Finite Group Databasebeta

← C31×#18195 #21474954

Subgroups — C31×#18195 #21474954

Order 7936.

1454 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3496C211yesmaximal, characteristic
2C2×C31×#448C212yesmaximal, complemented
3C31×#2409C211yesmaximal, characteristic, complemented
4C31×#1604C214yesmaximal, complemented
5C31×#1601C214yesmaximal, complemented
6C31×#1519C212yesmaximal, complemented
7C31×#1459C211yesmaximal, characteristic, complemented
8C2×C31×H(5,𝔽2)—21no—
9C23×Q8×C31C2211yescharacteristic, complemented
10C22×SD16×C31—22no—
11C31×#547—22no—
12C31×#546—24no—
13D42×C31—22no—
14C31×#537C2212yes—
15C31×#534—21no—
16C31×#534C2212yes—
17C31×#532—21no—
18C2×C31×C22⋊Q8—21no—
19C2×C31×#121—21no—
20C2×C31×#121C2212yes—
21C2×C4×D4×C31C2211yescharacteristic
22C31×#448—24no—
23C31×#448C2218yescomplemented
24C2×C31×#103—22no—
25C2×C31×C22⋊C8C2212yescomplemented
26C31×#395C2212yescomplemented
27C31×#393C2212yescomplemented
28C31×#391—21no—
29C31×#386—21no—
30C31×#385—21no—
31C31×#380C2211yescharacteristic, complemented
32C31×#327C2218yescomplemented
33C31×#324C2212yescomplemented
34C31×#323C2212yescomplemented
35C31×H(5,𝔽2)—44no—
36C31×H(5,𝔽2)—44no—
37C2×#41×C31—24no—
38C2×#41×C31—22no—
39C22×Q8×C31—21no—
40C22×Q8×C31—22no—
41C22×Q8×C31—22no—
42C22×Q8×C31—44no—
43C22×Q8×C31C2312yes—
44C22×D4×C31—24no—
45C22×D4×C31—22no—
46C22×D4×C31—22no—
47C22×D4×C31—21no—
48C22×D4×C31D411yescharacteristic
49C22×D4×C31D412yes—
50C23×C4×C31—21no—
51C23×C4×C31D411yescharacteristic
52C23×C4×C31C2311yescharacteristic
53C2×SD16×C31—28no—
54C2×SD16×C31—48no—
55C22×C8×C31—22no—
56C31×C4⋊Q8—21no—
57C31×#127—21no—
58C31×#124—22no—
59C31×#123—24no—
60C31×#123—24no—
61C31×C22⋊Q8—24no—
62C31×C22⋊Q8—44no—
63C31×#121—22no—
64C31×#121—22no—
65C31×#121—24no—
66C31×#121—22no—
67C31×#121—22no—
68C31×#121—22no—
69C31×#121—22no—
70C31×#121D414yes—
71C31×#121C2314yes—
72C31×#120—24no—
73C31×#120—24no—
74C4×D4×C31—22no—
75C4×D4×C31—22no—
76C4×D4×C31—22no—
77C4×D4×C31—22no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—22no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31D412yes—
82C2×C4⋊C4×C31C2311yescharacteristic
83C2×C22⋊C4×C31—22no—
84C2×C22⋊C4×C31—22no—
85C2×C22⋊C4×C31—22no—
86C2×C22⋊C4×C31—21no—
87C2×C22⋊C4×C31D411yescharacteristic
88C2×C22⋊C4×C31C2311yescharacteristic
89C2×C42×C31D411yescharacteristic
90C31×#103—28no—
91C31×C22⋊C8—22no—
92C31×C22⋊C8C2314yes—
93C31×#95—22no—
94C31×#95—22no—
95C31×#95C2312yes—
96C24×C31—22no—
97C24×C31—22no—
98C24×C31C2×D411yescharacteristic
99C24×C31C2×D411yescharacteristic
100#41×C31—44no—
101#41×C31—416no—
102#41×C31—44no—
103C2×Q8×C31—22no—
104C2×Q8×C31—24no—
105C2×Q8×C31—28no—
106C2×Q8×C31—24no—
107C2×Q8×C31—44no—
108C2×Q8×C31—42no—
109C2×Q8×C31—44no—
110C2×Q8×C31—48no—
111C2×Q8×C31—41no—
112C2×Q8×C31—41no—
113C2×D4×C31—22no—
114C2×D4×C31—22no—
115C2×D4×C31—24no—
116C2×D4×C31—22no—
117C2×D4×C31—24no—
118C2×D4×C31—24no—
119C2×D4×C31—22no—
120C2×D4×C31—22no—
121C2×D4×C31—22no—
122C2×D4×C31—44no—
123C2×D4×C31—416no—
124C2×D4×C31—44no—
125C2×D4×C31—44no—
126C2×D4×C31—44no—
127C2×D4×C31—44no—
128C2×D4×C31—82no—
129C2×D4×C31SD1614yescomplemented
130C2×D4×C31C2×D414yes—
131C22×C4×C31—22no—
132C22×C4×C31—24no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—21no—
136C22×C4×C31—21no—
137C22×C4×C31—22no—
138C22×C4×C31—42no—
139C22×C4×C31—44no—
140C22×C4×C31—42no—
141C22×C4×C31—42no—
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D412yes—
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D412yes—
148C22×C4×C31C2411yescharacteristic
149SD16×C31—416no—
150C2×C8×C31—24no—
151C2×C8×C31—44no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—42no—
157C4⋊C4×C31—44no—
158C4⋊C4×C31C2×D414yes—
159C22⋊C4×C31—24no—
160C22⋊C4×C31—24no—
161C22⋊C4×C31—22no—
162C22⋊C4×C31—24no—
163C22⋊C4×C31—24no—
164C22⋊C4×C31—22no—
165C22⋊C4×C31—44no—
166C22⋊C4×C31—42no—
167C42×C31—21no—
168C42×C31—21no—
169#18195C3111yesmaximal, characteristic, direct product factor, Sylow
170C23×C31—21no—
171C23×C31—21no—
172C23×C31—24no—
173C23×C31—41no—
174C23×C31—41no—
175C23×C31—41no—
176C23×C31—41no—
177C23×C31—44no—
178C23×C31—44no—
179C23×C31—82no—
180C23×C31—82no—
181C23×C31—82no—
182C23×C31—82no—
183C23×C31#12012yes—
184C23×C31C2×SD1612yes—
185C23×C31C2×SD1612yescomplemented
186C23×C31AGL(1,ℤ/8)12yes—
187C23×C31C22×D411yescharacteristic
188Q8×C31—28no—
189Q8×C31—44no—
190Q8×C31—44no—
191Q8×C31—44no—
192Q8×C31—44no—
193D4×C31—28no—
194D4×C31—44no—
195D4×C31—44no—
196D4×C31—48no—
197D4×C31—44no—
198D4×C31—44no—
199D4×C31—48no—
200D4×C31—44no—
201D4×C31—82no—
202D4×C31—82no—
203C2×C4×C31—21no—
204C2×C4×C31—21no—
205C2×C4×C31—22no—
206C2×C4×C31—24no—
207C2×C4×C31—24no—
208C2×C4×C31—21no—
209C2×C4×C31—21no—
210C2×C4×C31—24no—
211C2×C4×C31—42no—
212C2×C4×C31—41no—
213C2×C4×C31—48no—
214C2×C4×C31—42no—
215C2×C4×C31—44no—
216C2×C4×C31—48no—
217C2×C4×C31—42no—
218C2×C4×C31—44no—
219C2×C4×C31—42no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—81no—
226C2×C4×C31#12011yescharacteristic
227C2×C4×C31#12011yescharacteristic
228C2×C4×C31C2×SD1612yescomplemented
229C2×C4×C31C22×D412yes—
230C8×C31—44no—
231#3496C2×C3111yescharacteristic
232C2×#448C2×C3112yescomplemented
233#2409C2×C3111yescharacteristic, complemented
234#1604C2×C3114yescomplemented
235#1601C2×C3114yescomplemented
236#1519C2×C3112yescomplemented
237#1459C2×C3111yescharacteristic, complemented
238C22×C31—21no—
239C22×C31—22no—
240C22×C31—21no—
241C22×C31—22no—
242C22×C31—22no—
243C22×C31—42no—
244C22×C31—42no—
245C22×C31—42no—
246C22×C31—41no—
247C22×C31—41no—
248C22×C31—44no—
249C22×C31—44no—
250C22×C31—44no—
251C22×C31—81no—
252C22×C31—164no—
253C22×C31—164no—
254C22×C31#45014yes—
255C22×C31C2×#12011yescharacteristic
256C22×C31C22×SD1611yescharacteristic
257C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
258C4×C31—22no—
259C4×C31—22no—
260C4×C31—21no—
261C4×C31—44no—
262C4×C31—41no—
263C4×C31—44no—
264C4×C31—41no—
265C4×C31—82no—
266C4×C31B(4,𝔽2)12yes—
267C2×H(5,𝔽2)—21no—
268C23×Q8C22×C3111yescharacteristic, complemented
269C22×SD16—22no—
270#547—22no—
271#546—24no—
272D42—22no—
273#537C22×C3112yes—
274#534—21no—
275#534C22×C3112yes—
276#532—21no—
277C2×C22⋊Q8—21no—
278C2×#121—21no—
279C2×#121C22×C3112yes—
280C2×C4×D4C22×C3111yescharacteristic
281#448—24no—
282#448C22×C3118yescomplemented
283C2×#103—22no—
284C2×C22⋊C8C22×C3112yescomplemented
285#395C22×C3112yescomplemented
286#393C22×C3112yescomplemented
287#391—21no—
288#386—21no—
289#385—21no—
290#380C22×C3111yescharacteristic, complemented
291#327C22×C3118yescomplemented
292#324C22×C3112yescomplemented
293#323C22×C3112yescomplemented
294C2×C31—21no—
295C2×C31—21no—
296C2×C31—24no—
297C2×C31—42no—
298C2×C31—84no—
299C2×C31C2×#45011yescharacteristic
300C2×C31#302211yescharacteristic
301C2×C31C2×B(4,𝔽2)11yescharacteristic
302H(5,𝔽2)—44no—
303H(5,𝔽2)—44no—
304C2×#41—24no—
305C2×#41—22no—
306C22×Q8—21no—
307C22×Q8—22no—
308C22×Q8—22no—
309C22×Q8—44no—
310C22×Q8C23×C3112yes—
311C22×D4—24no—
312C22×D4—22no—
313C22×D4—22no—
314C22×D4—21no—
315C22×D4D4×C3111yescharacteristic
316C22×D4D4×C3112yes—
317C23×C4—21no—
318C23×C4D4×C3111yescharacteristic
319C23×C4C23×C3111yescharacteristic
320C2×SD16—28no—
321C2×SD16—48no—
322C22×C8—22no—
323C4⋊Q8—21no—
324#127—21no—
325#124—22no—
326#123—24no—
327#123—24no—
328C22⋊Q8—24no—
329C22⋊Q8—44no—
330#121—22no—
331#121—22no—
332#121—24no—
333#121—22no—
334#121—22no—
335#121—22no—
336#121—22no—
337#121D4×C3114yes—
338#121C23×C3114yes—
339#120—24no—
340#120—24no—
341C4×D4—22no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4D4×C3112yes—
349C2×C4⋊C4C23×C3111yescharacteristic
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4D4×C3111yescharacteristic
355C2×C22⋊C4C23×C3111yescharacteristic
356C2×C42D4×C3111yescharacteristic
357#103—28no—
358C22⋊C8—22no—
359C22⋊C8C23×C3114yes—
360#95—22no—
361#95—22no—
362#95C23×C3112yes—
363C31#1819511yescharacteristic, direct product factor, Sylow
364C24—22no—
365C24—22no—
366C24C2×D4×C3111yescharacteristic
367C24C2×D4×C3111yescharacteristic
368#41—44no—
369#41—416no—
370#41—44no—
371C2×Q8—22no—
372C2×Q8—24no—
373C2×Q8—28no—
374C2×Q8—24no—
375C2×Q8—44no—
376C2×Q8—42no—
377C2×Q8—44no—
378C2×Q8—48no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—24no—
384C2×D4—22no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—44no—
391C2×D4—416no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—82no—
397C2×D4SD16×C3114yescomplemented
398C2×D4C2×D4×C3114yes—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4C2×D4×C3111yescharacteristic
411C22×C4C2×D4×C3112yes—
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4C2×D4×C3111yescharacteristic
414C22×C4C2×D4×C3111yescharacteristic
415C22×C4C2×D4×C3112yes—
416C22×C4C24×C3111yescharacteristic
417SD16—416no—
418C2×C8—24no—
419C2×C8—44no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4C2×D4×C3114yes—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—22no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—22no—
433C22⋊C4—44no—
434C22⋊C4—42no—
435C42—21no—
436C42—21no—
437C23—21no—
438C23—21no—
439C23—24no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—44no—
445C23—44no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23C31×#12012yes—
451C23C2×SD16×C3112yes—
452C23C2×SD16×C3112yescomplemented
453C23C31×AGL(1,ℤ/8)12yes—
454C23C22×D4×C3111yescharacteristic
455Q8—28no—
456Q8—44no—
457Q8—44no—
458Q8—44no—
459Q8—44no—
460D4—28no—
461D4—44no—
462D4—44no—
463D4—48no—
464D4—48no—
465D4—44no—
466D4—44no—
467D4—44no—
468D4—82no—
469D4—82no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—24no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—48no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—48no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—81no—
493C2×C4C31×#12011yescharacteristic
494C2×C4C31×#12011yescharacteristic
495C2×C4C2×SD16×C3112yescomplemented
496C2×C4C22×D4×C3112yes—
497C8—44no—
498C22—21no—
499C22—22no—
500C22—21no—
501C22—22no—
502C22—22no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—81no—
512C22—164no—
513C22—164no—
514C22C31×#45014yes—
515C22C2×C31×#12011yescharacteristic
516C22C22×SD16×C3111yescharacteristic
517C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
518C4—22no—
519C4—22no—
520C4—21no—
521C4—44no—
522C4—41no—
523C4—44no—
524C4—41no—
525C4—82no—
526C4C31×B(4,𝔽2)12yes—
527C2—21no—
528C2—21no—
529C2—24no—
530C2—42no—
531C2—84no—
532C2C2×C31×#45011yescharacteristic
533C2C31×#302211yescharacteristic
534C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3496C2
LCSG ▷ C22×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. C2×C31×#448C2semidirect
3. C31×#2409C2semidirect
4. C31×#1604C2semidirect
5. C31×#1601C2semidirect
6. C31×#1519C2semidirect
7. C31×#1459C2semidirect
9. C23×Q8×C31C22semidirect
23. C31×#448C22semidirect
25. C2×C31×C22⋊C8C22semidirect
26. C31×#395C22semidirect
27. C31×#393C22semidirect
31. C31×#380C22semidirect
32. C31×#327C22semidirect
33. C31×#324C22semidirect
34. C31×#323C22semidirect
129. C2×D4×C31SD16semidirect
169. #18195C31direct (N × H)
185. C23×C31C2×SD16semidirect
228. C2×C4×C31C2×SD16semidirect
232. C2×#448C2×C31semidirect
233. #2409C2×C31semidirect
234. #1604C2×C31semidirect
235. #1601C2×C31semidirect
236. #1519C2×C31semidirect
237. #1459C2×C31semidirect
268. C23×Q8C22×C31semidirect
282. #448C22×C31semidirect
284. C2×C22⋊C8C22×C31semidirect
285. #395C22×C31semidirect
286. #393C22×C31semidirect
290. #380C22×C31semidirect
291. #327C22×C31semidirect
292. #324C22×C31semidirect
293. #323C22×C31semidirect
363. C31#18195direct (N × H)
397. C2×D4SD16×C31semidirect
452. C23C2×SD16×C31semidirect
495. C2×C4C2×SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34962
C2×C31×#4482
C31×#24092
C31×#16042
C31×#16012
C31×#15192
C31×#14592
#1819531