Finite Group Databasebeta

← C31×#23643 #21480402

Subgroups — C31×#23643 #21480402

Order 7936.

1014 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#428C211yesmaximal, characteristic, complemented
2C2×C31×#408C211yesmaximal, characteristic
3C31×#2311C211yesmaximal, characteristic
4C31×#2311C211yesmaximal, characteristic
5C31×#2077C212yesmaximal, complemented
6C31×#2077C212yesmaximal, complemented
7C31×#2076C212yesmaximal, complemented
8C31×#2075C212yesmaximal, complemented
9C31×#1736C211yesmaximal, characteristic, complemented
10C31×#1735C211yesmaximal, characteristic, complemented
11C2×C31×#340C211yesmaximal, characteristic, complemented
12C22×#41×C31C2211yescharacteristic
13C22×M16×C31C2211yescharacteristic
14C23×C8×C31C2211yescharacteristic
15C2×C31×#117C2211yescharacteristic
16C2×C31×#117C2211yescharacteristic
17C22×C4⋊C4×C31C2211yescharacteristic
18C22×C4⋊C4×C31C2211yescharacteristic
19C31×#428—24no—
20C2×C31×#107C2212yes—
21C2×C31×#106C2212yes—
22C31×#408—24no—
23C2×C31×C22⋊C8C2212yes—
24C2×C31×C22⋊C8C2212yes—
25C31×#391—21no—
26C31×#391—21no—
27C31×#390—21no—
28C31×#390—21no—
29C31×#386—22no—
30C31×#386—22no—
31C31×#380C2212yes—
32C31×#380C2212yes—
33C31×#380C2212yes—
34C31×#380C2212yes—
35C31×#340C2211yescharacteristic
36C31×#340C2212yes—
37C31×#340C2212yes—
38C31×#340C2211yescharacteristic
39C31×#340C2212yes—
40C31×#340C2212yes—
41C31×#340C2211yescharacteristic
42C31×#340C2211yescharacteristic
43C2×#41×C31—22no—
44C2×#41×C31—22no—
45C2×#41×C31—22no—
46C2×#41×C31—22no—
47C2×#41×C31—22no—
48C2×#41×C31—22no—
49C22×Q8×C31D411yescharacteristic
50C22×D4×C31D412yes—
51C22×D4×C31D411yescharacteristic
52C23×C4×C31—21no—
53C23×C4×C31—21no—
54C23×C4×C31C2312yes—
55C23×C4×C31C2311yescharacteristic
56C2×M16×C31—22no—
57C2×M16×C31—22no—
58C2×M16×C31D414yes—
59C22×C8×C31—22no—
60C22×C8×C31—22no—
61C22×C8×C31—22no—
62C22×C8×C31C2311yescharacteristic
63C22×C8×C31C2311yescharacteristic
64C22×C8×C31C2312yes—
65C31×#117—22no—
66C31×#117—22no—
67C31×#117—22no—
68C31×#117—22no—
69C2×C4⋊C4×C31—22no—
70C2×C4⋊C4×C31—22no—
71C2×C4⋊C4×C31—22no—
72C2×C4⋊C4×C31—22no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31C2311yescharacteristic
76C2×C4⋊C4×C31C2312yes—
77C2×C4⋊C4×C31C2311yescharacteristic
78C2×C4⋊C4×C31C2312yes—
79C2×C4⋊C4×C31C2311yescharacteristic
80C2×C4⋊C4×C31C2311yescharacteristic
81C2×C22⋊C4×C31—21no—
82C2×C22⋊C4×C31—21no—
83C2×C22⋊C4×C31—22no—
84C2×C22⋊C4×C31—22no—
85C2×C22⋊C4×C31—22no—
86C2×C22⋊C4×C31—22no—
87C2×C42×C31—21no—
88C2×C42×C31—21no—
89C31×#107—24no—
90C31×#106—24no—
91C31×C22⋊C8—24no—
92C31×C22⋊C8—24no—
93C31×#95—22no—
94C31×#95—22no—
95C31×#95—22no—
96C31×#95—22no—
97C24×C31C2×D412yes—
98C24×C31C2×D411yescharacteristic
99#41×C31—44no—
100#41×C31—44no—
101#41×C31—44no—
102#41×C31—44no—
103C2×Q8×C31—22no—
104C2×Q8×C31—22no—
105C2×Q8×C31—22no—
106C2×D4×C31—24no—
107C2×D4×C31—24no—
108C2×D4×C31—24no—
109C2×D4×C31—22no—
110C2×D4×C31—22no—
111C2×D4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—22no—
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—21no—
121C22×C4×C31—21no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—42no—
125C22×C4×C31—42no—
126C22×C4×C31—81no—
127C22×C4×C31—81no—
128C22×C4×C31—81no—
129C22×C4×C31—81no—
130C22×C4×C31C2×D412yes—
131C22×C4×C31C2×D411yescharacteristic
132C22×C4×C31C2×D412yes—
133C22×C4×C31#4112yes—
134C22×C4×C31#4112yes—
135C22×C4×C31#4112yes—
136C22×C4×C31#4112yes—
137C22×C4×C31#4112yes—
138C22×C4×C31C2411yescharacteristic
139M16×C31—28no—
140C2×C8×C31—22no—
141C2×C8×C31—22no—
142C2×C8×C31—21no—
143C2×C8×C31—21no—
144C2×C8×C31—21no—
145C2×C8×C31—21no—
146C2×C8×C31—21no—
147C2×C8×C31—21no—
148C2×C8×C31—44no—
149C2×C8×C31C2×D414yes—
150C4⋊C4×C31—21no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—21no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
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—42no—
163C4⋊C4×C31—42no—
164C22⋊C4×C31—42no—
165C22⋊C4×C31—42no—
166C22⋊C4×C31—82no—
167C22⋊C4×C31—82no—
168C22⋊C4×C31—82no—
169C22⋊C4×C31—82no—
170C42×C31—42no—
171C42×C31—42no—
172#23643C3111yesmaximal, characteristic, direct product factor, Sylow
173C23×C31—21no—
174C23×C31—21no—
175C23×C31—21no—
176C23×C31—21no—
177C23×C31—22no—
178C23×C31—22no—
179C23×C31—22no—
180C23×C31—22no—
181C23×C31—22no—
182C23×C31—22no—
183C23×C31—22no—
184C23×C31#12112yes—
185C23×C31#12312yes—
186C23×C31#12312yes—
187C23×C31C22×D411yescharacteristic
188Q8×C31—44no—
189D4×C31—48no—
190D4×C31—44no—
191C2×C4×C31—22no—
192C2×C4×C31—22no—
193C2×C4×C31—22no—
194C2×C4×C31—42no—
195C2×C4×C31—42no—
196C2×C4×C31—42no—
197C2×C4×C31—42no—
198C2×C4×C31—41no—
199C2×C4×C31—41no—
200C2×C4×C31—42no—
201C2×C4×C31—41no—
202C2×C4×C31—41no—
203C2×C4×C31—42no—
204C2×C4×C31—41no—
205C2×C4×C31—41no—
206C2×C4×C31—42no—
207C2×C4×C31—42no—
208C2×C4×C31—41no—
209C2×C4×C31—41no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—41no—
213C2×C4×C31—41no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—84no—
217C2×C4×C31—84no—
218C2×C4×C31#12112yes—
219C2×C4×C31#12314yes—
220C2×C4×C31#12312yes—
221C2×C4×C31#12312yes—
222C2×C4×C31#12414yes—
223C2×C4×C31#12614yes—
224C2×C4×C31#12614yes—
225C2×C4×C31C22×D411yescharacteristic
226C2×C4×C31C2×#4111yescharacteristic
227C2×C4×C31C2×#4111yescharacteristic
228C2×C4×C31C2×#4111yescharacteristic
229C2×C4×C31C2×#4111yescharacteristic
230C2×C4×C31C2×#4111yescharacteristic
231C8×C31—24no—
232C8×C31—41no—
233C8×C31—41no—
234C2×#428C2×C3111yescharacteristic, complemented
235C2×#408C2×C3111yescharacteristic
236#2311C2×C3111yescharacteristic
237#2311C2×C3111yescharacteristic
238#2077C2×C3112yescomplemented
239#2077C2×C3112yescomplemented
240#2076C2×C3112yescomplemented
241#2075C2×C3112yescomplemented
242#1736C2×C3111yescharacteristic, complemented
243#1735C2×C3111yescharacteristic, complemented
244C2×#340C2×C3111yescharacteristic, complemented
245C22×C31—22no—
246C22×C31—22no—
247C22×C31—22no—
248C22×C31—22no—
249C22×C31—22no—
250C22×C31—22no—
251C22×C31—42no—
252C22×C31—42no—
253C22×C31—42no—
254C22×C31—42no—
255C22×C31—42no—
256C22×C31—42no—
257C22×C31—42no—
258C22×C31#39714yes—
259C22×C31C2×#12111yescharacteristic
260C22×C31C2×#12311yescharacteristic
261C22×C31C2×#12311yescharacteristic
262C22×C31#57611yescharacteristic
263C22×C31#57711yescharacteristic
264C22×C31#57711yescharacteristic
265C22×C31#57811yescharacteristic
266C4×C31—42no—
267C4×C31—82no—
268C4×C31—81no—
269C4×C31—81no—
270C4×C31—81no—
271C4×C31—82no—
272C4×C31—81no—
273C4×C31#39714yes—
274C4×C31C2×#12311yescharacteristic
275C4×C31C2×#12411yescharacteristic
276C4×C31C2×#12611yescharacteristic
277C4×C31C2×#12611yescharacteristic
278C22×#41C22×C3111yescharacteristic
279C22×M16C22×C3111yescharacteristic
280C23×C8C22×C3111yescharacteristic
281C2×#117C22×C3111yescharacteristic
282C2×#117C22×C3111yescharacteristic
283C22×C4⋊C4C22×C3111yescharacteristic
284C22×C4⋊C4C22×C3111yescharacteristic
285#428—24no—
286C2×#107C22×C3112yes—
287C2×#106C22×C3112yes—
288#408—24no—
289C2×C22⋊C8C22×C3112yes—
290C2×C22⋊C8C22×C3112yes—
291#391—21no—
292#391—21no—
293#390—21no—
294#390—21no—
295#386—22no—
296#386—22no—
297#380C22×C3112yes—
298#380C22×C3112yes—
299#380C22×C3112yes—
300#380C22×C3112yes—
301#340C22×C3111yescharacteristic
302#340C22×C3112yes—
303#340C22×C3112yes—
304#340C22×C3111yescharacteristic
305#340C22×C3112yes—
306#340C22×C3112yes—
307#340C22×C3111yescharacteristic
308#340C22×C3111yescharacteristic
309C2×C31—24no—
310C2×C31—82no—
311C2×C31C2×#39711yescharacteristic
312C2×C31#306211yescharacteristic
313C2×C31#306311yescharacteristic
314C2×C31#309611yescharacteristic
315C2×C31#309711yescharacteristic
316C2×C31#309811yescharacteristic
317C2×C31#309911yescharacteristic
318C2×#41—22no—
319C2×#41—22no—
320C2×#41—22no—
321C2×#41—22no—
322C2×#41—22no—
323C2×#41—22no—
324C22×Q8D4×C3111yescharacteristic
325C22×D4D4×C3112yes—
326C22×D4D4×C3111yescharacteristic
327C23×C4—21no—
328C23×C4—21no—
329C23×C4C23×C3112yes—
330C23×C4C23×C3111yescharacteristic
331C2×M16—22no—
332C2×M16—22no—
333C2×M16D4×C3114yes—
334C22×C8—22no—
335C22×C8—22no—
336C22×C8—22no—
337C22×C8C23×C3111yescharacteristic
338C22×C8C23×C3111yescharacteristic
339C22×C8C23×C3112yes—
340#117—22no—
341#117—22no—
342#117—22no—
343#117—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4C23×C3112yes—
351C2×C4⋊C4C23×C3112yes—
352C2×C4⋊C4C23×C3111yescharacteristic
353C2×C4⋊C4C23×C3111yescharacteristic
354C2×C4⋊C4C23×C3111yescharacteristic
355C2×C4⋊C4C23×C3111yescharacteristic
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C42—21no—
363C2×C42—21no—
364#107—24no—
365#106—24no—
366C22⋊C8—24no—
367C22⋊C8—24no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372C31#2364311yescharacteristic, direct product factor, Sylow
373C24C2×D4×C3112yes—
374C24C2×D4×C3111yescharacteristic
375#41—44no—
376#41—44no—
377#41—44no—
378#41—44no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4C2×D4×C3112yes—
407C22×C4C2×D4×C3111yescharacteristic
408C22×C4C2×D4×C3112yes—
409C22×C4#41×C3112yes—
410C22×C4#41×C3112yes—
411C22×C4#41×C3112yes—
412C22×C4#41×C3112yes—
413C22×C4#41×C3112yes—
414C22×C4C24×C3111yescharacteristic
415M16—28no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—44no—
425C2×C8C2×D4×C3114yes—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—82no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C42—42no—
447C42—42no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23C31×#12112yes—
460C23C31×#12312yes—
461C23C31×#12312yes—
462C23C22×D4×C3111yescharacteristic
463Q8—44no—
464D4—48no—
465D4—44no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—84no—
492C2×C4—84no—
493C2×C4C31×#12112yes—
494C2×C4C31×#12312yes—
495C2×C4C31×#12312yes—
496C2×C4C31×#12314yes—
497C2×C4C31×#12414yes—
498C2×C4C31×#12614yes—
499C2×C4C31×#12614yes—
500C2×C4C22×D4×C3111yescharacteristic
501C2×C4C2×#41×C3111yescharacteristic
502C2×C4C2×#41×C3111yescharacteristic
503C2×C4C2×#41×C3111yescharacteristic
504C2×C4C2×#41×C3111yescharacteristic
505C2×C4C2×#41×C3111yescharacteristic
506C8—24no—
507C8—41no—
508C8—41no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22C31×#39714yes—
523C22C2×C31×#12111yescharacteristic
524C22C2×C31×#12311yescharacteristic
525C22C2×C31×#12311yescharacteristic
526C22C31×#57611yescharacteristic
527C22C31×#57711yescharacteristic
528C22C31×#57711yescharacteristic
529C22C31×#57811yescharacteristic
530C4—42no—
531C4—82no—
532C4—81no—
533C4—81no—
534C4—82no—
535C4—81no—
536C4—81no—
537C4C31×#39714yes—
538C4C2×C31×#12311yescharacteristic
539C4C2×C31×#12411yescharacteristic
540C4C2×C31×#12611yescharacteristic
541C4C2×C31×#12611yescharacteristic
542C2—24no—
543C2—82no—
544C2C2×C31×#39711yescharacteristic
545C2C31×#306211yescharacteristic
546C2C31×#306311yescharacteristic
547C2C31×#309611yescharacteristic
548C2C31×#309711yescharacteristic
549C2C31×#309811yescharacteristic
550C2C31×#309911yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×#41×C31C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#428C2semidirect
5. C31×#2077C2semidirect
6. C31×#2077C2semidirect
7. C31×#2076C2semidirect
8. C31×#2075C2semidirect
9. C31×#1736C2semidirect
10. C31×#1735C2semidirect
11. C2×C31×#340C2semidirect
172. #23643C31direct (N × H)
234. C2×#428C2×C31semidirect
238. #2077C2×C31semidirect
239. #2077C2×C31semidirect
240. #2076C2×C31semidirect
241. #2075C2×C31semidirect
242. #1736C2×C31semidirect
243. #1735C2×C31semidirect
244. C2×#340C2×C31semidirect
372. C31#23643direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4282
C2×C31×#4082
C31×#23112
C31×#20772
C31×#20762
C31×#20752
C31×#17362
C31×#17352
C2×C31×#3402
#2364331