Finite Group Databasebeta

← C31×#13085 #21469844

Subgroups — C31×#13085 #21469844

Order 7936.

724 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#410C211yesmaximal, characteristic
2C31×#2406C211yesmaximal, characteristic
3C2×C31×#352C212yesmaximal, complemented
4C31×#1785C211yesmaximal, characteristic
5C31×#1390C212yesmaximal, complemented
6C22×#41×C31C2211yescharacteristic
7C2×C31×#123C411yescharacteristic, complemented
8C2×C31×#123C2211yescharacteristic
9C2×C31×#117—21no—
10C22×C4⋊C4×C31C411yescharacteristic, complemented
11C2×C31×#101—21no—
12C2×C31×#100C2212yes—
13C31×#410—24no—
14C2×C31×#99—21no—
15C2×C31×#99C2212yes—
16C31×#407—21no—
17C31×#395—21no—
18C31×#394—21no—
19C31×#386C411yescharacteristic, complemented
20C31×#386C2211yescharacteristic
21C31×#385—21no—
22C31×#385—21no—
23C31×#381C411yescharacteristic, complemented
24C31×#352—24no—
25C2×#41×C31—22no—
26C2×#41×C31—22no—
27C2×#41×C31—41no—
28C2×#41×C31—41no—
29C2×#41×C31—42no—
30C22×Q8×C31D411yescharacteristic
31C22×D4×C31C2×C411yescharacteristic
32C22×D4×C31D411yescharacteristic
33C22×D4×C31C2311yescharacteristic
34C23×C4×C31C2×C411yescharacteristic
35C23×C4×C31C2×C411yescharacteristic
36C23×C4×C31C2×C411yescharacteristic
37C23×C4×C31D411yescharacteristic
38C23×C4×C31D411yescharacteristic
39C2×M16×C31—22no—
40C31×#123—22no—
41C31×#123—22no—
42C31×#123—42no—
43C31×#117—24no—
44C31×#117—42no—
45C2×C4⋊C4×C31—21no—
46C2×C4⋊C4×C31—21no—
47C2×C4⋊C4×C31—21no—
48C2×C4⋊C4×C31—21no—
49C2×C4⋊C4×C31—41no—
50C2×C4⋊C4×C31—41no—
51C2×C4⋊C4×C31—41no—
52C2×C22⋊C4×C31—22no—
53C2×C22⋊C4×C31—21no—
54C2×C22⋊C4×C31—21no—
55C2×C22⋊C4×C31—21no—
56C2×C22⋊C4×C31—21no—
57C2×C22⋊C4×C31C2×C411yescharacteristic
58C2×C22⋊C4×C31C2×C411yescharacteristic
59C2×C42×C31—41no—
60C31×#101—42no—
61C31×#100—22no—
62C31×#100—22no—
63C31×#99—24no—
64C31×#99—22no—
65C31×#99D414yes—
66C31×C22⋊C8—41no—
67C31×C22⋊C8—41no—
68C31×#95—21no—
69C31×#95—21no—
70C31×#95—21no—
71C31×#95—21no—
72C24×C31C22⋊C411yescharacteristic
73C24×C31C22×C411yescharacteristic
74C24×C31C2×D411yescharacteristic
75#41×C31—44no—
76#41×C31—44no—
77#41×C31—44no—
78#41×C31—44no—
79C2×Q8×C31—22no—
80C2×Q8×C31—22no—
81C2×Q8×C31—41no—
82C2×D4×C31—22no—
83C2×D4×C31—21no—
84C2×D4×C31—21no—
85C2×D4×C31—22no—
86C2×D4×C31—41no—
87C2×D4×C31—41no—
88C2×D4×C31—41no—
89C2×D4×C31—41no—
90C2×D4×C31—41no—
91C2×D4×C31C2×D412yes—
92C2×D4×C31#4112yes—
93C22×C4×C31—21no—
94C22×C4×C31—21no—
95C22×C4×C31—21no—
96C22×C4×C31—21no—
97C22×C4×C31—22no—
98C22×C4×C31—21no—
99C22×C4×C31—22no—
100C22×C4×C31—21no—
101C22×C4×C31—21no—
102C22×C4×C31—21no—
103C22×C4×C31—21no—
104C22×C4×C31—22no—
105C22×C4×C31—21no—
106C22×C4×C31—21no—
107C22×C4×C31—21no—
108C22×C4×C31—22no—
109C22×C4×C31—41no—
110C22×C4×C31—41no—
111C22×C4×C31—42no—
112C22×C4×C31—41no—
113C22×C4×C31—42no—
114C22×C4×C31—41no—
115C22×C4×C31—41no—
116C22×C4×C31C22⋊C411yescharacteristic
117C22×C4×C31C22⋊C411yescharacteristic
118C22×C4×C31C22⋊C411yescharacteristic
119C22×C4×C31C2×D411yescharacteristic
120M16×C31—42no—
121M16×C31—42no—
122C2×C8×C31—42no—
123C4⋊C4×C31—42no—
124C4⋊C4×C31—44no—
125C4⋊C4×C31—41no—
126C4⋊C4×C31—41no—
127C4⋊C4×C31—42no—
128C4⋊C4×C31—42no—
129C22⋊C4×C31—22no—
130C22⋊C4×C31—24no—
131C22⋊C4×C31—42no—
132C22⋊C4×C31—42no—
133C22⋊C4×C31—42no—
134C22⋊C4×C31—42no—
135C22⋊C4×C31—41no—
136C22⋊C4×C31—41no—
137C22⋊C4×C31—41no—
138C42×C31—44no—
139#13085C3111yesmaximal, characteristic, direct product factor, Sylow
140C23×C31—21no—
141C23×C31—21no—
142C23×C31—21no—
143C23×C31—21no—
144C23×C31—21no—
145C23×C31—21no—
146C23×C31—21no—
147C23×C31—41no—
148C23×C31—41no—
149C23×C31—41no—
150C23×C31—41no—
151C23×C31—41no—
152C23×C31—41no—
153C23×C31C2×C22⋊C411yescharacteristic
154C23×C31C4×D412yes—
155C23×C31#12011yescharacteristic
156C23×C31#12311yescharacteristic
157Q8×C31—44no—
158D4×C31—42no—
159D4×C31—42no—
160D4×C31—44no—
161D4×C31—42no—
162D4×C31—42no—
163C2×C4×C31—22no—
164C2×C4×C31—22no—
165C2×C4×C31—22no—
166C2×C4×C31—22no—
167C2×C4×C31—21no—
168C2×C4×C31—21no—
169C2×C4×C31—21no—
170C2×C4×C31—21no—
171C2×C4×C31—21no—
172C2×C4×C31—21no—
173C2×C4×C31—22no—
174C2×C4×C31—21no—
175C2×C4×C31—21no—
176C2×C4×C31—21no—
177C2×C4×C31—42no—
178C2×C4×C31—42no—
179C2×C4×C31—42no—
180C2×C4×C31—42no—
181C2×C4×C31—41no—
182C2×C4×C31—42no—
183C2×C4×C31—41no—
184C2×C4×C31—42no—
185C2×C4×C31—44no—
186C2×C4×C31—41no—
187C2×C4×C31—41no—
188C2×C4×C31—41no—
189C2×C4×C31—41no—
190C2×C4×C31—41no—
191C2×C4×C31—41no—
192C2×C4×C31—41no—
193C2×C4×C31—41no—
194C2×C4×C31—41no—
195C2×C4×C31—82no—
196C2×C4×C31—82no—
197C2×C4×C31—82no—
198C2×C4×C31#9912yes—
199C2×C4×C31#9911yescharacteristic
200C2×C4×C31#9911yescharacteristic
201C2×C4×C31#12112yes—
202C8×C31—82no—
203C2×#410C2×C3111yescharacteristic
204#2406C2×C3111yescharacteristic
205C2×#352C2×C3112yescomplemented
206#1785C2×C3111yescharacteristic
207#1390C2×C3112yescomplemented
208C22×C31—21no—
209C22×C31—21no—
210C22×C31—21no—
211C22×C31—21no—
212C22×C31—22no—
213C22×C31—42no—
214C22×C31—41no—
215C22×C31—41no—
216C22×C31—42no—
217C22×C31—42no—
218C22×C31—41no—
219C22×C31—41no—
220C22×C31—41no—
221C22×C31—41no—
222C22×C31—41no—
223C22×C31—41no—
224C22×C31—41no—
225C22×C31—81no—
226C22×C31—81no—
227C22×C31#38611yescharacteristic
228C22×C31C2×#9911yescharacteristic
229C22×C31C2×#9911yescharacteristic
230C4×C31—22no—
231C4×C31—21no—
232C4×C31—21no—
233C4×C31—22no—
234C4×C31—42no—
235C4×C31—41no—
236C4×C31—81no—
237C4×C31—81no—
238C4×C31—84no—
239C4×C31—81no—
240C22×#41C22×C3111yescharacteristic
241C2×#123C4×C3111yescharacteristic, complemented
242C2×#123C22×C3111yescharacteristic
243C2×#117—21no—
244C22×C4⋊C4C4×C3111yescharacteristic, complemented
245C2×#101—21no—
246C2×#100C22×C3112yes—
247#410—24no—
248C2×#99—21no—
249C2×#99C22×C3112yes—
250#407—21no—
251#395—21no—
252#394—21no—
253#386C4×C3111yescharacteristic, complemented
254#386C22×C3111yescharacteristic
255#385—21no—
256#385—21no—
257#381C4×C3111yescharacteristic, complemented
258#352—24no—
259C2×C31—21no—
260C2×C31—21no—
261C2×C31—41no—
262C2×C31—42no—
263C2×C31—41no—
264C2×C31—81no—
265C2×C31#189811yescharacteristic
266C2×C31#212011yescharacteristic
267C2×C31#212311yescharacteristic
268C2×#41—22no—
269C2×#41—22no—
270C2×#41—41no—
271C2×#41—41no—
272C2×#41—42no—
273C22×Q8D4×C3111yescharacteristic
274C22×D4C2×C4×C3111yescharacteristic
275C22×D4D4×C3111yescharacteristic
276C22×D4C23×C3111yescharacteristic
277C23×C4C2×C4×C3111yescharacteristic
278C23×C4C2×C4×C3111yescharacteristic
279C23×C4C2×C4×C3111yescharacteristic
280C23×C4D4×C3111yescharacteristic
281C23×C4D4×C3111yescharacteristic
282C2×M16—22no—
283#123—22no—
284#123—22no—
285#123—42no—
286#117—24no—
287#117—42no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—41no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—41no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4C2×C4×C3111yescharacteristic
301C2×C22⋊C4C2×C4×C3111yescharacteristic
302C2×C42—41no—
303#101—42no—
304#100—22no—
305#100—22no—
306#99—24no—
307#99—22no—
308#99D4×C3114yes—
309C22⋊C8—41no—
310C22⋊C8—41no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315C31#1308511yescharacteristic, direct product factor, Sylow
316C24C22⋊C4×C3111yescharacteristic
317C24C22×C4×C3111yescharacteristic
318C24C2×D4×C3111yescharacteristic
319#41—44no—
320#41—44no—
321#41—44no—
322#41—44no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—22no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4C2×D4×C3112yes—
336C2×D4#41×C3112yes—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4C22⋊C4×C3111yescharacteristic
361C22×C4C22⋊C4×C3111yescharacteristic
362C22×C4C22⋊C4×C3111yescharacteristic
363C22×C4C2×D4×C3111yescharacteristic
364M16—42no—
365M16—42no—
366C2×C8—42no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C22⋊C4—22no—
374C22⋊C4—24no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C42—44no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23C2×C22⋊C4×C3111yescharacteristic
397C23C4×D4×C3112yes—
398C23C31×#12011yescharacteristic
399C23C31×#12311yescharacteristic
400Q8—44no—
401D4—42no—
402D4—42no—
403D4—44no—
404D4—42no—
405D4—42no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4C31×#9911yescharacteristic
442C2×C4C31×#9911yescharacteristic
443C2×C4C31×#9912yes—
444C2×C4C31×#12112yes—
445C8—82no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—22no—
451C22—42no—
452C22—41no—
453C22—41no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—81no—
464C22—81no—
465C22C31×#38611yescharacteristic
466C22C2×C31×#9911yescharacteristic
467C22C2×C31×#9911yescharacteristic
468C4—22no—
469C4—21no—
470C4—21no—
471C4—22no—
472C4—42no—
473C4—41no—
474C4—81no—
475C4—81no—
476C4—84no—
477C4—81no—
478C2—21no—
479C2—21no—
480C2—41no—
481C2—42no—
482C2—41no—
483C2—81no—
484C2C31×#189811yescharacteristic
485C2C31×#212011yescharacteristic
486C2C31×#212311yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC24C22×C4×C31
Frattini, cosocleC22×D4C23×C31
prime radicalC22×#41×C31C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C22×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C31×#352C2semidirect
5. C31×#1390C2semidirect
7. C2×C31×#123C4semidirect
10. C22×C4⋊C4×C31C4semidirect
19. C31×#386C4semidirect
23. C31×#381C4semidirect
139. #13085C31direct (N × H)
205. C2×#352C2×C31semidirect
207. #1390C2×C31semidirect
241. C2×#123C4×C31semidirect
244. C22×C4⋊C4C4×C31semidirect
253. #386C4×C31semidirect
257. #381C4×C31semidirect
315. C31#13085direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4102
C31×#24062
C2×C31×#3522
C31×#17852
C31×#13902
#1308531