Finite Group Databasebeta

← C31×#18223 #21474982

Subgroups — C31×#18223 #21474982

Order 7936.

994 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3506C211yesmaximal, characteristic
2C31×#3104C211yesmaximal, characteristic, complemented
3C31×#3103C211yesmaximal, characteristic, complemented
4C31×#2580C211yesmaximal, characteristic, complemented
5C31×#1618C214yesmaximal, complemented
6C31×#1606C214yesmaximal, complemented
7C31×#1518C211yesmaximal, characteristic, complemented
8C31×#1517C211yesmaximal, characteristic, complemented
9C31×#1460C211yesmaximal, characteristic, complemented
10C31×#559—22no—
11C31×#556—21no—
12C31×#555—21no—
13C31×#553—22no—
14C31×#550—21no—
15C31×#547—21no—
16C31×#538C2214yes—
17C31×#535—21no—
18C31×#533C2211yescharacteristic
19C31×#532C2211yescharacteristic
20C31×#525—22no—
21C31×#520—21no—
22C2×C4×Q8×C31C2211yescharacteristic
23C2×C4×D4×C31C2211yescharacteristic
24C31×#484C2212yes—
25C31×#481C2212yes—
26C31×#477—21no—
27C31×#475—21no—
28C31×#462—21no—
29C31×#460—21no—
30C31×#451C2212yes—
31C31×#448C2212yes—
32C4×Q16×C31—22no—
33C4×SD16×C31—22no—
34C31×#431C2211yescharacteristic
35C31×#431C2211yescharacteristic
36C31×#399C2212yes—
37C31×#393C2212yes—
38C31×#390—21no—
39C31×#389—21no—
40C31×#385—22no—
41C31×#382C2211yescharacteristic
42C31×#328C2214yes—
43C31×#327C2214yes—
44C31×#323C2214yes—
45C2×#41×C31—22no—
46C2×#41×C31—21no—
47C22×Q8×C31C2311yescharacteristic
48C22×D4×C31D411yescharacteristic
49C23×C4×C31D412yes—
50C2×Q16×C31—22no—
51C2×SD16×C31—22no—
52C31×C4⋊Q8D411yescharacteristic
53C31×C4⋊Q8D411yescharacteristic
54C31×C4⋊Q8D411yescharacteristic
55C31×#127—22no—
56C31×#127D411yescharacteristic
57C31×#126—24no—
58C31×#125—21no—
59C31×#124—22no—
60C31×#124—24no—
61C31×#124—21no—
62C31×#123—24no—
63C31×#123—24no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—21no—
66C31×C22⋊Q8C2312yes—
67C31×#121—22no—
68C31×#121—22no—
69C31×#121—22no—
70C31×#121—24no—
71C31×#121—21no—
72C31×#121C2312yes—
73C31×#120—24no—
74C4×Q8×C31—22no—
75C4×Q8×C31—21no—
76C4×Q8×C31—21no—
77C4×Q8×C31—21no—
78C4×Q8×C31—21no—
79C4×Q8×C31—21no—
80C4×Q8×C31D412yes—
81C4×D4×C31—22no—
82C4×D4×C31—24no—
83C4×D4×C31—24no—
84C4×D4×C31—21no—
85C4×D4×C31—21no—
86C4×D4×C31—21no—
87C4×D4×C31—21no—
88C4×D4×C31—21no—
89C4×D4×C31D412yes—
90C31×#117—22no—
91C31×#117—22no—
92C31×#117—21no—
93C2×C4⋊C4×C31—22no—
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×C31—22no—
101C2×C22⋊C4×C31C2312yes—
102C2×C42×C31—21no—
103C2×C42×C31C2311yescharacteristic
104C31×#107—22no—
105C31×#106—22no—
106C31×C4⋊C8—21no—
107C31×C4⋊C8—21no—
108C31×#103—22no—
109C31×#103—22no—
110C31×#103—22no—
111C31×#102—22no—
112C31×C22⋊C8C2312yes—
113C31×C22⋊C8C2312yes—
114C4×C8×C31—21no—
115C4×C8×C31—21no—
116C31×#95—21no—
117C31×#95—22no—
118C31×#95C2312yes—
119C24×C31C2×D412yes—
120#41×C31—42no—
121#41×C31—42no—
122#41×C31—84no—
123C2×Q8×C31—21no—
124C2×Q8×C31—21no—
125C2×Q8×C31—21no—
126C2×Q8×C31—21no—
127C2×Q8×C31—42no—
128C2×Q8×C31C2×D412yes—
129C2×D4×C31—22no—
130C2×D4×C31—24no—
131C2×D4×C31—24no—
132C2×D4×C31—21no—
133C2×D4×C31—21no—
134C2×D4×C31—21no—
135C2×D4×C31—21no—
136C2×D4×C31—22no—
137C2×D4×C31—22no—
138C2×D4×C31C2×D412yes—
139C22×C4×C31—21no—
140C22×C4×C31—24no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—21no—
146C22×C4×C31—82no—
147C22×C4×C31C2×D412yes—
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2411yescharacteristic
151Q16×C31—44no—
152SD16×C31—44no—
153C2×C8×C31—22no—
154C2×C8×C31—22no—
155C4⋊C4×C31—24no—
156C4⋊C4×C31—21no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—41no—
167C4⋊C4×C31—41no—
168C4⋊C4×C31—42no—
169C4⋊C4×C31—82no—
170C4⋊C4×C31C2×D412yes—
171C4⋊C4×C31C2×D412yes—
172C22⋊C4×C31—24no—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—24no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—22no—
180C22⋊C4×C31—44no—
181C22⋊C4×C31—82no—
182C42×C31—24no—
183C42×C31—22no—
184C42×C31—41no—
185C42×C31—41no—
186C42×C31C2×D411yescharacteristic
187C42×C31C2×D411yescharacteristic
188C42×C31C2×D411yescharacteristic
189C42×C31C2×D411yescharacteristic
190#18223C3111yesmaximal, characteristic, direct product factor, Sylow
191C23×C31—22no—
192C23×C31—22no—
193C23×C31—22no—
194C23×C31—22no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31C22×D411yescharacteristic
198Q8×C31—22no—
199Q8×C31—22no—
200Q8×C31—42no—
201Q8×C31—41no—
202Q8×C31—41no—
203Q8×C31—42no—
204Q8×C31—42no—
205D4×C31—42no—
206D4×C31—42no—
207D4×C31—42no—
208D4×C31—42no—
209D4×C31—42no—
210D4×C31—42no—
211D4×C31—42no—
212D4×C31—84no—
213D4×C31—84no—
214C2×C4×C31—22no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—41no—
220C2×C4×C31—41no—
221C2×C4×C31—44no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—44no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—41no—
228C2×C4×C31—42no—
229C2×C4×C31—41no—
230C2×C4×C31—41no—
231C2×C4×C31—42no—
232C2×C4×C31—82no—
233C2×C4×C31—84no—
234C2×C4×C31#12011yescharacteristic
235C2×C4×C31#12011yescharacteristic
236C2×C4×C31#12011yescharacteristic
237C2×C4×C31#12011yescharacteristic
238C2×C4×C31#13512yes—
239C2×C4×C31#13712yes—
240C2×C4×C31C22×D411yescharacteristic
241C2×C4×C31C22×D411yescharacteristic
242C8×C31—42no—
243C8×C31—42no—
244#3506C2×C3111yescharacteristic
245#3104C2×C3111yescharacteristic, complemented
246#3103C2×C3111yescharacteristic, complemented
247#2580C2×C3111yescharacteristic, complemented
248#1618C2×C3114yescomplemented
249#1606C2×C3114yescomplemented
250#1518C2×C3111yescharacteristic, complemented
251#1517C2×C3111yescharacteristic, complemented
252#1460C2×C3111yescharacteristic, complemented
253C22×C31—21no—
254C22×C31—21no—
255C22×C31—44no—
256C22×C31—42no—
257C22×C31—42no—
258C22×C31—42no—
259C22×C31—42no—
260C22×C31—42no—
261C22×C31—42no—
262C22×C31—82no—
263C22×C31C2×#12011yescharacteristic
264C22×C31C2×#13511yescharacteristic
265C22×C31C2×#13711yescharacteristic
266C4×C31—21no—
267C4×C31—21no—
268C4×C31—21no—
269C4×C31—21no—
270C4×C31—42no—
271C4×C31—42no—
272C4×C31—42no—
273C4×C31—42no—
274C4×C31—42no—
275C4×C31—82no—
276C4×C31—81no—
277C4×C31#45212yes—
278C4×C31#45212yes—
279#559—22no—
280#556—21no—
281#555—21no—
282#553—22no—
283#550—21no—
284#547—21no—
285#538C22×C3114yes—
286#535—21no—
287#533C22×C3111yescharacteristic
288#532C22×C3111yescharacteristic
289#525—22no—
290#520—21no—
291C2×C4×Q8C22×C3111yescharacteristic
292C2×C4×D4C22×C3111yescharacteristic
293#484C22×C3112yes—
294#481C22×C3112yes—
295#477—21no—
296#475—21no—
297#462—21no—
298#460—21no—
299#451C22×C3112yes—
300#448C22×C3112yes—
301C4×Q16—22no—
302C4×SD16—22no—
303#431C22×C3111yescharacteristic
304#431C22×C3111yescharacteristic
305#399C22×C3112yes—
306#393C22×C3112yes—
307#390—21no—
308#389—21no—
309#385—22no—
310#382C22×C3111yescharacteristic
311#328C22×C3114yes—
312#327C22×C3114yes—
313#323C22×C3114yes—
314C2×C31—21no—
315C2×C31—21no—
316C2×C31—82no—
317C2×C31—82no—
318C2×C31C2×#45211yescharacteristic
319C2×C31#301911yescharacteristic
320C2×C31#302511yescharacteristic
321C2×#41—22no—
322C2×#41—21no—
323C22×Q8C23×C3111yescharacteristic
324C22×D4D4×C3111yescharacteristic
325C23×C4D4×C3112yes—
326C2×Q16—22no—
327C2×SD16—22no—
328C4⋊Q8D4×C3111yescharacteristic
329C4⋊Q8D4×C3111yescharacteristic
330C4⋊Q8D4×C3111yescharacteristic
331#127—22no—
332#127D4×C3111yescharacteristic
333#126—24no—
334#125—21no—
335#124—22no—
336#124—24no—
337#124—21no—
338#123—24no—
339#123—24no—
340C22⋊Q8—22no—
341C22⋊Q8—21no—
342C22⋊Q8C23×C3112yes—
343#121—22no—
344#121—22no—
345#121—22no—
346#121—24no—
347#121—21no—
348#121C23×C3112yes—
349#120—24no—
350C4×Q8—22no—
351C4×Q8—21no—
352C4×Q8—21no—
353C4×Q8—21no—
354C4×Q8—21no—
355C4×Q8—21no—
356C4×Q8D4×C3112yes—
357C4×D4—22no—
358C4×D4—24no—
359C4×D4—24no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4—21no—
364C4×D4—21no—
365C4×D4D4×C3112yes—
366#117—22no—
367#117—22no—
368#117—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4D4×C3111yescharacteristic
374C2×C4⋊C4C23×C3111yescharacteristic
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4C23×C3112yes—
378C2×C42—21no—
379C2×C42C23×C3111yescharacteristic
380#107—22no—
381#106—22no—
382C4⋊C8—21no—
383C4⋊C8—21no—
384#103—22no—
385#103—22no—
386#103—22no—
387#102—22no—
388C22⋊C8C23×C3112yes—
389C22⋊C8C23×C3112yes—
390C4×C8—21no—
391C4×C8—21no—
392#95—21no—
393#95—22no—
394#95C23×C3112yes—
395C31#1822311yescharacteristic, direct product factor, Sylow
396C24C2×D4×C3112yes—
397#41—42no—
398#41—42no—
399#41—84no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—42no—
405C2×Q8C2×D4×C3112yes—
406C2×D4—22no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4C2×D4×C3112yes—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—82no—
424C22×C4C2×D4×C3112yes—
425C22×C4C2×D4×C3111yescharacteristic
426C22×C4C2×D4×C3111yescharacteristic
427C22×C4C24×C3111yescharacteristic
428Q16—44no—
429SD16—44no—
430C2×C8—22no—
431C2×C8—22no—
432C4⋊C4—24no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—42no—
446C4⋊C4—82no—
447C4⋊C4C2×D4×C3112yes—
448C4⋊C4C2×D4×C3112yes—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—44no—
458C22⋊C4—82no—
459C42—24no—
460C42—22no—
461C42—41no—
462C42—41no—
463C42C2×D4×C3111yescharacteristic
464C42C2×D4×C3111yescharacteristic
465C42C2×D4×C3111yescharacteristic
466C42C2×D4×C3111yescharacteristic
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—42no—
472C23—42no—
473C23C22×D4×C3111yescharacteristic
474Q8—22no—
475Q8—22no—
476Q8—42no—
477Q8—41no—
478Q8—41no—
479Q8—42no—
480Q8—42no—
481D4—42no—
482D4—42no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—84no—
489D4—84no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—82no—
509C2×C4—84no—
510C2×C4C31×#12011yescharacteristic
511C2×C4C31×#12011yescharacteristic
512C2×C4C31×#12011yescharacteristic
513C2×C4C31×#12011yescharacteristic
514C2×C4C31×#13512yes—
515C2×C4C31×#13712yes—
516C2×C4C22×D4×C3111yescharacteristic
517C2×C4C22×D4×C3111yescharacteristic
518C8—42no—
519C8—42no—
520C22—21no—
521C22—21no—
522C22—44no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—82no—
530C22C2×C31×#12011yescharacteristic
531C22C2×C31×#13511yescharacteristic
532C22C2×C31×#13711yescharacteristic
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—82no—
543C4—81no—
544C4C31×#45212yes—
545C4C31×#45212yes—
546C2—21no—
547C2—21no—
548C2—82no—
549C2—82no—
550C2C2×C31×#45211yescharacteristic
551C2C31×#301911yescharacteristic
552C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3506C2
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×#3104C2semidirect
3. C31×#3103C2semidirect
4. C31×#2580C2semidirect
5. C31×#1618C2semidirect
6. C31×#1606C2semidirect
7. C31×#1518C2semidirect
8. C31×#1517C2semidirect
9. C31×#1460C2semidirect
190. #18223C31direct (N × H)
245. #3104C2×C31semidirect
246. #3103C2×C31semidirect
247. #2580C2×C31semidirect
248. #1618C2×C31semidirect
249. #1606C2×C31semidirect
250. #1518C2×C31semidirect
251. #1517C2×C31semidirect
252. #1460C2×C31semidirect
395. C31#18223direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35062
C31×#31042
C31×#31032
C31×#25802
C31×#16182
C31×#16062
C31×#15182
C31×#15172
C31×#14602
#1822331