Finite Group Databasebeta

← C31×#17357 #21474116

Subgroups — C31×#17357 #21474116

Order 7936.

1322 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
1C22×C31×#117C211yesmaximal, characteristic
2C31×#2920C212yesmaximal, complemented
3C31×#1464C214yesmaximal, complemented
4C31×#1463C212yesmaximal, complemented
5C31×#1462C212yesmaximal
6C31×#1458C214yesmaximal, complemented
7C24×C4×C31C411yescharacteristic
8C2×C31×#117—22no—
9C2×C31×#117—22no—
10C2×C31×#117—22no—
11C2×C31×#117—22no—
12C2×C31×#117C412yes—
13C2×C31×#117C412yes—
14C2×C31×#117C2212yes—
15C2×C31×#117C2212yes—
16C22×C4⋊C4×C31C411yescharacteristic
17C22×C4⋊C4×C31C2211yescharacteristic
18C22×C22⋊C4×C31C411yescharacteristic
19C22×C22⋊C4×C31C2211yescharacteristic
20C22×C42×C31C411yescharacteristic
21C22×C42×C31C2211yescharacteristic
22C31×#432—22no—
23C31×#432C2214yescomplemented
24C31×#431—22no—
25C31×#431C2214yescomplemented
26C31×#423—22no—
27C31×#407C2214yes—
28C4×M16×C31—22no—
29C31×#324C2218yescomplemented
30C31×#323C2218yes—
31C25×C31C2×C411yescharacteristic
32C23×C4×C31—21no—
33C23×C4×C31—21no—
34C23×C4×C31—21no—
35C23×C4×C31—21no—
36C23×C4×C31—28no—
37C23×C4×C31C2×C411yescharacteristic
38C23×C4×C31C2×C412yes—
39C23×C4×C31C2×C411yescharacteristic
40C23×C4×C31C2×C412yes—
41C23×C4×C31C2×C411yescharacteristic
42C23×C4×C31C2×C411yescharacteristic
43C23×C4×C31C2×C411yescharacteristic
44C23×C4×C31C2311yescharacteristic
45C2×M16×C31—24no—
46C31×#117—24no—
47C31×#117—24no—
48C31×#117—24no—
49C31×#117—24no—
50C31×#117—24no—
51C31×#117—24no—
52C31×#117—24no—
53C31×#117—24no—
54C2×C4⋊C4×C31—22no—
55C2×C4⋊C4×C31—22no—
56C2×C4⋊C4×C31—22no—
57C2×C4⋊C4×C31—22no—
58C2×C4⋊C4×C31C2×C412yes—
59C2×C4⋊C4×C31C2×C412yes—
60C2×C4⋊C4×C31C2×C412yes—
61C2×C4⋊C4×C31C2312yes—
62C2×C22⋊C4×C31—22no—
63C2×C22⋊C4×C31—22no—
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31—22no—
66C2×C22⋊C4×C31C2×C412yes—
67C2×C22⋊C4×C31C2×C412yes—
68C2×C22⋊C4×C31C2×C412yes—
69C2×C22⋊C4×C31C2312yes—
70C2×C42×C31—21no—
71C2×C42×C31—21no—
72C2×C42×C31—21no—
73C2×C42×C31—21no—
74C2×C42×C31C2×C412yes—
75C2×C42×C31C2×C412yes—
76C2×C42×C31C2×C411yescharacteristic
77C2×C42×C31C2×C411yescharacteristic
78C2×C42×C31D412yes—
79C2×C42×C31D412yes—
80C2×C42×C31D412yes—
81C2×C42×C31D412yes—
82C2×C42×C31C2311yescharacteristic
83C2×C42×C31C2311yescharacteristic
84C31×C4⋊C8—24no—
85C31×C4⋊C8—24no—
86C31×C22⋊C8—24no—
87C31×C22⋊C8C2318yes—
88C31×#97—24no—
89C4×C8×C31—24no—
90C24×C31—21no—
91C24×C31—21no—
92C24×C31—24no—
93C24×C31—42no—
94C24×C31—42no—
95C24×C31C22×C411yescharacteristic
96C24×C31C22×C412yes—
97C22×C4×C31—28no—
98C22×C4×C31—28no—
99C22×C4×C31—28no—
100C22×C4×C31—28no—
101C22×C4×C31—21no—
102C22×C4×C31—21no—
103C22×C4×C31—24no—
104C22×C4×C31—22no—
105C22×C4×C31—22no—
106C22×C4×C31—22no—
107C22×C4×C31—22no—
108C22×C4×C31—24no—
109C22×C4×C31—24no—
110C22×C4×C31—24no—
111C22×C4×C31—24no—
112C22×C4×C31—21no—
113C22×C4×C31—21no—
114C22×C4×C31—21no—
115C22×C4×C31—21no—
116C22×C4×C31—42no—
117C22×C4×C31—42no—
118C22×C4×C31—42no—
119C22×C4×C31—42no—
120C22×C4×C31C22⋊C411yescharacteristic
121C22×C4×C31C22⋊C411yescharacteristic
122C22×C4×C31C22⋊C412yes—
123C22×C4×C31C22⋊C411yescharacteristic
124C22×C4×C31C22⋊C411yescharacteristic
125C22×C4×C31C22⋊C412yes—
126C22×C4×C31M1618yescomplemented
127C22×C4×C31C22×C411yescharacteristic
128C22×C4×C31C22×C412yes—
129C22×C4×C31C22×C412yes—
130C22×C4×C31C22×C412yes—
131C22×C4×C31C22×C411yescharacteristic
132C22×C4×C31C22×C411yescharacteristic
133C22×C4×C31C22×C411yescharacteristic
134C22×C4×C31C22×C411yescharacteristic
135C22×C4×C31C2×D412yes—
136C22×C4×C31C2×D412yes—
137C22×C4×C31C2411yescharacteristic
138M16×C31—48no—
139C2×C8×C31—28no—
140C4⋊C4×C31—24no—
141C4⋊C4×C31—24no—
142C4⋊C4×C31—24no—
143C4⋊C4×C31—24no—
144C22⋊C4×C31—24no—
145C22⋊C4×C31—24no—
146C22⋊C4×C31—24no—
147C22⋊C4×C31—24no—
148C42×C31—22no—
149C42×C31—22no—
150C42×C31—22no—
151C42×C31—22no—
152C42×C31C22⋊C412yes—
153C42×C31C22⋊C412yes—
154C42×C31C22⋊C412yes—
155C42×C31C22⋊C412yes—
156C42×C31C2×D412yes—
157C42×C31C2×D412yes—
158C42×C31C2×D412yes—
159C42×C31C2×D412yes—
160#17357C3111yesmaximal, characteristic, direct product factor, Sylow
161C23×C31—22no—
162C23×C31—22no—
163C23×C31—22no—
164C23×C31—24no—
165C23×C31—24no—
166C23×C31—22no—
167C23×C31—22no—
168C23×C31—42no—
169C23×C31—42no—
170C23×C31—416no—
171C23×C31—42no—
172C23×C31—42no—
173C23×C31—42no—
174C23×C31—42no—
175C23×C31C2×C22⋊C411yescharacteristic
176C23×C31C2×C22⋊C411yescharacteristic
177C23×C31C2×M1614yes—
178C23×C31C23×C411yescharacteristic
179C2×C4×C31—28no—
180C2×C4×C31—28no—
181C2×C4×C31—24no—
182C2×C4×C31—24no—
183C2×C4×C31—22no—
184C2×C4×C31—22no—
185C2×C4×C31—22no—
186C2×C4×C31—24no—
187C2×C4×C31—24no—
188C2×C4×C31—22no—
189C2×C4×C31—22no—
190C2×C4×C31—22no—
191C2×C4×C31—22no—
192C2×C4×C31—28no—
193C2×C4×C31—28no—
194C2×C4×C31—22no—
195C2×C4×C31—22no—
196C2×C4×C31—42no—
197C2×C4×C31—41no—
198C2×C4×C31—41no—
199C2×C4×C31—41no—
200C2×C4×C31—42no—
201C2×C4×C31—41no—
202C2×C4×C31—42no—
203C2×C4×C31—42no—
204C2×C4×C31—42no—
205C2×C4×C31—42no—
206C2×C4×C31—42no—
207C2×C4×C31—42no—
208C2×C4×C31C2×C22⋊C411yescharacteristic
209C2×C4×C31C2×C22⋊C411yescharacteristic
210C2×C4×C31C2×C22⋊C412yes—
211C2×C4×C31C2×C22⋊C412yes—
212C2×C4×C31C2×C22⋊C412yes—
213C2×C4×C31C2×C22⋊C411yescharacteristic
214C2×C4×C31C2×C22⋊C411yescharacteristic
215C2×C4×C31C2×M1614yes—
216C2×C4×C31C2×M1614yes—
217C2×C4×C31C22×D412yes—
218C8×C31—48no—
219C22×#117C2×C3111yescharacteristic
220#2920C2×C3112yescomplemented
221#1464C2×C3114yescomplemented
222#1463C2×C3112yescomplemented
223#1462C2×C3112yes—
224#1458C2×C3114yescomplemented
225C22×C31—28no—
226C22×C31—21no—
227C22×C31—21no—
228C22×C31—21no—
229C22×C31—21no—
230C22×C31—21no—
231C22×C31—21no—
232C22×C31—24no—
233C22×C31—44no—
234C22×C31—44no—
235C22×C31—48no—
236C22×C31—48no—
237C22×C31—42no—
238C22×C31—42no—
239C22×C31#41012yes—
240C22×C31#41312yes—
241C22×C31C22×C22⋊C411yescharacteristic
242C22×C31C22×M1612yes—
243C4×C31—21no—
244C4×C31—21no—
245C4×C31—24no—
246C4×C31—22no—
247C4×C31—22no—
248C4×C31—22no—
249C4×C31—22no—
250C4×C31—42no—
251C4×C31—42no—
252C4×C31#40712yes—
253C4×C31#40712yes—
254C24×C4C4×C3111yescharacteristic
255C2×#117—22no—
256C2×#117—22no—
257C2×#117—22no—
258C2×#117—22no—
259C2×#117C4×C3112yes—
260C2×#117C4×C3112yes—
261C2×#117C22×C3112yes—
262C2×#117C22×C3112yes—
263C22×C4⋊C4C4×C3111yescharacteristic
264C22×C4⋊C4C22×C3111yescharacteristic
265C22×C22⋊C4C4×C3111yescharacteristic
266C22×C22⋊C4C22×C3111yescharacteristic
267C22×C42C4×C3111yescharacteristic
268C22×C42C22×C3111yescharacteristic
269#432—22no—
270#432C22×C3114yescomplemented
271#431—22no—
272#431C22×C3114yescomplemented
273#423—22no—
274#407C22×C3114yes—
275C4×M16—22no—
276#324C22×C3118yescomplemented
277#323C22×C3118yes—
278C2×C31—22no—
279C2×C31—22no—
280C2×C31—22no—
281C2×C31—44no—
282C2×C31C2×#40711yescharacteristic
283C2×C31C2×#41011yescharacteristic
284C2×C31C2×#41311yescharacteristic
285C25C2×C4×C3111yescharacteristic
286C23×C4—21no—
287C23×C4—21no—
288C23×C4—21no—
289C23×C4—21no—
290C23×C4—28no—
291C23×C4C2×C4×C3111yescharacteristic
292C23×C4C2×C4×C3112yes—
293C23×C4C2×C4×C3111yescharacteristic
294C23×C4C2×C4×C3112yes—
295C23×C4C2×C4×C3111yescharacteristic
296C23×C4C2×C4×C3111yescharacteristic
297C23×C4C2×C4×C3111yescharacteristic
298C23×C4C23×C3111yescharacteristic
299C2×M16—24no—
300#117—24no—
301#117—24no—
302#117—24no—
303#117—24no—
304#117—24no—
305#117—24no—
306#117—24no—
307#117—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4C2×C4×C3112yes—
313C2×C4⋊C4C2×C4×C3112yes—
314C2×C4⋊C4C2×C4×C3112yes—
315C2×C4⋊C4C23×C3112yes—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4C2×C4×C3112yes—
321C2×C22⋊C4C2×C4×C3112yes—
322C2×C22⋊C4C2×C4×C3112yes—
323C2×C22⋊C4C23×C3112yes—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42C2×C4×C3112yes—
329C2×C42C2×C4×C3112yes—
330C2×C42C2×C4×C3111yescharacteristic
331C2×C42C2×C4×C3111yescharacteristic
332C2×C42D4×C3112yes—
333C2×C42D4×C3112yes—
334C2×C42D4×C3112yes—
335C2×C42D4×C3112yes—
336C2×C42C23×C3111yescharacteristic
337C2×C42C23×C3111yescharacteristic
338C4⋊C8—24no—
339C4⋊C8—24no—
340C22⋊C8—24no—
341C22⋊C8C23×C3118yes—
342#97—24no—
343C4×C8—24no—
344C31#1735711yescharacteristic, direct product factor, Sylow
345C24—21no—
346C24—21no—
347C24—24no—
348C24—42no—
349C24—42no—
350C24C22×C4×C3111yescharacteristic
351C24C22×C4×C3112yes—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4C22⋊C4×C3111yescharacteristic
376C22×C4C22⋊C4×C3111yescharacteristic
377C22×C4C22⋊C4×C3111yescharacteristic
378C22×C4C22⋊C4×C3111yescharacteristic
379C22×C4C22⋊C4×C3112yes—
380C22×C4C22⋊C4×C3112yes—
381C22×C4M16×C3118yescomplemented
382C22×C4C22×C4×C3111yescharacteristic
383C22×C4C22×C4×C3112yes—
384C22×C4C22×C4×C3112yes—
385C22×C4C22×C4×C3112yes—
386C22×C4C22×C4×C3111yescharacteristic
387C22×C4C22×C4×C3111yescharacteristic
388C22×C4C22×C4×C3111yescharacteristic
389C22×C4C22×C4×C3111yescharacteristic
390C22×C4C2×D4×C3112yes—
391C22×C4C2×D4×C3112yes—
392C22×C4C24×C3111yescharacteristic
393M16—48no—
394C2×C8—28no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42C22⋊C4×C3112yes—
408C42C22⋊C4×C3112yes—
409C42C22⋊C4×C3112yes—
410C42C22⋊C4×C3112yes—
411C42C2×D4×C3112yes—
412C42C2×D4×C3112yes—
413C42C2×D4×C3112yes—
414C42C2×D4×C3112yes—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—24no—
419C23—24no—
420C23—22no—
421C23—22no—
422C23—42no—
423C23—42no—
424C23—416no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23C2×C22⋊C4×C3111yescharacteristic
430C23C2×C22⋊C4×C3111yescharacteristic
431C23C2×M16×C3114yes—
432C23C23×C4×C3111yescharacteristic
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—28no—
447C2×C4—28no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4C2×C22⋊C4×C3111yescharacteristic
463C2×C4C2×C22⋊C4×C3111yescharacteristic
464C2×C4C2×C22⋊C4×C3112yes—
465C2×C4C2×C22⋊C4×C3112yes—
466C2×C4C2×C22⋊C4×C3112yes—
467C2×C4C2×C22⋊C4×C3111yescharacteristic
468C2×C4C2×C22⋊C4×C3111yescharacteristic
469C2×C4C2×M16×C3114yes—
470C2×C4C2×M16×C3114yes—
471C2×C4C22×D4×C3112yes—
472C8—48no—
473C22—28no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—24no—
481C22—44no—
482C22—48no—
483C22—48no—
484C22—44no—
485C22—42no—
486C22—42no—
487C22C31×#41012yes—
488C22C31×#41312yes—
489C22C22×C22⋊C4×C3111yescharacteristic
490C22C22×M16×C3112yes—
491C4—21no—
492C4—21no—
493C4—24no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—42no—
499C4—42no—
500C4C31×#40712yes—
501C4C31×#40712yes—
502C2—22no—
503C2—22no—
504C2—22no—
505C2—44no—
506C2C2×C31×#40711yescharacteristic
507C2C2×C31×#41011yescharacteristic
508C2C2×C31×#41311yescharacteristic
subgroupG/H
centerC2×C4×C31C2×C22⋊C4
commutatorC23C23×C4×C31
Frattini, cosocleC22×C4C24×C31
socleC22×C31C22×C22⋊C4
prime radicalC25×C31C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2920C2semidirect
3. C31×#1464C2semidirect
4. C31×#1463C2semidirect
6. C31×#1458C2semidirect
23. C31×#432C22semidirect
25. C31×#431C22semidirect
29. C31×#324C22semidirect
126. C22×C4×C31M16semidirect
160. #17357C31direct (N × H)
220. #2920C2×C31semidirect
221. #1464C2×C31semidirect
222. #1463C2×C31semidirect
224. #1458C2×C31semidirect
270. #432C22×C31semidirect
272. #431C22×C31semidirect
276. #324C22×C31semidirect
344. C31#17357direct (N × H)
381. C22×C4M16×C31semidirect

Maximal subgroups

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

subgroupindex
C22×C31×#1172
C31×#29202
C31×#14642
C31×#14632
C31×#14622
C31×#14582
#1735731