Finite Group Databasebeta

← C31×#13222 #21469981

Subgroups — C31×#13222 #21469981

Order 7936.

628 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2534C211yesmaximal, characteristic
2C2×C31×#352C212yesmaximal, complemented
3C2×C31×AΣL(1,𝔽16)C212yesmaximal
4C31×#1800C211yesmaximal, characteristic, complemented
5C4×C31×#99C211yesmaximal, characteristic, complemented
6C2×C31×#123C2211yescharacteristic
7C2×C31×#120C2211yescharacteristic
8C2×C4×D4×C31C2211yescharacteristic
9C22×C22⋊C4×C31C411yescharacteristic, complemented
10C31×#423—21no—
11C2×C31×#100C2212yes—
12C2×C31×#99C2212yes—
13C31×#396—21no—
14C31×#394—21no—
15C31×#388—21no—
16C31×#386—21no—
17C31×#381C411yescharacteristic, complemented
18C31×#380C411yescharacteristic, complemented
19C31×#379C411yescharacteristic, complemented
20C4×C22⋊C4×C31—21no—
21C31×#352—24no—
22C31×AΣL(1,𝔽16)—24no—
23C31×#343—21no—
24C31×#342—21no—
25C25×C31C2×C411yescharacteristic
26C22×D4×C31—21no—
27C22×D4×C31C2311yescharacteristic
28C23×C4×C31C2×C411yescharacteristic
29C23×C4×C31C2×C411yescharacteristic
30C23×C4×C31D411yescharacteristic
31C2×M16×C31—22no—
32C31×#123—22no—
33C31×#123—22no—
34C31×#120—22no—
35C31×#120—22no—
36C4×D4×C31—41no—
37C4×D4×C31—41no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31D411yescharacteristic
41C2×C22⋊C4×C31—22no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—21no—
48C2×C22⋊C4×C31—41no—
49C2×C22⋊C4×C31—41no—
50C2×C22⋊C4×C31C2×C411yescharacteristic
51C2×C22⋊C4×C31C2×C411yescharacteristic
52C2×C22⋊C4×C31C2×C411yescharacteristic
53C2×C22⋊C4×C31D411yescharacteristic
54C2×C42×C31—41no—
55C2×C42×C31D411yescharacteristic
56C31×C4⋊C8—41no—
57C31×C4⋊C8—41no—
58C31×#100—22no—
59C31×#100—22no—
60C31×#99—22no—
61C31×#99D414yes—
62C31×#95—21no—
63C31×#95—21no—
64C31×#95—21no—
65C31×#95—41no—
66C24×C31—21no—
67C24×C31—22no—
68C24×C31—41no—
69C24×C31—42no—
70C24×C31—41no—
71C24×C31—81no—
72C24×C31C22×C411yescharacteristic
73C24×C31C2×D411yescharacteristic
74C2×D4×C31—41no—
75C2×D4×C31—41no—
76C2×D4×C31—42no—
77C2×D4×C31—42no—
78C2×D4×C31—81no—
79C2×D4×C31C2×D412yes—
80C2×D4×C31#4112yes—
81C22×C4×C31—21no—
82C22×C4×C31—21no—
83C22×C4×C31—21no—
84C22×C4×C31—21no—
85C22×C4×C31—21no—
86C22×C4×C31—21no—
87C22×C4×C31—21no—
88C22×C4×C31—22no—
89C22×C4×C31—42no—
90C22×C4×C31—41no—
91C22×C4×C31—41no—
92C22×C4×C31—81no—
93C22×C4×C31—81no—
94C22×C4×C31C22⋊C411yescharacteristic
95C22×C4×C31C22⋊C411yescharacteristic
96C22×C4×C31C22⋊C411yescharacteristic
97C22×C4×C31C22⋊C411yescharacteristic
98C22×C4×C31C2×D411yescharacteristic
99M16×C31—42no—
100M16×C31—42no—
101C2×C8×C31—42no—
102C4⋊C4×C31—41no—
103C4⋊C4×C31—41no—
104C4⋊C4×C31—41no—
105C4⋊C4×C31—81no—
106C22⋊C4×C31—22no—
107C22⋊C4×C31—22no—
108C22⋊C4×C31—24no—
109C22⋊C4×C31—41no—
110C22⋊C4×C31—41no—
111C22⋊C4×C31—42no—
112C22⋊C4×C31—42no—
113C22⋊C4×C31—42no—
114C22⋊C4×C31—42no—
115C22⋊C4×C31—41no—
116C22⋊C4×C31—41no—
117C22⋊C4×C31—41no—
118C22⋊C4×C31—41no—
119C22⋊C4×C31—81no—
120C42×C31—21no—
121C42×C31—21no—
122C42×C31—82no—
123#13222C3111yesmaximal, characteristic, direct product factor, Sylow
124C23×C31—21no—
125C23×C31—21no—
126C23×C31—21no—
127C23×C31—21no—
128C23×C31—22no—
129C23×C31—41no—
130C23×C31—41no—
131C23×C31—41no—
132C23×C31—41no—
133C23×C31—41no—
134C23×C31—41no—
135C23×C31—41no—
136C23×C31—42no—
137C23×C31—42no—
138C23×C31—41no—
139C23×C31—41no—
140C23×C31—41no—
141C23×C31—82no—
142C23×C31—84no—
143C23×C31—84no—
144C23×C31—82no—
145C23×C31C2×C22⋊C411yescharacteristic
146C23×C31C4×D412yes—
147C23×C31#12112yes—
148D4×C31—42no—
149D4×C31—42no—
150D4×C31—84no—
151C2×C4×C31—21no—
152C2×C4×C31—21no—
153C2×C4×C31—21no—
154C2×C4×C31—21no—
155C2×C4×C31—21no—
156C2×C4×C31—41no—
157C2×C4×C31—42no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—44no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—41no—
169C2×C4×C31—41no—
170C2×C4×C31—41no—
171C2×C4×C31—81no—
172C2×C4×C31—81no—
173C2×C4×C31—81no—
174C2×C4×C31—82no—
175C2×C4×C31—82no—
176C2×C4×C31—82no—
177C2×C4×C31#9911yescharacteristic
178C2×C4×C31#9911yescharacteristic
179C2×C4×C31#12411yescharacteristic
180C2×C4×C31#12711yescharacteristic
181C8×C31—82no—
182#2534C2×C3111yescharacteristic
183C2×#352C2×C3112yescomplemented
184C2×AΣL(1,𝔽16)C2×C3112yes—
185#1800C2×C3111yescharacteristic, complemented
186C4×#99C2×C3111yescharacteristic, complemented
187C22×C31—21no—
188C22×C31—21no—
189C22×C31—22no—
190C22×C31—22no—
191C22×C31—42no—
192C22×C31—42no—
193C22×C31—42no—
194C22×C31—42no—
195C22×C31—42no—
196C22×C31—41no—
197C22×C31—41no—
198C22×C31—41no—
199C22×C31—41no—
200C22×C31—42no—
201C22×C31—81no—
202C22×C31—81no—
203C22×C31—81no—
204C22×C31—84no—
205C22×C31—84no—
206C22×C31—81no—
207C22×C31—81no—
208C22×C31#39011yescharacteristic
209C22×C31C2×#9911yescharacteristic
210C22×C31#41011yescharacteristic
211C4×C31—21no—
212C4×C31—21no—
213C4×C31—21no—
214C4×C31—21no—
215C4×C31—41no—
216C4×C31—41no—
217C4×C31—41no—
218C4×C31—81no—
219C4×C31—81no—
220C4×C31—84no—
221C4×C31—81no—
222C2×#123C22×C3111yescharacteristic
223C2×#120C22×C3111yescharacteristic
224C2×C4×D4C22×C3111yescharacteristic
225C22×C22⋊C4C4×C3111yescharacteristic, complemented
226#423—21no—
227C2×#100C22×C3112yes—
228C2×#99C22×C3112yes—
229#396—21no—
230#394—21no—
231#388—21no—
232#386—21no—
233#381C4×C3111yescharacteristic, complemented
234#380C4×C3111yescharacteristic, complemented
235#379C4×C3111yescharacteristic, complemented
236C4×C22⋊C4—21no—
237#352—24no—
238AΣL(1,𝔽16)—24no—
239#343—21no—
240#342—21no—
241C2×C31—21no—
242C2×C31—21no—
243C2×C31—42no—
244C2×C31—42no—
245C2×C31—82no—
246C2×C31#196411yescharacteristic
247C2×C31#212011yescharacteristic
248C2×C31#212111yescharacteristic
249C25C2×C4×C3111yescharacteristic
250C22×D4—21no—
251C22×D4C23×C3111yescharacteristic
252C23×C4C2×C4×C3111yescharacteristic
253C23×C4C2×C4×C3111yescharacteristic
254C23×C4D4×C3111yescharacteristic
255C2×M16—22no—
256#123—22no—
257#123—22no—
258#120—22no—
259#120—22no—
260C4×D4—41no—
261C4×D4—41no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4D4×C3111yescharacteristic
265C2×C22⋊C4—22no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4C2×C4×C3111yescharacteristic
275C2×C22⋊C4C2×C4×C3111yescharacteristic
276C2×C22⋊C4C2×C4×C3111yescharacteristic
277C2×C22⋊C4D4×C3111yescharacteristic
278C2×C42—41no—
279C2×C42D4×C3111yescharacteristic
280C4⋊C8—41no—
281C4⋊C8—41no—
282#100—22no—
283#100—22no—
284#99—22no—
285#99D4×C3114yes—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—41no—
290C31#1322211yescharacteristic, direct product factor, Sylow
291C24—21no—
292C24—22no—
293C24—41no—
294C24—42no—
295C24—41no—
296C24—81no—
297C24C22×C4×C3111yescharacteristic
298C24C2×D4×C3111yescharacteristic
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—81no—
304C2×D4C2×D4×C3112yes—
305C2×D4#41×C3112yes—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4C22⋊C4×C3111yescharacteristic
320C22×C4C22⋊C4×C3111yescharacteristic
321C22×C4C22⋊C4×C3111yescharacteristic
322C22×C4C22⋊C4×C3111yescharacteristic
323C22×C4C2×D4×C3111yescharacteristic
324M16—42no—
325M16—42no—
326C2×C8—42no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—81no—
331C22⋊C4—22no—
332C22⋊C4—22no—
333C22⋊C4—24no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—81no—
345C42—21no—
346C42—21no—
347C42—82no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—22no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—42no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—82no—
366C23—84no—
367C23—84no—
368C23—82no—
369C23C2×C22⋊C4×C3111yescharacteristic
370C23C4×D4×C3112yes—
371C23C31×#12112yes—
372D4—42no—
373D4—42no—
374D4—84no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—44no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4C31×#9911yescharacteristic
402C2×C4C31×#9911yescharacteristic
403C2×C4C31×#12411yescharacteristic
404C2×C4C31×#12711yescharacteristic
405C8—82no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—42no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—84no—
424C22—84no—
425C22—81no—
426C22—81no—
427C22C31×#39011yescharacteristic
428C22C2×C31×#9911yescharacteristic
429C22C31×#41011yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—81no—
438C4—81no—
439C4—84no—
440C4—81no—
441C2—21no—
442C2—21no—
443C2—42no—
444C2—42no—
445C2—82no—
446C2C31×#196411yescharacteristic
447C2C31×#212011yescharacteristic
448C2C31×#212111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC24C22×C4×C31
Frattini, cosocleC22×D4C23×C31
prime radicalC2×C31×#120C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#352C2semidirect
4. C31×#1800C2semidirect
5. C4×C31×#99C2semidirect
9. C22×C22⋊C4×C31C4semidirect
17. C31×#381C4semidirect
18. C31×#380C4semidirect
19. C31×#379C4semidirect
123. #13222C31direct (N × H)
183. C2×#352C2×C31semidirect
185. #1800C2×C31semidirect
186. C4×#99C2×C31semidirect
225. C22×C22⋊C4C4×C31semidirect
233. #381C4×C31semidirect
234. #380C4×C31semidirect
235. #379C4×C31semidirect
290. C31#13222direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25342
C2×C31×#3522
C2×C31×AΣL(1,𝔽16)2
C31×#18002
C4×C31×#992
#1322231