Finite Group Databasebeta

← C31×#9965 #21466724

Subgroups — C31×#9965 #21466724

Order 7936.

830 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#451C211yesmaximal, characteristic, complemented
2C2×C31×#448C211yesmaximal, characteristic
3C31×#2639C211yesmaximal, characteristic
4C2×C31×#328C211yesmaximal, characteristic, complemented
5C2×C31×#327C211yesmaximal, characteristic
6C31×#1446C211yesmaximal, characteristic, complemented
7C31×#1280C211yesmaximal, characteristic, complemented
8C23×Q8×C31C2211yescharacteristic
9C22×Q16×C31—21no—
10C22×SD16×C31—21no—
11C2×C31×C22⋊Q8C2211yescharacteristic
12C2×C31×#121C2211yescharacteristic
13C31×#451—22no—
14C31×#451—22no—
15C31×#448—22no—
16C31×#448—22no—
17C2×C31×#103—21no—
18C2×C31×#103—21no—
19C2×C31×C22⋊C8C2211yescharacteristic
20C2×C31×C22⋊C8C2211yescharacteristic
21C31×#400—21no—
22C31×#397—21no—
23C31×#394—21no—
24C31×#393—21no—
25C31×#386—22no—
26C2×C31×#95—22no—
27C2×C31×#95C2211yescharacteristic
28C2×C31×#95C2211yescharacteristic
29C31×#328—22no—
30C31×#327—22no—
31C22×Q8×C31—21no—
32C22×Q8×C31—21no—
33C22×Q8×C31—21no—
34C22×Q8×C31—21no—
35C22×Q8×C31—21no—
36C22×Q8×C31—44no—
37C22×Q8×C31D412yes—
38C22×Q8×C31D411yescharacteristic
39C22×D4×C31—21no—
40C22×D4×C31D411yescharacteristic
41C23×C4×C31—21no—
42C23×C4×C31—21no—
43C23×C4×C31D412yes—
44C23×C4×C31C2311yescharacteristic
45C2×Q16×C31—24no—
46C2×Q16×C31—44no—
47C2×SD16×C31—24no—
48C2×SD16×C31—44no—
49C22×C8×C31—21no—
50C22×C8×C31—21no—
51C31×C22⋊Q8—21no—
52C31×C22⋊Q8—21no—
53C31×C22⋊Q8—21no—
54C31×C22⋊Q8—21no—
55C31×#121—21no—
56C31×#121—21no—
57C31×#121—21no—
58C31×#121—21no—
59C2×C4⋊C4×C31—21no—
60C2×C4⋊C4×C31D411yescharacteristic
61C2×C4⋊C4×C31D411yescharacteristic
62C2×C22⋊C4×C31—21no—
63C2×C22⋊C4×C31—21no—
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31—22no—
66C31×#103—24no—
67C31×#103—24no—
68C31×C22⋊C8—21no—
69C31×C22⋊C8—21no—
70C31×C22⋊C8D412yes—
71C31×C22⋊C8D412yes—
72C31×#95—22no—
73C31×#95—22no—
74C31×#95—21no—
75C31×#95—21no—
76C31×#95—21no—
77C31×#95—21no—
78C31×#95—44no—
79C24×C31—21no—
80C24×C31C2×D411yescharacteristic
81C2×Q8×C31—24no—
82C2×Q8×C31—24no—
83C2×Q8×C31—22no—
84C2×Q8×C31—22no—
85C2×Q8×C31—21no—
86C2×Q8×C31—21no—
87C2×Q8×C31—44no—
88C2×Q8×C31—42no—
89C2×Q8×C31—42no—
90C2×Q8×C31—44no—
91C2×Q8×C31—42no—
92C2×Q8×C31—44no—
93C2×Q8×C31—44no—
94C2×Q8×C31—41no—
95C2×Q8×C31—41no—
96C2×D4×C31—21no—
97C2×D4×C31—21no—
98C2×D4×C31—42no—
99C2×D4×C31—41no—
100C2×D4×C31—41no—
101C2×D4×C31—42no—
102C2×D4×C31—42no—
103C22×C4×C31—24no—
104C22×C4×C31—22no—
105C22×C4×C31—22no—
106C22×C4×C31—22no—
107C22×C4×C31—21no—
108C22×C4×C31—21no—
109C22×C4×C31—21no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—42no—
113C22×C4×C31—42no—
114C22×C4×C31—41no—
115C22×C4×C31—41no—
116C22×C4×C31—42no—
117C22×C4×C31—42no—
118C22×C4×C31—42no—
119C22×C4×C31—41no—
120C22×C4×C31—41no—
121C22×C4×C31—42no—
122C22×C4×C31C2×D411yescharacteristic
123C22×C4×C31C2×D411yescharacteristic
124C22×C4×C31C2×D411yescharacteristic
125C22×C4×C31C2×D411yescharacteristic
126C22×C4×C31C2×D411yescharacteristic
127C22×C4×C31#4111yescharacteristic
128Q16×C31—48no—
129SD16×C31—48no—
130C2×C8×C31—22no—
131C2×C8×C31—22no—
132C2×C8×C31—42no—
133C2×C8×C31—42no—
134C4⋊C4×C31—21no—
135C4⋊C4×C31—21no—
136C4⋊C4×C31—21no—
137C4⋊C4×C31—21no—
138C4⋊C4×C31—41no—
139C4⋊C4×C31—41no—
140C22⋊C4×C31—41no—
141C22⋊C4×C31—41no—
142C22⋊C4×C31—41no—
143C22⋊C4×C31—41no—
144C22⋊C4×C31—82no—
145C22⋊C4×C31—82no—
146#9965C3111yesmaximal, characteristic, direct product factor, Sylow
147C23×C31—21no—
148C23×C31—21no—
149C23×C31—22no—
150C23×C31—41no—
151C23×C31—42no—
152C23×C31—41no—
153C23×C31—41no—
154C23×C31—42no—
155C23×C31#12011yescharacteristic
156C23×C31#12111yescharacteristic
157C23×C31#12711yescharacteristic
158C23×C31#13512yes—
159C23×C31#13712yes—
160Q8×C31—24no—
161Q8×C31—24no—
162Q8×C31—42no—
163Q8×C31—42no—
164Q8×C31—44no—
165Q8×C31—44no—
166Q8×C31—42no—
167Q8×C31—81no—
168D4×C31—42no—
169D4×C31—81no—
170D4×C31—82no—
171D4×C31—81no—
172D4×C31—81no—
173C2×C4×C31—22no—
174C2×C4×C31—22no—
175C2×C4×C31—24no—
176C2×C4×C31—41no—
177C2×C4×C31—41no—
178C2×C4×C31—41no—
179C2×C4×C31—41no—
180C2×C4×C31—42no—
181C2×C4×C31—44no—
182C2×C4×C31—42no—
183C2×C4×C31—44no—
184C2×C4×C31—44no—
185C2×C4×C31—42no—
186C2×C4×C31—44no—
187C2×C4×C31—41no—
188C2×C4×C31—41no—
189C2×C4×C31—41no—
190C2×C4×C31—42no—
191C2×C4×C31—42no—
192C2×C4×C31—41no—
193C2×C4×C31—42no—
194C2×C4×C31—41no—
195C2×C4×C31—42no—
196C2×C4×C31—42no—
197C2×C4×C31—82no—
198C2×C4×C31—81no—
199C2×C4×C31—81no—
200C2×C4×C31—82no—
201C2×C4×C31#12011yescharacteristic
202C2×C4×C31#12011yescharacteristic
203C2×C4×C31#12111yescharacteristic
204C2×C4×C31#12111yescharacteristic
205C8×C31—42no—
206C8×C31—42no—
207C2×#451C2×C3111yescharacteristic, complemented
208C2×#448C2×C3111yescharacteristic
209#2639C2×C3111yescharacteristic
210C2×#328C2×C3111yescharacteristic, complemented
211C2×#327C2×C3111yescharacteristic
212#1446C2×C3111yescharacteristic, complemented
213#1280C2×C3111yescharacteristic, complemented
214C22×C31—22no—
215C22×C31—22no—
216C22×C31—22no—
217C22×C31—22no—
218C22×C31—21no—
219C22×C31—21no—
220C22×C31—21no—
221C22×C31—21no—
222C22×C31—82no—
223C22×C31—82no—
224C22×C31—81no—
225C22×C31—81no—
226C22×C31—81no—
227C22×C31#39211yescharacteristic
228C22×C31#45212yes—
229C22×C31#45511yescharacteristic
230C22×C31#45611yescharacteristic
231C22×C31B(4,𝔽2)12yes—
232C22×C31#46412yes—
233C22×C31#49511yescharacteristic
234C22×C31#49711yescharacteristic
235C4×C31—21no—
236C4×C31—21no—
237C4×C31—21no—
238C4×C31—21no—
239C4×C31—44no—
240C4×C31—82no—
241C4×C31—82no—
242C4×C31—81no—
243C4×C31—81no—
244C4×C31—81no—
245C23×Q8C22×C3111yescharacteristic
246C22×Q16—21no—
247C22×SD16—21no—
248C2×C22⋊Q8C22×C3111yescharacteristic
249C2×#121C22×C3111yescharacteristic
250#451—22no—
251#451—22no—
252#448—22no—
253#448—22no—
254C2×#103—21no—
255C2×#103—21no—
256C2×C22⋊C8C22×C3111yescharacteristic
257C2×C22⋊C8C22×C3111yescharacteristic
258#400—21no—
259#397—21no—
260#394—21no—
261#393—21no—
262#386—22no—
263C2×#95—22no—
264C2×#95C22×C3111yescharacteristic
265C2×#95C22×C3111yescharacteristic
266#328—22no—
267#327—22no—
268C2×C31—21no—
269C2×C31—22no—
270C2×C31—21no—
271C2×C31—161no—
272C2×C31#160312yes—
273C2×C31#160612yes—
274C2×C31#200711yescharacteristic
275C2×C31#201511yescharacteristic
276C2×C31#202111yescharacteristic
277C22×Q8—21no—
278C22×Q8—21no—
279C22×Q8—21no—
280C22×Q8—21no—
281C22×Q8—21no—
282C22×Q8—44no—
283C22×Q8D4×C3111yescharacteristic
284C22×Q8D4×C3112yes—
285C22×D4—21no—
286C22×D4D4×C3111yescharacteristic
287C23×C4—21no—
288C23×C4—21no—
289C23×C4D4×C3112yes—
290C23×C4C23×C3111yescharacteristic
291C2×Q16—24no—
292C2×Q16—44no—
293C2×SD16—24no—
294C2×SD16—44no—
295C22×C8—21no—
296C22×C8—21no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C22⋊Q8—21no—
300C22⋊Q8—21no—
301#121—21no—
302#121—21no—
303#121—21no—
304#121—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4D4×C3111yescharacteristic
307C2×C4⋊C4D4×C3111yescharacteristic
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312#103—24no—
313#103—24no—
314C22⋊C8—21no—
315C22⋊C8—21no—
316C22⋊C8D4×C3112yes—
317C22⋊C8D4×C3112yes—
318#95—22no—
319#95—22no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—44no—
325C31#996511yescharacteristic, direct product factor, Sylow
326C24—21no—
327C24C2×D4×C3111yescharacteristic
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—44no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—44no—
338C2×Q8—42no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—42no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—42no—
349C2×D4—42no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4C2×D4×C3111yescharacteristic
370C22×C4C2×D4×C3111yescharacteristic
371C22×C4C2×D4×C3111yescharacteristic
372C22×C4C2×D4×C3111yescharacteristic
373C22×C4C2×D4×C3111yescharacteristic
374C22×C4#41×C3111yescharacteristic
375Q16—48no—
376SD16—48no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—42no—
380C2×C8—42no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C23—21no—
394C23—21no—
395C23—22no—
396C23—42no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—41no—
401C23C31×#12011yescharacteristic
402C23C31×#12111yescharacteristic
403C23C31×#12711yescharacteristic
404C23C31×#13512yes—
405C23C31×#13712yes—
406Q8—24no—
407Q8—24no—
408Q8—42no—
409Q8—42no—
410Q8—44no—
411Q8—44no—
412Q8—42no—
413Q8—81no—
414D4—42no—
415D4—81no—
416D4—82no—
417D4—81no—
418D4—81no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4C31×#12011yescharacteristic
448C2×C4C31×#12011yescharacteristic
449C2×C4C31×#12111yescharacteristic
450C2×C4C31×#12111yescharacteristic
451C8—42no—
452C8—42no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—82no—
462C22—82no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22C31×#39211yescharacteristic
467C22C31×#45212yes—
468C22C31×#45511yescharacteristic
469C22C31×#45611yescharacteristic
470C22C31×B(4,𝔽2)12yes—
471C22C31×#46412yes—
472C22C31×#49511yescharacteristic
473C22C31×#49711yescharacteristic
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—44no—
479C4—82no—
480C4—82no—
481C4—81no—
482C4—81no—
483C4—81no—
484C2—21no—
485C2—22no—
486C2—21no—
487C2—161no—
488C2C31×#160312yes—
489C2C31×#160612yes—
490C2C31×#200711yescharacteristic
491C2C31×#201511yescharacteristic
492C2C31×#202111yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC2×C31×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#451C2semidirect
4. C2×C31×#328C2semidirect
6. C31×#1446C2semidirect
7. C31×#1280C2semidirect
146. #9965C31direct (N × H)
207. C2×#451C2×C31semidirect
210. C2×#328C2×C31semidirect
212. #1446C2×C31semidirect
213. #1280C2×C31semidirect
325. C31#9965direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4512
C2×C31×#4482
C31×#26392
C2×C31×#3282
C2×C31×#3272
C31×#14462
C31×#12802
#996531