Finite Group Databasebeta

← C31×#34522 #21491281

Subgroups — C31×#34522 #21491281

Order 7936.

2216 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
1C2×C31×#562C211yesmaximal, characteristic
2C31×#2382C211yesmaximal, characteristic, complemented
3C31×#2379C212yesmaximal, complemented
4C31×#2368C214yesmaximal, complemented
5C31×#2363C214yesmaximal, complemented
6C31×#2359C214yesmaximal, complemented
7C31×#2335C214yesmaximal, complemented
8C31×#2325C214yesmaximal, complemented
9C31×#2323C214yesmaximal, complemented
10C4×C31×#125C211yesmaximal, characteristic, complemented
11C4×C31×#124C212yesmaximal, complemented
12C31×#562C4116yescomplemented
13C2×C31×#125C2211yescharacteristic
14C2×C31×#124C2212yes—
15C2×C31×#123C2214yes—
16C2×C31×C22⋊Q8C2214yes—
17C2×C31×#121C2214yes—
18C2×C4×Q8×C31C2212yes—
19C2×C4×D4×C31C2212yes—
20C2×C4×D4×C31C2214yes—
21C2×C31×#117C2214yes—
22C31×#391C2214yes—
23C31×#391C2212yes—
24C31×#390C2214yes—
25C31×#390C2212yes—
26C31×#389C2214yes—
27C31×#389C2214yes—
28C31×#389C2212yes—
29C31×#388C2214yes—
30C31×#388C2214yes—
31C31×#388C2214yes—
32C31×#388C2214yes—
33C31×#388C2214yes—
34C31×#388C2214yes—
35C31×#387C2214yes—
36C31×#387C2214yes—
37C31×#387C2214yes—
38C31×#387C2214yes—
39C31×#387C2214yes—
40C31×#386C2214yes—
41C31×#386C2214yes—
42C31×#386C2214yes—
43C31×#385C2214yes—
44C31×#385C2214yes—
45C31×#385C2214yes—
46C31×#382C2211yescharacteristic
47C31×#382C2212yes—
48C31×#380C2214yes—
49C4×C4⋊C4×C31C2214yes—
50C4×C4⋊C4×C31C2212yes—
51C4×C4⋊C4×C31C2214yes—
52C4×C22⋊C4×C31C2214yes—
53C4×C22⋊C4×C31C2214yes—
54C4×C22⋊C4×C31C2214yes—
55C4×C22⋊C4×C31C2214yes—
56C2×C31×#95C2214yes—
57C43×C31C2211yescharacteristic
58C22×Q8×C31C2312yes—
59C22×D4×C31C2314yes—
60C22×D4×C31C2312yes—
61C23×C4×C31C2314yes—
62C23×C4×C31C2314yes—
63C23×C4×C31C2314yes—
64C31×#125C2×C418yescomplemented
65C31×#124C2×C4116yescomplemented
66C31×#123C2×C4132yescomplemented
67C31×C22⋊Q8C2×C4132yescomplemented
68C31×#121C2×C4132yescomplemented
69C4×Q8×C31—44no—
70C4×D4×C31—44no—
71C4×D4×C31—48no—
72C31×#117—48no—
73C2×C4⋊C4×C31C2314yes—
74C2×C4⋊C4×C31C2314yes—
75C2×C4⋊C4×C31C2312yes—
76C2×C4⋊C4×C31C2314yes—
77C2×C4⋊C4×C31C2312yes—
78C2×C4⋊C4×C31C2314yes—
79C2×C4⋊C4×C31C2312yes—
80C2×C4⋊C4×C31C2314yes—
81C2×C4⋊C4×C31C2314yes—
82C2×C4⋊C4×C31C2312yes—
83C2×C22⋊C4×C31C2314yes—
84C2×C22⋊C4×C31C2314yes—
85C2×C22⋊C4×C31C2314yes—
86C2×C22⋊C4×C31C2314yes—
87C2×C22⋊C4×C31C2314yes—
88C2×C22⋊C4×C31C2314yes—
89C2×C22⋊C4×C31C2314yes—
90C2×C22⋊C4×C31C2314yes—
91C2×C22⋊C4×C31C2314yes—
92C2×C42×C31C2314yes—
93C2×C42×C31C2312yes—
94C2×C42×C31C2311yescharacteristic
95C2×C42×C31C2314yes—
96C2×C42×C31C2314yes—
97C2×C42×C31C2311yescharacteristic
98C2×C42×C31C2312yes—
99C2×C42×C31C2314yes—
100C2×C42×C31C2311yescharacteristic
101C31×#95C2314yes—
102C31×#95C2314yes—
103C31×#95C2314yes—
104C31×#95C2314yes—
105C31×#95C2314yes—
106C31×#95C2314yes—
107C31×#95C2314yes—
108C31×#95C2314yes—
109C31×#95C2314yes—
110C31×#95C2314yes—
111C31×#95C2314yes—
112C24×C31C2414yes—
113C2×Q8×C31—42no—
114C2×Q8×C31—42no—
115C2×Q8×C31C22×C418yes—
116C2×D4×C31—42no—
117C2×D4×C31—44no—
118C2×D4×C31—44no—
119C2×D4×C31—42no—
120C2×D4×C31C22×C4116yes—
121C2×D4×C31C22×C418yes—
122C22×C4×C31—28no—
123C22×C4×C31—44no—
124C22×C4×C31—44no—
125C22×C4×C31—44no—
126C22×C4×C31—44no—
127C22×C4×C31—44no—
128C22×C4×C31—44no—
129C22×C4×C31—44no—
130C22×C4×C31C22×C4116yes—
131C22×C4×C31C2414yes—
132C22×C4×C31C2414yes—
133C22×C4×C31C2411yescharacteristic
134C22×C4×C31C2414yes—
135C22×C4×C31C2411yescharacteristic
136C22×C4×C31C2412yes—
137C22×C4×C31C2414yes—
138C22×C4×C31C2414yes—
139C22×C4×C31C2412yes—
140C22×C4×C31C2411yescharacteristic
141C4⋊C4×C31—28no—
142C4⋊C4×C31—42no—
143C4⋊C4×C31—44no—
144C4⋊C4×C31—44no—
145C4⋊C4×C31—42no—
146C4⋊C4×C31—44no—
147C4⋊C4×C31—42no—
148C4⋊C4×C31C22×C4116yes—
149C4⋊C4×C31C22×C4116yes—
150C4⋊C4×C31C22×C418yes—
151C22⋊C4×C31—28no—
152C22⋊C4×C31—28no—
153C22⋊C4×C31—44no—
154C22⋊C4×C31—44no—
155C22⋊C4×C31—44no—
156C22⋊C4×C31—44no—
157C22⋊C4×C31C22×C4116yes—
158C22⋊C4×C31C22×C4116yes—
159C22⋊C4×C31C22×C4116yes—
160C42×C31—28no—
161C42×C31—24no—
162C42×C31—22no—
163C42×C31—42no—
164C42×C31—44no—
165C42×C31—41no—
166C42×C31—44no—
167C42×C31—44no—
168C42×C31C22×C414yes—
169#34522C3111yesmaximal, characteristic, direct product factor, Sylow
170C23×C31—24no—
171C23×C31—24no—
172C23×C31—24no—
173C23×C31—24no—
174C23×C31—24no—
175C23×C31—24no—
176C23×C31C23×C418yes—
177C23×C31C2511yescharacteristic
178Q8×C31—48no—
179D4×C31—416no—
180D4×C31—48no—
181C2×C4×C31—24no—
182C2×C4×C31—22no—
183C2×C4×C31—24no—
184C2×C4×C31—21no—
185C2×C4×C31—22no—
186C2×C4×C31—21no—
187C2×C4×C31—22no—
188C2×C4×C31—22no—
189C2×C4×C31—24no—
190C2×C4×C31—22no—
191C2×C4×C31—24no—
192C2×C4×C31—22no—
193C2×C4×C31—24no—
194C2×C4×C31—24no—
195C2×C4×C31—24no—
196C2×C4×C31—24no—
197C2×C4×C31—21no—
198C2×C4×C31—24no—
199C2×C4×C31—24no—
200C2×C4×C31—24no—
201C2×C4×C31—24no—
202C2×C4×C31—24no—
203C2×C4×C31—416no—
204C2×C4×C31—88no—
205C2×C4×C31—88no—
206C2×C4×C31C23×C412yes—
207C2×C4×C31C23×C414yes—
208C2×C4×C31C23×C418yes—
209C2×C4×C31C23×C418yes—
210C2×C4×C31H(5,𝔽2)14yes—
211C2×C4×C312−1+412yes—
212C2×#562C2×C3111yescharacteristic
213#2382C2×C3111yescharacteristic, complemented
214#2379C2×C3112yescomplemented
215#2368C2×C3114yescomplemented
216#2363C2×C3114yescomplemented
217#2359C2×C3114yescomplemented
218#2335C2×C3114yescomplemented
219#2325C2×C3114yescomplemented
220#2323C2×C3114yescomplemented
221C4×#125C2×C3111yescharacteristic, complemented
222C4×#124C2×C3112yescomplemented
223C22×C31—28no—
224C22×C31—28no—
225C22×C31—28no—
226C22×C31—44no—
227C22×C31—44no—
228C22×C31—44no—
229C22×C31—44no—
230C22×C31C24×C411yescharacteristic
231C22×C31C2×H(5,𝔽2)12yes—
232C22×C31C2×2−1+411yescharacteristic
233C22×C31#58512yes—
234C22×C31#58511yescharacteristic
235C4×C31—28no—
236C4×C31—28no—
237C4×C31—24no—
238C4×C31—22no—
239C4×C31—24no—
240C4×C31—22no—
241C4×C31—44no—
242C4×C31—44no—
243C4×C31—41no—
244C4×C31—44no—
245#562C4×C31116yescomplemented
246C2×#125C22×C3111yescharacteristic
247C2×#124C22×C3112yes—
248C2×#123C22×C3114yes—
249C2×C22⋊Q8C22×C3114yes—
250C2×#121C22×C3114yes—
251C2×C4×Q8C22×C3112yes—
252C2×C4×D4C22×C3112yes—
253C2×C4×D4C22×C3114yes—
254C2×#117C22×C3114yes—
255#391C22×C3114yes—
256#391C22×C3112yes—
257#390C22×C3114yes—
258#390C22×C3112yes—
259#389C22×C3112yes—
260#389C22×C3114yes—
261#389C22×C3114yes—
262#388C22×C3114yes—
263#388C22×C3114yes—
264#388C22×C3114yes—
265#388C22×C3114yes—
266#388C22×C3114yes—
267#388C22×C3114yes—
268#387C22×C3114yes—
269#387C22×C3114yes—
270#387C22×C3114yes—
271#387C22×C3114yes—
272#387C22×C3114yes—
273#386C22×C3114yes—
274#386C22×C3114yes—
275#386C22×C3114yes—
276#385C22×C3114yes—
277#385C22×C3114yes—
278#385C22×C3114yes—
279#382C22×C3111yescharacteristic
280#382C22×C3112yes—
281#380C22×C3114yes—
282C4×C4⋊C4C22×C3112yes—
283C4×C4⋊C4C22×C3114yes—
284C4×C4⋊C4C22×C3114yes—
285C4×C22⋊C4C22×C3114yes—
286C4×C22⋊C4C22×C3114yes—
287C4×C22⋊C4C22×C3114yes—
288C4×C22⋊C4C22×C3114yes—
289C2×#95C22×C3114yes—
290C43C22×C3111yescharacteristic
291C2×C31—48no—
292C2×C31C4×H(5,𝔽2)12yes—
293C2×C31C4×2−1+411yescharacteristic
294C2×C31#354512yes—
295C2×C31#354811yescharacteristic
296C2×C31#355311yescharacteristic
297C22×Q8C23×C3112yes—
298C22×D4C23×C3114yes—
299C22×D4C23×C3112yes—
300C23×C4C23×C3114yes—
301C23×C4C23×C3114yes—
302C23×C4C23×C3114yes—
303#125C2×C4×C3118yescomplemented
304#124C2×C4×C31116yescomplemented
305#123C2×C4×C31132yescomplemented
306C22⋊Q8C2×C4×C31132yescomplemented
307#121C2×C4×C31132yescomplemented
308C4×Q8—44no—
309C4×D4—44no—
310C4×D4—48no—
311#117—48no—
312C2×C4⋊C4C23×C3114yes—
313C2×C4⋊C4C23×C3114yes—
314C2×C4⋊C4C23×C3112yes—
315C2×C4⋊C4C23×C3114yes—
316C2×C4⋊C4C23×C3112yes—
317C2×C4⋊C4C23×C3114yes—
318C2×C4⋊C4C23×C3112yes—
319C2×C4⋊C4C23×C3114yes—
320C2×C4⋊C4C23×C3114yes—
321C2×C4⋊C4C23×C3112yes—
322C2×C22⋊C4C23×C3114yes—
323C2×C22⋊C4C23×C3114yes—
324C2×C22⋊C4C23×C3114yes—
325C2×C22⋊C4C23×C3114yes—
326C2×C22⋊C4C23×C3114yes—
327C2×C22⋊C4C23×C3114yes—
328C2×C22⋊C4C23×C3114yes—
329C2×C22⋊C4C23×C3114yes—
330C2×C22⋊C4C23×C3114yes—
331C2×C42C23×C3114yes—
332C2×C42C23×C3112yes—
333C2×C42C23×C3111yescharacteristic
334C2×C42C23×C3114yes—
335C2×C42C23×C3114yes—
336C2×C42C23×C3111yescharacteristic
337C2×C42C23×C3112yes—
338C2×C42C23×C3114yes—
339C2×C42C23×C3111yescharacteristic
340#95C23×C3114yes—
341#95C23×C3114yes—
342#95C23×C3114yes—
343#95C23×C3114yes—
344#95C23×C3114yes—
345#95C23×C3114yes—
346#95C23×C3114yes—
347#95C23×C3114yes—
348#95C23×C3114yes—
349#95C23×C3114yes—
350#95C23×C3114yes—
351C31#3452211yescharacteristic, direct product factor, Sylow
352C24C24×C3114yes—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8C22×C4×C3118yes—
356C2×D4—42no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—42no—
360C2×D4C22×C4×C31116yes—
361C2×D4C22×C4×C3118yes—
362C22×C4—28no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4C22×C4×C31116yes—
371C22×C4C24×C3114yes—
372C22×C4C24×C3114yes—
373C22×C4C24×C3111yescharacteristic
374C22×C4C24×C3114yes—
375C22×C4C24×C3111yescharacteristic
376C22×C4C24×C3112yes—
377C22×C4C24×C3114yes—
378C22×C4C24×C3114yes—
379C22×C4C24×C3112yes—
380C22×C4C24×C3111yescharacteristic
381C4⋊C4—28no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4C22×C4×C31116yes—
389C4⋊C4C22×C4×C31116yes—
390C4⋊C4C22×C4×C3118yes—
391C22⋊C4—28no—
392C22⋊C4—28no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4C22×C4×C31116yes—
398C22⋊C4C22×C4×C31116yes—
399C22⋊C4C22×C4×C31116yes—
400C42—28no—
401C42—24no—
402C42—22no—
403C42—42no—
404C42—44no—
405C42—41no—
406C42—44no—
407C42—44no—
408C42C22×C4×C3114yes—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23C23×C4×C3118yes—
416C23C25×C3111yescharacteristic
417Q8—48no—
418D4—416no—
419D4—48no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—416no—
443C2×C4—88no—
444C2×C4—88no—
445C2×C4C23×C4×C3112yes—
446C2×C4C23×C4×C3114yes—
447C2×C4C23×C4×C3118yes—
448C2×C4C23×C4×C3118yes—
449C2×C4C31×H(5,𝔽2)14yes—
450C2×C4C31×2−1+412yes—
451C22—28no—
452C22—28no—
453C22—28no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22—44no—
458C22C24×C4×C3111yescharacteristic
459C22C2×C31×H(5,𝔽2)12yes—
460C22C2×C31×2−1+411yescharacteristic
461C22C31×#58512yes—
462C22C31×#58511yescharacteristic
463C4—28no—
464C4—28no—
465C4—24no—
466C4—22no—
467C4—24no—
468C4—22no—
469C4—44no—
470C4—44no—
471C4—41no—
472C4—44no—
473C2—48no—
474C2C4×C31×H(5,𝔽2)12yes—
475C2C4×C31×2−1+411yescharacteristic
476C2C31×#354512yes—
477C2C31×#354811yescharacteristic
478C2C31×#355311yescharacteristic
subgroupG/H
center, socleC23×C31C25
commutatorC22C24×C4×C31
Frattini, cosocleC23C25×C31
prime radicalC2×C31×#562C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2382C2semidirect
3. C31×#2379C2semidirect
4. C31×#2368C2semidirect
5. C31×#2363C2semidirect
6. C31×#2359C2semidirect
7. C31×#2335C2semidirect
8. C31×#2325C2semidirect
9. C31×#2323C2semidirect
10. C4×C31×#125C2semidirect
11. C4×C31×#124C2semidirect
12. C31×#562C4semidirect
64. C31×#125C2×C4semidirect
65. C31×#124C2×C4semidirect
66. C31×#123C2×C4semidirect
67. C31×C22⋊Q8C2×C4semidirect
68. C31×#121C2×C4semidirect
169. #34522C31direct (N × H)
213. #2382C2×C31semidirect
214. #2379C2×C31semidirect
215. #2368C2×C31semidirect
216. #2363C2×C31semidirect
217. #2359C2×C31semidirect
218. #2335C2×C31semidirect
219. #2325C2×C31semidirect
220. #2323C2×C31semidirect
221. C4×#125C2×C31semidirect
222. C4×#124C2×C31semidirect
245. #562C4×C31semidirect
303. #125C2×C4×C31semidirect
304. #124C2×C4×C31semidirect
305. #123C2×C4×C31semidirect
306. C22⋊Q8C2×C4×C31semidirect
307. #121C2×C4×C31semidirect
351. C31#34522direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5622
C31×#23822
C31×#23792
C31×#23682
C31×#23632
C31×#23592
C31×#23352
C31×#23252
C31×#23232
C4×C31×#1252
C4×C31×#1242
#3452231