Finite Group Databasebeta

← C31×#19564 #21476323

Subgroups — C31×#19564 #21476323

Order 7936.

626 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3255C211yesmaximal, characteristic, complemented
2C31×#3208C211yesmaximal, characteristic
3C31×#3208C211yesmaximal, characteristic
4C31×#2986C211yesmaximal, characteristic, complemented
5C31×#1702C212yesmaximal, complemented
6C31×#1686C212yesmaximal, complemented
7C31×#1682C212yesmaximal, complemented
8C31×#1666C212yesmaximal, complemented
9C31×#1574C211yesmaximal, characteristic, complemented
10C31×#1545C211yesmaximal, characteristic
11C31×#1544C211yesmaximal, characteristic, complemented
12C31×#533C2211yescharacteristic
13C31×#533C2211yescharacteristic
14C2×C31×#127C2211yescharacteristic
15C31×#492—21no—
16C31×#492C2212yes—
17C31×#486—21no—
18C31×#486C2212yes—
19C31×#475—21no—
20C31×#475C2212yescomplemented
21C31×#470—22no—
22C31×#462—21no—
23C31×#462C2212yescomplemented
24C31×#460—21no—
25C31×#460—21no—
26C31×#460C2212yes—
27C31×#460C2212yes—
28C31×#450—22no—
29C31×#450—22no—
30C31×#446—21no—
31C31×#446C2212yescomplemented
32C8×Q8×C31—21no—
33C8×Q8×C31C2212yescomplemented
34C31×#433—22no—
35C31×#431C2211yescharacteristic
36C31×#431C2211yescharacteristic
37C31×#431C2211yescharacteristic
38C31×#423C2211yescharacteristic
39C31×#335C2212yescomplemented
40C31×#334C2212yescomplemented
41C31×#332C2211yescharacteristic, complemented
42C31×#332C2211yescharacteristic, complemented
43C31×#332C2211yescharacteristic, complemented
44C31×#332C2211yescharacteristic, complemented
45C31×#331C2212yes—
46C31×#331C2212yes—
47C22×D4×C31—21no—
48C22×D4×C31—21no—
49C22×D4×C31—21no—
50C2×Q16×C31—22no—
51C2×SD16×C31—22no—
52C2×SD16×C31—22no—
53C2×SD16×C31—22no—
54C2×SD16×C31—22no—
55C2×M16×C31—21no—
56C31×C4⋊Q8D411yescharacteristic
57C31×C4⋊Q8D411yescharacteristic
58C31×C4⋊Q8C2311yescharacteristic
59C31×C4⋊Q8C2311yescharacteristic
60C31×#127—21no—
61C31×#127—21no—
62C31×#127—21no—
63C31×#127C2312yes—
64C31×#125—21no—
65C31×#125—21no—
66C31×C22⋊Q8—21no—
67C31×C22⋊Q8—21no—
68C31×C22⋊Q8—21no—
69C31×C22⋊Q8D412yes—
70C4×Q8×C31—21no—
71C4×Q8×C31—21no—
72C4×Q8×C31—21no—
73C4×Q8×C31C2312yes—
74C31×#117—21no—
75C31×#117—21no—
76C2×C42×C31C2311yescharacteristic
77C31×#106—22no—
78C31×C4⋊C8—22no—
79C31×C4⋊C8—22no—
80C31×C4⋊C8—21no—
81C31×C4⋊C8D411yescharacteristic
82C31×C4⋊C8D411yescharacteristic
83C31×C4⋊C8C2312yes—
84C31×C4⋊C8C2311yescharacteristic
85C31×C4⋊C8C2311yescharacteristic
86C31×C4⋊C8C2311yescharacteristic
87C31×C4⋊C8C2311yescharacteristic
88C31×#103—22no—
89C31×#103—22no—
90C31×#102—22no—
91C31×#102—22no—
92C31×#102—22no—
93C31×#102—22no—
94C31×C22⋊C8—21no—
95C31×C22⋊C8—21no—
96C31×C22⋊C8D412yes—
97C31×#97—22no—
98C4×C8×C31—22no—
99C4×C8×C31—21no—
100C4×C8×C31—21no—
101C4×C8×C31C2312yes—
102C24×C31—41no—
103C2×Q8×C31—21no—
104C2×Q8×C31—21no—
105C2×Q8×C31—21no—
106C2×Q8×C31C2×D412yes—
107C2×D4×C31—22no—
108C2×D4×C31—22no—
109C2×D4×C31—22no—
110C2×D4×C31—41no—
111C2×D4×C31—41no—
112C2×D4×C31—41no—
113C2×D4×C31—81no—
114C2×D4×C31—81no—
115C2×D4×C31—81no—
116C2×D4×C31—81no—
117C2×D4×C31—82no—
118C22×C4×C31C2×D411yescharacteristic
119C22×C4×C31C2×D411yescharacteristic
120C22×C4×C31#4111yescharacteristic
121Q16×C31—82no—
122SD16×C31—82no—
123SD16×C31—82no—
124SD16×C31—82no—
125SD16×C31—82no—
126M16×C31—81no—
127C2×C8×C31—21no—
128C2×C8×C31—21no—
129C2×C8×C31—21no—
130C2×C8×C31—21no—
131C2×C8×C31—22no—
132C2×C8×C31C2×D412yes—
133C2×C8×C31C2×D412yes—
134C4⋊C4×C31—21no—
135C4⋊C4×C31—21no—
136C4⋊C4×C31—21no—
137C4⋊C4×C31—22no—
138C4⋊C4×C31—22no—
139C4⋊C4×C31—21no—
140C4⋊C4×C31—21no—
141C4⋊C4×C31—21no—
142C4⋊C4×C31—21no—
143C4⋊C4×C31—41no—
144C4⋊C4×C31—41no—
145C4⋊C4×C31C2×D412yes—
146C22⋊C4×C31—22no—
147C22⋊C4×C31—41no—
148C42×C31—22no—
149C42×C31—41no—
150C42×C31C2×D411yescharacteristic
151C42×C31C2×D411yescharacteristic
152C42×C31#4111yescharacteristic
153C42×C31C2411yescharacteristic
154#19564C3111yesmaximal, characteristic, direct product factor, Sylow
155C23×C31—42no—
156C23×C31—82no—
157C23×C31—82no—
158C23×C31—82no—
159C23×C31#12111yescharacteristic
160Q8×C31—22no—
161Q8×C31—22no—
162Q8×C31—42no—
163Q8×C31—41no—
164Q8×C31—41no—
165Q8×C31—41no—
166Q8×C31—41no—
167D4×C31—42no—
168D4×C31—42no—
169D4×C31—42no—
170D4×C31—42no—
171D4×C31—82no—
172D4×C31—82no—
173D4×C31—82no—
174D4×C31—82no—
175C2×C4×C31—21no—
176C2×C4×C31—21no—
177C2×C4×C31—21no—
178C2×C4×C31—21no—
179C2×C4×C31—22no—
180C2×C4×C31—22no—
181C2×C4×C31—22no—
182C2×C4×C31—41no—
183C2×C4×C31—41no—
184C2×C4×C31#12111yescharacteristic
185C2×C4×C31#12111yescharacteristic
186C2×C4×C31#12111yescharacteristic
187C2×C4×C31C22×D411yescharacteristic
188C2×C4×C31C22×D411yescharacteristic
189C2×C4×C31C2×#4111yescharacteristic
190C8×C31—22no—
191C8×C31—22no—
192C8×C31—42no—
193C8×C31—41no—
194C8×C31—41no—
195C8×C31—41no—
196C8×C31—41no—
197#3255C2×C3111yescharacteristic, complemented
198#3208C2×C3111yescharacteristic
199#3208C2×C3111yescharacteristic
200#2986C2×C3111yescharacteristic, complemented
201#1702C2×C3112yescomplemented
202#1686C2×C3112yescomplemented
203#1682C2×C3112yescomplemented
204#1666C2×C3112yescomplemented
205#1574C2×C3111yescharacteristic, complemented
206#1545C2×C3111yescharacteristic
207#1544C2×C3111yescharacteristic, complemented
208C22×C31—21no—
209C22×C31—21no—
210C22×C31—21no—
211C22×C31—82no—
212C22×C31—82no—
213C22×C31—82no—
214C22×C31—164no—
215C22×C31C2×#12111yescharacteristic
216C4×C31—21no—
217C4×C31—21no—
218C4×C31—21no—
219C4×C31—42no—
220C4×C31—42no—
221C4×C31—81no—
222C4×C31—81no—
223C4×C31#47112yes—
224C4×C31#57611yescharacteristic
225C4×C31#57711yescharacteristic
226C4×C31#57711yescharacteristic
227C4×C31#57711yescharacteristic
228#533C22×C3111yescharacteristic
229#533C22×C3111yescharacteristic
230C2×#127C22×C3111yescharacteristic
231#492—21no—
232#492C22×C3112yes—
233#486—21no—
234#486C22×C3112yes—
235#475—21no—
236#475C22×C3112yescomplemented
237#470—22no—
238#462—21no—
239#462C22×C3112yescomplemented
240#460—21no—
241#460—21no—
242#460C22×C3112yes—
243#460C22×C3112yes—
244#450—22no—
245#450—22no—
246#446—21no—
247#446C22×C3112yescomplemented
248C8×Q8—21no—
249C8×Q8C22×C3112yescomplemented
250#433—22no—
251#431C22×C3111yescharacteristic
252#431C22×C3111yescharacteristic
253#431C22×C3111yescharacteristic
254#423C22×C3111yescharacteristic
255#335C22×C3112yescomplemented
256#334C22×C3112yescomplemented
257#332C22×C3111yescharacteristic, complemented
258#332C22×C3111yescharacteristic, complemented
259#332C22×C3111yescharacteristic, complemented
260#332C22×C3111yescharacteristic, complemented
261#331C22×C3112yes—
262#331C22×C3112yes—
263C2×C31—41no—
264C2×C31—162no—
265C2×C31#304311yescharacteristic
266C2×C31#306211yescharacteristic
267C2×C31C2×#47111yescharacteristic
268C22×D4—21no—
269C22×D4—21no—
270C22×D4—21no—
271C2×Q16—22no—
272C2×SD16—22no—
273C2×SD16—22no—
274C2×SD16—22no—
275C2×SD16—22no—
276C2×M16—21no—
277C4⋊Q8D4×C3111yescharacteristic
278C4⋊Q8D4×C3111yescharacteristic
279C4⋊Q8C23×C3111yescharacteristic
280C4⋊Q8C23×C3111yescharacteristic
281#127—21no—
282#127—21no—
283#127—21no—
284#127C23×C3112yes—
285#125—21no—
286#125—21no—
287C22⋊Q8—21no—
288C22⋊Q8—21no—
289C22⋊Q8—21no—
290C22⋊Q8D4×C3112yes—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8C23×C3112yes—
295#117—21no—
296#117—21no—
297C2×C42C23×C3111yescharacteristic
298#106—22no—
299C4⋊C8—22no—
300C4⋊C8—22no—
301C4⋊C8—21no—
302C4⋊C8D4×C3111yescharacteristic
303C4⋊C8D4×C3111yescharacteristic
304C4⋊C8C23×C3111yescharacteristic
305C4⋊C8C23×C3111yescharacteristic
306C4⋊C8C23×C3112yes—
307C4⋊C8C23×C3111yescharacteristic
308C4⋊C8C23×C3111yescharacteristic
309#103—22no—
310#103—22no—
311#102—22no—
312#102—22no—
313#102—22no—
314#102—22no—
315C22⋊C8—21no—
316C22⋊C8—21no—
317C22⋊C8D4×C3112yes—
318#97—22no—
319C4×C8—22no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8C23×C3112yes—
323C31#1956411yescharacteristic, direct product factor, Sylow
324C24—41no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8C2×D4×C3112yes—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—81no—
336C2×D4—81no—
337C2×D4—81no—
338C2×D4—81no—
339C2×D4—82no—
340C22×C4C2×D4×C3111yescharacteristic
341C22×C4C2×D4×C3111yescharacteristic
342C22×C4#41×C3111yescharacteristic
343Q16—82no—
344SD16—82no—
345SD16—82no—
346SD16—82no—
347SD16—82no—
348M16—81no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—22no—
354C2×C8C2×D4×C3112yes—
355C2×C8C2×D4×C3112yes—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4C2×D4×C3112yes—
368C22⋊C4—22no—
369C22⋊C4—41no—
370C42—22no—
371C42—41no—
372C42C2×D4×C3111yescharacteristic
373C42C2×D4×C3111yescharacteristic
374C42#41×C3111yescharacteristic
375C42C24×C3111yescharacteristic
376C23—42no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23C31×#12111yescharacteristic
381Q8—22no—
382Q8—22no—
383Q8—42no—
384Q8—41no—
385Q8—41no—
386Q8—41no—
387Q8—41no—
388D4—42no—
389D4—42no—
390D4—42no—
391D4—42no—
392D4—82no—
393D4—82no—
394D4—82no—
395D4—82no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4C31×#12111yescharacteristic
406C2×C4C31×#12111yescharacteristic
407C2×C4C31×#12111yescharacteristic
408C2×C4C22×D4×C3111yescharacteristic
409C2×C4C22×D4×C3111yescharacteristic
410C2×C4C2×#41×C3111yescharacteristic
411C8—22no—
412C8—22no—
413C8—42no—
414C8—41no—
415C8—41no—
416C8—41no—
417C8—41no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22—164no—
425C22C2×C31×#12111yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—42no—
431C4—81no—
432C4—81no—
433C4C31×#47112yes—
434C4C31×#57611yescharacteristic
435C4C31×#57711yescharacteristic
436C4C31×#57711yescharacteristic
437C4C31×#57711yescharacteristic
438C2—41no—
439C2—162no—
440C2C31×#304311yescharacteristic
441C2C31×#306211yescharacteristic
442C2C2×C31×#47111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#121
commutator, Frattini, cosocleC42C24×C31
prime radicalC2×C31×#127C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3255C2semidirect
4. C31×#2986C2semidirect
5. C31×#1702C2semidirect
6. C31×#1686C2semidirect
7. C31×#1682C2semidirect
8. C31×#1666C2semidirect
9. C31×#1574C2semidirect
11. C31×#1544C2semidirect
20. C31×#475C22semidirect
23. C31×#462C22semidirect
31. C31×#446C22semidirect
33. C8×Q8×C31C22semidirect
39. C31×#335C22semidirect
40. C31×#334C22semidirect
41. C31×#332C22semidirect
42. C31×#332C22semidirect
43. C31×#332C22semidirect
44. C31×#332C22semidirect
154. #19564C31direct (N × H)
197. #3255C2×C31semidirect
200. #2986C2×C31semidirect
201. #1702C2×C31semidirect
202. #1686C2×C31semidirect
203. #1682C2×C31semidirect
204. #1666C2×C31semidirect
205. #1574C2×C31semidirect
207. #1544C2×C31semidirect
236. #475C22×C31semidirect
239. #462C22×C31semidirect
247. #446C22×C31semidirect
249. C8×Q8C22×C31semidirect
255. #335C22×C31semidirect
256. #334C22×C31semidirect
257. #332C22×C31semidirect
258. #332C22×C31semidirect
259. #332C22×C31semidirect
260. #332C22×C31semidirect
323. C31#19564direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32552
C31×#32082
C31×#29862
C31×#17022
C31×#16862
C31×#16822
C31×#16662
C31×#15742
C31×#15452
C31×#15442
#1956431