Finite Group Databasebeta

← C31×#18367 #21475126

Subgroups — C31×#18367 #21475126

Order 7936.

928 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3493C211yesmaximal, characteristic
2C31×#3218C212yesmaximal, complemented
3C31×#2628C211yesmaximal, characteristic, complemented
4C31×#1606C214yesmaximal, complemented
5C31×#1605C212yesmaximal, complemented
6C31×#1604C212yesmaximal, complemented
7C31×#1512C212yesmaximal, complemented
8C31×#1464C211yesmaximal, characteristic, complemented
9C31×#556—22no—
10C31×#555—22no—
11C31×#553—22no—
12C31×#552—22no—
13C31×#544C2211yescharacteristic
14C31×#538—21no—
15C31×#538C2212yes—
16C31×#537C2211yescharacteristic
17C2×C31×C4⋊Q8C2211yescharacteristic
18C31×#529—21no—
19C31×#529C2212yes—
20C31×#525—21no—
21C31×#520—21no—
22C2×C31×#117C2211yescharacteristic
23C31×#497—22no—
24C31×#491—22no—
25C31×#462—22no—
26C31×#461—22no—
27C31×#451C2214yes—
28C31×#448C2214yes—
29C31×#432C2212yes—
30C31×#400—21no—
31C31×#399—21no—
32C31×#395C2212yes—
33C31×#393C2212yes—
34C31×#389—21no—
35C31×#382C2211yescharacteristic
36C31×#380—21no—
37C31×#328C2214yes—
38C31×#327C2214yes—
39C31×#324C2212yes—
40C31×#323C2212yes—
41C2×#41×C31—21no—
42C22×Q8×C31C2312yes—
43C23×C4×C31D411yescharacteristic
44C2×Q16×C31—24no—
45C2×SD16×C31—24no—
46C31×C4⋊Q8—22no—
47C31×C4⋊Q8—22no—
48C31×C4⋊Q8—22no—
49C31×#126—22no—
50C31×#126—22no—
51C31×#126—22no—
52C31×#126—22no—
53C31×#125—21no—
54C31×#125—22no—
55C31×#125D412yes—
56C31×#124—21no—
57C31×#124—22no—
58C31×#124—22no—
59C31×#124D412yes—
60C31×#123—22no—
61C31×#123—22no—
62C31×#123—22no—
63C31×C22⋊Q8—21no—
64C31×C22⋊Q8—21no—
65C31×C22⋊Q8—21no—
66C31×C22⋊Q8—22no—
67C31×C22⋊Q8C2312yes—
68C31×#121—21no—
69C31×#121—21no—
70C31×#121—22no—
71C31×#121—22no—
72C31×#121—22no—
73C31×#121—21no—
74C31×#121C2312yes—
75C31×#120—22no—
76C4×Q8×C31—22no—
77C4×Q8×C31—21no—
78C4×Q8×C31D412yes—
79C4×D4×C31—22no—
80C4×D4×C31—22no—
81C4×D4×C31—22no—
82C4×D4×C31—21no—
83C4×D4×C31—22no—
84C4×D4×C31—22no—
85C4×D4×C31D412yes—
86C31×#117—22no—
87C31×#117—22no—
88C31×#117—21no—
89C31×#117—22no—
90C31×#117—22no—
91C31×#117—22no—
92C2×C4⋊C4×C31—22no—
93C2×C4⋊C4×C31—21no—
94C2×C4⋊C4×C31—21no—
95C2×C4⋊C4×C31—21no—
96C2×C4⋊C4×C31—22no—
97C2×C4⋊C4×C31D411yescharacteristic
98C2×C4⋊C4×C31C2311yescharacteristic
99C2×C22⋊C4×C31—22no—
100C2×C22⋊C4×C31D411yescharacteristic
101C2×C22⋊C4×C31C2311yescharacteristic
102C2×C42×C31D411yescharacteristic
103C2×C42×C31C2311yescharacteristic
104C31×C4⋊C8—22no—
105C31×#103—24no—
106C31×#103—24no—
107C31×C22⋊C8C2314yes—
108C31×#97—22no—
109C31×#95—22no—
110C31×#95—21no—
111C31×#95—22no—
112C31×#95C2312yes—
113C24×C31C2×D411yescharacteristic
114#41×C31—42no—
115#41×C31—42no—
116C2×Q8×C31—21no—
117C2×Q8×C31—21no—
118C2×Q8×C31—24no—
119C2×Q8×C31—82no—
120C2×Q8×C31C2×D412yes—
121C2×D4×C31—21no—
122C2×D4×C31—22no—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—21no—
126C2×D4×C31C2×D412yes—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—41no—
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31C2411yescharacteristic
142Q16×C31—84no—
143SD16×C31—84no—
144C2×C8×C31—24no—
145C4⋊C4×C31—21no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—21no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31—21no—
150C4⋊C4×C31—21no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—42no—
158C4⋊C4×C31—42no—
159C4⋊C4×C31—41no—
160C4⋊C4×C31—41no—
161C4⋊C4×C31—82no—
162C4⋊C4×C31C2×D412yes—
163C4⋊C4×C31C2×D412yes—
164C22⋊C4×C31—22no—
165C22⋊C4×C31—22no—
166C22⋊C4×C31—22no—
167C22⋊C4×C31—22no—
168C22⋊C4×C31—22no—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—82no—
176C42×C31—22no—
177C42×C31—22no—
178C42×C31—21no—
179C42×C31—21no—
180C42×C31—22no—
181C42×C31C2×D412yes—
182C42×C31C2×D412yes—
183#18367C3111yesmaximal, characteristic, direct product factor, Sylow
184C23×C31—21no—
185C23×C31—22no—
186C23×C31—21no—
187C23×C31—21no—
188C23×C31—41no—
189C23×C31—41no—
190C23×C31C22×D411yescharacteristic
191Q8×C31—42no—
192Q8×C31—42no—
193Q8×C31—44no—
194Q8×C31—44no—
195Q8×C31—81no—
196D4×C31—42no—
197D4×C31—42no—
198D4×C31—42no—
199D4×C31—42no—
200D4×C31—82no—
201D4×C31—81no—
202D4×C31—82no—
203C2×C4×C31—21no—
204C2×C4×C31—21no—
205C2×C4×C31—21no—
206C2×C4×C31—21no—
207C2×C4×C31—21no—
208C2×C4×C31—22no—
209C2×C4×C31—21no—
210C2×C4×C31—22no—
211C2×C4×C31—22no—
212C2×C4×C31—41no—
213C2×C4×C31—41no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
216C2×C4×C31—44no—
217C2×C4×C31—41no—
218C2×C4×C31—41no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—82no—
226C2×C4×C31—82no—
227C2×C4×C31#12011yescharacteristic
228C2×C4×C31#12011yescharacteristic
229C2×C4×C31#12012yes—
230C2×C4×C31#13712yes—
231C2×C4×C31C22×D412yes—
232C8×C31—44no—
233#3493C2×C3111yescharacteristic
234#3218C2×C3112yescomplemented
235#2628C2×C3111yescharacteristic, complemented
236#1606C2×C3114yescomplemented
237#1605C2×C3112yescomplemented
238#1604C2×C3112yescomplemented
239#1512C2×C3112yescomplemented
240#1464C2×C3111yescharacteristic, complemented
241C22×C31—21no—
242C22×C31—21no—
243C22×C31—42no—
244C22×C31—41no—
245C22×C31—42no—
246C22×C31—42no—
247C22×C31—41no—
248C22×C31—42no—
249C22×C31—41no—
250C22×C31—81no—
251C22×C31C2×#12011yescharacteristic
252C22×C31C2×#13711yescharacteristic
253C22×C31#57511yescharacteristic
254C4×C31—22no—
255C4×C31—21no—
256C4×C31—21no—
257C4×C31—22no—
258C4×C31—22no—
259C4×C31—42no—
260C4×C31—41no—
261C4×C31—42no—
262C4×C31—41no—
263C4×C31—42no—
264C4×C31—82no—
265C4×C31—82no—
266#556—22no—
267#555—22no—
268#553—22no—
269#552—22no—
270#544C22×C3111yescharacteristic
271#538—21no—
272#538C22×C3112yes—
273#537C22×C3111yescharacteristic
274C2×C4⋊Q8C22×C3111yescharacteristic
275#529—21no—
276#529C22×C3112yes—
277#525—21no—
278#520—21no—
279C2×#117C22×C3111yescharacteristic
280#497—22no—
281#491—22no—
282#462—22no—
283#461—22no—
284#451C22×C3114yes—
285#448C22×C3114yes—
286#432C22×C3112yes—
287#400—21no—
288#399—21no—
289#395C22×C3112yes—
290#393C22×C3112yes—
291#389—21no—
292#382C22×C3111yescharacteristic
293#380—21no—
294#328C22×C3114yes—
295#327C22×C3114yes—
296#324C22×C3112yes—
297#323C22×C3112yes—
298C2×C31—21no—
299C2×C31—21no—
300C2×C31—82no—
301C2×C31—81no—
302C2×C31#300711yescharacteristic
303C2×C31#302211yescharacteristic
304C2×C31#302711yescharacteristic
305C2×#41—21no—
306C22×Q8C23×C3112yes—
307C23×C4D4×C3111yescharacteristic
308C2×Q16—24no—
309C2×SD16—24no—
310C4⋊Q8—22no—
311C4⋊Q8—22no—
312C4⋊Q8—22no—
313#126—22no—
314#126—22no—
315#126—22no—
316#126—22no—
317#125—21no—
318#125—22no—
319#125D4×C3112yes—
320#124—21no—
321#124—22no—
322#124—22no—
323#124D4×C3112yes—
324#123—22no—
325#123—22no—
326#123—22no—
327C22⋊Q8—21no—
328C22⋊Q8—21no—
329C22⋊Q8—21no—
330C22⋊Q8—22no—
331C22⋊Q8C23×C3112yes—
332#121—21no—
333#121—21no—
334#121—22no—
335#121—22no—
336#121—22no—
337#121—21no—
338#121C23×C3112yes—
339#120—22no—
340C4×Q8—22no—
341C4×Q8—21no—
342C4×Q8D4×C3112yes—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—22no—
346C4×D4—21no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4D4×C3112yes—
350#117—22no—
351#117—22no—
352#117—21no—
353#117—22no—
354#117—22no—
355#117—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4D4×C3111yescharacteristic
362C2×C4⋊C4C23×C3111yescharacteristic
363C2×C22⋊C4—22no—
364C2×C22⋊C4D4×C3111yescharacteristic
365C2×C22⋊C4C23×C3111yescharacteristic
366C2×C42D4×C3111yescharacteristic
367C2×C42C23×C3111yescharacteristic
368C4⋊C8—22no—
369#103—24no—
370#103—24no—
371C22⋊C8C23×C3114yes—
372#97—22no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95C23×C3112yes—
377C31#1836711yescharacteristic, direct product factor, Sylow
378C24C2×D4×C3111yescharacteristic
379#41—42no—
380#41—42no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—24no—
384C2×Q8—82no—
385C2×Q8C2×D4×C3112yes—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4C2×D4×C3112yes—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4C2×D4×C3111yescharacteristic
402C22×C4C2×D4×C3111yescharacteristic
403C22×C4C2×D4×C3111yescharacteristic
404C22×C4C2×D4×C3111yescharacteristic
405C22×C4C2×D4×C3111yescharacteristic
406C22×C4C24×C3111yescharacteristic
407Q16—84no—
408SD16—84no—
409C2×C8—24no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—42no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—42no—
426C4⋊C4—82no—
427C4⋊C4C2×D4×C3112yes—
428C4⋊C4C2×D4×C3112yes—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—82no—
441C42—22no—
442C42—22no—
443C42—21no—
444C42—21no—
445C42—22no—
446C42C2×D4×C3112yes—
447C42C2×D4×C3112yes—
448C23—21no—
449C23—22no—
450C23—21no—
451C23—21no—
452C23—41no—
453C23—41no—
454C23C22×D4×C3111yescharacteristic
455Q8—42no—
456Q8—42no—
457Q8—44no—
458Q8—44no—
459Q8—81no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—42no—
464D4—82no—
465D4—81no—
466D4—82no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4C31×#12011yescharacteristic
492C2×C4C31×#12011yescharacteristic
493C2×C4C31×#12012yes—
494C2×C4C31×#13712yes—
495C2×C4C22×D4×C3112yes—
496C8—44no—
497C22—21no—
498C22—21no—
499C22—42no—
500C22—41no—
501C22—42no—
502C22—42no—
503C22—41no—
504C22—42no—
505C22—41no—
506C22—81no—
507C22C2×C31×#12011yescharacteristic
508C22C2×C31×#13711yescharacteristic
509C22C31×#57511yescharacteristic
510C4—22no—
511C4—21no—
512C4—21no—
513C4—22no—
514C4—22no—
515C4—42no—
516C4—41no—
517C4—42no—
518C4—41no—
519C4—42no—
520C4—82no—
521C4—82no—
522C2—21no—
523C2—21no—
524C2—81no—
525C2—82no—
526C2C31×#300711yescharacteristic
527C2C31×#302211yescharacteristic
528C2C31×#302711yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3493C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3218C2semidirect
3. C31×#2628C2semidirect
4. C31×#1606C2semidirect
5. C31×#1605C2semidirect
6. C31×#1604C2semidirect
7. C31×#1512C2semidirect
8. C31×#1464C2semidirect
183. #18367C31direct (N × H)
234. #3218C2×C31semidirect
235. #2628C2×C31semidirect
236. #1606C2×C31semidirect
237. #1605C2×C31semidirect
238. #1604C2×C31semidirect
239. #1512C2×C31semidirect
240. #1464C2×C31semidirect
377. C31#18367direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34932
C31×#32182
C31×#26282
C31×#16062
C31×#16052
C31×#16042
C31×#15122
C31×#14642
#1836731