Finite Group Databasebeta

← C31×#33571 #21490330

Subgroups — C31×#33571 #21490330

Order 7936.

1084 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3550C211yesmaximal, characteristic, complemented
2C31×#3022C212yesmaximal, complemented
3C31×#3018C211yesmaximal, characteristic
4C31×#2205C214yesmaximal, complemented
5C31×#2203C214yesmaximal, complemented
6C31×#2135C212yesmaximal, complemented
7C31×#2132C211yesmaximal, characteristic, complemented
8C31×#585C2211yescharacteristic
9C31×#575—21no—
10C2×C31×#137—22no—
11C31×#563—21no—
12C31×#563C2212yescomplemented
13C31×#559—21no—
14Q82×C31C2212yescomplemented
15C31×#556—22no—
16D4×Q8×C31—22no—
17C31×#542—24no—
18C31×#536—21no—
19C31×#536C2212yes—
20C31×#533—21no—
21C31×#532C2211yescharacteristic
22C31×#456C2218yes—
23C31×#456C2212yes—
24C31×#455—21no—
25C31×#455—24no—
26C31×#454C2212yes—
27C31×#453—21no—
28C31×#421—22no—
29C2×C31×#104—21no—
30C31×#413C2212yes—
31C31×#413C2211yescharacteristic
32C31×#356C2212yescomplemented
33C31×#356C2218yescomplemented
34C31×#355C2212yescomplemented
35C31×2−1+4—41no—
36C31×2−1+4C2312yes—
37C31×H(5,𝔽2)—21no—
38C31×H(5,𝔽2)—22no—
39C31×H(5,𝔽2)—41no—
40C2×#41×C31—21no—
41C2×#41×C31—22no—
42C2×#41×C31—42no—
43C2×#41×C31D411yescharacteristic
44C2×#41×C31C2311yescharacteristic
45C22×Q8×C31—21no—
46C22×Q8×C31D412yes—
47C31×#137—28no—
48C31×#137—22no—
49C31×#137—41no—
50C31×#137—44no—
51C31×AGL(1,ℤ/8)—22no—
52C31×AGL(1,ℤ/8)—41no—
53C31×#135—42no—
54C2×Q16×C31—42no—
55C2×SD16×C31—42no—
56C2×M16×C31—21no—
57C2×M16×C31—22no—
58C31×C4⋊Q8—21no—
59C31×C4⋊Q8—21no—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8—24no—
62C31×C4⋊Q8C2311yescharacteristic
63C31×C4⋊Q8C2314yes—
64C31×#127D411yescharacteristic
65C31×#126—24no—
66C31×#126—24no—
67C31×#125—22no—
68C31×#124—22no—
69C31×#124—24no—
70C31×#124D411yescharacteristic
71C31×#124D414yes—
72C31×#124C2311yescharacteristic
73C31×#123—22no—
74C31×#123—22no—
75C31×#123—22no—
76C31×#123—22no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8—24no—
80C31×C22⋊Q8—24no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—22no—
83C31×#121—22no—
84C4×Q8×C31—22no—
85C4×Q8×C31—22no—
86C4×Q8×C31—24no—
87C4×D4×C31—24no—
88C4×D4×C31—22no—
89C31×#117—22no—
90C31×#117—21no—
91C31×#117D412yes—
92C2×C42×C31D411yescharacteristic
93C31×#104—22no—
94C31×#104—22no—
95C31×#104—28no—
96C31×#101C2312yes—
97C31×#101C2314yes—
98C31×#100—21no—
99C31×#100—22no—
100#41×C31—22no—
101#41×C31—24no—
102#41×C31—22no—
103#41×C31—24no—
104#41×C31—41no—
105#41×C31—41no—
106#41×C31—42no—
107#41×C31—41no—
108#41×C31—41no—
109#41×C31—41no—
110#41×C31—42no—
111#41×C31—42no—
112#41×C31—44no—
113#41×C31—41no—
114C2×Q8×C31—22no—
115C2×Q8×C31—22no—
116C2×Q8×C31—22no—
117C2×Q8×C31—22no—
118C2×Q8×C31—21no—
119C2×Q8×C31—42no—
120C2×Q8×C31—42no—
121C2×Q8×C31—42no—
122C2×Q8×C31—81no—
123C2×Q8×C31—81no—
124C2×Q8×C31C2×D414yes—
125C2×Q8×C31C2×D411yescharacteristic
126C2×Q8×C31C2411yescharacteristic
127C2×D4×C31—22no—
128C2×D4×C31—21no—
129C2×D4×C31—22no—
130C2×D4×C31—21no—
131C2×D4×C31—42no—
132C2×D4×C31—44no—
133C2×D4×C31—41no—
134C2×D4×C31—42no—
135C2×D4×C31C2×D411yescharacteristic
136C2×D4×C31C2×D412yes—
137C2×D4×C31C2×D411yescharacteristic
138C22×C4×C31—21no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—21no—
142C22×C4×C31—41no—
143C22×C4×C31—42no—
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D412yes—
146Q16×C31—48no—
147Q16×C31—42no—
148SD16×C31—48no—
149SD16×C31—42no—
150SD16×C31—42no—
151D8×C31—42no—
152M16×C31—22no—
153M16×C31—24no—
154M16×C31—41no—
155M16×C31—42no—
156C2×C8×C31—42no—
157C2×C8×C31—41no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—24no—
162C4⋊C4×C31—24no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—24no—
166C4⋊C4×C31—24no—
167C22⋊C4×C31—22no—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—24no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—24no—
173C22⋊C4×C31—22no—
174C42×C31—24no—
175C42×C31—22no—
176C42×C31—21no—
177C42×C31C2×D411yescharacteristic
178C42×C31C2×D411yescharacteristic
179C42×C31C2×D414yes—
180#33571C3111yesmaximal, characteristic, direct product factor, Sylow
181C23×C31—21no—
182C23×C31—42no—
183C23×C31—41no—
184C23×C31#12012yes—
185C23×C31#12011yescharacteristic
186Q8×C31—22no—
187Q8×C31—24no—
188Q8×C31—41no—
189Q8×C31—44no—
190Q8×C31—41no—
191Q8×C31—44no—
192Q8×C31—41no—
193Q8×C31—42no—
194Q8×C31—42no—
195Q8×C31—82no—
196D4×C31—22no—
197D4×C31—24no—
198D4×C31—41no—
199D4×C31—42no—
200D4×C31—41no—
201D4×C31—44no—
202D4×C31—44no—
203D4×C31—42no—
204D4×C31—42no—
205D4×C31—41no—
206D4×C31—81no—
207C2×C4×C31—21no—
208C2×C4×C31—21no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—21no—
212C2×C4×C31—24no—
213C2×C4×C31—21no—
214C2×C4×C31—21no—
215C2×C4×C31—24no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—44no—
219C2×C4×C31—42no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
222C2×C4×C31—41no—
223C2×C4×C31—42no—
224C2×C4×C31—82no—
225C2×C4×C31—82no—
226C2×C4×C31#12011yescharacteristic
227C2×C4×C31C22×D412yes—
228C2×C4×C31C22×D411yescharacteristic
229C8×C31—42no—
230C8×C31—44no—
231#3550C2×C3111yescharacteristic, complemented
232#3022C2×C3112yescomplemented
233#3018C2×C3111yescharacteristic
234#2205C2×C3114yescomplemented
235#2203C2×C3114yescomplemented
236#2135C2×C3112yescomplemented
237#2132C2×C3111yescharacteristic, complemented
238C22×C31—22no—
239C22×C31—21no—
240C22×C31—41no—
241C22×C31—42no—
242C22×C31—42no—
243C22×C31—42no—
244C22×C31—41no—
245C22×C31—162no—
246C22×C31—161no—
247C22×C31C2×#12011yescharacteristic
248C4×C31—22no—
249C4×C31—21no—
250C4×C31—22no—
251C4×C31—41no—
252C4×C31—44no—
253C4×C31—41no—
254C4×C31—44no—
255C4×C31—42no—
256C4×C31B(4,𝔽2)11yescharacteristic
257C4×C31B(4,𝔽2)11yescharacteristic
258#585C22×C3111yescharacteristic
259#575—21no—
260C2×#137—22no—
261#563—21no—
262#563C22×C3112yescomplemented
263#559—21no—
264Q82C22×C3112yescomplemented
265#556—22no—
266D4×Q8—22no—
267#542—24no—
268#536—21no—
269#536C22×C3112yes—
270#533—21no—
271#532C22×C3111yescharacteristic
272#456C22×C3118yes—
273#456C22×C3112yes—
274#455—21no—
275#455—24no—
276#454C22×C3112yes—
277#453—21no—
278#421—22no—
279C2×#104—21no—
280#413C22×C3112yes—
281#413C22×C3111yescharacteristic
282#356C22×C3112yescomplemented
283#356C22×C3118yescomplemented
284#355C22×C3112yescomplemented
285C2×C31—21no—
286C2×C31—42no—
287C2×C31—41no—
288C2×C31—81no—
289C2×C31—82no—
290C2×C31C2×B(4,𝔽2)11yescharacteristic
2912−1+4—41no—
2922−1+4C23×C3112yes—
293H(5,𝔽2)—21no—
294H(5,𝔽2)—22no—
295H(5,𝔽2)—41no—
296C2×#41—22no—
297C2×#41—21no—
298C2×#41—42no—
299C2×#41D4×C3111yescharacteristic
300C2×#41C23×C3111yescharacteristic
301C22×Q8—21no—
302C22×Q8D4×C3112yes—
303#137—28no—
304#137—22no—
305#137—41no—
306#137—44no—
307AGL(1,ℤ/8)—22no—
308AGL(1,ℤ/8)—41no—
309#135—42no—
310C2×Q16—42no—
311C2×SD16—42no—
312C2×M16—22no—
313C2×M16—21no—
314C4⋊Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8—22no—
317C4⋊Q8—24no—
318C4⋊Q8C23×C3111yescharacteristic
319C4⋊Q8C23×C3114yes—
320#127D4×C3111yescharacteristic
321#126—24no—
322#126—24no—
323#125—22no—
324#124—22no—
325#124—24no—
326#124D4×C3111yescharacteristic
327#124D4×C3114yes—
328#124C23×C3111yescharacteristic
329#123—22no—
330#123—22no—
331#123—22no—
332#123—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—24no—
336C22⋊Q8—24no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339#121—22no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8—24no—
343C4×D4—24no—
344C4×D4—22no—
345#117—22no—
346#117—21no—
347#117D4×C3112yes—
348C2×C42D4×C3111yescharacteristic
349#104—22no—
350#104—22no—
351#104—28no—
352#101C23×C3114yes—
353#101C23×C3112yes—
354#100—21no—
355#100—22no—
356C31#3357111yescharacteristic, direct product factor, Sylow
357#41—22no—
358#41—24no—
359#41—22no—
360#41—24no—
361#41—41no—
362#41—41no—
363#41—42no—
364#41—41no—
365#41—41no—
366#41—41no—
367#41—42no—
368#41—42no—
369#41—44no—
370#41—41no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—81no—
380C2×Q8—81no—
381C2×Q8C2×D4×C3114yes—
382C2×Q8C2×D4×C3111yescharacteristic
383C2×Q8C24×C3111yescharacteristic
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—42no—
389C2×D4—44no—
390C2×D4—42no—
391C2×D4—41no—
392C2×D4C2×D4×C3111yescharacteristic
393C2×D4C2×D4×C3112yes—
394C2×D4C2×D4×C3111yescharacteristic
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4C2×D4×C3111yescharacteristic
402C22×C4C2×D4×C3112yes—
403Q16—48no—
404Q16—42no—
405SD16—48no—
406SD16—42no—
407SD16—42no—
408D8—42no—
409M16—22no—
410M16—24no—
411M16—41no—
412M16—42no—
413C2×C8—42no—
414C2×C8—41no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C22⋊C4—22no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—24no—
430C22⋊C4—22no—
431C42—24no—
432C42—22no—
433C42—21no—
434C42C2×D4×C3111yescharacteristic
435C42C2×D4×C3111yescharacteristic
436C42C2×D4×C3114yes—
437C23—21no—
438C23—42no—
439C23—41no—
440C23C31×#12012yes—
441C23C31×#12011yescharacteristic
442Q8—22no—
443Q8—24no—
444Q8—41no—
445Q8—44no—
446Q8—41no—
447Q8—44no—
448Q8—41no—
449Q8—42no—
450Q8—42no—
451Q8—82no—
452D4—22no—
453D4—24no—
454D4—41no—
455D4—42no—
456D4—41no—
457D4—44no—
458D4—44no—
459D4—42no—
460D4—42no—
461D4—41no—
462D4—81no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—24no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4C31×#12011yescharacteristic
483C2×C4C22×D4×C3112yes—
484C2×C4C22×D4×C3111yescharacteristic
485C8—42no—
486C8—44no—
487C22—22no—
488C22—21no—
489C22—41no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—41no—
494C22—162no—
495C22—161no—
496C22C2×C31×#12011yescharacteristic
497C4—22no—
498C4—21no—
499C4—22no—
500C4—41no—
501C4—44no—
502C4—41no—
503C4—44no—
504C4—42no—
505C4C31×B(4,𝔽2)11yescharacteristic
506C4C31×B(4,𝔽2)11yescharacteristic
507C2—21no—
508C2—42no—
509C2—41no—
510C2—81no—
511C2—82no—
512C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C31C2×B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C24×C31
prime radicalC31×#3018C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3550C2semidirect
2. C31×#3022C2semidirect
4. C31×#2205C2semidirect
5. C31×#2203C2semidirect
6. C31×#2135C2semidirect
7. C31×#2132C2semidirect
12. C31×#563C22semidirect
14. Q82×C31C22semidirect
32. C31×#356C22semidirect
33. C31×#356C22semidirect
34. C31×#355C22semidirect
180. #33571C31direct (N × H)
231. #3550C2×C31semidirect
232. #3022C2×C31semidirect
234. #2205C2×C31semidirect
235. #2203C2×C31semidirect
236. #2135C2×C31semidirect
237. #2132C2×C31semidirect
262. #563C22×C31semidirect
264. Q82C22×C31semidirect
282. #356C22×C31semidirect
283. #356C22×C31semidirect
284. #355C22×C31semidirect
356. C31#33571direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35502
C31×#30222
C31×#30182
C31×#22052
C31×#22032
C31×#21352
C31×#21322
#3357131