Finite Group Databasebeta

← C31×#61906 #21518665

Subgroups — C31×#61906 #21518665

Order 7936.

1748 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3522C211yesmaximal, characteristic
2C2×C31×#557C211yesmaximal, characteristic, complemented
3C31×#3400C216yesmaximal, complemented
4C31×#3392C212yesmaximal, complemented
5C31×#3372C216yesmaximal, complemented
6C31×#3363C212yesmaximal, complemented
7C31×#3218C213yesmaximal, complemented
8C31×#3206C213yesmaximal, complemented
9C31×#3112C213yesmaximal, complemented
10C31×#3108C213yesmaximal, complemented
11C31×#2984C211yesmaximal, characteristic, complemented
12C31×#559C2213yes—
13Q82×C31C2211yescharacteristic
14C31×#557—21no—
15C31×#557—23no—
16C31×#557C2212yes—
17C31×#557C2216yes—
18C31×#554C2213yes—
19C31×#550C2211yescharacteristic
20C31×#548—21no—
21C31×#547—23no—
22C31×#542—23no—
23C31×#532C2213yes—
24C2×C31×C4⋊Q8C2213yes—
25C31×#529C2213yes—
26C2×C31×#125—23no—
27C31×#520—21no—
28C2×C4×Q8×C31—21no—
29C2×C4×Q8×C31C2213yes—
30C2×C4×Q8×C31C2211yescharacteristic
31C31×#497C2216yes—
32C31×#492C2216yes—
33C31×#491C2216yes—
34C31×#487C2216yes—
35C31×#483C2211yescharacteristic
36C31×#480C2213yes—
37C31×#479C2216yes—
38C31×#477C2216yes—
39C31×#464C2216yes—
40C31×#462C2216yes—
41C31×#461C2216yes—
42C31×#461C22112yes—
43C31×#460C2216yes—
44C31×#451C2213yes—
45C31×#448C2213yes—
46C31×#448C2216yes—
47C31×#446C2216yes—
48C8×Q8×C31C2212yes—
49C31×#440C22112yes—
50C4×Q16×C31C2216yes—
51C4×SD16×C31C2216yes—
52C31×#432C2213yes—
53C31×#431C2213yes—
54C2×#41×C31—21no—
55C2×#41×C31—23no—
56C22×Q8×C31D411yescharacteristic
57C22×Q8×C31C2313yes—
58C2×Q16×C31C2316yes—
59C2×SD16×C31C2316yes—
60C2×SD16×C31C23112yes—
61C31×C4⋊Q8—23no—
62C31×C4⋊Q8—23no—
63C31×C4⋊Q8—23no—
64C31×C4⋊Q8C2316yes—
65C31×C4⋊Q8C2316yes—
66C31×C4⋊Q8C2313yes—
67C31×#127C2313yes—
68C31×#126—23no—
69C31×#125—26no—
70C31×#125—26no—
71C31×#125—212no—
72C31×#125C2313yes—
73C31×#124—23no—
74C31×#124—23no—
75C31×#124C2313yes—
76C31×#123—23no—
77C31×#123—26no—
78C31×C22⋊Q8—23no—
79C31×C22⋊Q8—26no—
80C31×C22⋊Q8C2311yescharacteristic
81C31×#121—23no—
82C31×#121—23no—
83C31×#121C2313yes—
84C4×Q8×C31—26no—
85C4×Q8×C31—22no—
86C4×Q8×C31—21no—
87C4×Q8×C31—23no—
88C4×Q8×C31—23no—
89C4×Q8×C31D411yescharacteristic
90C4×Q8×C31D413yes—
91C4×Q8×C31C2316yes—
92C4×Q8×C31C2313yes—
93C4×Q8×C31C2316yes—
94C4×Q8×C31C2311yescharacteristic
95C4×Q8×C31C2313yes—
96C4×D4×C31—23no—
97C4×D4×C31—23no—
98C4×D4×C31—23no—
99C4×D4×C31C2313yes—
100C4×D4×C31C2316yes—
101C31×#117—23no—
102C2×C4⋊C4×C31—23no—
103C2×C4⋊C4×C31—23no—
104C2×C4⋊C4×C31—23no—
105C2×C4⋊C4×C31—23no—
106C2×C4⋊C4×C31D413yes—
107C2×C4⋊C4×C31C2311yescharacteristic
108C2×C42×C31—23no—
109C2×C42×C31C2313yes—
110C31×#107C2316yes—
111C31×#106C2312yes—
112C31×C4⋊C8C2316yes—
113C31×C4⋊C8C2316yes—
114C31×#103C2316yes—
115C31×#103C2312yes—
116C31×#103C2316yes—
117C31×#103C23112yes—
118C31×#102C2316yes—
119C31×C22⋊C8C2311yescharacteristic
120C31×C22⋊C8C2313yes—
121C31×#97C2316yes—
122C4×C8×C31C2316yes—
123#41×C31—44no—
124#41×C31—86no—
125C2×Q8×C31—24no—
126C2×Q8×C31—23no—
127C2×Q8×C31—23no—
128C2×Q8×C31—46no—
129C2×Q8×C31C2×D413yes—
130C2×Q8×C31C2×D411yescharacteristic
131C2×Q8×C31C2416yes—
132C2×Q8×C31C2411yescharacteristic
133C2×D4×C31—23no—
134C2×D4×C31—23no—
135C2×D4×C31—23no—
136C2×D4×C31C2413yes—
137C22×C4×C31—21no—
138C22×C4×C31—23no—
139C22×C4×C31—23no—
140C22×C4×C31—26no—
141C22×C4×C31C2×D413yes—
142C22×C4×C31C2×D413yes—
143C22×C4×C31C2411yescharacteristic
144Q16×C31—46no—
145SD16×C31—46no—
146SD16×C31—412no—
147C2×C8×C31C2412yes—
148C2×C8×C31C2416yes—
149C4⋊C4×C31—23no—
150C4⋊C4×C31—23no—
151C4⋊C4×C31—26no—
152C4⋊C4×C31—26no—
153C4⋊C4×C31—26no—
154C4⋊C4×C31—26no—
155C4⋊C4×C31—26no—
156C4⋊C4×C31—26no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—23no—
160C4⋊C4×C31—23no—
161C4⋊C4×C31C2×D413yes—
162C4⋊C4×C31C2×D413yes—
163C4⋊C4×C31C2×D416yes—
164C4⋊C4×C31C2412yes—
165C4⋊C4×C31C2411yescharacteristic
166C4⋊C4×C31C2413yes—
167C22⋊C4×C31—21no—
168C22⋊C4×C31—23no—
169C22⋊C4×C31—23no—
170C22⋊C4×C31—23no—
171C22⋊C4×C31—23no—
172C22⋊C4×C31—26no—
173C42×C31—26no—
174C42×C31—26no—
175C42×C31—23no—
176C42×C31C2×D413yes—
177C42×C31C2×D413yes—
178C42×C31C2413yes—
179C42×C31C2413yes—
180#61906C3111yesmaximal, characteristic, direct product factor, Sylow
181C23×C31—23no—
182C23×C31C22×D411yescharacteristic
183Q8×C31—26no—
184Q8×C31—26no—
185Q8×C31—26no—
186Q8×C31—22no—
187Q8×C31—46no—
188Q8×C31—83no—
189Q8×C31#13714yes—
190D4×C31—26no—
191D4×C31—46no—
192D4×C31—46no—
193D4×C31—83no—
194C2×C4×C31—26no—
195C2×C4×C31—26no—
196C2×C4×C31—22no—
197C2×C4×C31—22no—
198C2×C4×C31—21no—
199C2×C4×C31—26no—
200C2×C4×C31—23no—
201C2×C4×C31—21no—
202C2×C4×C31—46no—
203C2×C4×C31—42no—
204C2×C4×C31—46no—
205C2×C4×C31—86no—
206C2×C4×C31C22×D413yes—
207C2×C4×C31C22×D411yescharacteristic
208C2×C4×C31C22×D413yes—
209C2×C4×C31C22×D413yes—
210C2×C4×C31C22×D413yes—
211C2×C4×C31C2511yescharacteristic
212C8×C31—22no—
213C8×C31—26no—
214#3522C2×C3111yescharacteristic
215C2×#557C2×C3111yescharacteristic, complemented
216#3400C2×C3116yescomplemented
217#3392C2×C3112yescomplemented
218#3372C2×C3116yescomplemented
219#3363C2×C3112yescomplemented
220#3218C2×C3113yescomplemented
221#3206C2×C3113yescomplemented
222#3112C2×C3113yescomplemented
223#3108C2×C3113yescomplemented
224#2984C2×C3111yescharacteristic, complemented
225C22×C31—21no—
226C22×C31—21no—
227C22×C31—43no—
228C22×C31—43no—
229C22×C31—43no—
230C22×C31#57712yes—
231C22×C31C23×D411yescharacteristic
232C4×C31—21no—
233C4×C31—23no—
234C4×C31—23no—
235C4×C31—23no—
236C4×C31—46no—
237C4×C31—41no—
238C4×C31—41no—
239C4×C31—42no—
240C4×C31—43no—
241C4×C31C2×#13716yes—
242C4×C31#58512yes—
243#559C22×C3113yes—
244Q82C22×C3111yescharacteristic
245#557—21no—
246#557—23no—
247#557C22×C3112yes—
248#557C22×C3116yes—
249#554C22×C3113yes—
250#550C22×C3111yescharacteristic
251#548—21no—
252#547—23no—
253#542—23no—
254#532C22×C3113yes—
255C2×C4⋊Q8C22×C3113yes—
256#529C22×C3113yes—
257C2×#125—23no—
258#520—21no—
259C2×C4×Q8—21no—
260C2×C4×Q8C22×C3113yes—
261C2×C4×Q8C22×C3111yescharacteristic
262#497C22×C3116yes—
263#492C22×C3116yes—
264#491C22×C3116yes—
265#487C22×C3116yes—
266#483C22×C3111yescharacteristic
267#480C22×C3113yes—
268#479C22×C3116yes—
269#477C22×C3116yes—
270#464C22×C3116yes—
271#462C22×C3116yes—
272#461C22×C3116yes—
273#461C22×C31112yes—
274#460C22×C3116yes—
275#451C22×C3113yes—
276#448C22×C3113yes—
277#448C22×C3116yes—
278#446C22×C3116yes—
279C8×Q8C22×C3112yes—
280#440C22×C31112yes—
281C4×Q16C22×C3116yes—
282C4×SD16C22×C3116yes—
283#432C22×C3113yes—
284#431C22×C3113yes—
285C2×C31—22no—
286C2×C31—83no—
287C2×C31#348711yescharacteristic
288C2×C31C22×#13711yescharacteristic
289C2×C31C2×#57711yescharacteristic
290C2×#41—21no—
291C2×#41—23no—
292C22×Q8D4×C3111yescharacteristic
293C22×Q8C23×C3113yes—
294C2×Q16C23×C3116yes—
295C2×SD16C23×C3116yes—
296C2×SD16C23×C31112yes—
297C4⋊Q8—23no—
298C4⋊Q8—23no—
299C4⋊Q8—23no—
300C4⋊Q8C23×C3116yes—
301C4⋊Q8C23×C3116yes—
302C4⋊Q8C23×C3113yes—
303#127C23×C3113yes—
304#126—23no—
305#125—26no—
306#125—26no—
307#125—212no—
308#125C23×C3113yes—
309#124—23no—
310#124—23no—
311#124C23×C3113yes—
312#123—23no—
313#123—26no—
314C22⋊Q8—23no—
315C22⋊Q8—26no—
316C22⋊Q8C23×C3111yescharacteristic
317#121—23no—
318#121—23no—
319#121C23×C3113yes—
320C4×Q8—26no—
321C4×Q8—22no—
322C4×Q8—21no—
323C4×Q8—23no—
324C4×Q8—23no—
325C4×Q8D4×C3111yescharacteristic
326C4×Q8D4×C3113yes—
327C4×Q8C23×C3116yes—
328C4×Q8C23×C3116yes—
329C4×Q8C23×C3113yes—
330C4×Q8C23×C3111yescharacteristic
331C4×Q8C23×C3113yes—
332C4×D4—23no—
333C4×D4—23no—
334C4×D4—23no—
335C4×D4C23×C3113yes—
336C4×D4C23×C3116yes—
337#117—23no—
338C2×C4⋊C4—23no—
339C2×C4⋊C4—23no—
340C2×C4⋊C4—23no—
341C2×C4⋊C4—23no—
342C2×C4⋊C4D4×C3113yes—
343C2×C4⋊C4C23×C3111yescharacteristic
344C2×C42—23no—
345C2×C42C23×C3113yes—
346#107C23×C3116yes—
347#106C23×C3112yes—
348C4⋊C8C23×C3116yes—
349C4⋊C8C23×C3116yes—
350#103C23×C3116yes—
351#103C23×C3112yes—
352#103C23×C3116yes—
353#103C23×C31112yes—
354#102C23×C3116yes—
355C22⋊C8C23×C3111yescharacteristic
356C22⋊C8C23×C3113yes—
357#97C23×C3116yes—
358C4×C8C23×C3116yes—
359C31#6190611yescharacteristic, direct product factor, Sylow
360#41—44no—
361#41—86no—
362C2×Q8—24no—
363C2×Q8—23no—
364C2×Q8—23no—
365C2×Q8—46no—
366C2×Q8C2×D4×C3113yes—
367C2×Q8C2×D4×C3111yescharacteristic
368C2×Q8C24×C3116yes—
369C2×Q8C24×C3111yescharacteristic
370C2×D4—23no—
371C2×D4—23no—
372C2×D4—23no—
373C2×D4C24×C3113yes—
374C22×C4—21no—
375C22×C4—23no—
376C22×C4—23no—
377C22×C4—26no—
378C22×C4C2×D4×C3113yes—
379C22×C4C2×D4×C3113yes—
380C22×C4C24×C3111yescharacteristic
381Q16—46no—
382SD16—46no—
383SD16—412no—
384C2×C8C24×C3112yes—
385C2×C8C24×C3116yes—
386C4⋊C4—23no—
387C4⋊C4—23no—
388C4⋊C4—26no—
389C4⋊C4—26no—
390C4⋊C4—26no—
391C4⋊C4—26no—
392C4⋊C4—26no—
393C4⋊C4—26no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—23no—
397C4⋊C4—23no—
398C4⋊C4C2×D4×C3113yes—
399C4⋊C4C2×D4×C3113yes—
400C4⋊C4C2×D4×C3116yes—
401C4⋊C4C24×C3112yes—
402C4⋊C4C24×C3111yescharacteristic
403C4⋊C4C24×C3113yes—
404C22⋊C4—21no—
405C22⋊C4—23no—
406C22⋊C4—23no—
407C22⋊C4—23no—
408C22⋊C4—23no—
409C22⋊C4—26no—
410C42—26no—
411C42—26no—
412C42—23no—
413C42C2×D4×C3113yes—
414C42C2×D4×C3113yes—
415C42C24×C3113yes—
416C42C24×C3113yes—
417C23—23no—
418C23C22×D4×C3111yescharacteristic
419Q8—26no—
420Q8—26no—
421Q8—26no—
422Q8—22no—
423Q8—46no—
424Q8—83no—
425Q8C31×#13714yes—
426D4—26no—
427D4—46no—
428D4—46no—
429D4—83no—
430C2×C4—26no—
431C2×C4—26no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—26no—
436C2×C4—23no—
437C2×C4—21no—
438C2×C4—46no—
439C2×C4—42no—
440C2×C4—46no—
441C2×C4—86no—
442C2×C4C22×D4×C3113yes—
443C2×C4C22×D4×C3111yescharacteristic
444C2×C4C22×D4×C3113yes—
445C2×C4C22×D4×C3113yes—
446C2×C4C22×D4×C3113yes—
447C2×C4C25×C3111yescharacteristic
448C8—22no—
449C8—26no—
450C22—21no—
451C22—21no—
452C22—43no—
453C22—43no—
454C22—43no—
455C22C31×#57712yes—
456C22C23×D4×C3111yescharacteristic
457C4—21no—
458C4—23no—
459C4—23no—
460C4—23no—
461C4—46no—
462C4—41no—
463C4—41no—
464C4—42no—
465C4—43no—
466C4C2×C31×#13716yes—
467C4C31×#58512yes—
468C2—22no—
469C2—83no—
470C2C31×#348711yescharacteristic
471C2C22×C31×#13711yescharacteristic
472C2C2×C31×#57711yescharacteristic
subgroupG/H
center, socleC22×C31C23×D4
commutator, Frattini, cosocleC2×C4C25×C31
prime radicalC31×#3522C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C4×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#557C2semidirect
3. C31×#3400C2semidirect
4. C31×#3392C2semidirect
5. C31×#3372C2semidirect
6. C31×#3363C2semidirect
7. C31×#3218C2semidirect
8. C31×#3206C2semidirect
9. C31×#3112C2semidirect
10. C31×#3108C2semidirect
11. C31×#2984C2semidirect
180. #61906C31direct (N × H)
215. C2×#557C2×C31semidirect
216. #3400C2×C31semidirect
217. #3392C2×C31semidirect
218. #3372C2×C31semidirect
219. #3363C2×C31semidirect
220. #3218C2×C31semidirect
221. #3206C2×C31semidirect
222. #3112C2×C31semidirect
223. #3108C2×C31semidirect
224. #2984C2×C31semidirect
359. C31#61906direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35222
C2×C31×#5572
C31×#34002
C31×#33922
C31×#33722
C31×#33632
C31×#32182
C31×#32062
C31×#31122
C31×#31082
C31×#29842
#6190631