Finite Group Databasebeta

← C2×C31×#1599 #21474869

Subgroups — C2×C31×#1599 #21474869

Order 7936.

1372 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#544C211yesmaximal, characteristic, complemented
2C2×C31×#450C212yesmaximal
3C2×C31×#394C211yesmaximal, characteristic
4C31×#1599C218yesmaximal, direct product factor
5C2×C31×#328C212yesmaximal, complemented
6C2×C31×#324C211yesmaximal, characteristic, complemented
7C23×D4×C31C2211yescharacteristic
8C22×SD16×C31—22no—
9C31×#544—24no—
10C31×#544—22no—
11C31×#544C2214yescomplemented
12C2×C31×C4⋊Q8—22no—
13C2×C31×#125—22no—
14C2×C31×C22⋊Q8—21no—
15C2×C31×C22⋊Q8C2212yes—
16C2×C31×#117—21no—
17C22×C4⋊C4×C31C2211yescharacteristic
18C22×C22⋊C4×C31—21no—
19C22×C22⋊C4×C31—21no—
20C31×#450—24no—
21C31×#450C2218yes—
22C2×C31×#102—22no—
23C2×C31×C22⋊C8C2212yes—
24C31×#394—22no—
25C31×#394C2214yes—
26C2×C31×#95C2211yescharacteristic
27C31×#328C2218yescomplemented
28C31×#324C2214yescomplemented
29C25×C31—21no—
30C22×Q8×C31—21no—
31C22×Q8×C31D412yes—
32C22×D4×C31—22no—
33C22×D4×C31—22no—
34C22×D4×C31—21no—
35C22×D4×C31—44no—
36C22×D4×C31C2312yes—
37C23×C4×C31—21no—
38C23×C4×C31D411yescharacteristic
39C23×C4×C31D411yescharacteristic
40C23×C4×C31C2311yescharacteristic
41C2×SD16×C31—28no—
42C2×SD16×C31—48no—
43C22×C8×C31—22no—
44C31×C4⋊Q8—28no—
45C31×C4⋊Q8—28no—
46C31×#125—28no—
47C31×#125—28no—
48C31×C22⋊Q8—24no—
49C31×C22⋊Q8—24no—
50C31×C22⋊Q8—22no—
51C31×C22⋊Q8—22no—
52C31×C22⋊Q8D414yes—
53C31×C22⋊Q8C2314yes—
54C31×#117—24no—
55C31×#117—24no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31—22no—
63C2×C4⋊C4×C31D412yes—
64C2×C4⋊C4×C31D412yes—
65C2×C4⋊C4×C31C2312yes—
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31—24no—
68C2×C22⋊C4×C31—24no—
69C2×C22⋊C4×C31—44no—
70C2×C22⋊C4×C31—44no—
71C2×C42×C31—22no—
72C31×#102—28no—
73C31×C22⋊C8—22no—
74C31×C22⋊C8C2314yes—
75C31×#95—21no—
76C31×#95C2312yes—
77C24×C31—24no—
78C24×C31—22no—
79C24×C31—42no—
80C24×C31—42no—
81C24×C31—42no—
82C24×C31—41no—
83C24×C31—41no—
84C24×C31—42no—
85C24×C31—42no—
86C24×C31C2×D411yescharacteristic
87C2×Q8×C31—24no—
88C2×Q8×C31—22no—
89C2×Q8×C31—44no—
90C2×Q8×C31—82no—
91C2×Q8×C31C2×D414yes—
92C2×D4×C31—28no—
93C2×D4×C31—24no—
94C2×D4×C31—44no—
95C2×D4×C31—44no—
96C2×D4×C31—48no—
97C2×D4×C31—44no—
98C2×D4×C31—42no—
99C22×C4×C31—21no—
100C22×C4×C31—21no—
101C22×C4×C31—22no—
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—21no—
105C22×C4×C31—21no—
106C22×C4×C31—22no—
107C22×C4×C31—42no—
108C22×C4×C31—42no—
109C22×C4×C31—42no—
110C22×C4×C31—42no—
111C22×C4×C31—42no—
112C22×C4×C31SD1614yescomplemented
113C22×C4×C31C2×D412yes—
114C22×C4×C31C2×D412yes—
115C22×C4×C31C2×D412yes—
116C22×C4×C31C2×D411yescharacteristic
117C22×C4×C31C2×D412yes—
118C22×C4×C31C2411yescharacteristic
119SD16×C31—416no—
120C2×C8×C31—24no—
121C2×C8×C31—44no—
122C4⋊C4×C31—24no—
123C4⋊C4×C31—24no—
124C4⋊C4×C31—24no—
125C4⋊C4×C31—28no—
126C4⋊C4×C31—24no—
127C4⋊C4×C31—24no—
128C4⋊C4×C31—22no—
129C4⋊C4×C31—24no—
130C4⋊C4×C31—24no—
131C4⋊C4×C31C2×D414yes—
132C22⋊C4×C31—24no—
133C22⋊C4×C31—24no—
134C22⋊C4×C31—48no—
135C22⋊C4×C31—48no—
136C42×C31—28no—
137C2×#1599C3111yesmaximal, characteristic, direct product factor, Sylow
138C23×C31—22no—
139C23×C31—22no—
140C23×C31—24no—
141C23×C31—42no—
142C23×C31—44no—
143C23×C31—48no—
144C23×C31—48no—
145C23×C31—42no—
146C23×C31—42no—
147C23×C31—42no—
148C23×C31—42no—
149C23×C31—42no—
150C23×C31—44no—
151C23×C31—84no—
152C23×C31—84no—
153C23×C31—82no—
154C23×C31—82no—
155C23×C31—82no—
156C23×C31—82no—
157C23×C31#12011yescharacteristic
158C23×C31#12011yescharacteristic
159C23×C31C2×SD1612yes—
160C23×C31#13712yes—
161C23×C31C22×D411yescharacteristic
162Q8×C31—44no—
163Q8×C31—82no—
164Q8×C31—84no—
165D4×C31—28no—
166D4×C31—44no—
167D4×C31—48no—
168C2×C4×C31—22no—
169C2×C4×C31—22no—
170C2×C4×C31—24no—
171C2×C4×C31—24no—
172C2×C4×C31—24no—
173C2×C4×C31—24no—
174C2×C4×C31—24no—
175C2×C4×C31—22no—
176C2×C4×C31—42no—
177C2×C4×C31—41no—
178C2×C4×C31—41no—
179C2×C4×C31—44no—
180C2×C4×C31—42no—
181C2×C4×C31—44no—
182C2×C4×C31—42no—
183C2×C4×C31—42no—
184C2×C4×C31—42no—
185C2×C4×C31—42no—
186C2×C4×C31—81no—
187C2×C4×C31—84no—
188C2×C4×C31#12012yes—
189C2×C4×C31C2×SD1614yescomplemented
190C2×C4×C31C22×D412yes—
191C8×C31—44no—
192C2×#544C2×C3111yescharacteristic, complemented
193C2×#450C2×C3112yes—
194C2×#394C2×C3111yescharacteristic
195#1599C2×C3118yesdirect product factor
196C2×#328C2×C3112yescomplemented
197C2×#324C2×C3111yescharacteristic, complemented
198C22×C31—24no—
199C22×C31—24no—
200C22×C31—21no—
201C22×C31—21no—
202C22×C31—21no—
203C22×C31—21no—
204C22×C31—44no—
205C22×C31—44no—
206C22×C31—44no—
207C22×C31—82no—
208C22×C31—82no—
209C22×C31—88no—
210C22×C31—88no—
211C22×C31—82no—
212C22×C31—82no—
213C22×C31#44812yes—
214C22×C31#44812yes—
215C22×C31#45412yes—
216C22×C31B(4,𝔽2)12yes—
217C22×C31C2×#12011yescharacteristic
218C22×C31C22×SD1611yescharacteristic
219C22×C31C2×#13711yescharacteristic
220C4×C31—24no—
221C4×C31—22no—
222C4×C31—22no—
223C4×C31—44no—
224C4×C31—42no—
225C4×C31—44no—
226C4×C31—82no—
227C23×D4C22×C3111yescharacteristic
228C22×SD16—22no—
229#544—24no—
230#544—22no—
231#544C22×C3114yescomplemented
232C2×C4⋊Q8—22no—
233C2×#125—22no—
234C2×C22⋊Q8—21no—
235C2×C22⋊Q8C22×C3112yes—
236C2×#117—21no—
237C22×C4⋊C4C22×C3111yescharacteristic
238C22×C22⋊C4—21no—
239C22×C22⋊C4—21no—
240#450—24no—
241#450C22×C3118yes—
242C2×#102—22no—
243C2×C22⋊C8C22×C3112yes—
244#394—22no—
245#394C22×C3114yes—
246C2×#95C22×C3111yescharacteristic
247#328C22×C3118yescomplemented
248#324C22×C3114yescomplemented
249C2×C31—21no—
250C2×C31—21no—
251C2×C31—22no—
252C2×C31—84no—
253C2×C31#159914yesdirect product factor
254C2×C31C2×#44811yescharacteristic
255C2×C31C2×#45411yescharacteristic
256C2×C31C2×B(4,𝔽2)11yescharacteristic
257C25—21no—
258C22×Q8—21no—
259C22×Q8D4×C3112yes—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—21no—
263C22×D4—44no—
264C22×D4C23×C3112yes—
265C23×C4—21no—
266C23×C4D4×C3111yescharacteristic
267C23×C4D4×C3111yescharacteristic
268C23×C4C23×C3111yescharacteristic
269C2×SD16—28no—
270C2×SD16—48no—
271C22×C8—22no—
272C4⋊Q8—28no—
273C4⋊Q8—28no—
274#125—28no—
275#125—28no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—22no—
279C22⋊Q8—22no—
280C22⋊Q8D4×C3114yes—
281C22⋊Q8C23×C3114yes—
282#117—24no—
283#117—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4D4×C3112yes—
292C2×C4⋊C4D4×C3112yes—
293C2×C4⋊C4C23×C3112yes—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C42—22no—
300#102—28no—
301C22⋊C8—22no—
302C22⋊C8C23×C3114yes—
303#95—21no—
304#95C23×C3112yes—
305C31C2×#159911yescharacteristic, direct product factor, Sylow
306C24—24no—
307C24—22no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—41no—
312C24—41no—
313C24—42no—
314C24—42no—
315C24C2×D4×C3111yescharacteristic
316C2×Q8—24no—
317C2×Q8—22no—
318C2×Q8—44no—
319C2×Q8—82no—
320C2×Q8C2×D4×C3114yes—
321C2×D4—28no—
322C2×D4—24no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—48no—
326C2×D4—44no—
327C2×D4—42no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4SD16×C3114yescomplemented
342C22×C4C2×D4×C3112yes—
343C22×C4C2×D4×C3112yes—
344C22×C4C2×D4×C3112yes—
345C22×C4C2×D4×C3111yescharacteristic
346C22×C4C2×D4×C3112yes—
347C22×C4C24×C3111yescharacteristic
348SD16—416no—
349C2×C8—24no—
350C2×C8—44no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—28no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4C2×D4×C3114yes—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—48no—
364C22⋊C4—48no—
365C42—28no—
366C23—22no—
367C23—22no—
368C23—24no—
369C23—42no—
370C23—44no—
371C23—48no—
372C23—48no—
373C23—42no—
374C23—42no—
375C23—44no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—84no—
380C23—84no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23C31×#12011yescharacteristic
386C23C31×#12011yescharacteristic
387C23C2×SD16×C3112yes—
388C23C31×#13712yes—
389C23C22×D4×C3111yescharacteristic
390Q8—44no—
391Q8—82no—
392Q8—84no—
393D4—28no—
394D4—44no—
395D4—48no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—81no—
415C2×C4—84no—
416C2×C4C31×#12012yes—
417C2×C4C2×SD16×C3114yescomplemented
418C2×C4C22×D4×C3112yes—
419C8—44no—
420C22—24no—
421C22—24no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—44no—
427C22—44no—
428C22—44no—
429C22—82no—
430C22—82no—
431C22—88no—
432C22—88no—
433C22—82no—
434C22—82no—
435C22C31×#44812yes—
436C22C31×#44812yes—
437C22C31×#45412yes—
438C22C31×B(4,𝔽2)12yes—
439C22C2×C31×#12011yescharacteristic
440C22C22×SD16×C3111yescharacteristic
441C22C2×C31×#13711yescharacteristic
442C4—24no—
443C4—22no—
444C4—22no—
445C4—44no—
446C4—42no—
447C4—44no—
448C4—82no—
449C2—21no—
450C2—21no—
451C2—22no—
452C2—84no—
453C2C31×#159914yesdirect product factor
454C2C2×C31×#44811yescharacteristic
455C2C2×C31×#45411yescharacteristic
456C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC23×D4×C31C22
LCSG ▷ C22×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×#544C2semidirect
4. C31×#1599C2direct (N × H)
5. C2×C31×#328C2semidirect
6. C2×C31×#324C2semidirect
11. C31×#544C22semidirect
27. C31×#328C22semidirect
28. C31×#324C22semidirect
112. C22×C4×C31SD16semidirect
137. C2×#1599C31direct (N × H)
189. C2×C4×C31C2×SD16semidirect
192. C2×#544C2×C31semidirect
195. #1599C2×C31direct (N × H)
196. C2×#328C2×C31semidirect
197. C2×#324C2×C31semidirect
231. #544C22×C31semidirect
247. #328C22×C31semidirect
248. #324C22×C31semidirect
253. C2×C31#1599direct (N × H)
305. C31C2×#1599direct (N × H)
341. C22×C4SD16×C31semidirect
417. C2×C4C2×SD16×C31semidirect
453. C2C31×#1599direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5442
C2×C31×#4502
C2×C31×#3942
C31×#15992
C2×C31×#3282
C2×C31×#3242
C2×#159931