Finite Group Databasebeta

← C137×#23901 #46873287

Subgroups — C137×#23901 #46873287

Order 35072.

2056 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3484×C137C211yesmaximal, characteristic
2C22×#123×C137C211yesmaximal, characteristic, complemented
3C2×#458×C137C212yesmaximal, complemented
4#2849×C137C212yesmaximal, complemented
5#2022×C137C214yesmaximal, complemented
6#1898×C137C212yesmaximal, complemented
7#1890×C137C212yesmaximal, complemented
8#1781×C137C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C137C2211yescharacteristic
10C23×D4×C137C2211yescharacteristic
11C24×C4×C137C2211yescharacteristic, complemented
12#546×C137—24no—
13D42×C137—24no—
14#535×C137—21no—
15#534×C137—21no—
16#534×C137C2212yes—
17#525×C137—21no—
18C2×#123×C137—21no—
19C2×#123×C137—21no—
20C2×#123×C137—28no—
21C2×#123×C137C2212yes—
22C2×#123×C137C2212yescomplemented
23C2×#123×C137C2214yescomplemented
24C2×#120×C137—21no—
25C2×#120×C137C2212yes—
26#518×C137C2211yescharacteristic
27C22×C4⋊C4×C137—21no—
28C22×C22⋊C4×C137—21no—
29C22×C22⋊C4×C137C2211yescharacteristic, complemented
30#458×C137—28no—
31C2×#99×C137C2212yes—
32C2×#99×C137C2214yescomplemented
33C2×#99×C137C2212yescomplemented
34#386×C137—24no—
35#385×C137—24no—
36#381×C137C2214yescomplemented
37#379×C137C2212yescomplemented
38#342×C137C2214yescomplemented
39C25×C137D411yescharacteristic
40C25×C137C2311yescharacteristic
41H(5,𝔽2)×C137—24no—
42H(5,𝔽2)×C137—42no—
43C2×#41×C137—21no—
44C2×#41×C137—22no—
45C22×D4×C137—21no—
46C22×D4×C137—22no—
47C22×D4×C137—24no—
48C22×D4×C137—22no—
49C22×D4×C137—24no—
50C22×D4×C137—42no—
51C22×D4×C137D411yescharacteristic
52C22×D4×C137D411yescharacteristic
53C22×D4×C137D414yes—
54C22×D4×C137C2312yes—
55C22×D4×C137C2311yescharacteristic
56C22×D4×C137C2311yescharacteristic
57C23×C4×C137—21no—
58C23×C4×C137—21no—
59C23×C4×C137—21no—
60C23×C4×C137—24no—
61C23×C4×C137—24no—
62C23×C4×C137—42no—
63C23×C4×C137D411yescharacteristic
64C23×C4×C137C2312yes—
65#127×C137—22no—
66#124×C137—22no—
67#123×C137—28no—
68#123×C137—28no—
69#123×C137—22no—
70#123×C137—24no—
71#123×C137—22no—
72#123×C137—22no—
73#123×C137—24no—
74#123×C137—24no—
75#123×C137—28no—
76#123×C137—44no—
77#123×C137D418yescomplemented
78C22⋊Q8×C137—22no—
79#121×C137—22no—
80#121×C137—24no—
81#121×C137—24no—
82#121×C137—24no—
83#120×C137—24no—
84#120×C137—22no—
85#120×C137—22no—
86#120×C137—22no—
87#120×C137—22no—
88#120×C137—24no—
89#120×C137—22no—
90C4×D4×C137—24no—
91C4×D4×C137—24no—
92#117×C137—21no—
93C2×C4⋊C4×C137—28no—
94C2×C4⋊C4×C137—22no—
95C2×C4⋊C4×C137—22no—
96C2×C4⋊C4×C137—24no—
97C2×C22⋊C4×C137—28no—
98C2×C22⋊C4×C137—22no—
99C2×C22⋊C4×C137—22no—
100C2×C22⋊C4×C137—22no—
101C2×C22⋊C4×C137—21no—
102C2×C22⋊C4×C137—21no—
103C2×C22⋊C4×C137—24no—
104C2×C22⋊C4×C137—24no—
105C2×C22⋊C4×C137—24no—
106C2×C22⋊C4×C137D414yes—
107C2×C22⋊C4×C137C2314yes—
108C2×C22⋊C4×C137C2312yes—
109C2×C22⋊C4×C137C2312yes—
110#99×C137—24no—
111#99×C137—28no—
112#99×C137—42no—
113#95×C137—28no—
114C24×C137—22no—
115C24×C137—22no—
116C24×C137—21no—
117C24×C137—22no—
118C24×C137—22no—
119C24×C137—44no—
120C24×C137—41no—
121C24×C137—44no—
122C24×C137C2×D411yescharacteristic
123C24×C137C2×D414yes—
124C24×C137C2×D411yescharacteristic
125C24×C137C2×D412yes—
126C24×C137C2×D411yescharacteristic
127C24×C137C2×D411yescharacteristic
128C24×C137C2411yescharacteristic
129#41×C137—48no—
130#41×C137—42no—
131#41×C137—42no—
132C2×Q8×C137—21no—
133C2×D4×C137—22no—
134C2×D4×C137—22no—
135C2×D4×C137—22no—
136C2×D4×C137—21no—
137C2×D4×C137—24no—
138C2×D4×C137—22no—
139C2×D4×C137—22no—
140C2×D4×C137—22no—
141C2×D4×C137—24no—
142C2×D4×C137—22no—
143C2×D4×C137—22no—
144C2×D4×C137—22no—
145C2×D4×C137—22no—
146C2×D4×C137—22no—
147C2×D4×C137—24no—
148C2×D4×C137—24no—
149C2×D4×C137—24no—
150C2×D4×C137—22no—
151C2×D4×C137—41no—
152C2×D4×C137—41no—
153C2×D4×C137—416no—
154C2×D4×C137—48no—
155C2×D4×C137—42no—
156C2×D4×C137—42no—
157C2×D4×C137—48no—
158C2×D4×C137—84no—
159C2×D4×C137C2×D414yes—
160C22×C4×C137—24no—
161C22×C4×C137—24no—
162C22×C4×C137—21no—
163C22×C4×C137—22no—
164C22×C4×C137—24no—
165C22×C4×C137—22no—
166C22×C4×C137—24no—
167C22×C4×C137—22no—
168C22×C4×C137—22no—
169C22×C4×C137—22no—
170C22×C4×C137—24no—
171C22×C4×C137—22no—
172C22×C4×C137—22no—
173C22×C4×C137—24no—
174C22×C4×C137—41no—
175C22×C4×C137—416no—
176C22×C4×C137—44no—
177C22×C4×C137—42no—
178C22×C4×C137—42no—
179C22×C4×C137—42no—
180C22×C4×C137—42no—
181C22×C4×C137C2×D412yes—
182C22×C4×C137C2×D411yescharacteristic
183C22×C4×C137C2×D411yescharacteristic
184C22×C4×C137C2×D414yes—
185C22×C4×C137C2×D414yescomplemented
186C4⋊C4×C137—28no—
187C4⋊C4×C137—22no—
188C4⋊C4×C137—24no—
189C4⋊C4×C137—28no—
190C4⋊C4×C137—48no—
191C22⋊C4×C137—24no—
192C22⋊C4×C137—28no—
193C22⋊C4×C137—22no—
194C22⋊C4×C137—24no—
195C22⋊C4×C137—24no—
196C22⋊C4×C137—28no—
197C22⋊C4×C137—24no—
198C22⋊C4×C137—24no—
199C22⋊C4×C137—22no—
200C22⋊C4×C137—22no—
201C22⋊C4×C137—22no—
202C22⋊C4×C137—24no—
203C22⋊C4×C137—24no—
204C22⋊C4×C137—48no—
205C22⋊C4×C137—48no—
206C22⋊C4×C137C2×D414yescomplemented
207C42×C137—22no—
208C23×C137—21no—
209C23×C137—21no—
210C23×C137—24no—
211C23×C137—21no—
212C23×C137—21no—
213C23×C137—21no—
214C23×C137—28no—
215C23×C137—28no—
216C23×C137—21no—
217C23×C137—21no—
218C23×C137—21no—
219C23×C137—22no—
220C23×C137—21no—
221C23×C137—24no—
222C23×C137—44no—
223C23×C137—42no—
224C23×C137—42no—
225C23×C137—44no—
226C23×C137—48no—
227C23×C137—48no—
228C23×C137—42no—
229C23×C137—42no—
230C23×C137—48no—
231C23×C137—42no—
232C23×C137—44no—
233C23×C137—44no—
234C23×C137—82no—
235C23×C137—84no—
236C23×C137—82no—
237C23×C137#12012yes—
238C23×C137#12012yes—
239C23×C137#12012yes—
240C23×C137C22×D411yescharacteristic
241C23×C137C22×D412yes—
242C23×C137C22×D412yes—
243C23×C137H(5,𝔽2)12yes—
244Q8×C137—42no—
245D4×C137—28no—
246D4×C137—44no—
247D4×C137—42no—
248D4×C137—44no—
249D4×C137—44no—
250D4×C137—42no—
251D4×C137—42no—
252D4×C137—44no—
253D4×C137—44no—
254D4×C137—48no—
255D4×C137—84no—
256D4×C137—84no—
257C2×C4×C137—21no—
258C2×C4×C137—21no—
259C2×C4×C137—24no—
260C2×C4×C137—24no—
261C2×C4×C137—21no—
262C2×C4×C137—24no—
263C2×C4×C137—22no—
264C2×C4×C137—21no—
265C2×C4×C137—21no—
266C2×C4×C137—28no—
267C2×C4×C137—24no—
268C2×C4×C137—44no—
269C2×C4×C137—44no—
270C2×C4×C137—48no—
271C2×C4×C137—44no—
272C2×C4×C137—48no—
273C2×C4×C137—48no—
274C2×C4×C137—42no—
275C2×C4×C137—42no—
276C2×C4×C137—42no—
277C2×C4×C137—44no—
278C2×C4×C137—44no—
279C2×C4×C137—48no—
280C2×C4×C137—44no—
281C2×C4×C137—82no—
282C2×C4×C137#12012yes—
283C22×C137—24no—
284C22×C137—24no—
285C22×C137—22no—
286C22×C137—24no—
287C22×C137—22no—
288C22×C137—22no—
289C22×C137—22no—
290C22×C137—22no—
291C22×C137—24no—
292C22×C137—42no—
293C22×C137—42no—
294C22×C137—416no—
295C22×C137—42no—
296C22×C137—42no—
297C22×C137—42no—
298C22×C137—42no—
299C22×C137—42no—
300C22×C137—42no—
301C22×C137—41no—
302C22×C137—44no—
303C22×C137—42no—
304C22×C137—81no—
305C22×C137—88no—
306C22×C137—84no—
307C22×C137—81no—
308C22×C137—162no—
309C22×C137#45814yescomplemented
310C22×C137C2×#12011yescharacteristic
311C22×C137C2×#12011yescharacteristic
312C22×C137#53411yescharacteristic
313C22×C137D4214yes—
314C4×C137—22no—
315C4×C137—44no—
316C4×C137—44no—
317C4×C137—41no—
318C4×C137—42no—
319C4×C137—44no—
320C4×C137—84no—
321C2×C137—21no—
322C2×C137—24no—
323C2×C137—21no—
324C2×C137—24no—
325C2×C137—42no—
326C2×C137—42no—
327C2×C137—81no—
328C2×C137—82no—
329C2×C137#240311yescharacteristic
330C2×C137C2×#45812yes—
331#23901C13711yesmaximal, characteristic, direct product factor, Sylow
332C137#2390111yescharacteristic, direct product factor, Sylow
333#3484C2×C13711yescharacteristic
334C22×#123C2×C13711yescharacteristic, complemented
335C2×#458C2×C13712yescomplemented
336#2849C2×C13712yescomplemented
337#2022C2×C13714yescomplemented
338#1898C2×C13712yescomplemented
339#1890C2×C13712yescomplemented
340#1781C2×C13711yescharacteristic, complemented
341C2×H(5,𝔽2)C22×C13711yescharacteristic
342C23×D4C22×C13711yescharacteristic
343C24×C4C22×C13711yescharacteristic, complemented
344#546—24no—
345D42—24no—
346#535—21no—
347#534—21no—
348#534C22×C13712yes—
349#525—21no—
350C2×#123—21no—
351C2×#123—21no—
352C2×#123—28no—
353C2×#123C22×C13712yes—
354C2×#123C22×C13712yescomplemented
355C2×#123C22×C13714yescomplemented
356C2×#120—21no—
357C2×#120C22×C13712yes—
358#518C22×C13711yescharacteristic
359C22×C4⋊C4—21no—
360C22×C22⋊C4—21no—
361C22×C22⋊C4C22×C13711yescharacteristic, complemented
362#458—28no—
363C2×#99C22×C13712yes—
364C2×#99C22×C13714yescomplemented
365C2×#99C22×C13712yescomplemented
366#386—24no—
367#385—24no—
368#381C22×C13714yescomplemented
369#379C22×C13712yescomplemented
370#342C22×C13714yescomplemented
371C25D4×C13711yescharacteristic
372C25C23×C13711yescharacteristic
373H(5,𝔽2)—24no—
374H(5,𝔽2)—42no—
375C2×#41—21no—
376C2×#41—22no—
377C22×D4—21no—
378C22×D4—22no—
379C22×D4—24no—
380C22×D4—22no—
381C22×D4—24no—
382C22×D4—42no—
383C22×D4D4×C13711yescharacteristic
384C22×D4D4×C13711yescharacteristic
385C22×D4D4×C13714yes—
386C22×D4C23×C13712yes—
387C22×D4C23×C13711yescharacteristic
388C22×D4C23×C13711yescharacteristic
389C23×C4—21no—
390C23×C4—21no—
391C23×C4—21no—
392C23×C4—24no—
393C23×C4—24no—
394C23×C4—42no—
395C23×C4D4×C13711yescharacteristic
396C23×C4C23×C13712yes—
397#127—22no—
398#124—22no—
399#123—28no—
400#123—28no—
401#123—22no—
402#123—24no—
403#123—22no—
404#123—22no—
405#123—24no—
406#123—24no—
407#123—28no—
408#123—44no—
409#123D4×C13718yescomplemented
410C22⋊Q8—22no—
411#121—22no—
412#121—24no—
413#121—24no—
414#121—24no—
415#120—24no—
416#120—22no—
417#120—22no—
418#120—22no—
419#120—22no—
420#120—24no—
421#120—22no—
422C4×D4—24no—
423C4×D4—24no—
424#117—21no—
425C2×C4⋊C4—28no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—24no—
429C2×C22⋊C4—28no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—22no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—24no—
436C2×C22⋊C4—24no—
437C2×C22⋊C4—24no—
438C2×C22⋊C4D4×C13714yes—
439C2×C22⋊C4C23×C13714yes—
440C2×C22⋊C4C23×C13712yes—
441C2×C22⋊C4C23×C13712yes—
442#99—24no—
443#99—28no—
444#99—42no—
445#95—28no—
446C24—22no—
447C24—22no—
448C24—21no—
449C24—22no—
450C24—22no—
451C24—44no—
452C24—41no—
453C24—44no—
454C24C2×D4×C13711yescharacteristic
455C24C2×D4×C13714yes—
456C24C2×D4×C13711yescharacteristic
457C24C2×D4×C13712yes—
458C24C2×D4×C13711yescharacteristic
459C24C2×D4×C13711yescharacteristic
460C24C24×C13711yescharacteristic
461#41—48no—
462#41—42no—
463#41—42no—
464C2×Q8—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—24no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—24no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—24no—
482C2×D4—22no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—416no—
486C2×D4—48no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—48no—
490C2×D4—84no—
491C2×D4C2×D4×C13714yes—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—24no—
506C22×C4—41no—
507C22×C4—416no—
508C22×C4—44no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4C2×D4×C13712yes—
514C22×C4C2×D4×C13711yescharacteristic
515C22×C4C2×D4×C13711yescharacteristic
516C22×C4C2×D4×C13714yes—
517C22×C4C2×D4×C13714yescomplemented
518C4⋊C4—28no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—28no—
522C4⋊C4—48no—
523C22⋊C4—24no—
524C22⋊C4—28no—
525C22⋊C4—22no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—28no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4C2×D4×C13714yescomplemented
539C42—22no—
540C23—24no—
541C23—21no—
542C23—21no—
543C23—28no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—28no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—21no—
553C23—24no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—48no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—48no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—82no—
567C23—84no—
568C23—82no—
569C23#120×C13712yes—
570C23#120×C13712yes—
571C23#120×C13712yes—
572C23C22×D4×C13711yescharacteristic
573C23C22×D4×C13712yes—
574C23C22×D4×C13712yes—
575C23H(5,𝔽2)×C13712yes—
576Q8—42no—
577D4—28no—
578D4—44no—
579D4—42no—
580D4—44no—
581D4—44no—
582D4—48no—
583D4—42no—
584D4—42no—
585D4—44no—
586D4—44no—
587D4—84no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—82no—
614C2×C4#120×C13712yes—
615C22—24no—
616C22—24no—
617C22—22no—
618C22—24no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—416no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—41no—
634C22—44no—
635C22—42no—
636C22—81no—
637C22—88no—
638C22—81no—
639C22—84no—
640C22—162no—
641C22#458×C13714yescomplemented
642C22C2×#120×C13711yescharacteristic
643C22C2×#120×C13711yescharacteristic
644C22#534×C13711yescharacteristic
645C22D42×C13714yes—
646C4—22no—
647C4—44no—
648C4—44no—
649C4—41no—
650C4—42no—
651C4—44no—
652C4—84no—
653C2—21no—
654C2—24no—
655C2—24no—
656C2—21no—
657C2—42no—
658C2—42no—
659C2—81no—
660C2—82no—
661C2#2403×C13711yescharacteristic
662C2C2×#458×C13712yes—
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC24C24×C137
prime radical#3484×C137C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C25×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#123×C137C2semidirect
3. C2×#458×C137C2semidirect
4. #2849×C137C2semidirect
5. #2022×C137C2semidirect
6. #1898×C137C2semidirect
7. #1890×C137C2semidirect
8. #1781×C137C2semidirect
11. C24×C4×C137C22semidirect
22. C2×#123×C137C22semidirect
23. C2×#123×C137C22semidirect
29. C22×C22⋊C4×C137C22semidirect
32. C2×#99×C137C22semidirect
33. C2×#99×C137C22semidirect
36. #381×C137C22semidirect
37. #379×C137C22semidirect
38. #342×C137C22semidirect
77. #123×C137D4semidirect
185. C22×C4×C137C2×D4semidirect
206. C22⋊C4×C137C2×D4semidirect
309. C22×C137#458semidirect
331. #23901C137direct (N × H)
332. C137#23901direct (N × H)
334. C22×#123C2×C137semidirect
335. C2×#458C2×C137semidirect
336. #2849C2×C137semidirect
337. #2022C2×C137semidirect
338. #1898C2×C137semidirect
339. #1890C2×C137semidirect
340. #1781C2×C137semidirect
343. C24×C4C22×C137semidirect
354. C2×#123C22×C137semidirect
355. C2×#123C22×C137semidirect
361. C22×C22⋊C4C22×C137semidirect
364. C2×#99C22×C137semidirect
365. C2×#99C22×C137semidirect
368. #381C22×C137semidirect
369. #379C22×C137semidirect
370. #342C22×C137semidirect
409. #123D4×C137semidirect
517. C22×C4C2×D4×C137semidirect
538. C22⋊C4C2×D4×C137semidirect
641. C22#458×C137semidirect

Maximal subgroups

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

subgroupindex
#3484×C1372
C22×#123×C1372
C2×#458×C1372
#2849×C1372
#2022×C1372
#1898×C1372
#1890×C1372
#1781×C1372
#23901137