Finite Group Databasebeta

← C31×#27139 #21483898

Subgroups — C31×#27139 #21483898

Order 7936.

670 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2512C211yesmaximal, characteristic
2C31×#2370C211yesmaximal, characteristic, complemented
3C31×#2100C211yesmaximal, characteristic, complemented
4C31×#2065C211yesmaximal, characteristic
5C31×#2059C211yesmaximal, characteristic, complemented
6C31×#2025C211yesmaximal, characteristic
7C31×#1948C211yesmaximal, characteristic, complemented
8C31×#1934C211yesmaximal, characteristic
9C31×#1926C211yesmaximal, characteristic
10C31×#1926C211yesmaximal, characteristic
11C31×#1925C211yesmaximal, characteristic
12C31×#1920C211yesmaximal, characteristic, complemented
13C31×#1918C211yesmaximal, characteristic, complemented
14C31×#1916C211yesmaximal, characteristic, complemented
15C4×C31×#106C211yesmaximal, characteristic, complemented
16C2×C31×#126—21no—
17C2×C31×#125C2211yescharacteristic
18C2×C31×#107C2211yescharacteristic
19C2×C31×#106C2211yescharacteristic
20C2×C31×#106C2211yescharacteristic
21C2×C31×#106C2211yescharacteristic
22C2×C31×C4⋊C8C2211yescharacteristic
23C2×C31×#102C2211yescharacteristic
24C2×C31×#102C2211yescharacteristic
25C2×C31×#102C2211yescharacteristic
26C2×C31×#102C2211yescharacteristic
27C2×C4×C8×C31C2211yescharacteristic
28C31×#397—21no—
29C31×#396—21no—
30C31×#390C2211yescharacteristic
31C31×#390C2211yescharacteristic
32C31×#390C2211yescharacteristic
33C31×#389—21no—
34C31×#389C2211yescharacteristic
35C31×#389C2211yescharacteristic
36C31×#389C2211yescharacteristic
37C31×#388—21no—
38C31×#387—21no—
39C31×#387—21no—
40C31×#384C2211yescharacteristic
41C4×C4⋊C4×C31—21no—
42C4×C4⋊C4×C31C2211yescharacteristic
43C4×C4⋊C4×C31C2211yescharacteristic
44C4×C4⋊C4×C31C2211yescharacteristic
45C4×C4⋊C4×C31C2211yescharacteristic
46C4×C4⋊C4×C31C2211yescharacteristic
47C31×#340C2211yescharacteristic
48C31×#340C2211yescharacteristic
49C31×#340C2211yescharacteristic
50C31×#340C2211yescharacteristic
51C31×#340C2211yescharacteristic
52C31×#340C2211yescharacteristic
53C31×#340C2211yescharacteristic
54C31×#340C2211yescharacteristic
55C31×#336C2211yescharacteristic
56C31×#336C2211yescharacteristic
57C31×#336C2211yescharacteristic
58C31×#336C2211yescharacteristic
59C22×D4×C31C2311yescharacteristic
60C22×C8×C31C2311yescharacteristic
61C22×C8×C31C2311yescharacteristic
62C22×C8×C31C2311yescharacteristic
63C22×C8×C31C2311yescharacteristic
64C31×#126—44no—
65C31×#125—22no—
66C31×#125—22no—
67C2×C4⋊C4×C31—21no—
68C2×C4⋊C4×C31—21no—
69C2×C4⋊C4×C31—21no—
70C2×C4⋊C4×C31—21no—
71C2×C4⋊C4×C31—21no—
72C2×C4⋊C4×C31—21no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31D411yescharacteristic
75C2×C4⋊C4×C31D411yescharacteristic
76C2×C4⋊C4×C31C2311yescharacteristic
77C2×C4⋊C4×C31C2311yescharacteristic
78C2×C4⋊C4×C31C2311yescharacteristic
79C2×C4⋊C4×C31C2311yescharacteristic
80C2×C4⋊C4×C31C2311yescharacteristic
81C2×C4⋊C4×C31C2311yescharacteristic
82C2×C4⋊C4×C31C2311yescharacteristic
83C2×C22⋊C4×C31—21no—
84C2×C22⋊C4×C31—21no—
85C2×C22⋊C4×C31—21no—
86C2×C22⋊C4×C31—21no—
87C2×C22⋊C4×C31—21no—
88C2×C22⋊C4×C31—21no—
89C2×C42×C31—21no—
90C2×C42×C31—21no—
91C2×C42×C31—21no—
92C2×C42×C31—21no—
93C2×C42×C31C2311yescharacteristic
94C2×C42×C31C2311yescharacteristic
95C2×C42×C31C2311yescharacteristic
96C31×#107—22no—
97C31×#106—22no—
98C31×#106—22no—
99C31×#106—22no—
100C31×C4⋊C8—22no—
101C31×#102—22no—
102C31×#102—22no—
103C31×#102—22no—
104C31×#102—22no—
105C4×C8×C31—22no—
106C31×#95—21no—
107C31×#95—21no—
108C31×#95—21no—
109C31×#95—21no—
110C31×#95—21no—
111C31×#95—21no—
112C31×#95—21no—
113C31×#95D411yescharacteristic
114C31×#95D411yescharacteristic
115C24×C31—21no—
116C2×D4×C31—21no—
117C2×D4×C31—21no—
118C2×D4×C31—21no—
119C2×D4×C31—21no—
120C2×D4×C31—21no—
121C2×D4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31C2×D411yescharacteristic
130C22×C4×C31C2×D411yescharacteristic
131C22×C4×C31C2×D411yescharacteristic
132C22×C4×C31C2×D411yescharacteristic
133C22×C4×C31C2×D411yescharacteristic
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31C2411yescharacteristic
136C2×C8×C31—21no—
137C2×C8×C31—21no—
138C2×C8×C31—21no—
139C2×C8×C31—21no—
140C2×C8×C31—21no—
141C2×C8×C31—21no—
142C2×C8×C31—21no—
143C2×C8×C31—21no—
144C2×C8×C31—21no—
145C2×C8×C31—21no—
146C2×C8×C31—21no—
147C2×C8×C31#4112yes—
148C4⋊C4×C31—24no—
149C4⋊C4×C31—24no—
150C4⋊C4×C31—22no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—21no—
153C4⋊C4×C31—21no—
154C4⋊C4×C31—21no—
155C4⋊C4×C31—21no—
156C4⋊C4×C31—21no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—42no—
164C4⋊C4×C31—42no—
165C4⋊C4×C31—42no—
166C4⋊C4×C31—81no—
167C4⋊C4×C31—81no—
168C4⋊C4×C31#4112yes—
169C4⋊C4×C31#4112yes—
170C4⋊C4×C31#4112yes—
171C22⋊C4×C31—42no—
172C22⋊C4×C31—42no—
173C22⋊C4×C31—42no—
174C22⋊C4×C31—42no—
175C22⋊C4×C31—81no—
176C22⋊C4×C31—81no—
177C42×C31—21no—
178C42×C31—21no—
179C42×C31—21no—
180C42×C31—21no—
181C42×C31—21no—
182C42×C31—21no—
183C42×C31—42no—
184C42×C31—42no—
185C42×C31—81no—
186C42×C31—81no—
187#27139C3111yesmaximal, characteristic, direct product factor, Sylow
188C23×C31—41no—
189C23×C31—41no—
190C23×C31—41no—
191C23×C31—41no—
192C23×C31—41no—
193C23×C31—41no—
194C23×C31—41no—
195C23×C31C22×D411yescharacteristic
196D4×C31—41no—
197D4×C31—41no—
198D4×C31—41no—
199D4×C31—41no—
200C2×C4×C31—21no—
201C2×C4×C31—21no—
202C2×C4×C31—21no—
203C2×C4×C31—21no—
204C2×C4×C31—21no—
205C2×C4×C31—21no—
206C2×C4×C31—22no—
207C2×C4×C31—22no—
208C2×C4×C31—21no—
209C2×C4×C31—21no—
210C2×C4×C31—22no—
211C2×C4×C31—21no—
212C2×C4×C31—21no—
213C2×C4×C31—21no—
214C2×C4×C31—21no—
215C2×C4×C31—21no—
216C2×C4×C31—21no—
217C2×C4×C31—21no—
218C2×C4×C31—21no—
219C2×C4×C31—41no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—41no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—41no—
230C2×C4×C31—41no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31#13512yes—
238C2×C4×C31AGL(1,ℤ/8)12yes—
239C2×C4×C31C2×#4111yescharacteristic
240C2×C4×C31C2×#4111yescharacteristic
241C2×C4×C31C2×#4111yescharacteristic
242C2×C4×C31C2×#4111yescharacteristic
243C2×C4×C312−1+411yescharacteristic
244C2×C4×C312−1+411yescharacteristic
245C8×C31—22no—
246C8×C31—41no—
247C8×C31—41no—
248C8×C31—41no—
249#2512C2×C3111yescharacteristic
250#2370C2×C3111yescharacteristic, complemented
251#2100C2×C3111yescharacteristic, complemented
252#2065C2×C3111yescharacteristic
253#2059C2×C3111yescharacteristic, complemented
254#2025C2×C3111yescharacteristic
255#1948C2×C3111yescharacteristic, complemented
256#1934C2×C3111yescharacteristic
257#1926C2×C3111yescharacteristic
258#1926C2×C3111yescharacteristic
259#1925C2×C3111yescharacteristic
260#1920C2×C3111yescharacteristic, complemented
261#1918C2×C3111yescharacteristic, complemented
262#1916C2×C3111yescharacteristic, complemented
263C4×#106C2×C3111yescharacteristic, complemented
264C22×C31—81no—
265C22×C31—81no—
266C22×C31—81no—
267C22×C31—81no—
268C22×C31—81no—
269C22×C31—81no—
270C22×C31—81no—
271C22×C31#52511yescharacteristic
272C22×C31#54711yescharacteristic
273C22×C31#54811yescharacteristic
274C22×C31C2×#13511yescharacteristic
275C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C31#57711yescharacteristic
277C22×C31#57711yescharacteristic
278C4×C31—22no—
279C4×C31—22no—
280C4×C31—41no—
281C4×C31—41no—
282C4×C31—42no—
283C4×C31—41no—
284C4×C31—42no—
285C4×C31—41no—
286C4×C31—42no—
287C4×C31—81no—
288C4×C31—81no—
289C4×C31—81no—
290C4×C31—81no—
291C4×C31#52911yescharacteristic
292C4×C31#54111yescharacteristic
293C4×C31#55611yescharacteristic
294C4×C31#55611yescharacteristic
295C2×#126—21no—
296C2×#125C22×C3111yescharacteristic
297C2×#107C22×C3111yescharacteristic
298C2×#106C22×C3111yescharacteristic
299C2×#106C22×C3111yescharacteristic
300C2×#106C22×C3111yescharacteristic
301C2×C4⋊C8C22×C3111yescharacteristic
302C2×#102C22×C3111yescharacteristic
303C2×#102C22×C3111yescharacteristic
304C2×#102C22×C3111yescharacteristic
305C2×#102C22×C3111yescharacteristic
306C2×C4×C8C22×C3111yescharacteristic
307#397—21no—
308#396—21no—
309#390C22×C3111yescharacteristic
310#390C22×C3111yescharacteristic
311#390C22×C3111yescharacteristic
312#389—21no—
313#389C22×C3111yescharacteristic
314#389C22×C3111yescharacteristic
315#389C22×C3111yescharacteristic
316#388—21no—
317#387—21no—
318#387—21no—
319#384C22×C3111yescharacteristic
320C4×C4⋊C4—21no—
321C4×C4⋊C4C22×C3111yescharacteristic
322C4×C4⋊C4C22×C3111yescharacteristic
323C4×C4⋊C4C22×C3111yescharacteristic
324C4×C4⋊C4C22×C3111yescharacteristic
325C4×C4⋊C4C22×C3111yescharacteristic
326#340C22×C3111yescharacteristic
327#340C22×C3111yescharacteristic
328#340C22×C3111yescharacteristic
329#340C22×C3111yescharacteristic
330#340C22×C3111yescharacteristic
331#340C22×C3111yescharacteristic
332#340C22×C3111yescharacteristic
333#340C22×C3111yescharacteristic
334#336C22×C3111yescharacteristic
335#336C22×C3111yescharacteristic
336#336C22×C3111yescharacteristic
337#336C22×C3111yescharacteristic
338C2×C31—161no—
339C2×C31#246911yescharacteristic
340C2×C31#310611yescharacteristic
341C2×C31#312511yescharacteristic
342C2×C31#333711yescharacteristic
343C2×C31#334511yescharacteristic
344C2×C31#336011yescharacteristic
345C2×C31#336811yescharacteristic
346C22×D4C23×C3111yescharacteristic
347C22×C8C23×C3111yescharacteristic
348C22×C8C23×C3111yescharacteristic
349C22×C8C23×C3111yescharacteristic
350C22×C8C23×C3111yescharacteristic
351#126—44no—
352#125—22no—
353#125—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4D4×C3111yescharacteristic
362C2×C4⋊C4D4×C3111yescharacteristic
363C2×C4⋊C4C23×C3111yescharacteristic
364C2×C4⋊C4C23×C3111yescharacteristic
365C2×C4⋊C4C23×C3111yescharacteristic
366C2×C4⋊C4C23×C3111yescharacteristic
367C2×C4⋊C4C23×C3111yescharacteristic
368C2×C4⋊C4C23×C3111yescharacteristic
369C2×C4⋊C4C23×C3111yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42C23×C3111yescharacteristic
381C2×C42C23×C3111yescharacteristic
382C2×C42C23×C3111yescharacteristic
383#107—22no—
384#106—22no—
385#106—22no—
386#106—22no—
387C4⋊C8—22no—
388#102—22no—
389#102—22no—
390#102—22no—
391#102—22no—
392C4×C8—22no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95D4×C3111yescharacteristic
401#95D4×C3111yescharacteristic
402C31#2713911yescharacteristic, direct product factor, Sylow
403C24—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4C2×D4×C3111yescharacteristic
418C22×C4C2×D4×C3111yescharacteristic
419C22×C4C2×D4×C3111yescharacteristic
420C22×C4C2×D4×C3111yescharacteristic
421C22×C4C2×D4×C3111yescharacteristic
422C22×C4C2×D4×C3111yescharacteristic
423C22×C4C24×C3111yescharacteristic
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8#41×C3112yes—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4#41×C3112yes—
457C4⋊C4#41×C3112yes—
458C4⋊C4#41×C3112yes—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—42no—
472C42—42no—
473C42—81no—
474C42—81no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23C22×D4×C3111yescharacteristic
483D4—41no—
484D4—41no—
485D4—41no—
486D4—41no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4C31×#13512yes—
525C2×C4C31×AGL(1,ℤ/8)12yes—
526C2×C4C2×#41×C3111yescharacteristic
527C2×C4C2×#41×C3111yescharacteristic
528C2×C4C2×#41×C3111yescharacteristic
529C2×C4C2×#41×C3111yescharacteristic
530C2×C4C31×2−1+411yescharacteristic
531C2×C4C31×2−1+411yescharacteristic
532C8—22no—
533C8—41no—
534C8—41no—
535C8—41no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22C31×#52511yescharacteristic
544C22C31×#54711yescharacteristic
545C22C31×#54811yescharacteristic
546C22C2×C31×#13511yescharacteristic
547C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
548C22C31×#57711yescharacteristic
549C22C31×#57711yescharacteristic
550C4—22no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—42no—
555C4—41no—
556C4—42no—
557C4—41no—
558C4—42no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—81no—
563C4C31×#52911yescharacteristic
564C4C31×#54111yescharacteristic
565C4C31×#55611yescharacteristic
566C4C31×#55611yescharacteristic
567C2—161no—
568C2C31×#246911yescharacteristic
569C2C31×#310611yescharacteristic
570C2C31×#312511yescharacteristic
571C2C31×#333711yescharacteristic
572C2C31×#334511yescharacteristic
573C2C31×#336011yescharacteristic
574C2C31×#336811yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×D4×C31C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C4×C4⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2370C2semidirect
3. C31×#2100C2semidirect
5. C31×#2059C2semidirect
7. C31×#1948C2semidirect
12. C31×#1920C2semidirect
13. C31×#1918C2semidirect
14. C31×#1916C2semidirect
15. C4×C31×#106C2semidirect
187. #27139C31direct (N × H)
250. #2370C2×C31semidirect
251. #2100C2×C31semidirect
253. #2059C2×C31semidirect
255. #1948C2×C31semidirect
260. #1920C2×C31semidirect
261. #1918C2×C31semidirect
262. #1916C2×C31semidirect
263. C4×#106C2×C31semidirect
402. C31#27139direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25122
C31×#23702
C31×#21002
C31×#20652
C31×#20592
C31×#20252
C31×#19482
C31×#19342
C31×#19262
C31×#19252
C31×#19202
C31×#19182
C31×#19162
C4×C31×#1062
#2713931