Finite Group Databasebeta

← C31×#25948 #21482707

Subgroups — C31×#25948 #21482707

Order 7936.

1082 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×#3501C211yesmaximal, characteristic
2C31×#2503C212yesmaximal, complemented
3C31×#2486C211yesmaximal, characteristic
4C31×#2030C214yesmaximal, complemented
5C31×#1996C212yesmaximal, complemented
6C31×#1890C214yesmaximal
7C4×C31×#99C211yesmaximal, characteristic
8C31×#554—24no—
9C31×#554—22no—
10D4×Q8×C31—22no—
11C31×#544—21no—
12C31×#543—21no—
13C31×#543C2212yes—
14C31×#533—21no—
15C2×C31×#123C2212yes—
16C2×C31×C22⋊Q8—21no—
17C2×C31×C22⋊Q8C2212yes—
18C31×#518C2212yes—
19C2×C4×D4×C31C2211yescharacteristic
20C22×C4⋊C4×C31C2211yescharacteristic
21C2×C31×#99C2212yes—
22C31×#400—24no—
23C31×#398—21no—
24C31×#397—21no—
25C31×#391—22no—
26C31×#389—21no—
27C31×#382—22no—
28C31×#381C2218yes—
29C31×#381C2212yes—
30C31×#380—21no—
31C31×#380C2211yescharacteristic
32C4×C22⋊C4×C31C2212yes—
33C31×#342C2218yes—
34C31×#342C2212yes—
35C22×Q8×C31—21no—
36C22×Q8×C31D412yes—
37C22×D4×C31C2311yescharacteristic
38C23×C4×C31D411yescharacteristic
39C23×C4×C31C2312yes—
40C23×C4×C31C2311yescharacteristic
41C31×C4⋊Q8—21no—
42C31×C4⋊Q8—21no—
43C31×C4⋊Q8—24no—
44C31×#125—22no—
45C31×#125—24no—
46C31×#123—28no—
47C31×C22⋊Q8—22no—
48C31×C22⋊Q8—22no—
49C31×C22⋊Q8—24no—
50C31×C22⋊Q8—24no—
51C31×C22⋊Q8—22no—
52C31×C22⋊Q8—22no—
53C31×C22⋊Q8—22no—
54C31×C22⋊Q8—22no—
55C31×C22⋊Q8—22no—
56C31×C22⋊Q8—22no—
57C31×C22⋊Q8—24no—
58C31×C22⋊Q8—24no—
59C4×Q8×C31—22no—
60C4×Q8×C31—22no—
61C4×D4×C31—22no—
62C4×D4×C31—22no—
63C4×D4×C31—24no—
64C4×D4×C31—24no—
65C31×#117—22no—
66C2×C4⋊C4×C31—21no—
67C2×C4⋊C4×C31—22no—
68C2×C4⋊C4×C31—21no—
69C2×C4⋊C4×C31—22no—
70C2×C4⋊C4×C31—22no—
71C2×C4⋊C4×C31—21no—
72C2×C4⋊C4×C31—22no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31D411yescharacteristic
75C2×C4⋊C4×C31D414yes—
76C2×C4⋊C4×C31D412yes—
77C2×C22⋊C4×C31—22no—
78C2×C22⋊C4×C31—22no—
79C2×C22⋊C4×C31D411yescharacteristic
80C2×C22⋊C4×C31C2314yes—
81C2×C22⋊C4×C31C2312yes—
82C2×C22⋊C4×C31C2311yescharacteristic
83C2×C22⋊C4×C31C2314yes—
84C2×C42×C31—22no—
85C2×C42×C31D411yescharacteristic
86C31×#99—24no—
87C31×#95—21no—
88C31×#95—22no—
89C31×#95—22no—
90C31×#95—22no—
91C31×#95—28no—
92C31×#95—28no—
93C24×C31C2×D411yescharacteristic
94C24×C31C2411yescharacteristic
95C2×Q8×C31—22no—
96C2×Q8×C31—22no—
97C2×Q8×C31—22no—
98C2×Q8×C31—22no—
99C2×Q8×C31—44no—
100C2×Q8×C31—82no—
101C2×D4×C31—22no—
102C2×D4×C31—22no—
103C2×D4×C31#4114yes—
104C22×C4×C31—22no—
105C22×C4×C31—22no—
106C22×C4×C31—21no—
107C22×C4×C31—21no—
108C22×C4×C31—24no—
109C22×C4×C31—24no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—24no—
113C22×C4×C31—24no—
114C22×C4×C31—42no—
115C22×C4×C31—42no—
116C22×C4×C31C2×D412yes—
117C22×C4×C31C2×D411yescharacteristic
118C22×C4×C31C2×D411yescharacteristic
119C22×C4×C31C2×D411yescharacteristic
120C22×C4×C31C2×D414yes—
121C22×C4×C31C2×D414yes—
122C22×C4×C31C2×D411yescharacteristic
123C22×C4×C31C2×D411yescharacteristic
124C22×C4×C31C2×D412yes—
125C4⋊C4×C31—22no—
126C4⋊C4×C31—24no—
127C4⋊C4×C31—22no—
128C4⋊C4×C31—22no—
129C4⋊C4×C31—24no—
130C4⋊C4×C31—22no—
131C4⋊C4×C31—24no—
132C4⋊C4×C31—22no—
133C4⋊C4×C31—22no—
134C4⋊C4×C31—22no—
135C4⋊C4×C31—24no—
136C4⋊C4×C31—42no—
137C4⋊C4×C31—41no—
138C4⋊C4×C31—42no—
139C4⋊C4×C31—41no—
140C4⋊C4×C31—44no—
141C22⋊C4×C31—24no—
142C22⋊C4×C31—24no—
143C22⋊C4×C31—22no—
144C22⋊C4×C31—24no—
145C22⋊C4×C31—22no—
146C22⋊C4×C31—24no—
147C22⋊C4×C31—24no—
148C22⋊C4×C31—24no—
149C22⋊C4×C31—44no—
150C42×C31—24no—
151C42×C31—21no—
152C42×C31—21no—
153C42×C31—44no—
154#25948C3111yesmaximal, characteristic, direct product factor, Sylow
155C23×C31—21no—
156C23×C31—21no—
157C23×C31—41no—
158C23×C31—41no—
159C23×C31—41no—
160C23×C31—41no—
161C23×C31#12412yes—
162C23×C31C22×D411yescharacteristic
163C23×C31C22×D412yes—
164C23×C31C2×#4112yes—
165C23×C312−1+412yes—
166Q8×C31—44no—
167Q8×C31—82no—
168Q8×C31—82no—
169Q8×C31—82no—
170D4×C31—28no—
171C2×C4×C31—21no—
172C2×C4×C31—21no—
173C2×C4×C31—24no—
174C2×C4×C31—22no—
175C2×C4×C31—21no—
176C2×C4×C31—24no—
177C2×C4×C31—24no—
178C2×C4×C31—21no—
179C2×C4×C31—42no—
180C2×C4×C31—41no—
181C2×C4×C31—42no—
182C2×C4×C31—41no—
183C2×C4×C31—44no—
184C2×C4×C31—41no—
185C2×C4×C31—44no—
186C2×C4×C31—44no—
187C2×C4×C31—42no—
188C2×C4×C31—44no—
189C2×C4×C31—42no—
190C2×C4×C31—42no—
191C2×C4×C31—42no—
192C2×C4×C31—42no—
193C2×C4×C31—42no—
194C2×C4×C31—44no—
195C2×C4×C31—48no—
196C2×C4×C31—81no—
197C2×C4×C31—82no—
198C2×C4×C31—82no—
199C2×C4×C31#12011yescharacteristic
200C2×C4×C31#12012yes—
201C2×C4×C31#12011yescharacteristic
202C2×C4×C31#12412yes—
203#3501C2×C3111yescharacteristic
204#2503C2×C3112yescomplemented
205#2486C2×C3111yescharacteristic
206#2030C2×C3114yescomplemented
207#1996C2×C3112yescomplemented
208#1890C2×C3114yes—
209C4×#99C2×C3111yescharacteristic
210C22×C31—22no—
211C22×C31—21no—
212C22×C31—22no—
213C22×C31—21no—
214C22×C31—22no—
215C22×C31—42no—
216C22×C31—42no—
217C22×C31—42no—
218C22×C31—41no—
219C22×C31—41no—
220C22×C31—81no—
221C22×C31C2×#12011yescharacteristic
222C22×C31C2×#12411yescharacteristic
223C22×C31#53611yescharacteristic
224C22×C31#54814yes—
225C4×C31—22no—
226C4×C31—21no—
227C4×C31—22no—
228C4×C31—41no—
229C4×C31—44no—
230C4×C31—44no—
231C4×C31—41no—
232C4×C31—82no—
233C4×C31—84no—
234C4×C31—82no—
235C4×C31#45812yes—
236#554—22no—
237#554—24no—
238D4×Q8—22no—
239#544—21no—
240#543—21no—
241#543C22×C3112yes—
242#533—21no—
243C2×#123C22×C3112yes—
244C2×C22⋊Q8—21no—
245C2×C22⋊Q8C22×C3112yes—
246#518C22×C3112yes—
247C2×C4×D4C22×C3111yescharacteristic
248C22×C4⋊C4C22×C3111yescharacteristic
249C2×#99C22×C3112yes—
250#400—24no—
251#398—21no—
252#397—21no—
253#391—22no—
254#389—21no—
255#382—22no—
256#381C22×C3118yes—
257#381C22×C3112yes—
258#380—21no—
259#380C22×C3111yescharacteristic
260C4×C22⋊C4C22×C3112yes—
261#342C22×C3118yes—
262#342C22×C3112yes—
263C2×C31—21no—
264C2×C31—24no—
265C2×C31—21no—
266C2×C31—42no—
267C2×C31#243611yescharacteristic
268C2×C31C2×#45811yescharacteristic
269C2×C31#302811yescharacteristic
270C22×Q8—21no—
271C22×Q8D4×C3112yes—
272C22×D4C23×C3111yescharacteristic
273C23×C4D4×C3111yescharacteristic
274C23×C4C23×C3112yes—
275C23×C4C23×C3111yescharacteristic
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—24no—
279#125—22no—
280#125—24no—
281#123—28no—
282C22⋊Q8—22no—
283C22⋊Q8—22no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286C22⋊Q8—22no—
287C22⋊Q8—22no—
288C22⋊Q8—22no—
289C22⋊Q8—22no—
290C22⋊Q8—22no—
291C22⋊Q8—22no—
292C22⋊Q8—24no—
293C22⋊Q8—24no—
294C4×Q8—22no—
295C4×Q8—22no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—24no—
299C4×D4—24no—
300#117—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4D4×C3111yescharacteristic
310C2×C4⋊C4D4×C3112yes—
311C2×C4⋊C4D4×C3114yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4D4×C3111yescharacteristic
315C2×C22⋊C4C23×C3112yes—
316C2×C22⋊C4C23×C3114yes—
317C2×C22⋊C4C23×C3111yescharacteristic
318C2×C22⋊C4C23×C3114yes—
319C2×C42—22no—
320C2×C42D4×C3111yescharacteristic
321#99—24no—
322#95—22no—
323#95—22no—
324#95—21no—
325#95—28no—
326#95—22no—
327#95—28no—
328C31#2594811yescharacteristic, direct product factor, Sylow
329C24C2×D4×C3111yescharacteristic
330C24C24×C3111yescharacteristic
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—44no—
336C2×Q8—82no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4#41×C3114yes—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4C2×D4×C3112yes—
353C22×C4C2×D4×C3111yescharacteristic
354C22×C4C2×D4×C3111yescharacteristic
355C22×C4C2×D4×C3111yescharacteristic
356C22×C4C2×D4×C3114yes—
357C22×C4C2×D4×C3114yes—
358C22×C4C2×D4×C3111yescharacteristic
359C22×C4C2×D4×C3111yescharacteristic
360C22×C4C2×D4×C3112yes—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—44no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—22no—
380C22⋊C4—24no—
381C22⋊C4—22no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—44no—
386C42—24no—
387C42—21no—
388C42—21no—
389C42—44no—
390C23—21no—
391C23—21no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23C31×#12412yes—
397C23C22×D4×C3111yescharacteristic
398C23C22×D4×C3112yes—
399C23C2×#41×C3112yes—
400C23C31×2−1+412yes—
401Q8—44no—
402Q8—82no—
403Q8—82no—
404Q8—82no—
405D4—28no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—21no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—48no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4C31×#12011yescharacteristic
435C2×C4C31×#12012yes—
436C2×C4C31×#12011yescharacteristic
437C2×C4C31×#12412yes—
438C22—22no—
439C22—21no—
440C22—22no—
441C22—21no—
442C22—22no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—41no—
447C22—41no—
448C22—81no—
449C22C2×C31×#12011yescharacteristic
450C22C2×C31×#12411yescharacteristic
451C22C31×#53611yescharacteristic
452C22C31×#54814yes—
453C4—22no—
454C4—21no—
455C4—22no—
456C4—41no—
457C4—44no—
458C4—44no—
459C4—41no—
460C4—82no—
461C4—84no—
462C4—82no—
463C4C31×#45812yes—
464C2—21no—
465C2—24no—
466C2—21no—
467C2—42no—
468C2C31×#243611yescharacteristic
469C2C2×C31×#45811yescharacteristic
470C2C31×#302811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC24C24×C31
prime radicalC22×D4×C31C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2503C2semidirect
4. C31×#2030C2semidirect
5. C31×#1996C2semidirect
154. #25948C31direct (N × H)
204. #2503C2×C31semidirect
206. #2030C2×C31semidirect
207. #1996C2×C31semidirect
328. C31#25948direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35012
C31×#25032
C31×#24862
C31×#20302
C31×#19962
C31×#18902
C4×C31×#992
#2594831