Finite Group Databasebeta

← C31×#11725 #21468484

Subgroups — C31×#11725 #21468484

Order 7936.

648 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2388C211yesmaximal, characteristic
2C4×C31×#99C211yesmaximal, characteristic, complemented
3C31×#1460C211yesmaximal, characteristic, complemented
4C31×#1351C212yesmaximal, complemented
5C31×#1343C212yesmaximal, complemented
6C22×#41×C31C411yescharacteristic, complemented
7C2×C4×D4×C31—21no—
8C2×C4×D4×C31C2211yescharacteristic
9C31×#431—21no—
10C2×C31×#99—21no—
11C31×#395C411yescharacteristic, complemented
12C31×#394—21no—
13C31×#394C411yescharacteristic, complemented
14C31×#393C2211yescharacteristic
15C31×#392—21no—
16C31×#392C2211yescharacteristic
17C4×C22⋊C4×C31—21no—
18C4×C22⋊C4×C31—21no—
19C4×C22⋊C4×C31C411yescharacteristic, complemented
20C31×#342C2212yes—
21C31×#323—21no—
22C31×#323C2212yes—
23C2×#41×C31—44no—
24C2×#41×C31—41no—
25C2×#41×C31—41no—
26C22×Q8×C31C2×C411yescharacteristic
27C22×D4×C31—21no—
28C22×D4×C31—21no—
29C22×D4×C31C2×C411yescharacteristic
30C22×D4×C31D411yescharacteristic
31C23×C4×C31—21no—
32C23×C4×C31C2×C411yescharacteristic
33C23×C4×C31D411yescharacteristic
34C4×D4×C31—41no—
35C4×D4×C31—41no—
36C4×D4×C31—44no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31C2×C411yescharacteristic
39C2×C22⋊C4×C31—22no—
40C2×C22⋊C4×C31—21no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31D411yescharacteristic
46C2×C22⋊C4×C31C2311yescharacteristic
47C2×C42×C31—21no—
48C2×C42×C31—41no—
49C2×C42×C31D411yescharacteristic
50C31×C4⋊C8—41no—
51C31×#99—24no—
52C31×C22⋊C8—22no—
53C4×C8×C31—41no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—41no—
57C31×#95C2×C411yescharacteristic
58C31×#95C2×C411yescharacteristic
59C24×C31—21no—
60C24×C31C22⋊C411yescharacteristic
61C24×C31C2×D411yescharacteristic
62#41×C31—44no—
63#41×C31—44no—
64#41×C31—44no—
65#41×C31—44no—
66C2×Q8×C31—42no—
67C2×Q8×C31—41no—
68C2×D4×C31—22no—
69C2×D4×C31—22no—
70C2×D4×C31—41no—
71C2×D4×C31—42no—
72C2×D4×C31—42no—
73C2×D4×C31—41no—
74C2×D4×C31—42no—
75C2×D4×C31—42no—
76C2×D4×C31—42no—
77C2×D4×C31—42no—
78C2×D4×C31—42no—
79C22×C4×C31—24no—
80C22×C4×C31—21no—
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—42no—
88C22×C4×C31—42no—
89C22×C4×C31—42no—
90C22×C4×C31—41no—
91C22×C4×C31—41no—
92C22×C4×C31—42no—
93C22×C4×C31C22⋊C411yescharacteristic
94C22×C4×C31C22⋊C411yescharacteristic
95C22×C4×C31C22⋊C411yescharacteristic
96C22×C4×C31C22×C411yescharacteristic
97C22×C4×C31C2×D411yescharacteristic
98C2×C8×C31—42no—
99C4⋊C4×C31—41no—
100C4⋊C4×C31—42no—
101C22⋊C4×C31—24no—
102C22⋊C4×C31—41no—
103C22⋊C4×C31—41no—
104C22⋊C4×C31—42no—
105C22⋊C4×C31—42no—
106C22⋊C4×C31—42no—
107C22⋊C4×C31—42no—
108C22⋊C4×C31—42no—
109C22⋊C4×C31—42no—
110C42×C31—21no—
111C42×C31—21no—
112C42×C31—42no—
113C42×C31—82no—
114#11725C3111yesmaximal, characteristic, direct product factor, Sylow
115C23×C31—21no—
116C23×C31—21no—
117C23×C31—22no—
118C23×C31—21no—
119C23×C31—21no—
120C23×C31—21no—
121C23×C31—41no—
122C23×C31—41no—
123C23×C31—41no—
124C23×C31—41no—
125C23×C31—41no—
126C23×C31—41no—
127C23×C31—41no—
128C23×C31—41no—
129C23×C31—41no—
130C23×C31—41no—
131C23×C31C2×C22⋊C411yescharacteristic
132C23×C31#13512yes—
133Q8×C31—42no—
134Q8×C31—42no—
135D4×C31—44no—
136D4×C31—42no—
137D4×C31—48no—
138D4×C31—48no—
139D4×C31—42no—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31—22no—
143C2×C4×C31—22no—
144C2×C4×C31—22no—
145C2×C4×C31—21no—
146C2×C4×C31—21no—
147C2×C4×C31—21no—
148C2×C4×C31—44no—
149C2×C4×C31—42no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—42no—
155C2×C4×C31—44no—
156C2×C4×C31—42no—
157C2×C4×C31—41no—
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—81no—
165C2×C4×C31—82no—
166C2×C4×C31—82no—
167C2×C4×C31#9911yescharacteristic
168C2×C4×C31#9911yescharacteristic
169C2×C4×C31#10412yes—
170C8×C31—82no—
171#2388C2×C3111yescharacteristic
172C4×#99C2×C3111yescharacteristic, complemented
173#1460C2×C3111yescharacteristic, complemented
174#1351C2×C3112yescomplemented
175#1343C2×C3112yescomplemented
176C22×C31—21no—
177C22×C31—21no—
178C22×C31—22no—
179C22×C31—42no—
180C22×C31—42no—
181C22×C31—42no—
182C22×C31—41no—
183C22×C31—41no—
184C22×C31—41no—
185C22×C31—42no—
186C22×C31—41no—
187C22×C31—41no—
188C22×C31—81no—
189C22×C31—81no—
190C22×C31—81no—
191C22×C31—81no—
192C22×C31—81no—
193C22×C31—81no—
194C22×C31—81no—
195C22×C31C2×#9911yescharacteristic
196C22×C31#41611yescharacteristic
197C22×C31C2×#10411yescharacteristic
198C4×C31—21no—
199C4×C31—22no—
200C4×C31—42no—
201C4×C31—42no—
202C4×C31—41no—
203C4×C31—41no—
204C4×C31—41no—
205C4×C31—81no—
206C4×C31—84no—
207C4×C31AΣL(1,𝔽16)12yes—
208C22×#41C4×C3111yescharacteristic, complemented
209C2×C4×D4—21no—
210C2×C4×D4C22×C3111yescharacteristic
211#431—21no—
212C2×#99—21no—
213#395C4×C3111yescharacteristic, complemented
214#394—21no—
215#394C4×C3111yescharacteristic, complemented
216#393C22×C3111yescharacteristic
217#392—21no—
218#392C22×C3111yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4C4×C3111yescharacteristic, complemented
222#342C22×C3112yes—
223#323—21no—
224#323C22×C3112yes—
225C2×C31—21no—
226C2×C31—21no—
227C2×C31—42no—
228C2×C31—81no—
229C2×C31—82no—
230C2×C31#150111yescharacteristic
231C2×C31C2×AΣL(1,𝔽16)11yescharacteristic
232C2×C31#212611yescharacteristic
233C2×#41—44no—
234C2×#41—41no—
235C2×#41—41no—
236C22×Q8C2×C4×C3111yescharacteristic
237C22×D4—21no—
238C22×D4—21no—
239C22×D4C2×C4×C3111yescharacteristic
240C22×D4D4×C3111yescharacteristic
241C23×C4—21no—
242C23×C4C2×C4×C3111yescharacteristic
243C23×C4D4×C3111yescharacteristic
244C4×D4—41no—
245C4×D4—41no—
246C4×D4—44no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C2×C4×C3111yescharacteristic
249C2×C22⋊C4—22no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4D4×C3111yescharacteristic
256C2×C22⋊C4C23×C3111yescharacteristic
257C2×C42—21no—
258C2×C42—41no—
259C2×C42D4×C3111yescharacteristic
260C4⋊C8—41no—
261#99—24no—
262C22⋊C8—22no—
263C4×C8—41no—
264#95—21no—
265#95—21no—
266#95—41no—
267#95C2×C4×C3111yescharacteristic
268#95C2×C4×C3111yescharacteristic
269C31#1172511yescharacteristic, direct product factor, Sylow
270C24—21no—
271C24C22⋊C4×C3111yescharacteristic
272C24C2×D4×C3111yescharacteristic
273#41—44no—
274#41—44no—
275#41—44no—
276#41—44no—
277C2×Q8—42no—
278C2×Q8—41no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4C22⋊C4×C3111yescharacteristic
305C22×C4C22⋊C4×C3111yescharacteristic
306C22×C4C22⋊C4×C3111yescharacteristic
307C22×C4C22×C4×C3111yescharacteristic
308C22×C4C2×D4×C3111yescharacteristic
309C2×C8—42no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C22⋊C4—24no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C42—21no—
322C42—21no—
323C42—42no—
324C42—82no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23C2×C22⋊C4×C3111yescharacteristic
342C23C31×#13512yes—
343Q8—42no—
344Q8—42no—
345D4—44no—
346D4—42no—
347D4—48no—
348D4—48no—
349D4—42no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—44no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4C31×#9911yescharacteristic
378C2×C4C31×#9911yescharacteristic
379C2×C4C31×#10412yes—
380C8—82no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—42no—
391C22—41no—
392C22—41no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22C2×C31×#9911yescharacteristic
401C22C31×#41611yescharacteristic
402C22C2×C31×#10411yescharacteristic
403C4—21no—
404C4—22no—
405C4—42no—
406C4—42no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—81no—
411C4—84no—
412C4C31×AΣL(1,𝔽16)12yes—
413C2—21no—
414C2—21no—
415C2—42no—
416C2—82no—
417C2—81no—
418C2C31×#150111yescharacteristic
419C2C2×C31×AΣL(1,𝔽16)11yescharacteristic
420C2C31×#212611yescharacteristic
subgroupG/H
centerC2×C4×C31#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C22⋊C4C23×C31
socleC22×C31C2×#99
prime radicalC31×#2388C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C22×C4×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×C31×#99C2semidirect
3. C31×#1460C2semidirect
4. C31×#1351C2semidirect
5. C31×#1343C2semidirect
6. C22×#41×C31C4semidirect
11. C31×#395C4semidirect
13. C31×#394C4semidirect
19. C4×C22⋊C4×C31C4semidirect
114. #11725C31direct (N × H)
172. C4×#99C2×C31semidirect
173. #1460C2×C31semidirect
174. #1351C2×C31semidirect
175. #1343C2×C31semidirect
208. C22×#41C4×C31semidirect
213. #395C4×C31semidirect
215. #394C4×C31semidirect
221. C4×C22⋊C4C4×C31semidirect
269. C31#11725direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#23882
C4×C31×#992
C31×#14602
C31×#13512
C31×#13432
#1172531