Finite Group Databasebeta

← C31×#32777 #21489536

Subgroups — C31×#32777 #21489536

Order 7936.

704 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2857C211yesmaximal, characteristic
2C31×#2851C211yesmaximal, characteristic
3C31×#2097C212yesmaximal, complemented
4C31×#2084C211yesmaximal, characteristic, complemented
5C31×#2083C211yesmaximal, characteristic, complemented
6C31×#2043C211yesmaximal, characteristic, complemented
7C31×#2043C211yesmaximal, characteristic, complemented
8C31×#2036C212yesmaximal
9C31×#2035C212yesmaximal
10C31×#2026C212yesmaximal, complemented
11C31×#1817C211yesmaximal, characteristic
12C2×C31×C4⋊Q8C2211yescharacteristic
13C2×C31×#125C2211yescharacteristic
14C2×C31×C22⋊Q8C2212yes—
15C22×C42×C31C2211yescharacteristic
16C2×C31×#107C2212yes—
17C2×C31×#106C2212yes—
18C2×C31×#103C2212yes—
19C2×C31×#103C2212yes—
20C2×C31×#103C2212yes—
21C2×C31×#103C2212yes—
22C2×C31×C22⋊C8C2212yes—
23C31×#391C2212yes—
24C31×#389—21no—
25C31×#389—21no—
26C31×#389C2211yescharacteristic
27C31×#389C2211yescharacteristic
28C31×#387—21no—
29C31×#387—21no—
30C31×#385—21no—
31C31×#385—21no—
32C31×#384C2211yescharacteristic
33C31×#382C2211yescharacteristic
34C31×#380—21no—
35C31×#380—21no—
36C31×#380C2211yescharacteristic
37C31×#380C2211yescharacteristic
38C31×#340C2212yes—
39C31×#340C2212yes—
40C31×#336C2211yescharacteristic
41C31×#336C2211yescharacteristic
42C31×#336C2212yes—
43C22×Q8×C31C2312yes—
44C23×C4×C31D411yescharacteristic
45C23×C4×C31D411yescharacteristic
46C23×C4×C31C2311yescharacteristic
47C22×C8×C31C2312yes—
48C22×C8×C31C2312yes—
49C31×C4⋊Q8—42no—
50C31×#125—42no—
51C31×C22⋊Q8—42no—
52C31×C22⋊Q8—42no—
53C2×C4⋊C4×C31—22no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31—21no—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31—21no—
62C2×C4⋊C4×C31C2311yescharacteristic
63C2×C4⋊C4×C31C2311yescharacteristic
64C2×C4⋊C4×C31C2311yescharacteristic
65C2×C4⋊C4×C31C2311yescharacteristic
66C2×C4⋊C4×C31C2312yes—
67C2×C22⋊C4×C31—22no—
68C2×C22⋊C4×C31—21no—
69C2×C22⋊C4×C31—21no—
70C2×C42×C31—42no—
71C2×C42×C31D411yescharacteristic
72C2×C42×C31D411yescharacteristic
73C2×C42×C31C2311yescharacteristic
74C2×C42×C31C2311yescharacteristic
75C31×#107—42no—
76C31×#106—42no—
77C31×#103—42no—
78C31×#103—42no—
79C31×#103—42no—
80C31×#103—42no—
81C31×C22⋊C8—42no—
82C31×#95—22no—
83C31×#95—22no—
84C31×#95—21no—
85C31×#95—21no—
86C31×#95—21no—
87C31×#95—21no—
88C31×#95—21no—
89C31×#95—21no—
90C24×C31C2×D411yescharacteristic
91C2×Q8×C31—22no—
92C2×Q8×C31—22no—
93C2×Q8×C31—22no—
94C2×Q8×C31—22no—
95C2×Q8×C31—22no—
96C2×Q8×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—21no—
106C22×C4×C31—21no—
107C22×C4×C31—21no—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—41no—
111C22×C4×C31—41no—
112C22×C4×C31—41no—
113C22×C4×C31C2×D411yescharacteristic
114C22×C4×C31C2×D411yescharacteristic
115C22×C4×C31C2×D411yescharacteristic
116C22×C4×C31C2×D411yescharacteristic
117C22×C4×C31C2×D411yescharacteristic
118C22×C4×C31C2411yescharacteristic
119C2×C8×C31—22no—
120C2×C8×C31—22no—
121C2×C8×C31—22no—
122C2×C8×C31—22no—
123C2×C8×C31—22no—
124C2×C8×C31—22no—
125C4⋊C4×C31—21no—
126C4⋊C4×C31—21no—
127C4⋊C4×C31—22no—
128C4⋊C4×C31—22no—
129C4⋊C4×C31—21no—
130C4⋊C4×C31—22no—
131C4⋊C4×C31—21no—
132C4⋊C4×C31—22no—
133C4⋊C4×C31—42no—
134C4⋊C4×C31—44no—
135C4⋊C4×C31—42no—
136C4⋊C4×C31—42no—
137C4⋊C4×C31—42no—
138C4⋊C4×C31—42no—
139C4⋊C4×C31—42no—
140C4⋊C4×C31—42no—
141C4⋊C4×C31—42no—
142C22⋊C4×C31—42no—
143C22⋊C4×C31—44no—
144C22⋊C4×C31—42no—
145C42×C31—22no—
146C42×C31—22no—
147C42×C31—41no—
148C42×C31—41no—
149#32777C3111yesmaximal, characteristic, direct product factor, Sylow
150C23×C31—21no—
151C23×C31—21no—
152C23×C31—21no—
153C23×C31—21no—
154C23×C31—21no—
155C23×C31—21no—
156C23×C31#13712yes—
157C23×C31C22×D411yescharacteristic
158Q8×C31—42no—
159Q8×C31—42no—
160Q8×C31—42no—
161Q8×C31—42no—
162C2×C4×C31—21no—
163C2×C4×C31—21no—
164C2×C4×C31—21no—
165C2×C4×C31—21no—
166C2×C4×C31—21no—
167C2×C4×C31—21no—
168C2×C4×C31—21no—
169C2×C4×C31—21no—
170C2×C4×C31—21no—
171C2×C4×C31—21no—
172C2×C4×C31—21no—
173C2×C4×C31—21no—
174C2×C4×C31—21no—
175C2×C4×C31—44no—
176C2×C4×C31—44no—
177C2×C4×C31—42no—
178C2×C4×C31—42no—
179C2×C4×C31—41no—
180C2×C4×C31—41no—
181C2×C4×C31—42no—
182C2×C4×C31—42no—
183C2×C4×C31—42no—
184C2×C4×C31—41no—
185C2×C4×C31—41no—
186C2×C4×C31—42no—
187C2×C4×C31—41no—
188C2×C4×C31—41no—
189C2×C4×C31—42no—
190C2×C4×C31—41no—
191C2×C4×C31—41no—
192C2×C4×C31—41no—
193C2×C4×C31—41no—
194C2×C4×C31—41no—
195C2×C4×C31—41no—
196C2×C4×C31—42no—
197C2×C4×C31#13512yes—
198C2×C4×C31#13712yes—
199C2×C4×C31H(5,𝔽2)12yes—
200C2×C4×C31H(5,𝔽2)11yescharacteristic
201C2×C4×C31H(5,𝔽2)11yescharacteristic
202C2×C4×C312−1+412yes—
203C8×C31—42no—
204C8×C31—42no—
205#2857C2×C3111yescharacteristic
206#2851C2×C3111yescharacteristic
207#2097C2×C3112yescomplemented
208#2084C2×C3111yescharacteristic, complemented
209#2083C2×C3111yescharacteristic, complemented
210#2043C2×C3111yescharacteristic, complemented
211#2043C2×C3111yescharacteristic, complemented
212#2036C2×C3112yes—
213#2035C2×C3112yes—
214#2026C2×C3112yescomplemented
215#1817C2×C3111yescharacteristic
216C22×C31—22no—
217C22×C31—22no—
218C22×C31—22no—
219C22×C31—41no—
220C22×C31—41no—
221C22×C31—41no—
222C22×C31—41no—
223C22×C31#53411yescharacteristic
224C22×C31#53511yescharacteristic
225C22×C31#53611yescharacteristic
226C22×C31C2×#13511yescharacteristic
227C22×C31C2×#13711yescharacteristic
228C22×C31C2×#13711yescharacteristic
229C22×C31#57511yescharacteristic
230C4×C31—22no—
231C4×C31—22no—
232C4×C31—41no—
233C4×C31—41no—
234C4×C31—41no—
235C4×C31—82no—
236C4×C31—81no—
237C4×C31—81no—
238C4×C31—82no—
239C4×C31—81no—
240C4×C31—81no—
241C4×C31#56212yes—
242C4×C31#56212yes—
243C2×C4⋊Q8C22×C3111yescharacteristic
244C2×#125C22×C3111yescharacteristic
245C2×C22⋊Q8C22×C3112yes—
246C22×C42C22×C3111yescharacteristic
247C2×#107C22×C3112yes—
248C2×#106C22×C3112yes—
249C2×#103C22×C3112yes—
250C2×#103C22×C3112yes—
251C2×#103C22×C3112yes—
252C2×#103C22×C3112yes—
253C2×C22⋊C8C22×C3112yes—
254#391C22×C3112yes—
255#389—21no—
256#389—21no—
257#389C22×C3111yescharacteristic
258#389C22×C3111yescharacteristic
259#387—21no—
260#387—21no—
261#385—21no—
262#385—21no—
263#384C22×C3111yescharacteristic
264#382C22×C3111yescharacteristic
265#380—21no—
266#380—21no—
267#380C22×C3111yescharacteristic
268#380C22×C3111yescharacteristic
269#340C22×C3112yes—
270#340C22×C3112yes—
271#336C22×C3111yescharacteristic
272#336C22×C3111yescharacteristic
273#336C22×C3112yes—
274C2×C31—42no—
275C2×C31#281611yescharacteristic
276C2×C31#319011yescharacteristic
277C2×C31#319611yescharacteristic
278C2×C31#321211yescharacteristic
279C2×C31#321811yescharacteristic
280C2×C31#323111yescharacteristic
281C2×C31#323611yescharacteristic
282C22×Q8C23×C3112yes—
283C23×C4D4×C3111yescharacteristic
284C23×C4D4×C3111yescharacteristic
285C23×C4C23×C3111yescharacteristic
286C22×C8C23×C3112yes—
287C22×C8C23×C3112yes—
288C4⋊Q8—42no—
289#125—42no—
290C22⋊Q8—42no—
291C22⋊Q8—42no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4C23×C3112yes—
302C2×C4⋊C4C23×C3111yescharacteristic
303C2×C4⋊C4C23×C3111yescharacteristic
304C2×C4⋊C4C23×C3111yescharacteristic
305C2×C4⋊C4C23×C3111yescharacteristic
306C2×C22⋊C4—22no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C42—42no—
310C2×C42D4×C3111yescharacteristic
311C2×C42D4×C3111yescharacteristic
312C2×C42C23×C3111yescharacteristic
313C2×C42C23×C3111yescharacteristic
314#107—42no—
315#106—42no—
316#103—42no—
317#103—42no—
318#103—42no—
319#103—42no—
320C22⋊C8—42no—
321#95—22no—
322#95—22no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329C31#3277711yescharacteristic, direct product factor, Sylow
330C24C2×D4×C3111yescharacteristic
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4C2×D4×C3111yescharacteristic
354C22×C4C2×D4×C3111yescharacteristic
355C22×C4C2×D4×C3111yescharacteristic
356C22×C4C2×D4×C3111yescharacteristic
357C22×C4C2×D4×C3111yescharacteristic
358C22×C4C24×C3111yescharacteristic
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—42no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—44no—
384C22⋊C4—42no—
385C42—22no—
386C42—22no—
387C42—41no—
388C42—41no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23C31×#13712yes—
396C23C22×D4×C3111yescharacteristic
397Q8—42no—
398Q8—42no—
399Q8—42no—
400Q8—42no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4C31×#13512yes—
437C2×C4C31×#13712yes—
438C2×C4C31×H(5,𝔽2)12yes—
439C2×C4C31×H(5,𝔽2)11yescharacteristic
440C2×C4C31×H(5,𝔽2)11yescharacteristic
441C2×C4C31×2−1+412yes—
442C8—42no—
443C8—42no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22C31×#53411yescharacteristic
452C22C31×#53511yescharacteristic
453C22C31×#53611yescharacteristic
454C22C2×C31×#13511yescharacteristic
455C22C2×C31×#13711yescharacteristic
456C22C2×C31×#13711yescharacteristic
457C22C31×#57511yescharacteristic
458C4—22no—
459C4—22no—
460C4—41no—
461C4—41no—
462C4—41no—
463C4—81no—
464C4—81no—
465C4—82no—
466C4—81no—
467C4—81no—
468C4—82no—
469C4C31×#56212yes—
470C4C31×#56212yes—
471C2—42no—
472C2C31×#281611yescharacteristic
473C2C31×#319011yescharacteristic
474C2C31×#319611yescharacteristic
475C2C31×#321211yescharacteristic
476C2C31×#321811yescharacteristic
477C2C31×#323111yescharacteristic
478C2C31×#323611yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#2097C2semidirect
4. C31×#2084C2semidirect
5. C31×#2083C2semidirect
6. C31×#2043C2semidirect
7. C31×#2043C2semidirect
10. C31×#2026C2semidirect
149. #32777C31direct (N × H)
207. #2097C2×C31semidirect
208. #2084C2×C31semidirect
209. #2083C2×C31semidirect
210. #2043C2×C31semidirect
211. #2043C2×C31semidirect
214. #2026C2×C31semidirect
329. C31#32777direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28572
C31×#28512
C31×#20972
C31×#20842
C31×#20832
C31×#20432
C31×#20362
C31×#20352
C31×#20262
C31×#18172
#3277731