Finite Group Databasebeta

← C31×#18375 #21475134

Subgroups — C31×#18375 #21475134

Order 7936.

942 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3506C211yesmaximal, characteristic
2C31×#3236C211yesmaximal, characteristic, complemented
3C31×#3234C211yesmaximal, characteristic, complemented
4C31×#2633C211yesmaximal, characteristic, complemented
5C31×#1618C214yesmaximal, complemented
6C31×#1612C214yesmaximal, complemented
7C31×#1515C211yesmaximal, characteristic, complemented
8C31×#1514C211yesmaximal, characteristic, complemented
9C31×#1466C211yesmaximal, 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
20C2×C31×#125C2211yescharacteristic
21C31×#525—22no—
22C31×#520—21no—
23C2×C4×D4×C31C2211yescharacteristic
24C31×#501—21no—
25C31×#501—21no—
26C31×#491—21no—
27C31×#490—21no—
28C31×#484C2212yes—
29C31×#483C2212yes—
30C31×#481C2212yes—
31C31×#480C2212yes—
32C31×#479—22no—
33C31×#478—22no—
34C31×#432C2211yescharacteristic
35C31×#432C2211yescharacteristic
36C31×#399C2212yes—
37C31×#398—21no—
38C31×#397C2212yes—
39C31×#394—21no—
40C31×#385—22no—
41C31×#383C2211yescharacteristic
42C31×#328C2214yes—
43C31×#327C2214yes—
44C31×#323C2214yes—
45C2×#41×C31—22no—
46C2×#41×C31—21no—
47C22×D4×C31D411yescharacteristic
48C23×C4×C31D412yes—
49C31×C4⋊Q8D411yescharacteristic
50C31×C4⋊Q8D411yescharacteristic
51C31×C4⋊Q8D411yescharacteristic
52C31×#127—22no—
53C31×#127D411yescharacteristic
54C31×#126—24no—
55C31×#125—21no—
56C31×#125—21no—
57C31×#125—21no—
58C31×#125—21no—
59C31×#125—21no—
60C31×#124—22no—
61C31×#124—24no—
62C31×#124—21no—
63C31×#123—24no—
64C31×#123—24no—
65C31×C22⋊Q8—22no—
66C31×C22⋊Q8—21no—
67C31×C22⋊Q8C2312yes—
68C31×#121—22no—
69C31×#121—22no—
70C31×#121—22no—
71C31×#121—24no—
72C31×#121—21no—
73C31×#121C2312yes—
74C31×#120—24no—
75C4×Q8×C31—22no—
76C4×Q8×C31—21no—
77C4×Q8×C31D412yes—
78C4×D4×C31—22no—
79C4×D4×C31—24no—
80C4×D4×C31—24no—
81C4×D4×C31—21no—
82C4×D4×C31—21no—
83C4×D4×C31—21no—
84C4×D4×C31—21no—
85C4×D4×C31—21no—
86C4×D4×C31D412yes—
87C31×#117—22no—
88C31×#117—22no—
89C31×#117—21no—
90C2×C4⋊C4×C31—22no—
91C2×C4⋊C4×C31—21no—
92C2×C4⋊C4×C31—21no—
93C2×C4⋊C4×C31—22no—
94C2×C4⋊C4×C31D411yescharacteristic
95C2×C4⋊C4×C31C2311yescharacteristic
96C2×C4⋊C4×C31C2311yescharacteristic
97C2×C22⋊C4×C31—22no—
98C2×C22⋊C4×C31—22no—
99C2×C22⋊C4×C31C2312yes—
100C2×C42×C31C2311yescharacteristic
101C31×#107—22no—
102C31×#107—22no—
103C31×#106—22no—
104C31×#106—22no—
105C31×C4⋊C8—21no—
106C31×C4⋊C8—21no—
107C31×#103—22no—
108C31×#103—22no—
109C31×#102—22no—
110C31×#102—22no—
111C31×C22⋊C8C2312yes—
112C31×C22⋊C8C2312yes—
113C31×#97—21no—
114C31×#97—21no—
115C31×#95—22no—
116C31×#95—21no—
117C31×#95—21no—
118C31×#95C2312yes—
119C24×C31C2×D412yes—
120#41×C31—44no—
121#41×C31—84no—
122C2×Q8×C31—21no—
123C2×Q8×C31—21no—
124C2×Q8×C31C2×D412yes—
125C2×D4×C31—22no—
126C2×D4×C31—24no—
127C2×D4×C31—24no—
128C2×D4×C31—21no—
129C2×D4×C31—21no—
130C2×D4×C31—21no—
131C2×D4×C31—21no—
132C2×D4×C31—42no—
133C2×D4×C31C2×D412yes—
134C22×C4×C31—24no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—21no—
140C22×C4×C31—21no—
141C22×C4×C31—82no—
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D412yes—
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2411yescharacteristic
146C2×C8×C31—22no—
147C2×C8×C31—22no—
148C4⋊C4×C31—24no—
149C4⋊C4×C31—21no—
150C4⋊C4×C31—21no—
151C4⋊C4×C31—21no—
152C4⋊C4×C31—21no—
153C4⋊C4×C31—21no—
154C4⋊C4×C31—21no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—41no—
162C4⋊C4×C31—41no—
163C4⋊C4×C31—41no—
164C4⋊C4×C31—41no—
165C4⋊C4×C31—82no—
166C4⋊C4×C31C2×D412yes—
167C4⋊C4×C31C2×D412yes—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—24no—
170C22⋊C4×C31—24no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—44no—
177C22⋊C4×C31—82no—
178C42×C31—24no—
179C42×C31—22no—
180C42×C31C2×D411yescharacteristic
181C42×C31C2×D411yescharacteristic
182C42×C31C2×D411yescharacteristic
183C42×C31C2×D411yescharacteristic
184#18375C3111yesmaximal, characteristic, direct product factor, Sylow
185C23×C31—22no—
186C23×C31—22no—
187C23×C31—22no—
188C23×C31—22no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31C22×D411yescharacteristic
192Q8×C31—42no—
193Q8×C31—42no—
194Q8×C31—42no—
195D4×C31—42no—
196D4×C31—42no—
197D4×C31—42no—
198D4×C31—42no—
199D4×C31—42no—
200D4×C31—44no—
201D4×C31—84no—
202D4×C31—84no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—22no—
207C2×C4×C31—22no—
208C2×C4×C31—41no—
209C2×C4×C31—41no—
210C2×C4×C31—44no—
211C2×C4×C31—44no—
212C2×C4×C31—42no—
213C2×C4×C31—41no—
214C2×C4×C31—42no—
215C2×C4×C31—41no—
216C2×C4×C31—42no—
217C2×C4×C31—41no—
218C2×C4×C31—41no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—82no—
222C2×C4×C31—84no—
223C2×C4×C31#12011yescharacteristic
224C2×C4×C31#12011yescharacteristic
225C2×C4×C31#12011yescharacteristic
226C2×C4×C31#12011yescharacteristic
227C2×C4×C31C22×D411yescharacteristic
228C2×C4×C31C22×D411yescharacteristic
229C8×C31—42no—
230C8×C31—42no—
231#3506C2×C3111yescharacteristic
232#3236C2×C3111yescharacteristic, complemented
233#3234C2×C3111yescharacteristic, complemented
234#2633C2×C3111yescharacteristic, complemented
235#1618C2×C3114yescomplemented
236#1612C2×C3114yescomplemented
237#1515C2×C3111yescharacteristic, complemented
238#1514C2×C3111yescharacteristic, complemented
239#1466C2×C3111yescharacteristic, complemented
240C22×C31—21no—
241C22×C31—21no—
242C22×C31—44no—
243C22×C31—42no—
244C22×C31—42no—
245C22×C31—42no—
246C22×C31—42no—
247C22×C31—42no—
248C22×C31—42no—
249C22×C31—82no—
250C22×C31C2×#12011yescharacteristic
251C22×C31#57512yes—
252C4×C31—21no—
253C4×C31—21no—
254C4×C31—21no—
255C4×C31—21no—
256C4×C31—21no—
257C4×C31—21no—
258C4×C31—42no—
259C4×C31—42no—
260C4×C31—42no—
261C4×C31—42no—
262C4×C31—82no—
263C4×C31—81no—
264C4×C31—81no—
265#559—22no—
266#556—21no—
267#555—21no—
268#553—22no—
269#550—21no—
270#547—21no—
271#538C22×C3114yes—
272#535—21no—
273#533C22×C3111yescharacteristic
274#532C22×C3111yescharacteristic
275C2×#125C22×C3111yescharacteristic
276#525—22no—
277#520—21no—
278C2×C4×D4C22×C3111yescharacteristic
279#501—21no—
280#501—21no—
281#491—21no—
282#490—21no—
283#484C22×C3112yes—
284#483C22×C3112yes—
285#481C22×C3112yes—
286#480C22×C3112yes—
287#479—22no—
288#478—22no—
289#432C22×C3111yescharacteristic
290#432C22×C3111yescharacteristic
291#399C22×C3112yes—
292#398—21no—
293#397C22×C3112yes—
294#394—21no—
295#385—22no—
296#383C22×C3111yescharacteristic
297#328C22×C3114yes—
298#327C22×C3114yes—
299#323C22×C3114yes—
300C2×C31—22no—
301C2×C31—82no—
302C2×C31—82no—
303C2×C31#300511yescharacteristic
304C2×C31#301911yescharacteristic
305C2×C31#302811yescharacteristic
306C2×#41—22no—
307C2×#41—21no—
308C22×D4D4×C3111yescharacteristic
309C23×C4D4×C3112yes—
310C4⋊Q8D4×C3111yescharacteristic
311C4⋊Q8D4×C3111yescharacteristic
312C4⋊Q8D4×C3111yescharacteristic
313#127—22no—
314#127D4×C3111yescharacteristic
315#126—24no—
316#125—21no—
317#125—21no—
318#125—21no—
319#125—21no—
320#125—21no—
321#124—22no—
322#124—24no—
323#124—21no—
324#123—24no—
325#123—24no—
326C22⋊Q8—22no—
327C22⋊Q8—21no—
328C22⋊Q8C23×C3112yes—
329#121—22no—
330#121—22no—
331#121—22no—
332#121—24no—
333#121—21no—
334#121C23×C3112yes—
335#120—24no—
336C4×Q8—22no—
337C4×Q8—21no—
338C4×Q8D4×C3112yes—
339C4×D4—22no—
340C4×D4—24no—
341C4×D4—24no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4D4×C3112yes—
348#117—22no—
349#117—22no—
350#117—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4D4×C3111yescharacteristic
356C2×C4⋊C4C23×C3111yescharacteristic
357C2×C4⋊C4C23×C3111yescharacteristic
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4C23×C3112yes—
361C2×C42C23×C3111yescharacteristic
362#107—22no—
363#107—22no—
364#106—22no—
365#106—22no—
366C4⋊C8—21no—
367C4⋊C8—21no—
368#103—22no—
369#103—22no—
370#102—22no—
371#102—22no—
372C22⋊C8C23×C3112yes—
373C22⋊C8C23×C3112yes—
374#97—21no—
375#97—21no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95C23×C3112yes—
380C31#1837511yescharacteristic, direct product factor, Sylow
381C24C2×D4×C3112yes—
382#41—44no—
383#41—84no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8C2×D4×C3112yes—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—42no—
395C2×D4C2×D4×C3112yes—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—82no—
404C22×C4C2×D4×C3111yescharacteristic
405C22×C4C2×D4×C3112yes—
406C22×C4C2×D4×C3111yescharacteristic
407C22×C4C24×C3111yescharacteristic
408C2×C8—22no—
409C2×C8—22no—
410C4⋊C4—24no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—82no—
428C4⋊C4C2×D4×C3112yes—
429C4⋊C4C2×D4×C3112yes—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—44no—
439C22⋊C4—82no—
440C42—24no—
441C42—22no—
442C42C2×D4×C3111yescharacteristic
443C42C2×D4×C3111yescharacteristic
444C42C2×D4×C3111yescharacteristic
445C42C2×D4×C3111yescharacteristic
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—42no—
451C23—42no—
452C23C22×D4×C3111yescharacteristic
453Q8—42no—
454Q8—42no—
455Q8—42no—
456D4—42no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—44no—
462D4—84no—
463D4—84no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—82no—
483C2×C4—84no—
484C2×C4C31×#12011yescharacteristic
485C2×C4C31×#12011yescharacteristic
486C2×C4C31×#12011yescharacteristic
487C2×C4C31×#12011yescharacteristic
488C2×C4C22×D4×C3111yescharacteristic
489C2×C4C22×D4×C3111yescharacteristic
490C8—42no—
491C8—42no—
492C22—21no—
493C22—21no—
494C22—44no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—82no—
502C22C2×C31×#12011yescharacteristic
503C22C31×#57512yes—
504C4—21no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—42no—
514C4—82no—
515C4—81no—
516C4—81no—
517C2—22no—
518C2—82no—
519C2—82no—
520C2C31×#300511yescharacteristic
521C2C31×#301911yescharacteristic
522C2C31×#302811yescharacteristic
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×#3236C2semidirect
3. C31×#3234C2semidirect
4. C31×#2633C2semidirect
5. C31×#1618C2semidirect
6. C31×#1612C2semidirect
7. C31×#1515C2semidirect
8. C31×#1514C2semidirect
9. C31×#1466C2semidirect
184. #18375C31direct (N × H)
232. #3236C2×C31semidirect
233. #3234C2×C31semidirect
234. #2633C2×C31semidirect
235. #1618C2×C31semidirect
236. #1612C2×C31semidirect
237. #1515C2×C31semidirect
238. #1514C2×C31semidirect
239. #1466C2×C31semidirect
380. C31#18375direct (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×#32362
C31×#32342
C31×#26332
C31×#16182
C31×#16122
C31×#15152
C31×#15142
C31×#14662
#1837531