Finite Group Databasebeta

← C31×#10323 #21467082

Subgroups — C31×#10323 #21467082

Order 7936.

664 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3187C211yesmaximal, characteristic
2C31×#2855C211yesmaximal, characteristic, complemented
3C31×#2854C211yesmaximal, characteristic
4C31×#1817C211yesmaximal, characteristic, complemented
5C31×#1520C211yesmaximal, characteristic, complemented
6C31×#1519C211yesmaximal, characteristic
7C31×#1289C211yesmaximal, characteristic, complemented
8C2×C31×#123—21no—
9C2×C31×C22⋊Q8C2211yescharacteristic
10C2×C31×#121C2211yescharacteristic
11C22×C42×C31C2211yescharacteristic
12C31×#483—21no—
13C31×#481—21no—
14C31×#465—22no—
15C31×#450—21no—
16C31×#448—21no—
17C2×C31×C22⋊C8C2211yescharacteristic
18C2×C31×C22⋊C8C2211yescharacteristic
19C31×#391—21no—
20C31×#390—21no—
21C31×#389—21no—
22C31×#389—21no—
23C31×#388—22no—
24C31×#387—22no—
25C31×#386—21no—
26C31×#385—21no—
27C31×#381—21no—
28C31×#380C2211yescharacteristic
29C31×#380C2211yescharacteristic
30C31×#336—22no—
31C31×#328—22no—
32C31×#327—22no—
33C22×Q8×C31D411yescharacteristic
34C22×D4×C31—21no—
35C22×D4×C31D411yescharacteristic
36C23×C4×C31D411yescharacteristic
37C23×C4×C31D411yescharacteristic
38C23×C4×C31C2311yescharacteristic
39C2×SD16×C31—22no—
40C22×C8×C31—21no—
41C22×C8×C31D412yes—
42C31×#123—42no—
43C31×#123—42no—
44C31×C22⋊Q8—21no—
45C31×C22⋊Q8—21no—
46C31×C22⋊Q8—21no—
47C31×C22⋊Q8—21no—
48C31×#121—21no—
49C31×#121—21no—
50C31×#121—21no—
51C31×#121—21no—
52C2×C4⋊C4×C31—22no—
53C2×C4⋊C4×C31—22no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31D411yescharacteristic
58C2×C4⋊C4×C31D411yescharacteristic
59C2×C22⋊C4×C31—22no—
60C2×C22⋊C4×C31—22no—
61C2×C22⋊C4×C31—21no—
62C2×C22⋊C4×C31—21no—
63C2×C22⋊C4×C31—21no—
64C2×C22⋊C4×C31—21no—
65C2×C42×C31—21no—
66C2×C42×C31—21no—
67C2×C42×C31—21no—
68C2×C42×C31—21no—
69C2×C42×C31—21no—
70C2×C42×C31—21no—
71C31×#106—22no—
72C31×#103—22no—
73C31×#102—22no—
74C31×C22⋊C8—21no—
75C31×C22⋊C8—21no—
76C31×C22⋊C8—21no—
77C31×C22⋊C8—21no—
78C31×#95—22no—
79C31×#95—22no—
80C31×#95—22no—
81C31×#95—22no—
82C31×#95D412yes—
83C31×#95D412yes—
84C24×C31—21no—
85C24×C31C2×D411yescharacteristic
86C2×Q8×C31—21no—
87C2×Q8×C31—21no—
88C2×Q8×C31—42no—
89C2×D4×C31—21no—
90C2×D4×C31—21no—
91C2×D4×C31—42no—
92C2×D4×C31—41no—
93C2×D4×C31—41no—
94C2×D4×C31—41no—
95C2×D4×C31—41no—
96C2×D4×C31—41no—
97C2×D4×C31—41no—
98C22×C4×C31—21no—
99C22×C4×C31—21no—
100C22×C4×C31—21no—
101C22×C4×C31—21no—
102C22×C4×C31—21no—
103C22×C4×C31—21no—
104C22×C4×C31—21no—
105C22×C4×C31—21no—
106C22×C4×C31—21no—
107C22×C4×C31—21no—
108C22×C4×C31—22no—
109C22×C4×C31—21no—
110C22×C4×C31—21no—
111C22×C4×C31—22no—
112C22×C4×C31—42no—
113C22×C4×C31—42no—
114C22×C4×C31SD1612yescomplemented
115C22×C4×C31C2×D411yescharacteristic
116C22×C4×C31C2×D411yescharacteristic
117C22×C4×C31C2×D411yescharacteristic
118C22×C4×C31C2×D411yescharacteristic
119C22×C4×C31C2×D411yescharacteristic
120C22×C4×C31#4111yescharacteristic
121SD16×C31—82no—
122C2×C8×C31—22no—
123C2×C8×C31—42no—
124C2×C8×C31—42no—
125C2×C8×C31—41no—
126C4⋊C4×C31—21no—
127C4⋊C4×C31—21no—
128C4⋊C4×C31—21no—
129C4⋊C4×C31—21no—
130C4⋊C4×C31—42no—
131C4⋊C4×C31—42no—
132C4⋊C4×C31—44no—
133C4⋊C4×C31—44no—
134C4⋊C4×C31—41no—
135C4⋊C4×C31—41no—
136C22⋊C4×C31—44no—
137C22⋊C4×C31—44no—
138C22⋊C4×C31—41no—
139C22⋊C4×C31—41no—
140C22⋊C4×C31—41no—
141C22⋊C4×C31—41no—
142C22⋊C4×C31—41no—
143C22⋊C4×C31—41no—
144C22⋊C4×C31—41no—
145C22⋊C4×C31—41no—
146C42×C31—41no—
147C42×C31—41no—
148C42×C31—41no—
149C42×C31—41no—
150#10323C3111yesmaximal, characteristic, direct product factor, Sylow
151C23×C31—41no—
152C23×C31—41no—
153C23×C31—42no—
154C23×C31—42no—
155C23×C31—41no—
156C23×C31—41no—
157C23×C31—41no—
158C23×C31#12011yescharacteristic
159C23×C31#12011yescharacteristic
160C23×C31#12111yescharacteristic
161C23×C31C2×SD1611yescharacteristic
162C23×C31#13511yescharacteristic
163C23×C31AGL(1,ℤ/8)11yescharacteristic
164C23×C31#13711yescharacteristic
165Q8×C31—42no—
166Q8×C31—81no—
167D4×C31—42no—
168D4×C31—81no—
169D4×C31—81no—
170D4×C31—81no—
171D4×C31—81no—
172D4×C31—81no—
173C2×C4×C31—22no—
174C2×C4×C31—22no—
175C2×C4×C31—22no—
176C2×C4×C31—22no—
177C2×C4×C31—22no—
178C2×C4×C31—22no—
179C2×C4×C31—22no—
180C2×C4×C31—22no—
181C2×C4×C31—22no—
182C2×C4×C31—41no—
183C2×C4×C31—41no—
184C2×C4×C31—41no—
185C2×C4×C31—41no—
186C2×C4×C31—41no—
187C2×C4×C31—41no—
188C2×C4×C31—41no—
189C2×C4×C31—41no—
190C2×C4×C31—41no—
191C2×C4×C31—41no—
192C2×C4×C31—41no—
193C2×C4×C31—42no—
194C2×C4×C31—42no—
195C2×C4×C31—42no—
196C2×C4×C31—42no—
197C2×C4×C31—42no—
198C2×C4×C31—42no—
199C2×C4×C31—41no—
200C2×C4×C31—41no—
201C2×C4×C31—41no—
202C2×C4×C31—42no—
203C2×C4×C31—42no—
204C2×C4×C31—41no—
205C2×C4×C31—41no—
206C2×C4×C31#12011yescharacteristic
207C2×C4×C31#12111yescharacteristic
208C2×C4×C31#12111yescharacteristic
209C2×C4×C31#12711yescharacteristic
210C8×C31—42no—
211C8×C31—81no—
212#3187C2×C3111yescharacteristic
213#2855C2×C3111yescharacteristic, complemented
214#2854C2×C3111yescharacteristic
215#1817C2×C3111yescharacteristic, complemented
216#1520C2×C3111yescharacteristic, complemented
217#1519C2×C3111yescharacteristic
218#1289C2×C3111yescharacteristic, complemented
219C22×C31—21no—
220C22×C31—21no—
221C22×C31—21no—
222C22×C31—21no—
223C22×C31—21no—
224C22×C31—21no—
225C22×C31—41no—
226C22×C31—41no—
227C22×C31—41no—
228C22×C31—41no—
229C22×C31—82no—
230C22×C31—82no—
231C22×C31—81no—
232C22×C31—81no—
233C22×C31—81no—
234C22×C31#39211yescharacteristic
235C22×C31#44811yescharacteristic
236C22×C31#44911yescharacteristic
237C22×C31#45011yescharacteristic
238C22×C31#45211yescharacteristic
239C22×C31#46511yescharacteristic
240C22×C31#46811yescharacteristic
241C4×C31—21no—
242C4×C31—21no—
243C4×C31—21no—
244C4×C31—42no—
245C4×C31—42no—
246C4×C31—82no—
247C4×C31—82no—
248C4×C31—81no—
249C4×C31—81no—
250C4×C31—81no—
251C4×C31#45812yes—
252C2×#123—21no—
253C2×C22⋊Q8C22×C3111yescharacteristic
254C2×#121C22×C3111yescharacteristic
255C22×C42C22×C3111yescharacteristic
256#483—21no—
257#481—21no—
258#465—22no—
259#450—21no—
260#448—21no—
261C2×C22⋊C8C22×C3111yescharacteristic
262C2×C22⋊C8C22×C3111yescharacteristic
263#391—21no—
264#390—21no—
265#389—21no—
266#389—21no—
267#388—22no—
268#387—22no—
269#386—21no—
270#385—21no—
271#381—21no—
272#380C22×C3111yescharacteristic
273#380C22×C3111yescharacteristic
274#336—22no—
275#328—22no—
276#327—22no—
277C2×C31—21no—
278C2×C31—21no—
279C2×C31—21no—
280C2×C31—21no—
281C2×C31—21no—
282C2×C31—21no—
283C2×C31—161no—
284C2×C31#200011yescharacteristic
285C2×C31#201011yescharacteristic
286C2×C31#202211yescharacteristic
287C22×Q8D4×C3111yescharacteristic
288C22×D4—21no—
289C22×D4D4×C3111yescharacteristic
290C23×C4D4×C3111yescharacteristic
291C23×C4D4×C3111yescharacteristic
292C23×C4C23×C3111yescharacteristic
293C2×SD16—22no—
294C22×C8—21no—
295C22×C8D4×C3112yes—
296#123—42no—
297#123—42no—
298C22⋊Q8—21no—
299C22⋊Q8—21no—
300C22⋊Q8—21no—
301C22⋊Q8—21no—
302#121—21no—
303#121—21no—
304#121—21no—
305#121—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4D4×C3111yescharacteristic
312C2×C4⋊C4D4×C3111yescharacteristic
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325#106—22no—
326#103—22no—
327#102—22no—
328C22⋊C8—21no—
329C22⋊C8—21no—
330C22⋊C8—21no—
331C22⋊C8—21no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95D4×C3112yes—
337#95D4×C3112yes—
338C31#1032311yescharacteristic, direct product factor, Sylow
339C24—21no—
340C24C2×D4×C3111yescharacteristic
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—42no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—42no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4SD16×C3112yescomplemented
370C22×C4C2×D4×C3111yescharacteristic
371C22×C4C2×D4×C3111yescharacteristic
372C22×C4C2×D4×C3111yescharacteristic
373C22×C4C2×D4×C3111yescharacteristic
374C22×C4C2×D4×C3111yescharacteristic
375C22×C4#41×C3111yescharacteristic
376SD16—82no—
377C2×C8—22no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—41no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C23—41no—
406C23—41no—
407C23—42no—
408C23—42no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23C31×#12011yescharacteristic
413C23C31×#12011yescharacteristic
414C23C31×#12111yescharacteristic
415C23C2×SD16×C3111yescharacteristic
416C23C31×#13511yescharacteristic
417C23C31×AGL(1,ℤ/8)11yescharacteristic
418C23C31×#13711yescharacteristic
419Q8—42no—
420Q8—81no—
421D4—42no—
422D4—81no—
423D4—81no—
424D4—81no—
425D4—81no—
426D4—81no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4C31×#12011yescharacteristic
461C2×C4C31×#12111yescharacteristic
462C2×C4C31×#12111yescharacteristic
463C2×C4C31×#12711yescharacteristic
464C8—42no—
465C8—81no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—82no—
477C22—82no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22C31×#39211yescharacteristic
482C22C31×#44811yescharacteristic
483C22C31×#44911yescharacteristic
484C22C31×#45011yescharacteristic
485C22C31×#45211yescharacteristic
486C22C31×#46511yescharacteristic
487C22C31×#46811yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—42no—
492C4—42no—
493C4—82no—
494C4—82no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4C31×#45812yes—
499C2—21no—
500C2—21no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—161no—
506C2C31×#200011yescharacteristic
507C2C31×#201011yescharacteristic
508C2C31×#202211yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC2×C31×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2855C2semidirect
4. C31×#1817C2semidirect
5. C31×#1520C2semidirect
7. C31×#1289C2semidirect
114. C22×C4×C31SD16semidirect
150. #10323C31direct (N × H)
213. #2855C2×C31semidirect
215. #1817C2×C31semidirect
216. #1520C2×C31semidirect
218. #1289C2×C31semidirect
338. C31#10323direct (N × H)
369. C22×C4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31872
C31×#28552
C31×#28542
C31×#18172
C31×#15202
C31×#15192
C31×#12892
#1032331