Finite Group Databasebeta

← C31×#23985 #21480744

Subgroups — C31×#23985 #21480744

Order 7936.

1474 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3490C211yesmaximal, characteristic
2C31×#2849C212yesmaximal, complemented
3C2×C31×#381C211yesmaximal, characteristic, complemented
4C31×#2030C212yesmaximal, complemented
5C31×#2022C212yesmaximal, complemented
6C31×#1898C212yesmaximal, complemented
7C31×#1890C214yesmaximal, complemented
8C31×#1819C211yesmaximal, characteristic, complemented
9C24×C4×C31C2211yescharacteristic
10C31×#551—24no—
11C31×#546—24no—
12C31×#543C2211yescharacteristic
13C31×#538—22no—
14C31×#534C2211yescharacteristic
15C31×#525—22no—
16C2×C31×#123C2214yes—
17C2×C31×#120C2212yes—
18C31×#518C2212yes—
19C22×C22⋊C4×C31C2211yescharacteristic
20C22×C22⋊C4×C31C2211yescharacteristic
21C2×C31×#99C2214yes—
22C31×#386—24no—
23C31×#385—24no—
24C31×#381—22no—
25C31×#381C2214yes—
26C31×#381C2212yes—
27C31×#381C2212yes—
28C31×#379C2212yes—
29C2×C31×#95—22no—
30C31×#342C2212yes—
31C31×#342C2212yes—
32C31×#342C2214yes—
33C25×C31C2311yescharacteristic
34C2×#41×C31—22no—
35C22×D4×C31D414yes—
36C22×D4×C31C2312yes—
37C23×C4×C31—21no—
38C23×C4×C31—21no—
39C23×C4×C31—24no—
40C23×C4×C31—21no—
41C23×C4×C31—24no—
42C23×C4×C31—42no—
43C23×C4×C31D412yes—
44C23×C4×C31C2312yes—
45C31×#126—24no—
46C31×#124—24no—
47C31×#123—24no—
48C31×#123—24no—
49C31×#123—22no—
50C31×#123—28no—
51C31×#123—44no—
52C31×C22⋊Q8—22no—
53C31×C22⋊Q8—22no—
54C31×C22⋊Q8—24no—
55C31×#121—22no—
56C31×#121—24no—
57C31×#121—22no—
58C31×#120—22no—
59C31×#120—22no—
60C31×#120—24no—
61C31×#120—22no—
62C4×D4×C31—24no—
63C4×D4×C31—24no—
64C4×D4×C31—24no—
65C31×#117—22no—
66C31×#117—22no—
67C2×C4⋊C4×C31—24no—
68C2×C22⋊C4×C31—24no—
69C2×C22⋊C4×C31—22no—
70C2×C22⋊C4×C31—21no—
71C2×C22⋊C4×C31—24no—
72C2×C22⋊C4×C31—24no—
73C2×C22⋊C4×C31—42no—
74C2×C22⋊C4×C31D412yes—
75C2×C22⋊C4×C31D414yes—
76C2×C22⋊C4×C31C2312yes—
77C2×C22⋊C4×C31C2314yes—
78C2×C22⋊C4×C31C2311yescharacteristic
79C2×C22⋊C4×C31C2311yescharacteristic
80C2×C22⋊C4×C31C2312yes—
81C31×#99—44no—
82C31×#95—24no—
83C31×#95—24no—
84C31×#95—28no—
85C24×C31—22no—
86C24×C31—22no—
87C24×C31—44no—
88C24×C31C2×D411yescharacteristic
89C24×C31C2×D414yes—
90C24×C31C2×D412yes—
91C24×C31C2×D411yescharacteristic
92C24×C31C2411yescharacteristic
93#41×C31—84no—
94C2×Q8×C31—22no—
95C2×D4×C31—24no—
96C2×D4×C31—22no—
97C2×D4×C31—22no—
98C2×D4×C31—24no—
99C2×D4×C31—24no—
100C2×D4×C31—22no—
101C2×D4×C31—24no—
102C2×D4×C31—48no—
103C22×C4×C31—24no—
104C22×C4×C31—24no—
105C22×C4×C31—22no—
106C22×C4×C31—22no—
107C22×C4×C31—24no—
108C22×C4×C31—22no—
109C22×C4×C31—24no—
110C22×C4×C31—24no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—24no—
114C22×C4×C31—416no—
115C22×C4×C31—42no—
116C22×C4×C31—42no—
117C22×C4×C31—42no—
118C22×C4×C31—42no—
119C22×C4×C31—44no—
120C22×C4×C31—42no—
121C22×C4×C31—44no—
122C22×C4×C31C2×D412yes—
123C22×C4×C31C2×D412yes—
124C22×C4×C31C2×D414yes—
125C22×C4×C31C2×D412yes—
126C22×C4×C31#4114yescomplemented
127C4⋊C4×C31—24no—
128C4⋊C4×C31—22no—
129C4⋊C4×C31—22no—
130C4⋊C4×C31—24no—
131C4⋊C4×C31—48no—
132C22⋊C4×C31—24no—
133C22⋊C4×C31—24no—
134C22⋊C4×C31—24no—
135C22⋊C4×C31—24no—
136C22⋊C4×C31—24no—
137C22⋊C4×C31—22no—
138C22⋊C4×C31—22no—
139C22⋊C4×C31—22no—
140C22⋊C4×C31—22no—
141C22⋊C4×C31—24no—
142C22⋊C4×C31—24no—
143C22⋊C4×C31—24no—
144C22⋊C4×C31—48no—
145C22⋊C4×C31—48no—
146C22⋊C4×C31—42no—
147C42×C31—24no—
148#23985C3111yesmaximal, characteristic, direct product factor, Sylow
149C23×C31—21no—
150C23×C31—21no—
151C23×C31—24no—
152C23×C31—21no—
153C23×C31—21no—
154C23×C31—28no—
155C23×C31—28no—
156C23×C31—22no—
157C23×C31—24no—
158C23×C31—42no—
159C23×C31—44no—
160C23×C31—42no—
161C23×C31—48no—
162C23×C31—48no—
163C23×C31—44no—
164C23×C31#12012yes—
165C23×C31#12012yes—
166C23×C31C22×D411yescharacteristic
167C23×C31C22×D412yes—
168C23×C31C2×#4112yes—
169C23×C31H(5,𝔽2)12yes—
170Q8×C31—82no—
171D4×C31—42no—
172D4×C31—42no—
173D4×C31—44no—
174D4×C31—48no—
175D4×C31—84no—
176D4×C31—82no—
177C2×C4×C31—22no—
178C2×C4×C31—22no—
179C2×C4×C31—24no—
180C2×C4×C31—22no—
181C2×C4×C31—24no—
182C2×C4×C31—28no—
183C2×C4×C31—24no—
184C2×C4×C31—48no—
185C2×C4×C31—48no—
186C2×C4×C31—42no—
187C2×C4×C31—42no—
188C2×C4×C31—44no—
189C2×C4×C31—44no—
190C2×C4×C31—42no—
191C2×C4×C31—42no—
192C2×C4×C31—44no—
193C2×C4×C31—48no—
194C2×C4×C31—44no—
195C2×C4×C31—42no—
196C2×C4×C31—44no—
197C2×C4×C31—82no—
198C2×C4×C31—84no—
199C2×C4×C31—84no—
200#3490C2×C3111yescharacteristic
201#2849C2×C3112yescomplemented
202C2×#381C2×C3111yescharacteristic, complemented
203#2030C2×C3112yescomplemented
204#2022C2×C3112yescomplemented
205#1898C2×C3112yescomplemented
206#1890C2×C3114yescomplemented
207#1819C2×C3111yescharacteristic, complemented
208C22×C31—24no—
209C22×C31—24no—
210C22×C31—22no—
211C22×C31—22no—
212C22×C31—24no—
213C22×C31—22no—
214C22×C31—22no—
215C22×C31—24no—
216C22×C31—42no—
217C22×C31—42no—
218C22×C31—416no—
219C22×C31—42no—
220C22×C31—42no—
221C22×C31—42no—
222C22×C31—44no—
223C22×C31—84no—
224C22×C31#45814yes—
225C22×C31C2×#12011yescharacteristic
226C22×C31#52511yescharacteristic
227C22×C31#53511yescharacteristic
228C22×C31#54614yes—
229C4×C31—44no—
230C4×C31—42no—
231C4×C31—42no—
232C4×C31—42no—
233C4×C31—44no—
234C4×C31—84no—
235C4×C31—82no—
236C24×C4C22×C3111yescharacteristic
237#551—24no—
238#546—24no—
239#543C22×C3111yescharacteristic
240#538—22no—
241#534C22×C3111yescharacteristic
242#525—22no—
243C2×#123C22×C3114yes—
244C2×#120C22×C3112yes—
245#518C22×C3112yes—
246C22×C22⋊C4C22×C3111yescharacteristic
247C22×C22⋊C4C22×C3111yescharacteristic
248C2×#99C22×C3114yes—
249#386—24no—
250#385—24no—
251#381—22no—
252#381C22×C3114yes—
253#381C22×C3112yes—
254#381C22×C3112yes—
255#379C22×C3112yes—
256C2×#95—22no—
257#342C22×C3112yes—
258#342C22×C3112yes—
259#342C22×C3114yes—
260C2×C31—21no—
261C2×C31—24no—
262C2×C31—21no—
263C2×C31—24no—
264C2×C31—42no—
265C2×C31—82no—
266C2×C31#240411yescharacteristic
267C2×C31C2×#45812yes—
268C25C23×C3111yescharacteristic
269C2×#41—22no—
270C22×D4D4×C3114yes—
271C22×D4C23×C3112yes—
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—24no—
275C23×C4—21no—
276C23×C4—24no—
277C23×C4—42no—
278C23×C4D4×C3112yes—
279C23×C4C23×C3112yes—
280#126—24no—
281#124—24no—
282#123—24no—
283#123—24no—
284#123—22no—
285#123—28no—
286#123—44no—
287C22⋊Q8—22no—
288C22⋊Q8—22no—
289C22⋊Q8—24no—
290#121—22no—
291#121—24no—
292#121—22no—
293#120—22no—
294#120—22no—
295#120—24no—
296#120—22no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—24no—
300#117—22no—
301#117—22no—
302C2×C4⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4D4×C3112yes—
310C2×C22⋊C4D4×C3114yes—
311C2×C22⋊C4C23×C3112yes—
312C2×C22⋊C4C23×C3114yes—
313C2×C22⋊C4C23×C3111yescharacteristic
314C2×C22⋊C4C23×C3111yescharacteristic
315C2×C22⋊C4C23×C3112yes—
316#99—44no—
317#95—24no—
318#95—24no—
319#95—28no—
320C31#2398511yescharacteristic, direct product factor, Sylow
321C24—22no—
322C24—22no—
323C24—44no—
324C24C2×D4×C3111yescharacteristic
325C24C2×D4×C3114yes—
326C24C2×D4×C3112yes—
327C24C2×D4×C3111yescharacteristic
328C24C24×C3111yescharacteristic
329#41—84no—
330C2×Q8—22no—
331C2×D4—24no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—48no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—416no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4C2×D4×C3112yes—
359C22×C4C2×D4×C3112yes—
360C22×C4C2×D4×C3114yes—
361C22×C4C2×D4×C3112yes—
362C22×C4#41×C3114yescomplemented
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—48no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—48no—
381C22⋊C4—48no—
382C22⋊C4—42no—
383C42—24no—
384C23—24no—
385C23—21no—
386C23—21no—
387C23—28no—
388C23—21no—
389C23—21no—
390C23—28no—
391C23—22no—
392C23—24no—
393C23—42no—
394C23—44no—
395C23—42no—
396C23—48no—
397C23—48no—
398C23—44no—
399C23C31×#12012yes—
400C23C31×#12012yes—
401C23C22×D4×C3111yescharacteristic
402C23C22×D4×C3112yes—
403C23C2×#41×C3112yes—
404C23C31×H(5,𝔽2)12yes—
405Q8—82no—
406D4—48no—
407D4—42no—
408D4—42no—
409D4—44no—
410D4—84no—
411D4—82no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—28no—
418C2×C4—24no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—48no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—82no—
433C2×C4—84no—
434C2×C4—84no—
435C22—24no—
436C22—24no—
437C22—22no—
438C22—22no—
439C22—24no—
440C22—22no—
441C22—22no—
442C22—24no—
443C22—42no—
444C22—42no—
445C22—416no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—44no—
450C22—84no—
451C22C31×#45814yes—
452C22C2×C31×#12011yescharacteristic
453C22C31×#52511yescharacteristic
454C22C31×#53511yescharacteristic
455C22C31×#54614yes—
456C4—44no—
457C4—42no—
458C4—42no—
459C4—42no—
460C4—44no—
461C4—82no—
462C4—84no—
463C2—21no—
464C2—24no—
465C2—24no—
466C2—21no—
467C2—42no—
468C2—82no—
469C2C31×#240411yescharacteristic
470C2C2×C31×#45812yes—
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC24C24×C31
prime radicalC31×#3490C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C25×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2849C2semidirect
3. C2×C31×#381C2semidirect
4. C31×#2030C2semidirect
5. C31×#2022C2semidirect
6. C31×#1898C2semidirect
7. C31×#1890C2semidirect
8. C31×#1819C2semidirect
126. C22×C4×C31#41semidirect
148. #23985C31direct (N × H)
201. #2849C2×C31semidirect
202. C2×#381C2×C31semidirect
203. #2030C2×C31semidirect
204. #2022C2×C31semidirect
205. #1898C2×C31semidirect
206. #1890C2×C31semidirect
207. #1819C2×C31semidirect
320. C31#23985direct (N × H)
362. C22×C4#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34902
C31×#28492
C2×C31×#3812
C31×#20302
C31×#20222
C31×#18982
C31×#18902
C31×#18192
#2398531