Finite Group Databasebeta

← C131×#29669 #45715553

Subgroups — C131×#29669 #45715553

Order 33536.

1196 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#542×C131C211yesmaximal, characteristic, complemented
2C2×#460×C131C211yesmaximal, characteristic, complemented
3C2×#459×C131C211yesmaximal, characteristic, complemented
4#2619×C131C211yesmaximal, characteristic, complemented
5#2023×C131C212yesmaximal, complemented
6#2012×C131C212yesmaximal, complemented
7#2000×C131C212yesmaximal, complemented
8#1999×C131C212yesmaximal, complemented
9#1805×C131C211yesmaximal, characteristic, complemented
10#1804×C131C211yesmaximal, characteristic, complemented
11#1798×C131C211yesmaximal, characteristic, complemented
12C22×SD16×C131C2212yes—
13C22×D8×C131C2212yes—
14#542×C131—24no—
15#542×C131—24no—
16C2×C4⋊Q8×C131C2211yescharacteristic
17C2×#127×C131C2211yescharacteristic
18C2×#126×C131—21no—
19C2×#124×C131—21no—
20C2×#124×C131C2211yescharacteristic
21C2×#123×C131—21no—
22C2×C22⋊Q8×C131—21no—
23C2×C22⋊Q8×C131C2211yescharacteristic
24C2×#121×C131C2211yescharacteristic
25C2×#121×C131C2212yes—
26C2×C4×Q8×C131C2211yescharacteristic
27C2×C4×D4×C131C2211yescharacteristic
28C2×#117×C131C2211yescharacteristic
29#460×C131—24no—
30#459×C131—24no—
31C2×C4⋊C8×C131C2211yescharacteristic
32C2×C4⋊C8×C131C2211yescharacteristic
33C2×#103×C131C2212yes—
34C2×#102×C131C2212yes—
35C2×#102×C131C2212yes—
36C2×#102×C131C2212yes—
37C2×C22⋊C8×C131C2212yes—
38#394×C131—21no—
39#394×C131—21no—
40#392×C131—21no—
41#392×C131—21no—
42#390×C131C2212yes—
43#382×C131C2211yescharacteristic
44#340×C131C2212yes—
45#340×C131C2212yes—
46#336×C131C2212yes—
47C22×Q8×C131—21no—
48C22×Q8×C131C2311yescharacteristic
49C22×D4×C131—21no—
50C22×D4×C131—22no—
51C22×D4×C131—21no—
52C22×D4×C131—21no—
53C22×D4×C131C2312yes—
54C22×D4×C131C2311yescharacteristic
55C23×C4×C131—21no—
56C23×C4×C131C2311yescharacteristic
57C2×SD16×C131—24no—
58C2×SD16×C131—42no—
59C2×SD16×C131—42no—
60C2×D8×C131—24no—
61C2×D8×C131—42no—
62C2×D8×C131—42no—
63C22×C8×C131C2312yes—
64C22×C8×C131C2312yes—
65C4⋊Q8×C131—22no—
66C4⋊Q8×C131—22no—
67#127×C131—22no—
68#127×C131—22no—
69#126×C131—28no—
70#124×C131—28no—
71#124×C131—22no—
72#124×C131—22no—
73#123×C131—28no—
74C22⋊Q8×C131—28no—
75C22⋊Q8×C131—22no—
76C22⋊Q8×C131—22no—
77#121×C131—22no—
78#121×C131—22no—
79#121×C131—42no—
80#121×C131—42no—
81C4×Q8×C131—22no—
82C4×Q8×C131D414yescomplemented
83C4×D4×C131—22no—
84C4×D4×C131D414yescomplemented
85#117×C131—22no—
86#117×C131—22no—
87C2×C4⋊C4×C131—21no—
88C2×C4⋊C4×C131—21no—
89C2×C4⋊C4×C131—21no—
90C2×C4⋊C4×C131D411yescharacteristic
91C2×C4⋊C4×C131D411yescharacteristic
92C2×C4⋊C4×C131C2312yes—
93C2×C4⋊C4×C131C2311yescharacteristic
94C2×C4⋊C4×C131C2311yescharacteristic
95C2×C22⋊C4×C131—21no—
96C2×C22⋊C4×C131—21no—
97C2×C22⋊C4×C131—21no—
98C2×C22⋊C4×C131—21no—
99C2×C22⋊C4×C131—21no—
100C2×C22⋊C4×C131—22no—
101C2×C22⋊C4×C131—22no—
102C2×C22⋊C4×C131—22no—
103C2×C22⋊C4×C131D411yescharacteristic
104C2×C22⋊C4×C131D411yescharacteristic
105C2×C42×C131—21no—
106C2×C42×C131C2311yescharacteristic
107C2×C42×C131C2311yescharacteristic
108C4⋊C8×C131—22no—
109C4⋊C8×C131—22no—
110#103×C131—42no—
111#102×C131—24no—
112#102×C131—24no—
113#102×C131—42no—
114C22⋊C8×C131—42no—
115#95×C131—21no—
116#95×C131—21no—
117C24×C131—21no—
118C24×C131—22no—
119C24×C131C2×D411yescharacteristic
120C2×Q8×C131—24no—
121C2×Q8×C131—21no—
122C2×Q8×C131—21no—
123C2×Q8×C131—21no—
124C2×Q8×C131—21no—
125C2×Q8×C131—21no—
126C2×Q8×C131—82no—
127C2×Q8×C131C2×D412yes—
128C2×D4×C131—24no—
129C2×D4×C131—21no—
130C2×D4×C131—22no—
131C2×D4×C131—22no—
132C2×D4×C131—21no—
133C2×D4×C131—21no—
134C2×D4×C131—21no—
135C2×D4×C131—22no—
136C2×D4×C131—22no—
137C2×D4×C131—22no—
138C2×D4×C131—21no—
139C2×D4×C131—22no—
140C2×D4×C131—44no—
141C2×D4×C131—42no—
142C2×D4×C131—42no—
143C2×D4×C131—82no—
144C2×D4×C131—82no—
145C2×D4×C131—82no—
146C2×D4×C131—82no—
147C2×D4×C131—82no—
148C2×D4×C131C2×D412yes—
149C22×C4×C131—21no—
150C22×C4×C131—21no—
151C22×C4×C131—21no—
152C22×C4×C131—21no—
153C22×C4×C131—21no—
154C22×C4×C131—21no—
155C22×C4×C131—21no—
156C22×C4×C131—21no—
157C22×C4×C131—21no—
158C22×C4×C131—24no—
159C22×C4×C131—22no—
160C22×C4×C131—82no—
161C22×C4×C131C2×D411yescharacteristic
162C22×C4×C131C2×D411yescharacteristic
163C22×C4×C131C2×D411yescharacteristic
164C22×C4×C131C2×D411yescharacteristic
165C22×C4×C131C2×D411yescharacteristic
166C22×C4×C131C2411yescharacteristic
167SD16×C131—48no—
168D8×C131—48no—
169C2×C8×C131—22no—
170C2×C8×C131—22no—
171C2×C8×C131—22no—
172C2×C8×C131—22no—
173C2×C8×C131—22no—
174C2×C8×C131—22no—
175C4⋊C4×C131—24no—
176C4⋊C4×C131—24no—
177C4⋊C4×C131—24no—
178C4⋊C4×C131—22no—
179C4⋊C4×C131—22no—
180C4⋊C4×C131—21no—
181C4⋊C4×C131—21no—
182C4⋊C4×C131—22no—
183C4⋊C4×C131—22no—
184C4⋊C4×C131C2×D412yes—
185C4⋊C4×C131C2×D412yes—
186C22⋊C4×C131—24no—
187C22⋊C4×C131—24no—
188C22⋊C4×C131—24no—
189C22⋊C4×C131—24no—
190C22⋊C4×C131—24no—
191C22⋊C4×C131—22no—
192C22⋊C4×C131—22no—
193C22⋊C4×C131—44no—
194C22⋊C4×C131—82no—
195C22⋊C4×C131—82no—
196C42×C131—24no—
197C42×C131—21no—
198C42×C131—21no—
199C42×C131C2×D412yes—
200C42×C131C2×D412yes—
201C23×C131—21no—
202C23×C131—22no—
203C23×C131—21no—
204C23×C131—21no—
205C23×C131—22no—
206C23×C131—22no—
207C23×C131—41no—
208C23×C131—41no—
209C23×C131—42no—
210C23×C131—42no—
211C23×C131—42no—
212C23×C131—42no—
213C23×C131—42no—
214C23×C131—42no—
215C23×C131—42no—
216C23×C131—42no—
217C23×C131—42no—
218C23×C131C22×D411yescharacteristic
219Q8×C131—22no—
220Q8×C131—22no—
221Q8×C131—41no—
222Q8×C131—41no—
223Q8×C131—84no—
224D4×C131—22no—
225D4×C131—22no—
226D4×C131—41no—
227D4×C131—42no—
228D4×C131—42no—
229D4×C131—41no—
230D4×C131—42no—
231D4×C131—42no—
232D4×C131—84no—
233D4×C131—84no—
234D4×C131—84no—
235D4×C131—88no—
236C2×C4×C131—22no—
237C2×C4×C131—22no—
238C2×C4×C131—21no—
239C2×C4×C131—22no—
240C2×C4×C131—21no—
241C2×C4×C131—22no—
242C2×C4×C131—21no—
243C2×C4×C131—22no—
244C2×C4×C131—21no—
245C2×C4×C131—21no—
246C2×C4×C131—22no—
247C2×C4×C131—22no—
248C2×C4×C131—41no—
249C2×C4×C131—41no—
250C2×C4×C131—42no—
251C2×C4×C131—42no—
252C2×C4×C131—41no—
253C2×C4×C131—41no—
254C2×C4×C131—41no—
255C2×C4×C131—41no—
256C2×C4×C131—42no—
257C2×C4×C131—42no—
258C2×C4×C131—42no—
259C2×C4×C131—84no—
260C2×C4×C131#12012yes—
261C2×C4×C131#12012yes—
262C2×C4×C131C22×D411yescharacteristic
263C2×C4×C131C22×D411yescharacteristic
264C2×C4×C131H(5,𝔽2)11yescharacteristic
265C2×C4×C131H(5,𝔽2)11yescharacteristic
266C2×C4×C131H(5,𝔽2)11yescharacteristic
267C2×C4×C1312−1+411yescharacteristic
268C8×C131—42no—
269C8×C131—42no—
270C22×C131—41no—
271C22×C131—42no—
272C22×C131—42no—
273C22×C131—42no—
274C22×C131—42no—
275C22×C131—41no—
276C22×C131—41no—
277C22×C131—42no—
278C22×C131—82no—
279C22×C131—82no—
280C22×C131—82no—
281C22×C131—82no—
282C22×C131—82no—
283C22×C131—82no—
284C22×C131—82no—
285C22×C131—82no—
286C22×C131—82no—
287C22×C131C2×#12011yescharacteristic
288C22×C131#53511yescharacteristic
289C22×C131#53711yescharacteristic
290C22×C131#57612yes—
291C22×C131#57712yes—
292C4×C131—22no—
293C4×C131—22no—
294C4×C131—41no—
295C4×C131—42no—
296C4×C131—41no—
297C4×C131—42no—
298C4×C131—42no—
299C4×C131—41no—
300C4×C131—82no—
301C4×C131#53511yescharacteristic
302C4×C131#53511yescharacteristic
303C4×C131#53711yescharacteristic
304C4×C131#53711yescharacteristic
305C2×C131—81no—
306C2×C131—82no—
307C2×C131—162no—
308C2×C131#261511yescharacteristic
309C2×C131#300811yescharacteristic
310C2×C131#301011yescharacteristic
311C2×C131#322412yes—
312C2×C131#325712yes—
313#29669C13111yesmaximal, characteristic, direct product factor, Sylow
314C131#2966911yescharacteristic, direct product factor, Sylow
315C2×#542C2×C13111yescharacteristic, complemented
316C2×#460C2×C13111yescharacteristic, complemented
317C2×#459C2×C13111yescharacteristic, complemented
318#2619C2×C13111yescharacteristic, complemented
319#2023C2×C13112yescomplemented
320#2012C2×C13112yescomplemented
321#2000C2×C13112yescomplemented
322#1999C2×C13112yescomplemented
323#1805C2×C13111yescharacteristic, complemented
324#1804C2×C13111yescharacteristic, complemented
325#1798C2×C13111yescharacteristic, complemented
326C22×SD16C22×C13112yes—
327C22×D8C22×C13112yes—
328#542—24no—
329#542—24no—
330C2×C4⋊Q8C22×C13111yescharacteristic
331C2×#127C22×C13111yescharacteristic
332C2×#126—21no—
333C2×#124—21no—
334C2×#124C22×C13111yescharacteristic
335C2×#123—21no—
336C2×C22⋊Q8—21no—
337C2×C22⋊Q8C22×C13111yescharacteristic
338C2×#121C22×C13111yescharacteristic
339C2×#121C22×C13112yes—
340C2×C4×Q8C22×C13111yescharacteristic
341C2×C4×D4C22×C13111yescharacteristic
342C2×#117C22×C13111yescharacteristic
343#460—24no—
344#459—24no—
345C2×C4⋊C8C22×C13111yescharacteristic
346C2×C4⋊C8C22×C13111yescharacteristic
347C2×#103C22×C13112yes—
348C2×#102C22×C13112yes—
349C2×#102C22×C13112yes—
350C2×#102C22×C13112yes—
351C2×C22⋊C8C22×C13112yes—
352#394—21no—
353#394—21no—
354#392—21no—
355#392—21no—
356#390C22×C13112yes—
357#382C22×C13111yescharacteristic
358#340C22×C13112yes—
359#340C22×C13112yes—
360#336C22×C13112yes—
361C22×Q8—21no—
362C22×Q8C23×C13111yescharacteristic
363C22×D4—21no—
364C22×D4—22no—
365C22×D4—21no—
366C22×D4—21no—
367C22×D4C23×C13112yes—
368C22×D4C23×C13111yescharacteristic
369C23×C4—21no—
370C23×C4C23×C13111yescharacteristic
371C2×SD16—24no—
372C2×SD16—42no—
373C2×SD16—42no—
374C2×D8—24no—
375C2×D8—42no—
376C2×D8—42no—
377C22×C8C23×C13112yes—
378C22×C8C23×C13112yes—
379C4⋊Q8—22no—
380C4⋊Q8—22no—
381#127—22no—
382#127—22no—
383#126—28no—
384#124—28no—
385#124—22no—
386#124—22no—
387#123—28no—
388C22⋊Q8—28no—
389C22⋊Q8—22no—
390C22⋊Q8—22no—
391#121—22no—
392#121—22no—
393#121—42no—
394#121—42no—
395C4×Q8—22no—
396C4×Q8D4×C13114yescomplemented
397C4×D4—22no—
398C4×D4D4×C13114yescomplemented
399#117—22no—
400#117—22no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4D4×C13111yescharacteristic
405C2×C4⋊C4D4×C13111yescharacteristic
406C2×C4⋊C4C23×C13112yes—
407C2×C4⋊C4C23×C13111yescharacteristic
408C2×C4⋊C4C23×C13111yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—22no—
415C2×C22⋊C4—22no—
416C2×C22⋊C4—22no—
417C2×C22⋊C4D4×C13111yescharacteristic
418C2×C22⋊C4D4×C13111yescharacteristic
419C2×C42—21no—
420C2×C42C23×C13111yescharacteristic
421C2×C42C23×C13111yescharacteristic
422C4⋊C8—22no—
423C4⋊C8—22no—
424#103—42no—
425#102—24no—
426#102—24no—
427#102—42no—
428C22⋊C8—42no—
429#95—21no—
430#95—21no—
431C24—21no—
432C24—22no—
433C24C2×D4×C13111yescharacteristic
434C2×Q8—24no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—82no—
441C2×Q8C2×D4×C13112yes—
442C2×D4—24no—
443C2×D4—21no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—44no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—82no—
458C2×D4—82no—
459C2×D4—82no—
460C2×D4—82no—
461C2×D4—82no—
462C2×D4C2×D4×C13112yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—82no—
475C22×C4C2×D4×C13111yescharacteristic
476C22×C4C2×D4×C13111yescharacteristic
477C22×C4C2×D4×C13111yescharacteristic
478C22×C4C2×D4×C13111yescharacteristic
479C22×C4C2×D4×C13111yescharacteristic
480C22×C4C24×C13111yescharacteristic
481SD16—48no—
482D8—48no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4C2×D4×C13112yes—
499C4⋊C4C2×D4×C13112yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—44no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C42—24no—
511C42—21no—
512C42—21no—
513C42C2×D4×C13112yes—
514C42C2×D4×C13112yes—
515C23—21no—
516C23—22no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—22no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23C22×D4×C13111yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—41no—
536Q8—41no—
537Q8—84no—
538D4—22no—
539D4—22no—
540D4—41no—
541D4—42no—
542D4—42no—
543D4—41no—
544D4—42no—
545D4—42no—
546D4—84no—
547D4—84no—
548D4—84no—
549D4—88no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—84no—
574C2×C4#120×C13112yes—
575C2×C4#120×C13112yes—
576C2×C4C22×D4×C13111yescharacteristic
577C2×C4C22×D4×C13111yescharacteristic
578C2×C4H(5,𝔽2)×C13111yescharacteristic
579C2×C4H(5,𝔽2)×C13111yescharacteristic
580C2×C4H(5,𝔽2)×C13111yescharacteristic
581C2×C42−1+4×C13111yescharacteristic
582C8—42no—
583C8—42no—
584C22—42no—
585C22—42no—
586C22—41no—
587C22—41no—
588C22—42no—
589C22—41no—
590C22—42no—
591C22—42no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22C2×#120×C13111yescharacteristic
602C22#535×C13111yescharacteristic
603C22#537×C13111yescharacteristic
604C22#576×C13112yes—
605C22#577×C13112yes—
606C4—22no—
607C4—22no—
608C4—41no—
609C4—42no—
610C4—42no—
611C4—41no—
612C4—41no—
613C4—42no—
614C4—82no—
615C4#535×C13111yescharacteristic
616C4#535×C13111yescharacteristic
617C4#537×C13111yescharacteristic
618C4#537×C13111yescharacteristic
619C2—81no—
620C2—82no—
621C2—162no—
622C2#2615×C13111yescharacteristic
623C2#3008×C13111yescharacteristic
624C2#3010×C13111yescharacteristic
625C2#3224×C13112yes—
626C2#3257×C13112yes—
subgroupG/H
center, socleC23×C131C22×D4
commutator, Frattini, cosocleC22×C4C24×C131
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C131 ◁ C2×#117×C131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#542×C131C2semidirect
2. C2×#460×C131C2semidirect
3. C2×#459×C131C2semidirect
4. #2619×C131C2semidirect
5. #2023×C131C2semidirect
6. #2012×C131C2semidirect
7. #2000×C131C2semidirect
8. #1999×C131C2semidirect
9. #1805×C131C2semidirect
10. #1804×C131C2semidirect
11. #1798×C131C2semidirect
82. C4×Q8×C131D4semidirect
84. C4×D4×C131D4semidirect
313. #29669C131direct (N × H)
314. C131#29669direct (N × H)
315. C2×#542C2×C131semidirect
316. C2×#460C2×C131semidirect
317. C2×#459C2×C131semidirect
318. #2619C2×C131semidirect
319. #2023C2×C131semidirect
320. #2012C2×C131semidirect
321. #2000C2×C131semidirect
322. #1999C2×C131semidirect
323. #1805C2×C131semidirect
324. #1804C2×C131semidirect
325. #1798C2×C131semidirect
396. C4×Q8D4×C131semidirect
398. C4×D4D4×C131semidirect

Maximal subgroups

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

subgroupindex
C2×#542×C1312
C2×#460×C1312
C2×#459×C1312
#2619×C1312
#2023×C1312
#2012×C1312
#2000×C1312
#1999×C1312
#1805×C1312
#1804×C1312
#1798×C1312
#29669131