Finite Group Databasebeta

← C31×#11757 #21468516

Subgroups — C31×#11757 #21468516

Order 7936.

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

normal subgroups only

This group has 700 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×#2539C211yesmaximal, characteristic
2C2×C31×#342C211yesmaximal, characteristic, complemented
3C31×#1462C211yesmaximal, characteristic, complemented
4C31×#1352C212yesmaximal, complemented
5C31×#1343C212yesmaximal, complemented
6C2×C31×#123—21no—
7C2×C31×#121C411yescharacteristic, complemented
8C2×C31×#120—21no—
9C22×C22⋊C4×C31—21no—
10C22×C22⋊C4×C31C2211yescharacteristic
11C31×#407—21no—
12C31×#399C411yescharacteristic, complemented
13C31×#399C2211yescharacteristic
14C31×#392C411yescharacteristic, complemented
15C31×#392C2211yescharacteristic
16C31×#390—21no—
17C31×#390—21no—
18C4×C22⋊C4×C31C411yescharacteristic, complemented
19C31×#342—21no—
20C31×#342C2212yes—
21C31×#323—21no—
22C31×#323C2212yes—
23C25×C31D411yescharacteristic
24C22×D4×C31—21no—
25C22×D4×C31—21no—
26C22×D4×C31—21no—
27C22×D4×C31C2×C411yescharacteristic, complemented
28C23×C4×C31—41no—
29C23×C4×C31C2×C411yescharacteristic
30C23×C4×C31D411yescharacteristic
31C2×M16×C31—41no—
32C31×#123—44no—
33C31×#121—41no—
34C31×#121—41no—
35C31×#120—44no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31C2×C411yescharacteristic, complemented
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—24no—
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—41no—
48C2×C22⋊C4×C31—41no—
49C2×C22⋊C4×C31—41no—
50C2×C22⋊C4×C31—42no—
51C2×C22⋊C4×C31C2×C411yescharacteristic
52C2×C22⋊C4×C31D411yescharacteristic
53C2×C22⋊C4×C31D411yescharacteristic
54C2×C22⋊C4×C31C2311yescharacteristic
55C2×C42×C31—21no—
56C31×C22⋊C8—22no—
57C31×C22⋊C8—41no—
58C31×#95C2×C411yescharacteristic, complemented
59C31×#95C2×C411yescharacteristic, complemented
60C24×C31—21no—
61C24×C31—21no—
62C24×C31—22no—
63C24×C31—21no—
64C24×C31—21no—
65C24×C31—21no—
66C24×C31—42no—
67C24×C31—81no—
68C24×C31C22⋊C411yescharacteristic
69C24×C31C22⋊C411yescharacteristic
70C24×C31C2×D411yescharacteristic
71C2×D4×C31—42no—
72C2×D4×C31—42no—
73C2×D4×C31—41no—
74C2×D4×C31—42no—
75C2×D4×C31—42no—
76C2×D4×C31—41no—
77C2×D4×C31—41no—
78C2×D4×C31—42no—
79C2×D4×C31—41no—
80C2×D4×C31—41no—
81C2×D4×C31—81no—
82C2×D4×C31—81no—
83C2×D4×C31—81no—
84C22×C4×C31—21no—
85C22×C4×C31—21no—
86C22×C4×C31—21no—
87C22×C4×C31—21no—
88C22×C4×C31—21no—
89C22×C4×C31—21no—
90C22×C4×C31—21no—
91C22×C4×C31—42no—
92C22×C4×C31—44no—
93C22×C4×C31—42no—
94C22×C4×C31—81no—
95C22×C4×C31—82no—
96C22×C4×C31—82no—
97C22×C4×C31C22⋊C411yescharacteristic
98C22×C4×C31C22⋊C411yescharacteristic
99C22×C4×C31C22×C411yescharacteristic
100C22×C4×C31C2×D411yescharacteristic
101M16×C31—82no—
102C2×C8×C31—42no—
103C4⋊C4×C31—42no—
104C4⋊C4×C31—42no—
105C4⋊C4×C31—41no—
106C22⋊C4×C31—24no—
107C22⋊C4×C31—24no—
108C22⋊C4×C31—41no—
109C22⋊C4×C31—41no—
110C22⋊C4×C31—41no—
111C22⋊C4×C31—41no—
112C22⋊C4×C31—42no—
113C22⋊C4×C31—42no—
114C22⋊C4×C31—42no—
115C22⋊C4×C31—42no—
116C22⋊C4×C31—42no—
117C22⋊C4×C31—44no—
118C22⋊C4×C31—42no—
119C42×C31—81no—
120#11757C3111yesmaximal, characteristic, direct product factor, Sylow
121C23×C31—22no—
122C23×C31—22no—
123C23×C31—22no—
124C23×C31—22no—
125C23×C31—22no—
126C23×C31—21no—
127C23×C31—21no—
128C23×C31—21no—
129C23×C31—21no—
130C23×C31—21no—
131C23×C31—21no—
132C23×C31—41no—
133C23×C31—42no—
134C23×C31—41no—
135C23×C31—41no—
136C23×C31—41no—
137C23×C31—41no—
138C23×C31—41no—
139C23×C31—41no—
140C23×C31—41no—
141C23×C31—41no—
142C23×C31—42no—
143C23×C31—41no—
144C23×C31—41no—
145C23×C31—41no—
146C23×C31—81no—
147C23×C31—81no—
148C23×C31—84no—
149C23×C31—82no—
150C23×C31—82no—
151C23×C31#9911yescharacteristic
152C23×C31#9911yescharacteristic
153C23×C31#10412yes—
154C23×C31C2×C22⋊C411yescharacteristic
155C23×C31AGL(1,ℤ/8)11yescharacteristic
156C23×C31#13711yescharacteristic
157D4×C31—84no—
158D4×C31—82no—
159D4×C31—82no—
160D4×C31—82no—
161D4×C31—82no—
162D4×C31—82no—
163C2×C4×C31—22no—
164C2×C4×C31—22no—
165C2×C4×C31—21no—
166C2×C4×C31—21no—
167C2×C4×C31—21no—
168C2×C4×C31—21no—
169C2×C4×C31—41no—
170C2×C4×C31—41no—
171C2×C4×C31—42no—
172C2×C4×C31—41no—
173C2×C4×C31—41no—
174C2×C4×C31—41no—
175C2×C4×C31—41no—
176C2×C4×C31—41no—
177C2×C4×C31—41no—
178C2×C4×C31—44no—
179C2×C4×C31—41no—
180C2×C4×C31—41no—
181C2×C4×C31—41no—
182C2×C4×C31—81no—
183C2×C4×C31—81no—
184C2×C4×C31—81no—
185C2×C4×C31—88no—
186C2×C4×C31—82no—
187C2×C4×C31—82no—
188C8×C31—82no—
189#2539C2×C3111yescharacteristic
190C2×#342C2×C3111yescharacteristic, complemented
191#1462C2×C3111yescharacteristic, complemented
192#1352C2×C3112yescomplemented
193#1343C2×C3112yescomplemented
194C22×C31—21no—
195C22×C31—21no—
196C22×C31—21no—
197C22×C31—21no—
198C22×C31—21no—
199C22×C31—21no—
200C22×C31—21no—
201C22×C31—21no—
202C22×C31—22no—
203C22×C31—21no—
204C22×C31—42no—
205C22×C31—42no—
206C22×C31—42no—
207C22×C31—41no—
208C22×C31—41no—
209C22×C31—44no—
210C22×C31—44no—
211C22×C31—41no—
212C22×C31—41no—
213C22×C31—41no—
214C22×C31—41no—
215C22×C31—81no—
216C22×C31—81no—
217C22×C31—81no—
218C22×C31—81no—
219C22×C31—81no—
220C22×C31—81no—
221C22×C31—82no—
222C22×C31—82no—
223C22×C31—81no—
224C22×C31—81no—
225C22×C31—81no—
226C22×C31#35312yes—
227C22×C31C2×#9911yescharacteristic
228C22×C31#41711yescharacteristic
229C22×C31C2×#10411yescharacteristic
230C4×C31—42no—
231C4×C31—42no—
232C4×C31—41no—
233C4×C31—41no—
234C4×C31—41no—
235C4×C31—41no—
236C4×C31—84no—
237C4×C31—81no—
238C2×#123—21no—
239C2×#121C4×C3111yescharacteristic, complemented
240C2×#120—21no—
241C22×C22⋊C4—21no—
242C22×C22⋊C4C22×C3111yescharacteristic
243#407—21no—
244#399C4×C3111yescharacteristic, complemented
245#399C22×C3111yescharacteristic
246#392C4×C3111yescharacteristic, complemented
247#392C22×C3111yescharacteristic
248#390—21no—
249#390—21no—
250C4×C22⋊C4C4×C3111yescharacteristic, complemented
251#342—21no—
252#342C22×C3112yes—
253#323—21no—
254#323C22×C3112yes—
255C2×C31—21no—
256C2×C31—21no—
257C2×C31—22no—
258C2×C31—41no—
259C2×C31—41no—
260C2×C31—42no—
261C2×C31—41no—
262C2×C31—161no—
263C2×C31#150411yescharacteristic
264C2×C31#212111yescharacteristic
265C2×C31C2×#35311yescharacteristic
266C25D4×C3111yescharacteristic
267C22×D4—21no—
268C22×D4—21no—
269C22×D4—21no—
270C22×D4C2×C4×C3111yescharacteristic, complemented
271C23×C4—41no—
272C23×C4C2×C4×C3111yescharacteristic
273C23×C4D4×C3111yescharacteristic
274C2×M16—41no—
275#123—44no—
276#121—41no—
277#121—41no—
278#120—44no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4C2×C4×C3111yescharacteristic, complemented
282C2×C22⋊C4—21no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4C2×C4×C3111yescharacteristic
295C2×C22⋊C4D4×C3111yescharacteristic
296C2×C22⋊C4D4×C3111yescharacteristic
297C2×C22⋊C4C23×C3111yescharacteristic
298C2×C42—21no—
299C22⋊C8—22no—
300C22⋊C8—41no—
301#95C2×C4×C3111yescharacteristic, complemented
302#95C2×C4×C3111yescharacteristic, complemented
303C31#1175711yescharacteristic, direct product factor, Sylow
304C24—21no—
305C24—21no—
306C24—22no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—42no—
311C24—81no—
312C24C22⋊C4×C3111yescharacteristic
313C24C22⋊C4×C3111yescharacteristic
314C24C2×D4×C3111yescharacteristic
315C2×D4—41no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—42no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—81no—
326C2×D4—81no—
327C2×D4—81no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—42no—
336C22×C4—44no—
337C22×C4—42no—
338C22×C4—81no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4C22⋊C4×C3111yescharacteristic
342C22×C4C22⋊C4×C3111yescharacteristic
343C22×C4C22×C4×C3111yescharacteristic
344C22×C4C2×D4×C3111yescharacteristic
345M16—82no—
346C2×C8—42no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—44no—
362C22⋊C4—42no—
363C42—81no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—22no—
373C23—21no—
374C23—21no—
375C23—41no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—81no—
390C23—81no—
391C23—84no—
392C23—82no—
393C23—82no—
394C23C31×#9911yescharacteristic
395C23C31×#9911yescharacteristic
396C23C31×#10412yes—
397C23C2×C22⋊C4×C3111yescharacteristic
398C23C31×AGL(1,ℤ/8)11yescharacteristic
399C23C31×#13711yescharacteristic
400D4—84no—
401D4—82no—
402D4—82no—
403D4—82no—
404D4—82no—
405D4—82no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—88no—
429C2×C4—82no—
430C2×C4—82no—
431C8—82no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—22no—
441C22—21no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—41no—
446C22—41no—
447C22—44no—
448C22—44no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—82no—
460C22—82no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22C31×#35312yes—
465C22C2×C31×#9911yescharacteristic
466C22C31×#41711yescharacteristic
467C22C2×C31×#10411yescharacteristic
468C4—42no—
469C4—42no—
470C4—41no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—84no—
475C4—81no—
476C2—21no—
477C2—21no—
478C2—22no—
479C2—41no—
480C2—41no—
481C2—42no—
482C2—41no—
483C2—161no—
484C2C31×#150411yescharacteristic
485C2C31×#212111yescharacteristic
486C2C2×C31×#35311yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC31×#2539C2
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
2. C2×C31×#342C2semidirect
3. C31×#1462C2semidirect
4. C31×#1352C2semidirect
5. C31×#1343C2semidirect
7. C2×C31×#121C4semidirect
12. C31×#399C4semidirect
14. C31×#392C4semidirect
18. C4×C22⋊C4×C31C4semidirect
27. C22×D4×C31C2×C4semidirect
38. C2×C4⋊C4×C31C2×C4semidirect
58. C31×#95C2×C4semidirect
59. C31×#95C2×C4semidirect
120. #11757C31direct (N × H)
190. C2×#342C2×C31semidirect
191. #1462C2×C31semidirect
192. #1352C2×C31semidirect
193. #1343C2×C31semidirect
239. C2×#121C4×C31semidirect
244. #399C4×C31semidirect
246. #392C4×C31semidirect
250. C4×C22⋊C4C4×C31semidirect
270. C22×D4C2×C4×C31semidirect
281. C2×C4⋊C4C2×C4×C31semidirect
301. #95C2×C4×C31semidirect
302. #95C2×C4×C31semidirect
303. C31#11757direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25392
C2×C31×#3422
C31×#14622
C31×#13522
C31×#13432
#1175731