C4×C31C22×C411yescharacteristic
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31#4111yescharacteristic
144Q16×C31—42no—
145SD16×C31—42no—
146SD16×C31—42no—
147D8×C31—42no—
148M16×C31—41no—
149M16×C31—41no—
150M16×C31—81no—
151C2×C8×C31—21no—
152C2×C8×C31—21no—
153C2×C8×C31—21no—
154C2×C8×C31—21no—
155C2×C8×C31—21no—
156C2×C8×C31—21no—
157C2×C8×C31—41no—
158C2×C8×C31—41no—
159C2×C8×C31—41no—
160C2×C8×C31—41no—
161C2×C8×C31—41no—
162C2×C8×C31—41no—
163C2×C8×C31—41no—
164C2×C8×C31—81no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31C22⋊C411yescharacteristic
168C4⋊C4×C31C22⋊C411yescharacteristic
169C22⋊C4×C31—21no—
170C22⋊C4×C31—21no—
171C22⋊C4×C31—21no—
172C22⋊C4×C31—21no—
173C22⋊C4×C31—21no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—41no—
176C42×C31—21no—
177C42×C31—41no—
178C42×C31—41no—
179#9037C3111yesmaximal, characteristic, direct product factor, Sylow
180C23×C31—21no—
181C23×C31—21no—
182C23×C31—22no—
183C23×C31—21no—
184C23×C31—41no—
185C23×C31—41no—
186C23×C31—41no—
187C23×C31—41no—
188C23×C31—41no—
189C23×C31—41no—
190C23×C31—41no—
191C23×C31—41no—
192C23×C31C4×D411yescharacteristic
193C23×C31#12111yescharacteristic
194C23×C31#12311yescharacteristic
195Q8×C31—22no—
196Q8×C31—22no—
197Q8×C31—41no—
198Q8×C31—41no—
199Q8×C31—41no—
200D4×C31—22no—
201D4×C31—22no—
202D4×C31—41no—
203D4×C31—41no—
204D4×C31—41no—
205D4×C31—42no—
206D4×C31—42no—
207D4×C31—42no—
208D4×C31—42no—
209D4×C31—41no—
210D4×C31—41no—
211D4×C31—81no—
212D4×C31—81no—
213C2×C4×C31—21no—
214C2×C4×C31—21no—
215C2×C4×C31—21no—
216C2×C4×C31—21no—
217C2×C4×C31—21no—
218C2×C4×C31—21no—
219C2×C4×C31—22no—
220C2×C4×C31—21no—
221C2×C4×C31—21no—
222C2×C4×C31—21no—
223C2×C4×C31—41no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—41no—
229C2×C4×C31—41no—
230C2×C4×C31—41no—
231C2×C4×C31—41no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—81no—
236C2×C4×C31—81no—
237C2×C4×C31C2×C22⋊C411yescharacteristic
238C2×C4×C31C4×D411yescharacteristic
239C2×C4×C31#12011yescharacteristic
240C2×C4×C31#12111yescharacteristic
241C8×C31—41no—
242C8×C31—41no—
243C8×C31—41no—
244C8×C31—41no—
245C8×C31—41no—
246C8×C31—41no—
247C8×C31—81no—
248#3002C2×C3111yescharacteristic
249#2976C2×C3111yescharacteristic, complemented
250#1789C2×C3111yescharacteristic, complemented
251#1780C2×C3111yescharacteristic, complemented
252#1501C2×C3111yescharacteristic, complemented
253#1298C2×C3111yescharacteristic, complemented
254#1282C2×C3111yescharacteristic, complemented
255C22×C31—21no—
256C22×C31—21no—
257C22×C31—21no—
258C22×C31—21no—
259C22×C31—21no—
260C22×C31—21no—
261C22×C31—21no—
262C22×C31—21no—
263C22×C31—42no—
264C22×C31—42no—
265C22×C31—41no—
266C22×C31—41no—
267C22×C31—42no—
268C22×C31—41no—
269C22×C31—81no—
270C22×C31—81no—
271C22×C31—81no—
272C22×C31—81no—
273C22×C31—81no—
274C22×C31—81no—
275C22×C31#38611yescharacteristic
276C22×C31#44311yescharacteristic
277C22×C31#44411yescharacteristic
278C4×C31—21no—
279C4×C31—21no—
280C4×C31—21no—
281C4×C31—41no—
282C4×C31—41no—
283C4×C31—42no—
284C4×C31—41no—
285C4×C31—82no—
286C4×C31B(4,𝔽2)11yescharacteristic
287C4×C31B(4,𝔽2)11yescharacteristic
288C22×#41C22×C3111yescharacteristic
289C2×#135—21no—
290#525C22×C3111yescharacteristic
291#452—21no—
292#451C4×C3111yescharacteristic, complemented
293#450C4×C3111yescharacteristic, complemented
294#449—21no—
295#448C4×C3111yescharacteristic, complemented
296#447C4×C3111yescharacteristic, complemented
297#436—21no—
298#435—21no—
299#435—21no—
300C8×D4—21no—
301#433—21no—
302C2×#104—21no—
303C2×#104—21no—
304#416—21no—
305#408—21no—
306#407C22×C3111yescharacteristic
307C2×C22⋊C8—21no—
308C2×C22⋊C8—21no—
309C2×C22⋊C8C22×C3111yescharacteristic
310C2×C22⋊C8C22×C3111yescharacteristic
311C2×C22⋊C8C22×C3111yescharacteristic
312C4×C22⋊C4C22×C3111yescharacteristic
313#328—21no—
314#327—21no—
315C2×C31—21no—
316C2×C31—21no—
317C2×C31—41no—
318C2×C31—41no—
319C2×C31—81no—
320C2×C31—82no—
321C2×C31#187611yescharacteristic
322C2×C31#188111yescharacteristic
323C2×C31#188911yescharacteristic
324C2×#41—21no—
325C2×#41—21no—
326C2×#41—21no—
327C2×#41—21no—
328C2×#41—42no—
329C2×#41—42no—
330C2×#41D4×C3111yescharacteristic
331C22×Q8C2×C4×C3111yescharacteristic
332C22×D4—21no—
333C22×D4C2×C4×C3111yescharacteristic
334C23×C4—21no—
335C23×C4C23×C3111yescharacteristic
336#135—42no—
337#135—42no—
338C2×Q16—21no—
339C2×SD16—21no—
340C2×SD16—21no—
341C2×D8—21no—
342C2×M16—21no—
343C2×M16D4×C3111yescharacteristic
344C2×M16D4×C3111yescharacteristic
345C22×C8—21no—
346C22×C8—21no—
347C22×C8—21no—
348C22×C8D4×C3111yescharacteristic
349C22×C8D4×C3111yescharacteristic
350#123—21no—
351C22⋊Q8C2×C4×C3111yescharacteristic
352C22⋊Q8D4×C3111yescharacteristic
353#121C2×C4×C3111yescharacteristic
354#121D4×C3111yescharacteristic
355#120—21no—
356C4×D4—21no—
357C4×D4—21no—
358#117D4×C3111yescharacteristic
359C2×C22⋊C4—21no—
360C2×C42—21no—
361C4⋊C8—21no—
362C4⋊C8—21no—
363#104—42no—
364#104—42no—
365#103—21no—
366#103—21no—
367#102—21no—
368#102—21no—
369C22⋊C8—21no—
370C22⋊C8—21no—
371C22⋊C8—21no—
372C22⋊C8—21no—
373C22⋊C8—21no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376C22⋊C8—21no—
377C22⋊C8—21no—
378C22⋊C8—42no—
379C22⋊C8—42no—
380C22⋊C8C2×C4×C3111yescharacteristic
381C22⋊C8C2×C4×C3111yescharacteristic
382#97—21no—
383C4×C8—21no—
384#95—21no—
385C31#903711yescharacteristic, direct product factor, Sylow
386C24—21no—
387C24#41×C3111yescharacteristic
388#41—22no—
389#41—22no—
390#41—22no—
391#41—22no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—41no—
397#41—41no—
398#41—42no—
399#41—42no—
400#41—42no—
401#41—42no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—42no—
405C2×Q8C22⋊C4×C3111yescharacteristic
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4C22⋊C4×C3111yescharacteristic
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4C22×C4×C3111yescharacteristic
429C22×C4C2×D4×C3111yescharacteristic
430C22×C4C2×D4×C3111yescharacteristic
431C22×C4C2×D4×C3111yescharacteristic
432C22×C4C2×D4×C3111yescharacteristic
433C22×C4#41×C3111yescharacteristic
434Q16—42no—
435SD16—42no—
436SD16—42no—
437D8—42no—
438M16—41no—
439M16—41no—
440M16—81no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—81no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4C22⋊C4×C3111yescharacteristic
458C4⋊C4C22⋊C4×C3111yescharacteristic
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—21no—
462C22⋊C4—21no—
463C22⋊C4—21no—
464C22⋊C4—22no—
465C22⋊C4—41no—
466C42—21no—
467C42—41no—
468C42—41no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—21no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23C4×D4×C3111yescharacteristic
482C23C31×#12111yescharacteristic
483C23C31×#12311yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—41no—
487Q8—41no—
488Q8—41no—
489D4—22no—
490D4—22no—
491D4—41no—
492D4—41no—
493D4—41no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—42no—
498D4—41no—
499D4—41no—
500D4—81no—
501D4—81no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4C2×C22⋊C4×C3111yescharacteristic
527C2×C4C4×D4×C3111yescharacteristic
528C2×C4C31×#12011yescharacteristic
529C2×C4C31×#12111yescharacteristic
530C8—41no—
531C8—41no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—41no—
536C8—81no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—42no—
546C22—42no—
547C22—41no—
548C22—41no—
549C22—42no—
550C22—41no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C31×#38611yescharacteristic
558C22C31×#44311yescharacteristic
559C22C31×#44411yescharacteristic
560C4—21no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—42no—
566C4—41no—
567C4—82no—
568C4C31×B(4,𝔽2)11yescharacteristic
569C4C31×B(4,𝔽2)11yescharacteristic
570C2—21no—
571C2—21no—
572C2—41no—
573C2—41no—
574C2—82no—
575C2—81no—
576C2C31×#187611yescharacteristic
577C2C31×#188111yescharacteristic
578C2C31×#188911yescharacteristic
subgroupG/H
centerC2×C4×C31#120
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
socleC22×C31#386
prime radicalC31×#3002C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2976C2semidirect
3. C31×#1789C2semidirect
4. C31×#1780C2semidirect
5. C31×#1501C2semidirect
6. C31×#1298C2semidirect
7. C31×#1282C2semidirect
12. C31×#451C4semidirect
13. C31×#450C4semidirect
15. C31×#448C4semidirect
16. C31×#447C4semidirect
179. #9037C31direct (N × H)
249. #2976C2×C31semidirect
250. #1789C2×C31semidirect
251. #1780C2×C31semidirect
252. #1501C2×C31semidirect
253. #1298C2×C31semidirect
254. #1282C2×C31semidirect
292. #451C4×C31semidirect
293. #450C4×C31semidirect
295. #448C4×C31semidirect
296. #447C4×C31semidirect
385. C31#9037direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#30022
C31×#29762
C31×#17892
C31×#17802
C31×#15012
C31×#12982
C31×#12822
#903731