Finite Group Databasebeta

← C31×#10374 #21467133

Subgroups — C31×#10374 #21467133

Order 7936.

508 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2549C211yesmaximal, characteristic, complemented
2C31×#1964C211yesmaximal, characteristic
3C31×#1519C211yesmaximal, characteristic
4C2×C31×#327C211yesmaximal, characteristic
5C31×#1294C211yesmaximal, characteristic, complemented
6C31×#1289C211yesmaximal, characteristic, complemented
7C31×#1282C211yesmaximal, characteristic, complemented
8C2×C31×C22⋊Q8—21no—
9C2×C31×#121C2211yescharacteristic
10C2×C31×#99—21no—
11C2×C31×#99—21no—
12C2×C31×C22⋊C8C2211yescharacteristic
13C2×C31×C22⋊C8C2211yescharacteristic
14C31×#398—21no—
15C31×#395—21no—
16C31×#391—21no—
17C31×#389—21no—
18C31×#385—21no—
19C31×#380C2211yescharacteristic
20C31×#380C2211yescharacteristic
21C4×C22⋊C4×C31C2211yescharacteristic
22C2×C31×#95C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#327—21no—
26C31×#327—21no—
27C22×Q8×C31—21no—
28C22×D4×C31—21no—
29C22×D4×C31Q811yescharacteristic, complemented
30C23×C4×C31—21no—
31C23×C4×C31C2311yescharacteristic
32C22×C8×C31—21no—
33C22×C8×C31—21no—
34C31×C22⋊Q8—42no—
35C31×C22⋊Q8—42no—
36C31×#121—21no—
37C31×#121—21no—
38C31×#121—21no—
39C31×#121—21no—
40C2×C4⋊C4×C31—21no—
41C2×C4⋊C4×C31—21no—
42C2×C4⋊C4×C31—21no—
43C2×C4⋊C4×C31—21no—
44C2×C4⋊C4×C31—21no—
45C2×C4⋊C4×C31—21no—
46C2×C4⋊C4×C31—21no—
47C2×C4⋊C4×C31Q811yescharacteristic, complemented
48C2×C22⋊C4×C31—21no—
49C2×C22⋊C4×C31—21no—
50C2×C22⋊C4×C31—21no—
51C2×C22⋊C4×C31D411yescharacteristic
52C2×C22⋊C4×C31D411yescharacteristic
53C2×C42×C31—21no—
54C31×#99—42no—
55C31×#99—42no—
56C31×C22⋊C8—21no—
57C31×C22⋊C8—21no—
58C31×C22⋊C8—21no—
59C31×C22⋊C8—21no—
60C31×#95—21no—
61C31×#95—21no—
62C31×#95—21no—
63C31×#95—21no—
64C31×#95—21no—
65C31×#95D411yescharacteristic
66C31×#95D411yescharacteristic
67C31×#95D411yescharacteristic
68C31×#95D411yescharacteristic
69C24×C31—21no—
70C24×C31C2×D411yescharacteristic
71C2×Q8×C31—42no—
72C2×Q8×C31—42no—
73C2×Q8×C31—81no—
74C2×D4×C31—21no—
75C2×D4×C31—21no—
76C2×D4×C31—41no—
77C2×D4×C31—41no—
78C2×D4×C31—41no—
79C2×D4×C31—41no—
80C2×D4×C31—41no—
81C2×D4×C31—41no—
82C2×D4×C31—81no—
83C22×C4×C31—21no—
84C22×C4×C31—21no—
85C22×C4×C31—21no—
86C22×C4×C31—21no—
87C22×C4×C31—21no—
88C22×C4×C31—21no—
89C22×C4×C31—21no—
90C22×C4×C31—42no—
91C22×C4×C31—41no—
92C22×C4×C31—41no—
93C22×C4×C31—41no—
94C22×C4×C31—81no—
95C22×C4×C31—81no—
96C22×C4×C31C2×D411yescharacteristic
97C22×C4×C31C2×D411yescharacteristic
98C22×C4×C31C2×Q811yescharacteristic
99C22×C4×C31#4111yescharacteristic
100C22×C4×C31#4111yescharacteristic
101C22×C4×C31#4111yescharacteristic
102C2×C8×C31—41no—
103C2×C8×C31—41no—
104C2×C8×C31—81no—
105C2×C8×C31—81no—
106C4⋊C4×C31—21no—
107C4⋊C4×C31—21no—
108C4⋊C4×C31—42no—
109C4⋊C4×C31—42no—
110C4⋊C4×C31—42no—
111C4⋊C4×C31—42no—
112C4⋊C4×C31—81no—
113C4⋊C4×C31—81no—
114C4⋊C4×C31—81no—
115C22⋊C4×C31—21no—
116C22⋊C4×C31—21no—
117C22⋊C4×C31—41no—
118C22⋊C4×C31—41no—
119C22⋊C4×C31—41no—
120C22⋊C4×C31—41no—
121C22⋊C4×C31—41no—
122C22⋊C4×C31—41no—
123C22⋊C4×C31D812yes—
124C22⋊C4×C31SD1612yes—
125C42×C31—81no—
126#10374C3111yesmaximal, characteristic, direct product factor, Sylow
127C23×C31—41no—
128C23×C31—41no—
129C23×C31—41no—
130C23×C31—41no—
131C23×C31—41no—
132C23×C31—81no—
133C23×C31—81no—
134C23×C31#12011yescharacteristic
135C23×C31C22⋊Q811yescharacteristic
136C23×C31#12411yescharacteristic
137C23×C31C2×D811yescharacteristic
138C23×C31C2×SD1611yescharacteristic
139C23×C31AGL(1,ℤ/8)11yescharacteristic
140C23×C31#13711yescharacteristic
141Q8×C31—84no—
142D4×C31—81no—
143D4×C31—81no—
144D4×C31—81no—
145D4×C31—81no—
146D4×C31—81no—
147D4×C31—81no—
148C2×C4×C31—22no—
149C2×C4×C31—22no—
150C2×C4×C31—22no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—41no—
169C2×C4×C31—41no—
170C2×C4×C31—41no—
171C2×C4×C31—41no—
172C2×C4×C31—41no—
173C2×C4×C31—41no—
174C2×C4×C31—41no—
175C2×C4×C31—82no—
176C2×C4×C31—82no—
177C2×C4×C31—81no—
178C2×C4×C31—81no—
179C2×C4×C31—81no—
180C2×C4×C31C22⋊Q811yescharacteristic
181C2×C4×C31C22⋊Q811yescharacteristic
182C2×C4×C31#12411yescharacteristic
183C2×C4×C31#12411yescharacteristic
184C8×C31—81no—
185C8×C31—81no—
186#2549C2×C3111yescharacteristic, complemented
187#1964C2×C3111yescharacteristic
188#1519C2×C3111yescharacteristic
189C2×#327C2×C3111yescharacteristic
190#1294C2×C3111yescharacteristic, complemented
191#1289C2×C3111yescharacteristic, complemented
192#1282C2×C3111yescharacteristic, complemented
193C22×C31—21no—
194C22×C31—21no—
195C22×C31—21no—
196C22×C31—21no—
197C22×C31—21no—
198C22×C31—21no—
199C22×C31—41no—
200C22×C31—41no—
201C22×C31—41no—
202C22×C31—41no—
203C22×C31—81no—
204C22×C31—81no—
205C22×C31—81no—
206C22×C31—161no—
207C22×C31—161no—
208C22×C31#39311yescharacteristic
209C22×C31#44711yescharacteristic
210C22×C31#44811yescharacteristic
211C22×C31#47411yescharacteristic
212C22×C31#47611yescharacteristic
213C22×C31#48611yescharacteristic
214C22×C31#48911yescharacteristic
215C4×C31—21no—
216C4×C31—21no—
217C4×C31—21no—
218C4×C31—21no—
219C4×C31—42no—
220C4×C31—42no—
221C4×C31—42no—
222C4×C31—81no—
223C4×C31—81no—
224C4×C31—81no—
225C4×C31—81no—
226C4×C31—161no—
227C2×C22⋊Q8—21no—
228C2×#121C22×C3111yescharacteristic
229C2×#99—21no—
230C2×#99—21no—
231C2×C22⋊C8C22×C3111yescharacteristic
232C2×C22⋊C8C22×C3111yescharacteristic
233#398—21no—
234#395—21no—
235#391—21no—
236#389—21no—
237#385—21no—
238#380C22×C3111yescharacteristic
239#380C22×C3111yescharacteristic
240C4×C22⋊C4C22×C3111yescharacteristic
241C2×#95C22×C3111yescharacteristic
242#327—21no—
243#327—21no—
244#327—21no—
245#327—21no—
246C2×C31—21no—
247C2×C31—21no—
248C2×C31—21no—
249C2×C31—21no—
250C2×C31—21no—
251C2×C31—21no—
252C2×C31—161no—
253C2×C31#202311yescharacteristic
254C2×C31#203111yescharacteristic
255C2×C31#203211yescharacteristic
256C22×Q8—21no—
257C22×D4—21no—
258C22×D4Q8×C3111yescharacteristic, complemented
259C23×C4—21no—
260C23×C4C23×C3111yescharacteristic
261C22×C8—21no—
262C22×C8—21no—
263C22⋊Q8—42no—
264C22⋊Q8—42no—
265#121—21no—
266#121—21no—
267#121—21no—
268#121—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4Q8×C3111yescharacteristic, complemented
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4D4×C3111yescharacteristic
281C2×C22⋊C4D4×C3111yescharacteristic
282C2×C42—21no—
283#99—42no—
284#99—42no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95D4×C3111yescharacteristic
295#95D4×C3111yescharacteristic
296#95D4×C3111yescharacteristic
297#95D4×C3111yescharacteristic
298C31#1037411yescharacteristic, direct product factor, Sylow
299C24—21no—
300C24C2×D4×C3111yescharacteristic
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—81no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—81no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—81no—
325C22×C4—81no—
326C22×C4C2×D4×C3111yescharacteristic
327C22×C4C2×D4×C3111yescharacteristic
328C22×C4C2×Q8×C3111yescharacteristic
329C22×C4#41×C3111yescharacteristic
330C22×C4#41×C3111yescharacteristic
331C22×C4#41×C3111yescharacteristic
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—81no—
335C2×C8—81no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—81no—
343C4⋊C4—81no—
344C4⋊C4—81no—
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4D8×C3112yes—
354C22⋊C4SD16×C3112yes—
355C42—81no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—81no—
362C23—81no—
363C23C31×#12011yescharacteristic
364C23C31×C22⋊Q811yescharacteristic
365C23C31×#12411yescharacteristic
366C23C2×D8×C3111yescharacteristic
367C23C2×SD16×C3111yescharacteristic
368C23C31×AGL(1,ℤ/8)11yescharacteristic
369C23C31×#13711yescharacteristic
370Q8—84no—
371D4—81no—
372D4—81no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4C31×C22⋊Q811yescharacteristic
410C2×C4C31×C22⋊Q811yescharacteristic
411C2×C4C31×#12411yescharacteristic
412C2×C4C31×#12411yescharacteristic
413C8—81no—
414C8—81no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—161no—
429C22—161no—
430C22C31×#39311yescharacteristic
431C22C31×#44711yescharacteristic
432C22C31×#44811yescharacteristic
433C22C31×#47411yescharacteristic
434C22C31×#47611yescharacteristic
435C22C31×#48611yescharacteristic
436C22C31×#48911yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—42no—
443C4—42no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—161no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—21no—
454C2—21no—
455C2—161no—
456C2C31×#202311yescharacteristic
457C2C31×#203111yescharacteristic
458C2C31×#203211yescharacteristic
subgroupG/H
center, socleC22×C31#393
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC2×C31×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2549C2semidirect
5. C31×#1294C2semidirect
6. C31×#1289C2semidirect
7. C31×#1282C2semidirect
29. C22×D4×C31Q8semidirect
47. C2×C4⋊C4×C31Q8semidirect
126. #10374C31direct (N × H)
186. #2549C2×C31semidirect
190. #1294C2×C31semidirect
191. #1289C2×C31semidirect
192. #1282C2×C31semidirect
258. C22×D4Q8×C31semidirect
276. C2×C4⋊C4Q8×C31semidirect
298. C31#10374direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25492
C31×#19642
C31×#15192
C2×C31×#3272
C31×#12942
C31×#12892
C31×#12822
#1037431