Finite Group Databasebeta

← C31×#23949 #21480708

Subgroups — C31×#23949 #21480708

Order 7936.

1258 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3492C211yesmaximal, characteristic
2C31×#3004C212yesmaximal, complemented
3C2×C31×#391C211yesmaximal, characteristic, complemented
4C31×#2028C212yesmaximal, complemented
5C31×#2007C212yesmaximal, complemented
6C31×#1896C212yesmaximal, complemented
7C31×#1882C214yesmaximal, complemented
8C31×#1821C211yesmaximal, characteristic, complemented
9C23×Q8×C31C2211yescharacteristic
10C2×C31×#137C2214yes—
11C31×#554—22no—
12C31×#553—22no—
13C31×#552—22no—
14C31×#547—22no—
15C31×#544C2211yescharacteristic
16C31×#539—22no—
17C31×#537C2211yescharacteristic
18C31×#525—22no—
19C31×#525C2212yes—
20C31×#520C2212yes—
21C22×C4⋊C4×C31C2211yescharacteristic
22C22×C42×C31C2211yescharacteristic
23C31×#451—22no—
24C31×#451—22no—
25C31×#448—22no—
26C31×#448—22no—
27C2×C31×#104C2214yes—
28C31×#419C2214yes—
29C31×#407C2212yes—
30C31×#391—22no—
31C31×#391C2212yes—
32C31×#391C2212yes—
33C2×C31×#95—22no—
34C31×#343C2212yes—
35C31×#343C2212yes—
36C31×#339C2214yes—
37C2×#41×C31—22no—
38C2×#41×C31C2312yes—
39C22×Q8×C31—24no—
40C22×Q8×C31—21no—
41C22×Q8×C31—22no—
42C22×Q8×C31—22no—
43C22×Q8×C31—42no—
44C22×Q8×C31C2312yes—
45C23×C4×C31—21no—
46C23×C4×C31—21no—
47C23×C4×C31D412yes—
48C23×C4×C31C2311yescharacteristic
49C31×#137—28no—
50C31×#137—44no—
51C2×Q16×C31—24no—
52C2×SD16×C31—24no—
53C2×M16×C31C2314yes—
54C31×C4⋊Q8—22no—
55C31×#126—24no—
56C31×#125—24no—
57C31×#125—22no—
58C31×#123—24no—
59C31×#123—24no—
60C31×#123—22no—
61C31×C22⋊Q8—22no—
62C31×C22⋊Q8—24no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8D412yes—
66C31×C22⋊Q8D412yes—
67C31×#121—22no—
68C31×#121—22no—
69C31×#121—22no—
70C31×#121D412yes—
71C31×#121D412yes—
72C31×#120—22no—
73C4×Q8×C31—22no—
74C4×D4×C31—24no—
75C4×D4×C31—22no—
76C4×D4×C31—24no—
77C4×D4×C31—22no—
78C4×D4×C31—22no—
79C31×#117—22no—
80C31×#117—22no—
81C31×#117C2312yes—
82C2×C4⋊C4×C31—22no—
83C2×C4⋊C4×C31—22no—
84C2×C4⋊C4×C31—22no—
85C2×C4⋊C4×C31D412yes—
86C2×C4⋊C4×C31C2311yescharacteristic
87C2×C4⋊C4×C31C2311yescharacteristic
88C2×C42×C31—21no—
89C2×C42×C31—42no—
90C2×C42×C31C2312yes—
91C31×#104—44no—
92C31×#103—24no—
93C31×#103—24no—
94C31×C22⋊C8—22no—
95C31×C22⋊C8—22no—
96C31×#95—24no—
97C31×#95—24no—
98C24×C31C2×D411yescharacteristic
99#41×C31—24no—
100#41×C31—24no—
101#41×C31—84no—
102C2×Q8×C31—24no—
103C2×Q8×C31—22no—
104C2×Q8×C31—22no—
105C2×Q8×C31—24no—
106C2×Q8×C31—24no—
107C2×Q8×C31—48no—
108C2×Q8×C31—48no—
109C2×Q8×C31—42no—
110C2×Q8×C31—42no—
111C2×Q8×C31C2×D412yes—
112C2×Q8×C31#4114yescomplemented
113C2×D4×C31—22no—
114C2×D4×C31—22no—
115C2×D4×C31—22no—
116C2×D4×C31—22no—
117C2×D4×C31C2×D412yes—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—24no—
121C22×C4×C31—24no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—24no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—42no—
129C22×C4×C31—42no—
130C22×C4×C31—42no—
131C22×C4×C31—41no—
132C22×C4×C31—41no—
133C22×C4×C31—44no—
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31C2×D412yes—
136C22×C4×C31C2×D412yes—
137C22×C4×C31C2×D412yes—
138C22×C4×C31C2×D412yes—
139C22×C4×C31C2411yescharacteristic
140Q16×C31—48no—
141SD16×C31—48no—
142M16×C31—24no—
143M16×C31—24no—
144C2×C8×C31—24no—
145C4⋊C4×C31—24no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—24no—
149C4⋊C4×C31—24no—
150C4⋊C4×C31—22no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31C2×D412yes—
155C4⋊C4×C31C2×D412yes—
156C22⋊C4×C31—24no—
157C22⋊C4×C31—24no—
158C22⋊C4×C31—22no—
159C22⋊C4×C31—22no—
160C22⋊C4×C31—22no—
161C22⋊C4×C31—22no—
162C22⋊C4×C31—22no—
163C42×C31—24no—
164C42×C31—22no—
165C42×C31—24no—
166C42×C31—42no—
167#23949C3111yesmaximal, characteristic, direct product factor, Sylow
168C23×C31—21no—
169C23×C31—21no—
170C23×C31—22no—
171C23×C31—22no—
172C23×C31—22no—
173C23×C31#12011yescharacteristic
174C23×C31#12011yescharacteristic
175C23×C31C22×D411yescharacteristic
176Q8×C31—28no—
177Q8×C31—24no—
178Q8×C31—48no—
179Q8×C31—44no—
180Q8×C31—82no—
181D4×C31—24no—
182D4×C31—42no—
183D4×C31—42no—
184D4×C31—44no—
185D4×C31—44no—
186D4×C31—82no—
187C2×C4×C31—21no—
188C2×C4×C31—21no—
189C2×C4×C31—24no—
190C2×C4×C31—24no—
191C2×C4×C31—24no—
192C2×C4×C31—22no—
193C2×C4×C31—22no—
194C2×C4×C31—22no—
195C2×C4×C31—22no—
196C2×C4×C31—22no—
197C2×C4×C31—24no—
198C2×C4×C31—48no—
199C2×C4×C31—42no—
200C2×C4×C31—44no—
201C2×C4×C31—44no—
202C2×C4×C31—42no—
203C2×C4×C31—42no—
204C2×C4×C31—42no—
205C2×C4×C31—44no—
206C2×C4×C31—44no—
207C2×C4×C31—42no—
208C2×C4×C31—42no—
209C2×C4×C31—82no—
210C2×C4×C31—84no—
211C2×C4×C31#12012yes—
212C2×C4×C31C22×D412yes—
213C2×C4×C31C2×#4112yes—
214C2×C4×C31H(5,𝔽2)12yes—
215C8×C31—44no—
216#3492C2×C3111yescharacteristic
217#3004C2×C3112yescomplemented
218C2×#391C2×C3111yescharacteristic, complemented
219#2028C2×C3112yescomplemented
220#2007C2×C3112yescomplemented
221#1896C2×C3112yescomplemented
222#1882C2×C3114yescomplemented
223#1821C2×C3111yescharacteristic, complemented
224C22×C31—24no—
225C22×C31—24no—
226C22×C31—21no—
227C22×C31—21no—
228C22×C31—21no—
229C22×C31—21no—
230C22×C31—21no—
231C22×C31—21no—
232C22×C31—42no—
233C22×C31—42no—
234C22×C31—42no—
235C22×C31#45512yes—
236C22×C31#45612yes—
237C22×C31C2×#12011yescharacteristic
238C22×C31#52511yescharacteristic
239C22×C31#53511yescharacteristic
240C4×C31—22no—
241C4×C31—44no—
242C4×C31—44no—
243C4×C31—42no—
244C4×C31—42no—
245C4×C31—42no—
246C4×C31—42no—
247C4×C31—42no—
248C4×C31#54614yes—
249C23×Q8C22×C3111yescharacteristic
250C2×#137C22×C3114yes—
251#554—22no—
252#553—22no—
253#552—22no—
254#547—22no—
255#544C22×C3111yescharacteristic
256#539—22no—
257#537C22×C3111yescharacteristic
258#525—22no—
259#525C22×C3112yes—
260#520C22×C3112yes—
261C22×C4⋊C4C22×C3111yescharacteristic
262C22×C42C22×C3111yescharacteristic
263#451—22no—
264#451—22no—
265#448—22no—
266#448—22no—
267C2×#104C22×C3114yes—
268#419C22×C3114yes—
269#407C22×C3112yes—
270#391—22no—
271#391C22×C3112yes—
272#391C22×C3112yes—
273C2×#95—22no—
274#343C22×C3112yes—
275#343C22×C3112yes—
276#339C22×C3114yes—
277C2×C31—21no—
278C2×C31—21no—
279C2×C31—22no—
280C2×C31—22no—
281C2×C31—82no—
282C2×C31#240411yescharacteristic
283C2×C31C2×#45511yescharacteristic
284C2×C31C2×#45611yescharacteristic
285C2×#41—22no—
286C2×#41C23×C3112yes—
287C22×Q8—24no—
288C22×Q8—21no—
289C22×Q8—22no—
290C22×Q8—22no—
291C22×Q8—42no—
292C22×Q8C23×C3112yes—
293C23×C4—21no—
294C23×C4—21no—
295C23×C4D4×C3112yes—
296C23×C4C23×C3111yescharacteristic
297#137—28no—
298#137—44no—
299C2×Q16—24no—
300C2×SD16—24no—
301C2×M16C23×C3114yes—
302C4⋊Q8—22no—
303#126—24no—
304#125—24no—
305#125—22no—
306#123—24no—
307#123—24no—
308#123—22no—
309C22⋊Q8—22no—
310C22⋊Q8—24no—
311C22⋊Q8—22no—
312C22⋊Q8—22no—
313C22⋊Q8D4×C3112yes—
314C22⋊Q8D4×C3112yes—
315#121—22no—
316#121—22no—
317#121—22no—
318#121D4×C3112yes—
319#121D4×C3112yes—
320#120—22no—
321C4×Q8—22no—
322C4×D4—24no—
323C4×D4—22no—
324C4×D4—24no—
325C4×D4—22no—
326C4×D4—22no—
327#117—22no—
328#117—22no—
329#117C23×C3112yes—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4D4×C3112yes—
334C2×C4⋊C4C23×C3111yescharacteristic
335C2×C4⋊C4C23×C3111yescharacteristic
336C2×C42—21no—
337C2×C42—42no—
338C2×C42C23×C3112yes—
339#104—44no—
340#103—24no—
341#103—24no—
342C22⋊C8—22no—
343C22⋊C8—22no—
344#95—24no—
345#95—24no—
346C31#2394911yescharacteristic, direct product factor, Sylow
347C24C2×D4×C3111yescharacteristic
348#41—24no—
349#41—24no—
350#41—84no—
351C2×Q8—24no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—24no—
355C2×Q8—24no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8C2×D4×C3112yes—
361C2×Q8#41×C3114yescomplemented
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4C2×D4×C3112yes—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—44no—
383C22×C4C2×D4×C3111yescharacteristic
384C22×C4C2×D4×C3112yes—
385C22×C4C2×D4×C3112yes—
386C22×C4C2×D4×C3112yes—
387C22×C4C2×D4×C3112yes—
388C22×C4C24×C3111yescharacteristic
389Q16—48no—
390SD16—48no—
391M16—24no—
392M16—24no—
393C2×C8—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4C2×D4×C3112yes—
404C4⋊C4C2×D4×C3112yes—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C42—24no—
413C42—22no—
414C42—24no—
415C42—42no—
416C23—22no—
417C23—22no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23C31×#12011yescharacteristic
422C23C31×#12011yescharacteristic
423C23C22×D4×C3111yescharacteristic
424Q8—28no—
425Q8—24no—
426Q8—48no—
427Q8—44no—
428Q8—82no—
429D4—24no—
430D4—44no—
431D4—44no—
432D4—42no—
433D4—42no—
434D4—82no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—48no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—82no—
458C2×C4—84no—
459C2×C4C31×#12012yes—
460C2×C4C22×D4×C3112yes—
461C2×C4C2×#41×C3112yes—
462C2×C4C31×H(5,𝔽2)12yes—
463C8—44no—
464C22—24no—
465C22—24no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22C31×#45512yes—
476C22C31×#45612yes—
477C22C2×C31×#12011yescharacteristic
478C22C31×#52511yescharacteristic
479C22C31×#53511yescharacteristic
480C4—22no—
481C4—44no—
482C4—44no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4C31×#54614yes—
489C2—21no—
490C2—22no—
491C2—22no—
492C2—21no—
493C2—82no—
494C2C31×#240411yescharacteristic
495C2C2×C31×#45511yescharacteristic
496C2C2×C31×#45611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3492C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3004C2semidirect
3. C2×C31×#391C2semidirect
4. C31×#2028C2semidirect
5. C31×#2007C2semidirect
6. C31×#1896C2semidirect
7. C31×#1882C2semidirect
8. C31×#1821C2semidirect
112. C2×Q8×C31#41semidirect
167. #23949C31direct (N × H)
217. #3004C2×C31semidirect
218. C2×#391C2×C31semidirect
219. #2028C2×C31semidirect
220. #2007C2×C31semidirect
221. #1896C2×C31semidirect
222. #1882C2×C31semidirect
223. #1821C2×C31semidirect
346. C31#23949direct (N × H)
361. C2×Q8#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34922
C31×#30042
C2×C31×#3912
C31×#20282
C31×#20072
C31×#18962
C31×#18822
C31×#18212
#2394931