Finite Group Databasebeta

← C31×#19034 #21475793

Subgroups — C31×#19034 #21475793

Order 7936.

836 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3563C211yesmaximal, characteristic
2C31×#3218C211yesmaximal, characteristic
3C31×#3165C211yesmaximal, characteristic, complemented
4C2×C31×#501C211yesmaximal, characteristic, complemented
5C31×#1663C214yesmaximal, complemented
6C31×#1660C214yesmaximal, complemented
7C31×#1531C211yesmaximal, characteristic, complemented
8C31×#1528C211yesmaximal, characteristic, complemented
9C31×#1524C211yesmaximal, characteristic
10C31×#564C2212yes—
11C31×#563C2212yes—
12C31×#557—21no—
13C31×#554—21no—
14C31×#552—21no—
15C31×#552—22no—
16C31×#552—21no—
17C31×#544—21no—
18C31×#544—21no—
19C31×#541—21no—
20C31×#541—22no—
21C31×#539—21no—
22C31×#533C2211yescharacteristic
23C2×C31×C4⋊Q8C2211yescharacteristic
24C31×#529C2211yescharacteristic
25C2×C31×#125C2211yescharacteristic
26C31×#501—22no—
27C31×#501C2214yes—
28C31×#501C2214yes—
29C31×#500—21no—
30C31×#498—21no—
31C31×#497C2212yes—
32C31×#491C2212yes—
33C31×#462—21no—
34C31×#461—21no—
35C31×#451—21no—
36C31×#448—21no—
37C31×#432C2211yescharacteristic
38C31×#428—22no—
39C2×C31×#107—21no—
40C2×C31×#106—21no—
41C4×M16×C31C2211yescharacteristic
42C2×C31×#97C2211yescharacteristic
43C31×#330C2214yes—
44C31×#330C2214yes—
45C31×#330C2212yes—
46C31×#329C2212yes—
47C22×Q8×C31—21no—
48C2×Q16×C31—22no—
49C2×SD16×C31—22no—
50C2×M16×C31—21no—
51C22×C8×C31—21no—
52C31×C4⋊Q8—22no—
53C31×C4⋊Q8—21no—
54C31×C4⋊Q8—22no—
55C31×C4⋊Q8—21no—
56C31×C4⋊Q8—21no—
57C31×C4⋊Q8—21no—
58C31×C4⋊Q8D411yescharacteristic
59C31×C4⋊Q8D411yescharacteristic
60C31×C4⋊Q8C2312yes—
61C31×#126—22no—
62C31×#126—22no—
63C31×#126—22no—
64C31×#126—22no—
65C31×#126—21no—
66C31×#125—22no—
67C31×#125—22no—
68C31×#125—21no—
69C31×#125—22no—
70C31×#125—22no—
71C31×#125—22no—
72C31×#125—22no—
73C31×#125—22no—
74C31×#125—22no—
75C31×#125—21no—
76C31×#125—21no—
77C31×#125D412yes—
78C31×#125C2311yescharacteristic
79C31×#125C2312yes—
80C31×#125C2312yes—
81C31×#124C2311yescharacteristic
82C31×#123—22no—
83C31×#123—22no—
84C31×#123—21no—
85C31×C22⋊Q8—22no—
86C31×C22⋊Q8—21no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8—21no—
90C31×C22⋊Q8—21no—
91C31×C22⋊Q8—22no—
92C31×C22⋊Q8—22no—
93C31×C22⋊Q8D411yescharacteristic
94C31×#121D411yescharacteristic
95C4×Q8×C31—21no—
96C4×Q8×C31—22no—
97C4×Q8×C31—21no—
98C4×Q8×C31—21no—
99C4×Q8×C31—21no—
100C4×Q8×C31D411yescharacteristic
101C4×D4×C31—22no—
102C4×D4×C31—21no—
103C4×D4×C31D411yescharacteristic
104C31×#117—22no—
105C31×#117—21no—
106C31×#117—22no—
107C31×#117—21no—
108C31×#117D412yes—
109C2×C4⋊C4×C31—21no—
110C2×C4⋊C4×C31—21no—
111C2×C4⋊C4×C31—21no—
112C2×C4⋊C4×C31—21no—
113C2×C4⋊C4×C31—21no—
114C2×C4⋊C4×C31D411yescharacteristic
115C2×C4⋊C4×C31D411yescharacteristic
116C2×C42×C31C2311yescharacteristic
117C31×#107—24no—
118C31×#107—24no—
119C31×#106—24no—
120C31×#106—24no—
121C31×C4⋊C8—21no—
122C31×#103—22no—
123C31×#103—22no—
124C31×C22⋊C8—21no—
125C31×#97—21no—
126C31×#97C2312yes—
127C31×#97C2312yes—
128C31×#97C2312yes—
129C4×C8×C31—21no—
130C2×Q8×C31—22no—
131C2×Q8×C31—21no—
132C2×Q8×C31—21no—
133C2×Q8×C31—22no—
134C2×Q8×C31—41no—
135C2×Q8×C31—82no—
136C2×Q8×C31C2×D411yescharacteristic
137C2×D4×C31—21no—
138C2×D4×C31C2×D411yescharacteristic
139C22×C4×C31—21no—
140C22×C4×C31—22no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—41no—
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147Q16×C31—82no—
148SD16×C31—82no—
149M16×C31—24no—
150C2×C8×C31—24no—
151C2×C8×C31—22no—
152C2×C8×C31—22no—
153C2×C8×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—41no—
179C4⋊C4×C31—41no—
180C4⋊C4×C31C2×D411yescharacteristic
181C4⋊C4×C31C2×D411yescharacteristic
182C4⋊C4×C31C2×D412yes—
183C4⋊C4×C31C2×D412yes—
184C4⋊C4×C31C2×D412yes—
185C4⋊C4×C31C2×D412yes—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—21no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—21no—
192C22⋊C4×C31—21no—
193C22⋊C4×C31—21no—
194C22⋊C4×C31—21no—
195C42×C31—22no—
196C42×C31—22no—
197C42×C31—22no—
198C42×C31—22no—
199C42×C31—21no—
200C42×C31C2×D411yescharacteristic
201C42×C31C2×D411yescharacteristic
202C42×C31C2×D411yescharacteristic
203C42×C31C2411yescharacteristic
204#19034C3111yesmaximal, characteristic, direct product factor, Sylow
205C23×C31—21no—
206C23×C31#12011yescharacteristic
207Q8×C31—41no—
208Q8×C31—42no—
209Q8×C31—42no—
210Q8×C31—42no—
211Q8×C31—81no—
212Q8×C31—82no—
213Q8×C31—82no—
214D4×C31—41no—
215D4×C31—42no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—21no—
220C2×C4×C31—21no—
221C2×C4×C31—21no—
222C2×C4×C31—22no—
223C2×C4×C31—22no—
224C2×C4×C31—22no—
225C2×C4×C31—41no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—81no—
230C2×C4×C31—82no—
231C2×C4×C31—82no—
232C2×C4×C31#12011yescharacteristic
233C2×C4×C31#12011yescharacteristic
234C2×C4×C31#12011yescharacteristic
235C2×C4×C31C22×D411yescharacteristic
236C2×C4×C31C22×D411yescharacteristic
237C2×C4×C31C22×D411yescharacteristic
238C8×C31—24no—
239C8×C31—24no—
240C8×C31—42no—
241#3563C2×C3111yescharacteristic
242#3218C2×C3111yescharacteristic
243#3165C2×C3111yescharacteristic, complemented
244C2×#501C2×C3111yescharacteristic, complemented
245#1663C2×C3114yescomplemented
246#1660C2×C3114yescomplemented
247#1531C2×C3111yescharacteristic, complemented
248#1528C2×C3111yescharacteristic, complemented
249#1524C2×C3111yescharacteristic
250C22×C31—21no—
251C22×C31—21no—
252C22×C31—41no—
253C22×C31—41no—
254C22×C31—41no—
255C22×C31#45612yes—
256C22×C31C2×#12011yescharacteristic
257C4×C31—21no—
258C4×C31—21no—
259C4×C31—21no—
260C4×C31—21no—
261C4×C31—21no—
262C4×C31—42no—
263C4×C31—41no—
264C4×C31—41no—
265C4×C31—42no—
266C4×C31—42no—
267C4×C31—42no—
268C4×C31—41no—
269C4×C31—82no—
270C4×C31#45612yes—
271#564C22×C3112yes—
272#563C22×C3112yes—
273#557—21no—
274#554—21no—
275#552—21no—
276#552—22no—
277#552—21no—
278#544—21no—
279#544—21no—
280#541—21no—
281#541—22no—
282#539—21no—
283#533C22×C3111yescharacteristic
284C2×C4⋊Q8C22×C3111yescharacteristic
285#529C22×C3111yescharacteristic
286C2×#125C22×C3111yescharacteristic
287#501—22no—
288#501C22×C3114yes—
289#501C22×C3114yes—
290#500—21no—
291#498—21no—
292#497C22×C3112yes—
293#491C22×C3112yes—
294#462—21no—
295#461—21no—
296#451—21no—
297#448—21no—
298#432C22×C3111yescharacteristic
299#428—22no—
300C2×#107—21no—
301C2×#106—21no—
302C4×M16C22×C3111yescharacteristic
303C2×#97C22×C3111yescharacteristic
304#330C22×C3114yes—
305#330C22×C3114yes—
306#330C22×C3112yes—
307#329C22×C3112yes—
308C2×C31—22no—
309C2×C31—81no—
310C2×C31C2×#45611yescharacteristic
311C2×C31C2×#45611yescharacteristic
312C2×C31#302111yescharacteristic
313C22×Q8—21no—
314C2×Q16—22no—
315C2×SD16—22no—
316C2×M16—21no—
317C22×C8—21no—
318C4⋊Q8—22no—
319C4⋊Q8—21no—
320C4⋊Q8—22no—
321C4⋊Q8—21no—
322C4⋊Q8—21no—
323C4⋊Q8—21no—
324C4⋊Q8D4×C3111yescharacteristic
325C4⋊Q8D4×C3111yescharacteristic
326C4⋊Q8C23×C3112yes—
327#126—22no—
328#126—22no—
329#126—22no—
330#126—22no—
331#126—21no—
332#125—22no—
333#125—22no—
334#125—21no—
335#125—22no—
336#125—22no—
337#125—22no—
338#125—22no—
339#125—22no—
340#125—22no—
341#125—21no—
342#125—21no—
343#125D4×C3112yes—
344#125C23×C3112yes—
345#125C23×C3111yescharacteristic
346#125C23×C3112yes—
347#124C23×C3111yescharacteristic
348#123—22no—
349#123—22no—
350#123—21no—
351C22⋊Q8—22no—
352C22⋊Q8—21no—
353C22⋊Q8—21no—
354C22⋊Q8—22no—
355C22⋊Q8—21no—
356C22⋊Q8—21no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8D4×C3111yescharacteristic
360#121D4×C3111yescharacteristic
361C4×Q8—21no—
362C4×Q8—22no—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8D4×C3111yescharacteristic
367C4×D4—22no—
368C4×D4—21no—
369C4×D4D4×C3111yescharacteristic
370#117—22no—
371#117—21no—
372#117—22no—
373#117—21no—
374#117D4×C3112yes—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4D4×C3111yescharacteristic
381C2×C4⋊C4D4×C3111yescharacteristic
382C2×C42C23×C3111yescharacteristic
383#107—24no—
384#107—24no—
385#106—24no—
386#106—24no—
387C4⋊C8—21no—
388#103—22no—
389#103—22no—
390C22⋊C8—21no—
391#97—21no—
392#97C23×C3112yes—
393#97C23×C3112yes—
394#97C23×C3112yes—
395C4×C8—21no—
396C31#1903411yescharacteristic, direct product factor, Sylow
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—41no—
402C2×Q8—82no—
403C2×Q8C2×D4×C3111yescharacteristic
404C2×D4—21no—
405C2×D4C2×D4×C3111yescharacteristic
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—41no—
411C22×C4C2×D4×C3111yescharacteristic
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4C2×D4×C3111yescharacteristic
414Q16—82no—
415SD16—82no—
416M16—24no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4C2×D4×C3111yescharacteristic
448C4⋊C4C2×D4×C3111yescharacteristic
449C4⋊C4C2×D4×C3112yes—
450C4⋊C4C2×D4×C3112yes—
451C4⋊C4C2×D4×C3112yes—
452C4⋊C4C2×D4×C3112yes—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—21no—
457C22⋊C4—22no—
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—21no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—21no—
467C42C2×D4×C3111yescharacteristic
468C42C2×D4×C3111yescharacteristic
469C42C2×D4×C3111yescharacteristic
470C42C24×C3111yescharacteristic
471C23—21no—
472C23C31×#12011yescharacteristic
473Q8—41no—
474Q8—42no—
475Q8—42no—
476Q8—42no—
477Q8—81no—
478Q8—82no—
479Q8—82no—
480D4—41no—
481D4—42no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—81no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4C31×#12011yescharacteristic
499C2×C4C31×#12011yescharacteristic
500C2×C4C31×#12011yescharacteristic
501C2×C4C22×D4×C3111yescharacteristic
502C2×C4C22×D4×C3111yescharacteristic
503C2×C4C22×D4×C3111yescharacteristic
504C8—24no—
505C8—24no—
506C8—42no—
507C22—21no—
508C22—21no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22C31×#45612yes—
513C22C2×C31×#12011yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—42no—
520C4—41no—
521C4—41no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—41no—
526C4—82no—
527C4C31×#45612yes—
528C2—22no—
529C2—81no—
530C2C2×C31×#45611yescharacteristic
531C2C2×C31×#45611yescharacteristic
532C2C31×#302111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#3165C2semidirect
4. C2×C31×#501C2semidirect
5. C31×#1663C2semidirect
6. C31×#1660C2semidirect
7. C31×#1531C2semidirect
8. C31×#1528C2semidirect
204. #19034C31direct (N × H)
243. #3165C2×C31semidirect
244. C2×#501C2×C31semidirect
245. #1663C2×C31semidirect
246. #1660C2×C31semidirect
247. #1531C2×C31semidirect
248. #1528C2×C31semidirect
396. C31#19034direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35632
C31×#32182
C31×#31652
C2×C31×#5012
C31×#16632
C31×#16602
C31×#15312
C31×#15282
C31×#15242
#1903431