Finite Group Databasebeta

← C31×#19514 #21476273

Subgroups — C31×#19514 #21476273

Order 7936.

622 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3226C211yesmaximal, characteristic, complemented
2C31×#3214C211yesmaximal, characteristic
3C31×#3206C211yesmaximal, characteristic, complemented
4C31×#2981C211yesmaximal, characteristic, complemented
5C31×#1701C212yesmaximal, complemented
6C31×#1693C212yesmaximal, complemented
7C31×#1681C212yesmaximal, complemented
8C31×#1673C212yesmaximal, complemented
9C31×#1572C211yesmaximal, characteristic, complemented
10C31×#1556C211yesmaximal, characteristic, complemented
11C31×#1553C211yesmaximal, characteristic, complemented
12C31×#532C2211yescharacteristic
13C31×#532C2211yescharacteristic
14C2×C31×C4⋊Q8C2211yescharacteristic
15C31×#500C2211yescharacteristic
16C31×#500C2211yescharacteristic
17C31×#496—21no—
18C31×#492—21no—
19C31×#489—21no—
20C31×#487—21no—
21C31×#486C2211yescharacteristic
22C31×#486C2211yescharacteristic
23C31×#480—22no—
24C31×#474—22no—
25C31×#461C2212yes—
26C31×#461C2212yes—
27C31×#461C2212yes—
28C31×#459C2212yes—
29C31×#449—22no—
30C31×#448—22no—
31C31×#436C2212yes—
32C31×#435—22no—
33C8×D4×C31C2212yes—
34C31×#432C2211yescharacteristic
35C31×#432C2211yescharacteristic
36C31×#431C2211yescharacteristic
37C31×#431C2211yescharacteristic
38C31×#408—22no—
39C31×#335C2212yes—
40C31×#334C2212yes—
41C31×#332C2212yes—
42C31×#332C2212yes—
43C31×#332C2212yes—
44C31×#331C2212yes—
45C2×#41×C31—21no—
46C2×#41×C31—21no—
47C22×Q8×C31—21no—
48C2×SD16×C31—22no—
49C2×SD16×C31—22no—
50C2×SD16×C31—22no—
51C2×D8×C31—22no—
52C2×M16×C31—22no—
53C22×C8×C31—22no—
54C31×C4⋊Q8—21no—
55C31×C4⋊Q8—21no—
56C31×C4⋊Q8—21no—
57C31×C4⋊Q8D411yescharacteristic
58C31×C4⋊Q8C2311yescharacteristic
59C31×C4⋊Q8C2311yescharacteristic
60C31×C4⋊Q8C2311yescharacteristic
61C31×#127D411yescharacteristic
62C31×#127C2311yescharacteristic
63C31×#124—21no—
64C31×#124—21no—
65C31×#121—21no—
66C31×#121—21no—
67C31×#121—21no—
68C31×#121D412yes—
69C4×D4×C31—21no—
70C4×D4×C31—21no—
71C4×D4×C31—21no—
72C4×D4×C31C2312yes—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—21no—
75C2×C42×C31C2311yescharacteristic
76C31×#107—22no—
77C31×#107—22no—
78C31×C4⋊C8D412yes—
79C31×C4⋊C8C2312yes—
80C31×C4⋊C8C2312yes—
81C31×C4⋊C8C2312yes—
82C31×#103—22no—
83C31×#103—22no—
84C31×#103—22no—
85C31×#102—22no—
86C31×#102—22no—
87C31×#102—22no—
88C31×C22⋊C8—21no—
89C31×C22⋊C8—21no—
90C31×C22⋊C8—21no—
91C31×C22⋊C8—22no—
92C31×C22⋊C8—22no—
93C31×C22⋊C8D412yes—
94C31×#97—21no—
95C31×#97—21no—
96C4×C8×C31—21no—
97C4×C8×C31C2311yescharacteristic
98C4×C8×C31C2311yescharacteristic
99#41×C31—82no—
100#41×C31—82no—
101C2×Q8×C31—21no—
102C2×Q8×C31—21no—
103C2×Q8×C31—21no—
104C2×Q8×C31—21no—
105C2×Q8×C31—41no—
106C2×Q8×C31—82no—
107C2×D4×C31—22no—
108C2×D4×C31—21no—
109C2×D4×C31—21no—
110C2×D4×C31—21no—
111C2×D4×C31—21no—
112C2×D4×C31—21no—
113C2×D4×C31—41no—
114C2×D4×C31C2×D412yes—
115C22×C4×C31—22no—
116C22×C4×C31—41no—
117C22×C4×C31—41no—
118C22×C4×C31C2×D411yescharacteristic
119C22×C4×C31C2×D411yescharacteristic
120C22×C4×C31#4111yescharacteristic
121SD16×C31—82no—
122SD16×C31—82no—
123SD16×C31—82no—
124D8×C31—82no—
125M16×C31—82no—
126C2×C8×C31—22no—
127C2×C8×C31—22no—
128C2×C8×C31—22no—
129C2×C8×C31—82no—
130C2×C8×C31C2×D412yes—
131C2×C8×C31C2×D412yes—
132C4⋊C4×C31—21no—
133C4⋊C4×C31—21no—
134C4⋊C4×C31—21no—
135C4⋊C4×C31—21no—
136C4⋊C4×C31—21no—
137C4⋊C4×C31—21no—
138C4⋊C4×C31—21no—
139C4⋊C4×C31—41no—
140C4⋊C4×C31—41no—
141C4⋊C4×C31C2×D412yes—
142C22⋊C4×C31—22no—
143C22⋊C4×C31—22no—
144C22⋊C4×C31—41no—
145C22⋊C4×C31—41no—
146C42×C31C2×D411yescharacteristic
147C42×C31C2×D411yescharacteristic
148C42×C31#4111yescharacteristic
149C42×C31C2411yescharacteristic
150#19514C3111yesmaximal, characteristic, direct product factor, Sylow
151C23×C31—22no—
152C23×C31—41no—
153C23×C31#12111yescharacteristic
154Q8×C31—42no—
155Q8×C31—42no—
156Q8×C31—42no—
157Q8×C31—81no—
158Q8×C31—82no—
159D4×C31—22no—
160D4×C31—22no—
161D4×C31—42no—
162D4×C31—42no—
163D4×C31—81no—
164D4×C31—81no—
165D4×C31—81no—
166D4×C31—82no—
167D4×C31—82no—
168C2×C4×C31—22no—
169C2×C4×C31—22no—
170C2×C4×C31—22no—
171C2×C4×C31—41no—
172C2×C4×C31—42no—
173C2×C4×C31—42no—
174C2×C4×C31—41no—
175C2×C4×C31—41no—
176C2×C4×C31—41no—
177C2×C4×C31—82no—
178C2×C4×C31—82no—
179C2×C4×C31#12111yescharacteristic
180C2×C4×C31#12111yescharacteristic
181C2×C4×C31#12111yescharacteristic
182C2×C4×C31C22×D411yescharacteristic
183C2×C4×C31C22×D411yescharacteristic
184C2×C4×C31C2×#4111yescharacteristic
185C8×C31—22no—
186C8×C31—22no—
187C8×C31—42no—
188C8×C31—42no—
189C8×C31—42no—
190#3226C2×C3111yescharacteristic, complemented
191#3214C2×C3111yescharacteristic
192#3206C2×C3111yescharacteristic, complemented
193#2981C2×C3111yescharacteristic, complemented
194#1701C2×C3112yescomplemented
195#1693C2×C3112yescomplemented
196#1681C2×C3112yescomplemented
197#1673C2×C3112yescomplemented
198#1572C2×C3111yescharacteristic, complemented
199#1556C2×C3111yescharacteristic, complemented
200#1553C2×C3111yescharacteristic, complemented
201C22×C31—21no—
202C22×C31—21no—
203C22×C31—21no—
204C22×C31—42no—
205C22×C31—42no—
206C22×C31—42no—
207C22×C31—81no—
208C22×C31—81no—
209C22×C31—81no—
210C22×C31C2×#12111yescharacteristic
211C4×C31—21no—
212C4×C31—21no—
213C4×C31—21no—
214C4×C31—21no—
215C4×C31—42no—
216C4×C31—81no—
217C4×C31—81no—
218C4×C31—81no—
219C4×C31#57612yes—
220C4×C31#57712yes—
221#532C22×C3111yescharacteristic
222#532C22×C3111yescharacteristic
223C2×C4⋊Q8C22×C3111yescharacteristic
224#500C22×C3111yescharacteristic
225#500C22×C3111yescharacteristic
226#496—21no—
227#492—21no—
228#489—21no—
229#487—21no—
230#486C22×C3111yescharacteristic
231#486C22×C3111yescharacteristic
232#480—22no—
233#474—22no—
234#461C22×C3112yes—
235#461C22×C3112yes—
236#461C22×C3112yes—
237#459C22×C3112yes—
238#449—22no—
239#448—22no—
240#436C22×C3112yes—
241#435—22no—
242C8×D4C22×C3112yes—
243#432C22×C3111yescharacteristic
244#432C22×C3111yescharacteristic
245#431C22×C3111yescharacteristic
246#431C22×C3111yescharacteristic
247#408—22no—
248#335C22×C3112yes—
249#334C22×C3112yes—
250#332C22×C3112yes—
251#332C22×C3112yes—
252#332C22×C3112yes—
253#331C22×C3112yes—
254C2×C31—41no—
255C2×C31—82no—
256C2×C31—161no—
257C2×C31#304211yescharacteristic
258C2×C31#305811yescharacteristic
259C2×C31#306911yescharacteristic
260C2×#41—21no—
261C2×#41—21no—
262C22×Q8—21no—
263C2×SD16—22no—
264C2×SD16—22no—
265C2×SD16—22no—
266C2×D8—22no—
267C2×M16—22no—
268C22×C8—22no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8—21no—
272C4⋊Q8D4×C3111yescharacteristic
273C4⋊Q8C23×C3111yescharacteristic
274C4⋊Q8C23×C3111yescharacteristic
275C4⋊Q8C23×C3111yescharacteristic
276#127D4×C3111yescharacteristic
277#127C23×C3111yescharacteristic
278#124—21no—
279#124—21no—
280#121—21no—
281#121—21no—
282#121—21no—
283#121D4×C3112yes—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4C23×C3112yes—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C42C23×C3111yescharacteristic
291#107—22no—
292#107—22no—
293C4⋊C8D4×C3112yes—
294C4⋊C8C23×C3112yes—
295C4⋊C8C23×C3112yes—
296C4⋊C8C23×C3112yes—
297#103—22no—
298#103—22no—
299#103—22no—
300#102—22no—
301#102—22no—
302#102—22no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—22no—
307C22⋊C8—22no—
308C22⋊C8D4×C3112yes—
309#97—21no—
310#97—21no—
311C4×C8—21no—
312C4×C8C23×C3111yescharacteristic
313C4×C8C23×C3111yescharacteristic
314C31#1951411yescharacteristic, direct product factor, Sylow
315#41—82no—
316#41—82no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—41no—
322C2×Q8—82no—
323C2×D4—22no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—41no—
330C2×D4C2×D4×C3112yes—
331C22×C4—22no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4C2×D4×C3111yescharacteristic
335C22×C4C2×D4×C3111yescharacteristic
336C22×C4#41×C3111yescharacteristic
337SD16—82no—
338SD16—82no—
339SD16—82no—
340D8—82no—
341M16—82no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—82no—
346C2×C8C2×D4×C3112yes—
347C2×C8C2×D4×C3112yes—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4C2×D4×C3112yes—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C42C2×D4×C3111yescharacteristic
363C42C2×D4×C3111yescharacteristic
364C42#41×C3111yescharacteristic
365C42C24×C3111yescharacteristic
366C23—22no—
367C23—41no—
368C23C31×#12111yescharacteristic
369Q8—42no—
370Q8—42no—
371Q8—42no—
372Q8—81no—
373Q8—82no—
374D4—22no—
375D4—22no—
376D4—42no—
377D4—42no—
378D4—81no—
379D4—81no—
380D4—81no—
381D4—82no—
382D4—82no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4C31×#12111yescharacteristic
395C2×C4C31×#12111yescharacteristic
396C2×C4C31×#12111yescharacteristic
397C2×C4C22×D4×C3111yescharacteristic
398C2×C4C22×D4×C3111yescharacteristic
399C2×C4C2×#41×C3111yescharacteristic
400C8—22no—
401C8—22no—
402C8—42no—
403C8—42no—
404C8—42no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22C2×C31×#12111yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—42no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4C31×#57612yes—
424C4C31×#57712yes—
425C2—41no—
426C2—82no—
427C2—161no—
428C2C31×#304211yescharacteristic
429C2C31×#305811yescharacteristic
430C2C31×#306911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#121
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3214C2
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×#3226C2semidirect
3. C31×#3206C2semidirect
4. C31×#2981C2semidirect
5. C31×#1701C2semidirect
6. C31×#1693C2semidirect
7. C31×#1681C2semidirect
8. C31×#1673C2semidirect
9. C31×#1572C2semidirect
10. C31×#1556C2semidirect
11. C31×#1553C2semidirect
150. #19514C31direct (N × H)
190. #3226C2×C31semidirect
192. #3206C2×C31semidirect
193. #2981C2×C31semidirect
194. #1701C2×C31semidirect
195. #1693C2×C31semidirect
196. #1681C2×C31semidirect
197. #1673C2×C31semidirect
198. #1572C2×C31semidirect
199. #1556C2×C31semidirect
200. #1553C2×C31semidirect
314. C31#19514direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32262
C31×#32142
C31×#32062
C31×#29812
C31×#17012
C31×#16932
C31×#16812
C31×#16732
C31×#15722
C31×#15562
C31×#15532
#1951431