Finite Group Databasebeta

← C31×#11755 #21468514

Subgroups — C31×#11755 #21468514

Order 7936.

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

normal subgroups only

This group has 586 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×#2482C211yesmaximal, characteristic
2C2×C31×#342C211yesmaximal, characteristic
3C31×#1462C211yesmaximal, characteristic
4C31×#1352C212yesmaximal, complemented
5C31×#1344C212yesmaximal, complemented
6C2×C31×#117C411yescharacteristic, complemented
7C22×C22⋊C4×C31—21no—
8C22×C22⋊C4×C31C2211yescharacteristic
9C31×#407—21no—
10C31×#399C411yescharacteristic, complemented
11C31×#399C2211yescharacteristic
12C31×#397C411yescharacteristic, complemented
13C31×#397C2211yescharacteristic
14C31×#388—21no—
15C31×#388—21no—
16C31×#385—21no—
17C31×#381C411yescharacteristic, complemented
18C31×#379—21no—
19C31×#342—21no—
20C31×#342C2212yes—
21C31×#323—21no—
22C31×#323C2212yes—
23C25×C31D411yescharacteristic
24C23×C4×C31—41no—
25C23×C4×C31C2×C411yescharacteristic
26C23×C4×C31D411yescharacteristic
27C2×M16×C31—41no—
28C31×#117—41no—
29C31×#117—41no—
30C2×C4⋊C4×C31—21no—
31C2×C4⋊C4×C31—21no—
32C2×C4⋊C4×C31C2×C411yescharacteristic, complemented
33C2×C4⋊C4×C31C2×C411yescharacteristic, complemented
34C2×C22⋊C4×C31—24no—
35C2×C22⋊C4×C31—22no—
36C2×C22⋊C4×C31—21no—
37C2×C22⋊C4×C31—21no—
38C2×C22⋊C4×C31—21no—
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—41no—
43C2×C22⋊C4×C31—41no—
44C2×C22⋊C4×C31—41no—
45C2×C22⋊C4×C31—42no—
46C2×C22⋊C4×C31C2×C411yescharacteristic
47C2×C22⋊C4×C31D411yescharacteristic
48C2×C22⋊C4×C31D411yescharacteristic
49C2×C22⋊C4×C31C2311yescharacteristic
50C2×C42×C31—21no—
51C31×C22⋊C8—22no—
52C31×C22⋊C8—41no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95C2×C411yescharacteristic, complemented
57C31×#95C2×C411yescharacteristic, complemented
58C24×C31—21no—
59C24×C31—21no—
60C24×C31—22no—
61C24×C31—21no—
62C24×C31—21no—
63C24×C31—42no—
64C24×C31—81no—
65C24×C31C22⋊C411yescharacteristic
66C24×C31C22⋊C411yescharacteristic
67C24×C31C2×D411yescharacteristic
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—42no—
77C22×C4×C31—44no—
78C22×C4×C31—42no—
79C22×C4×C31—81no—
80C22×C4×C31—82no—
81C22×C4×C31—82no—
82C22×C4×C31C22⋊C411yescharacteristic
83C22×C4×C31C22⋊C411yescharacteristic
84C22×C4×C31C22×C411yescharacteristic
85C22×C4×C31C2×D411yescharacteristic
86M16×C31—82no—
87C2×C8×C31—42no—
88C4⋊C4×C31—41no—
89C4⋊C4×C31—41no—
90C4⋊C4×C31—81no—
91C4⋊C4×C31—81no—
92C22⋊C4×C31—24no—
93C22⋊C4×C31—41no—
94C22⋊C4×C31—41no—
95C22⋊C4×C31—41no—
96C22⋊C4×C31—41no—
97C22⋊C4×C31—42no—
98C22⋊C4×C31—44no—
99C22⋊C4×C31—41no—
100C22⋊C4×C31—41no—
101C22⋊C4×C31—42no—
102C22⋊C4×C31—41no—
103C22⋊C4×C31—42no—
104C22⋊C4×C31—41no—
105C22⋊C4×C31—81no—
106C22⋊C4×C31—81no—
107C42×C31—42no—
108#11755C3111yesmaximal, characteristic, direct product factor, Sylow
109C23×C31—22no—
110C23×C31—22no—
111C23×C31—22no—
112C23×C31—22no—
113C23×C31—22no—
114C23×C31—21no—
115C23×C31—21no—
116C23×C31—21no—
117C23×C31—21no—
118C23×C31—21no—
119C23×C31—21no—
120C23×C31—41no—
121C23×C31—42no—
122C23×C31—41no—
123C23×C31—41no—
124C23×C31—41no—
125C23×C31—41no—
126C23×C31—42no—
127C23×C31—81no—
128C23×C31—81no—
129C23×C31—84no—
130C23×C31—82no—
131C23×C31—82no—
132C23×C31#9911yescharacteristic
133C23×C31#9911yescharacteristic
134C23×C31#10412yes—
135C23×C31C2×C22⋊C411yescharacteristic
136C23×C31AGL(1,ℤ/8)11yescharacteristic
137C23×C31#13711yescharacteristic
138C2×C4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31—42no—
143C2×C4×C31—42no—
144C2×C4×C31—42no—
145C2×C4×C31—41no—
146C2×C4×C31—41no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
149C2×C4×C31—44no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—81no—
163C2×C4×C31—88no—
164C2×C4×C31—82no—
165C2×C4×C31—82no—
166C8×C31—82no—
167#2482C2×C3111yescharacteristic
168C2×#342C2×C3111yescharacteristic
169#1462C2×C3111yescharacteristic
170#1352C2×C3112yescomplemented
171#1344C2×C3112yescomplemented
172C22×C31—21no—
173C22×C31—21no—
174C22×C31—21no—
175C22×C31—21no—
176C22×C31—21no—
177C22×C31—21no—
178C22×C31—21no—
179C22×C31—21no—
180C22×C31—22no—
181C22×C31—21no—
182C22×C31—42no—
183C22×C31—42no—
184C22×C31—42no—
185C22×C31—41no—
186C22×C31—41no—
187C22×C31—44no—
188C22×C31—44no—
189C22×C31—41no—
190C22×C31—41no—
191C22×C31—41no—
192C22×C31—41no—
193C22×C31—81no—
194C22×C31—81no—
195C22×C31—82no—
196C22×C31—82no—
197C22×C31#35212yes—
198C22×C31C2×#9911yescharacteristic
199C22×C31#41711yescharacteristic
200C22×C31C2×#10411yescharacteristic
201C4×C31—41no—
202C4×C31—41no—
203C4×C31—41no—
204C4×C31—41no—
205C4×C31—81no—
206C4×C31—81no—
207C4×C31—84no—
208C4×C31—81no—
209C4×C31—81no—
210C2×#117C4×C3111yescharacteristic, complemented
211C22×C22⋊C4—21no—
212C22×C22⋊C4C22×C3111yescharacteristic
213#407—21no—
214#399C4×C3111yescharacteristic, complemented
215#399C22×C3111yescharacteristic
216#397C4×C3111yescharacteristic, complemented
217#397C22×C3111yescharacteristic
218#388—21no—
219#388—21no—
220#385—21no—
221#381C4×C3111yescharacteristic, complemented
222#379—21no—
223#342—21no—
224#342C22×C3112yes—
225#323—21no—
226#323C22×C3112yes—
227C2×C31—21no—
228C2×C31—21no—
229C2×C31—22no—
230C2×C31—41no—
231C2×C31—41no—
232C2×C31—42no—
233C2×C31—41no—
234C2×C31#150411yescharacteristic
235C2×C31C2×#35211yescharacteristic
236C2×C31#212711yescharacteristic
237C25D4×C3111yescharacteristic
238C23×C4—41no—
239C23×C4C2×C4×C3111yescharacteristic
240C23×C4D4×C3111yescharacteristic
241C2×M16—41no—
242#117—41no—
243#117—41no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4C2×C4×C3111yescharacteristic, complemented
247C2×C4⋊C4C2×C4×C3111yescharacteristic, complemented
248C2×C22⋊C4—24no—
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⋊C4—21no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4C2×C4×C3111yescharacteristic
261C2×C22⋊C4D4×C3111yescharacteristic
262C2×C22⋊C4D4×C3111yescharacteristic
263C2×C22⋊C4C23×C3111yescharacteristic
264C2×C42—21no—
265C22⋊C8—22no—
266C22⋊C8—41no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95C2×C4×C3111yescharacteristic, complemented
271#95C2×C4×C3111yescharacteristic, complemented
272C31#1175511yescharacteristic, direct product factor, Sylow
273C24—21no—
274C24—21no—
275C24—22no—
276C24—21no—
277C24—21no—
278C24—42no—
279C24—81no—
280C24C22⋊C4×C3111yescharacteristic
281C24C22⋊C4×C3111yescharacteristic
282C24C2×D4×C3111yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—81no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4C22⋊C4×C3111yescharacteristic
298C22×C4C22⋊C4×C3111yescharacteristic
299C22×C4C22×C4×C3111yescharacteristic
300C22×C4C2×D4×C3111yescharacteristic
301M16—82no—
302C2×C8—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C22⋊C4—24no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—42no—
313C22⋊C4—44no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—42no—
317C22⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—41no—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C42—42no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—81no—
342C23—81no—
343C23—84no—
344C23—82no—
345C23—82no—
346C23C31×#9911yescharacteristic
347C23C31×#9911yescharacteristic
348C23C31×#10412yes—
349C23C2×C22⋊C4×C3111yescharacteristic
350C23C31×AGL(1,ℤ/8)11yescharacteristic
351C23C31×#13711yescharacteristic
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—88no—
378C2×C4—82no—
379C2×C4—82no—
380C8—82no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—22no—
390C22—21no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—41no—
395C22—41no—
396C22—44no—
397C22—44no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22—81no—
404C22—82no—
405C22—82no—
406C22C31×#35212yes—
407C22C2×C31×#9911yescharacteristic
408C22C31×#41711yescharacteristic
409C22C2×C31×#10411yescharacteristic
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—81no—
415C4—81no—
416C4—84no—
417C4—81no—
418C4—81no—
419C2—21no—
420C2—21no—
421C2—22no—
422C2—41no—
423C2—41no—
424C2—42no—
425C2—41no—
426C2C31×#150411yescharacteristic
427C2C2×C31×#35211yescharacteristic
428C2C31×#212711yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC25×C31D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C24×C31 ◁ C22×C22⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#1352C2semidirect
5. C31×#1344C2semidirect
6. C2×C31×#117C4semidirect
10. C31×#399C4semidirect
12. C31×#397C4semidirect
17. C31×#381C4semidirect
32. C2×C4⋊C4×C31C2×C4semidirect
33. C2×C4⋊C4×C31C2×C4semidirect
56. C31×#95C2×C4semidirect
57. C31×#95C2×C4semidirect
108. #11755C31direct (N × H)
170. #1352C2×C31semidirect
171. #1344C2×C31semidirect
210. C2×#117C4×C31semidirect
214. #399C4×C31semidirect
216. #397C4×C31semidirect
221. #381C4×C31semidirect
246. C2×C4⋊C4C2×C4×C31semidirect
247. C2×C4⋊C4C2×C4×C31semidirect
270. #95C2×C4×C31semidirect
271. #95C2×C4×C31semidirect
272. C31#11755direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#24822
C2×C31×#3422
C31×#14622
C31×#13522
C31×#13442
#1175531