Finite Group Databasebeta

← C137×#18255 #46867641

Subgroups — C137×#18255 #46867641

Order 35072.

1260 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3537×C137C211yesmaximal, characteristic
2#3004×C137C211yesmaximal, characteristic, complemented
3C2×#448×C137C211yesmaximal, characteristic, complemented
4#2436×C137C211yesmaximal, characteristic, complemented
5#1606×C137C214yesmaximal, complemented
6#1601×C137C214yesmaximal, complemented
7#1507×C137C211yesmaximal, characteristic, complemented
8#1501×C137C211yesmaximal, characteristic, complemented
9#1471×C137C211yesmaximal, characteristic, complemented
10C23×Q8×C137C2211yescharacteristic, complemented
11C2×#137×C137—21no—
12C22×SD16×C137—21no—
13#560×C137—22no—
14#553×C137—21no—
15#553×C137—21no—
16D42×C137—22no—
17D42×C137—21no—
18D42×C137—21no—
19#540×C137—21no—
20#538×C137—21no—
21#538×C137C2212yes—
22#535×C137—21no—
23#534×C137—21no—
24#534×C137C2212yes—
25C2×#124×C137—21no—
26#525×C137C2211yescharacteristic
27C2×#121×C137C2211yescharacteristic
28C2×#121×C137C2211yescharacteristic
29#451×C137—21no—
30#451×C137C2212yescomplemented
31#448×C137—22no—
32#448×C137—21no—
33#448×C137C2212yescomplemented
34#448×C137C2214yescomplemented
35#419×C137—21no—
36C2×#103×C137—21no—
37#407×C137C2211yescharacteristic, complemented
38C2×C22⋊C8×C137C2211yescharacteristic, complemented
39#393×C137C2214yescomplemented
40#391×C137—21no—
41#386×C137—22no—
42C4×C22⋊C4×C137C2211yescharacteristic, complemented
43#328×C137C2212yescomplemented
44#327×C137C2212yescomplemented
45#327×C137C2214yescomplemented
46#323×C137C2214yescomplemented
47C2×#41×C137D411yescharacteristic
48C22×Q8×C137—21no—
49C22×Q8×C137—21no—
50C22×Q8×C137—21no—
51C22×Q8×C137—21no—
52C22×Q8×C137—21no—
53C22×Q8×C137—44no—
54C22×Q8×C137C2312yes—
55C22×D4×C137—22no—
56C22×D4×C137—21no—
57C22×D4×C137—21no—
58C22×D4×C137—22no—
59C22×D4×C137D412yes—
60C22×D4×C137D411yescharacteristic
61C22×D4×C137D411yescharacteristic
62C23×C4×C137—21no—
63C23×C4×C137C2311yescharacteristic
64#137×C137—44no—
65C2×Q16×C137—22no—
66C2×SD16×C137—24no—
67C2×SD16×C137—22no—
68C2×SD16×C137—44no—
69C2×M16×C137—21no—
70C22×C8×C137—21no—
71#127×C137—22no—
72#127×C137—21no—
73#126×C137—22no—
74#126×C137—22no—
75#125×C137—21no—
76#124×C137—22no—
77#124×C137—44no—
78#123×C137—22no—
79#123×C137—21no—
80#123×C137—22no—
81#123×C137—22no—
82#123×C137—22no—
83C22⋊Q8×C137D411yescharacteristic
84C22⋊Q8×C137C2311yescharacteristic
85#121×C137—22no—
86#121×C137—21no—
87#121×C137—21no—
88#121×C137—22no—
89#121×C137—22no—
90#121×C137—22no—
91#121×C137—22no—
92#121×C137—21no—
93#121×C137—21no—
94#121×C137—21no—
95#121×C137—21no—
96#121×C137—21no—
97#121×C137—21no—
98#121×C137—22no—
99#121×C137D412yes—
100#121×C137D411yescharacteristic
101#121×C137C2312yes—
102#121×C137C2311yescharacteristic
103#120×C137—22no—
104#120×C137—22no—
105#120×C137—22no—
106#120×C137—21no—
107#120×C137—22no—
108#120×C137—22no—
109#120×C137—22no—
110C4×D4×C137—22no—
111C4×D4×C137—21no—
112C4×D4×C137—21no—
113C4×D4×C137—21no—
114C4×D4×C137—21no—
115C4×D4×C137—22no—
116#117×C137—21no—
117#117×C137D411yescharacteristic
118C2×C4⋊C4×C137D411yescharacteristic
119C2×C4⋊C4×C137D411yescharacteristic
120C2×C22⋊C4×C137—21no—
121C2×C22⋊C4×C137—21no—
122C2×C22⋊C4×C137—24no—
123C2×C22⋊C4×C137C2312yes—
124C2×C42×C137—21no—
125#103×C137—24no—
126#103×C137—22no—
127#103×C137—22no—
128C22⋊C8×C137—21no—
129C22⋊C8×C137—21no—
130C22⋊C8×C137C2312yes—
131C22⋊C8×C137C2312yes—
132#95×C137—24no—
133#95×C137C2312yes—
134C24×C137—22no—
135C24×C137—21no—
136C24×C137C2×D411yescharacteristic
137C24×C137C2×D412yes—
138#41×C137—42no—
139C2×Q8×C137—24no—
140C2×Q8×C137—24no—
141C2×Q8×C137—24no—
142C2×Q8×C137—22no—
143C2×Q8×C137—22no—
144C2×Q8×C137—44no—
145C2×Q8×C137—42no—
146C2×Q8×C137—42no—
147C2×Q8×C137—42no—
148C2×Q8×C137—44no—
149C2×Q8×C137—44no—
150C2×Q8×C137—41no—
151C2×Q8×C137—41no—
152C2×Q8×C137C2×D411yescharacteristic
153C2×D4×C137—22no—
154C2×D4×C137—21no—
155C2×D4×C137—21no—
156C2×D4×C137—22no—
157C2×D4×C137—22no—
158C2×D4×C137—22no—
159C2×D4×C137—22no—
160C2×D4×C137—22no—
161C2×D4×C137—22no—
162C2×D4×C137—21no—
163C2×D4×C137—22no—
164C2×D4×C137—22no—
165C2×D4×C137—21no—
166C2×D4×C137—21no—
167C2×D4×C137—21no—
168C2×D4×C137—42no—
169C2×D4×C137—42no—
170C2×D4×C137—44no—
171C2×D4×C137—82no—
172C2×D4×C137—82no—
173C2×D4×C137—84no—
174C2×D4×C137—84no—
175C2×D4×C137C2×D411yescharacteristic
176C2×D4×C137C2×D412yes—
177C22×C4×C137—22no—
178C22×C4×C137—22no—
179C22×C4×C137—22no—
180C22×C4×C137—21no—
181C22×C4×C137—21no—
182C22×C4×C137—22no—
183C22×C4×C137—42no—
184C22×C4×C137—44no—
185C22×C4×C137—41no—
186C22×C4×C137—41no—
187C22×C4×C137C2×D411yescharacteristic
188C22×C4×C137C2×D411yescharacteristic
189C22×C4×C137C2×D411yescharacteristic
190C22×C4×C137C2×D412yes—
191C22×C4×C137C2×D411yescharacteristic
192C22×C4×C137C2×D411yescharacteristic
193C22×C4×C137C2411yescharacteristic
194Q16×C137—44no—
195SD16×C137—48no—
196SD16×C137—44no—
197M16×C137—42no—
198C2×C8×C137—22no—
199C2×C8×C137—22no—
200C2×C8×C137—42no—
201C4⋊C4×C137—22no—
202C4⋊C4×C137—22no—
203C4⋊C4×C137—21no—
204C4⋊C4×C137—21no—
205C4⋊C4×C137—21no—
206C4⋊C4×C137—21no—
207C4⋊C4×C137—21no—
208C4⋊C4×C137C2×D412yes—
209C4⋊C4×C137C2×D411yescharacteristic
210C4⋊C4×C137C2×D411yescharacteristic
211C22⋊C4×C137—22no—
212C22⋊C4×C137—22no—
213C22⋊C4×C137—22no—
214C22⋊C4×C137—22no—
215C22⋊C4×C137—22no—
216C22⋊C4×C137—21no—
217C22⋊C4×C137—21no—
218C22⋊C4×C137—21no—
219C22⋊C4×C137—22no—
220C22⋊C4×C137—21no—
221C22⋊C4×C137—22no—
222C22⋊C4×C137—22no—
223C22⋊C4×C137—22no—
224C22⋊C4×C137—21no—
225C22⋊C4×C137—48no—
226C22⋊C4×C137—42no—
227C42×C137—22no—
228C42×C137—21no—
229C42×C137—42no—
230C23×C137—22no—
231C23×C137—22no—
232C23×C137—22no—
233C23×C137—22no—
234C23×C137—21no—
235C23×C137—24no—
236C23×C137—41no—
237C23×C137—42no—
238C23×C137—42no—
239C23×C137—42no—
240C23×C137—81no—
241C23×C137—81no—
242C23×C137—82no—
243C23×C137—82no—
244C23×C137—82no—
245C23×C137#12011yescharacteristic
246C23×C137#12011yescharacteristic
247C23×C137C22×D411yescharacteristic
248Q8×C137—24no—
249Q8×C137—24no—
250Q8×C137—41no—
251Q8×C137—42no—
252Q8×C137—42no—
253Q8×C137—44no—
254Q8×C137—44no—
255D4×C137—42no—
256D4×C137—41no—
257D4×C137—44no—
258D4×C137—42no—
259D4×C137—44no—
260D4×C137—42no—
261D4×C137—42no—
262D4×C137—42no—
263D4×C137—81no—
264D4×C137—82no—
265D4×C137—82no—
266D4×C137—81no—
267D4×C137—82no—
268D4×C137—82no—
269D4×C137—82no—
270D4×C137—82no—
271D4×C137—82no—
272C2×C4×C137—22no—
273C2×C4×C137—22no—
274C2×C4×C137—22no—
275C2×C4×C137—24no—
276C2×C4×C137—21no—
277C2×C4×C137—22no—
278C2×C4×C137—22no—
279C2×C4×C137—21no—
280C2×C4×C137—21no—
281C2×C4×C137—41no—
282C2×C4×C137—41no—
283C2×C4×C137—41no—
284C2×C4×C137—42no—
285C2×C4×C137—44no—
286C2×C4×C137—44no—
287C2×C4×C137—42no—
288C2×C4×C137—42no—
289C2×C4×C137—42no—
290C2×C4×C137—42no—
291C2×C4×C137—42no—
292C2×C4×C137—42no—
293C2×C4×C137—81no—
294C2×C4×C137—82no—
295C2×C4×C137—82no—
296C2×C4×C137#12011yescharacteristic
297C2×C4×C137#12011yescharacteristic
298C2×C4×C137C22×D411yescharacteristic
299C2×C4×C137C22×D411yescharacteristic
300C8×C137—42no—
301C8×C137—42no—
302C22×C137—21no—
303C22×C137—21no—
304C22×C137—21no—
305C22×C137—21no—
306C22×C137—21no—
307C22×C137—21no—
308C22×C137—21no—
309C22×C137—44no—
310C22×C137—42no—
311C22×C137—42no—
312C22×C137—41no—
313C22×C137—42no—
314C22×C137—42no—
315C22×C137—41no—
316C22×C137—41no—
317C22×C137—42no—
318C22×C137—42no—
319C22×C137—84no—
320C22×C137—162no—
321C22×C137—164no—
322C22×C137#45512yes—
323C22×C137C2×#12011yescharacteristic
324C22×C137#57712yes—
325C4×C137—21no—
326C4×C137—21no—
327C4×C137—21no—
328C4×C137—42no—
329C4×C137—41no—
330C4×C137—42no—
331C4×C137—41no—
332C4×C137—44no—
333C4×C137—41no—
334C4×C137—82no—
335C4×C137B(4,𝔽2)12yes—
336C2×C137—22no—
337C2×C137—22no—
338C2×C137—41no—
339C2×C137—82no—
340C2×C137—82no—
341C2×C137—81no—
342C2×C137#300911yescharacteristic
343C2×C137C2×#45511yescharacteristic
344C2×C137C2×B(4,𝔽2)11yescharacteristic
345#18255C13711yesmaximal, characteristic, direct product factor, Sylow
346C137#1825511yescharacteristic, direct product factor, Sylow
347#3537C2×C13711yescharacteristic
348#3004C2×C13711yescharacteristic, complemented
349C2×#448C2×C13711yescharacteristic, complemented
350#2436C2×C13711yescharacteristic, complemented
351#1606C2×C13714yescomplemented
352#1601C2×C13714yescomplemented
353#1507C2×C13711yescharacteristic, complemented
354#1501C2×C13711yescharacteristic, complemented
355#1471C2×C13711yescharacteristic, complemented
356C23×Q8C22×C13711yescharacteristic, complemented
357C2×#137—21no—
358C22×SD16—21no—
359#560—22no—
360#553—21no—
361#553—21no—
362D42—22no—
363D42—21no—
364D42—21no—
365#540—21no—
366#538—21no—
367#538C22×C13712yes—
368#535—21no—
369#534—21no—
370#534C22×C13712yes—
371C2×#124—21no—
372#525C22×C13711yescharacteristic
373C2×#121C22×C13711yescharacteristic
374C2×#121C22×C13711yescharacteristic
375#451—21no—
376#451C22×C13712yescomplemented
377#448—22no—
378#448—21no—
379#448C22×C13712yescomplemented
380#448C22×C13714yescomplemented
381#419—21no—
382C2×#103—21no—
383#407C22×C13711yescharacteristic, complemented
384C2×C22⋊C8C22×C13711yescharacteristic, complemented
385#393C22×C13714yescomplemented
386#391—21no—
387#386—22no—
388C4×C22⋊C4C22×C13711yescharacteristic, complemented
389#328C22×C13712yescomplemented
390#327C22×C13712yescomplemented
391#327C22×C13714yescomplemented
392#323C22×C13714yescomplemented
393C2×#41D4×C13711yescharacteristic
394C22×Q8—21no—
395C22×Q8—21no—
396C22×Q8—21no—
397C22×Q8—21no—
398C22×Q8—21no—
399C22×Q8—44no—
400C22×Q8C23×C13712yes—
401C22×D4—22no—
402C22×D4—21no—
403C22×D4—21no—
404C22×D4—22no—
405C22×D4D4×C13711yescharacteristic
406C22×D4D4×C13712yes—
407C22×D4D4×C13711yescharacteristic
408C23×C4—21no—
409C23×C4C23×C13711yescharacteristic
410#137—44no—
411C2×Q16—22no—
412C2×SD16—24no—
413C2×SD16—22no—
414C2×SD16—44no—
415C2×M16—21no—
416C22×C8—21no—
417#127—22no—
418#127—21no—
419#126—22no—
420#126—22no—
421#125—21no—
422#124—22no—
423#124—44no—
424#123—22no—
425#123—21no—
426#123—22no—
427#123—22no—
428#123—22no—
429C22⋊Q8D4×C13711yescharacteristic
430C22⋊Q8C23×C13711yescharacteristic
431#121—22no—
432#121—21no—
433#121—21no—
434#121—22no—
435#121—22no—
436#121—22no—
437#121—22no—
438#121—21no—
439#121—21no—
440#121—21no—
441#121—21no—
442#121—21no—
443#121—21no—
444#121—22no—
445#121D4×C13712yes—
446#121D4×C13711yescharacteristic
447#121C23×C13712yes—
448#121C23×C13711yescharacteristic
449#120—22no—
450#120—22no—
451#120—22no—
452#120—21no—
453#120—22no—
454#120—22no—
455#120—22no—
456C4×D4—22no—
457C4×D4—21no—
458C4×D4—21no—
459C4×D4—21no—
460C4×D4—21no—
461C4×D4—22no—
462#117—21no—
463#117D4×C13711yescharacteristic
464C2×C4⋊C4D4×C13711yescharacteristic
465C2×C4⋊C4D4×C13711yescharacteristic
466C2×C22⋊C4—21no—
467C2×C22⋊C4—24no—
468C2×C22⋊C4—21no—
469C2×C22⋊C4C23×C13712yes—
470C2×C42—21no—
471#103—24no—
472#103—22no—
473#103—22no—
474C22⋊C8—21no—
475C22⋊C8—21no—
476C22⋊C8C23×C13712yes—
477C22⋊C8C23×C13712yes—
478#95—24no—
479#95C23×C13712yes—
480C24—22no—
481C24—21no—
482C24C2×D4×C13711yescharacteristic
483C24C2×D4×C13712yes—
484#41—42no—
485C2×Q8—24no—
486C2×Q8—24no—
487C2×Q8—24no—
488C2×Q8—22no—
489C2×Q8—22no—
490C2×Q8—44no—
491C2×Q8—42no—
492C2×Q8—42no—
493C2×Q8—42no—
494C2×Q8—44no—
495C2×Q8—44no—
496C2×Q8—41no—
497C2×Q8—41no—
498C2×Q8C2×D4×C13711yescharacteristic
499C2×D4—22no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—22no—
503C2×D4—22no—
504C2×D4—22no—
505C2×D4—22no—
506C2×D4—22no—
507C2×D4—22no—
508C2×D4—21no—
509C2×D4—22no—
510C2×D4—22no—
511C2×D4—21no—
512C2×D4—21no—
513C2×D4—21no—
514C2×D4—42no—
515C2×D4—42no—
516C2×D4—44no—
517C2×D4—82no—
518C2×D4—82no—
519C2×D4—84no—
520C2×D4—84no—
521C2×D4C2×D4×C13711yescharacteristic
522C2×D4C2×D4×C13712yes—
523C22×C4—22no—
524C22×C4—22no—
525C22×C4—22no—
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—22no—
529C22×C4—42no—
530C22×C4—44no—
531C22×C4—41no—
532C22×C4—41no—
533C22×C4C2×D4×C13711yescharacteristic
534C22×C4C2×D4×C13711yescharacteristic
535C22×C4C2×D4×C13711yescharacteristic
536C22×C4C2×D4×C13712yes—
537C22×C4C2×D4×C13711yescharacteristic
538C22×C4C2×D4×C13711yescharacteristic
539C22×C4C24×C13711yescharacteristic
540Q16—44no—
541SD16—48no—
542SD16—44no—
543M16—42no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—42no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4C2×D4×C13712yes—
555C4⋊C4C2×D4×C13711yescharacteristic
556C4⋊C4C2×D4×C13711yescharacteristic
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—22no—
562C22⋊C4—21no—
563C22⋊C4—21no—
564C22⋊C4—21no—
565C22⋊C4—22no—
566C22⋊C4—21no—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—21no—
571C22⋊C4—48no—
572C22⋊C4—42no—
573C42—22no—
574C42—21no—
575C42—42no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—21no—
581C23—24no—
582C23—41no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—81no—
587C23—81no—
588C23—82no—
589C23—82no—
590C23—82no—
591C23#120×C13711yescharacteristic
592C23#120×C13711yescharacteristic
593C23C22×D4×C13711yescharacteristic
594Q8—24no—
595Q8—24no—
596Q8—41no—
597Q8—42no—
598Q8—42no—
599Q8—44no—
600Q8—44no—
601D4—42no—
602D4—41no—
603D4—44no—
604D4—42no—
605D4—44no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—81no—
610D4—82no—
611D4—82no—
612D4—81no—
613D4—82no—
614D4—82no—
615D4—82no—
616D4—82no—
617D4—82no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—81no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4#120×C13711yescharacteristic
643C2×C4#120×C13711yescharacteristic
644C2×C4C22×D4×C13711yescharacteristic
645C2×C4C22×D4×C13711yescharacteristic
646C8—42no—
647C8—42no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—44no—
656C22—42no—
657C22—42no—
658C22—41no—
659C22—42no—
660C22—42no—
661C22—41no—
662C22—41no—
663C22—42no—
664C22—42no—
665C22—84no—
666C22—162no—
667C22—164no—
668C22#455×C13712yes—
669C22C2×#120×C13711yescharacteristic
670C22#577×C13712yes—
671C4—21no—
672C4—21no—
673C4—21no—
674C4—42no—
675C4—41no—
676C4—42no—
677C4—41no—
678C4—44no—
679C4—41no—
680C4—82no—
681C4B(4,𝔽2)×C13712yes—
682C2—22no—
683C2—22no—
684C2—41no—
685C2—82no—
686C2—82no—
687C2—81no—
688C2#3009×C13711yescharacteristic
689C2C2×#455×C13711yescharacteristic
690C2C2×B(4,𝔽2)×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC22×C4C24×C137
prime radical#3537×C137C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3004×C137C2semidirect
3. C2×#448×C137C2semidirect
4. #2436×C137C2semidirect
5. #1606×C137C2semidirect
6. #1601×C137C2semidirect
7. #1507×C137C2semidirect
8. #1501×C137C2semidirect
9. #1471×C137C2semidirect
10. C23×Q8×C137C22semidirect
30. #451×C137C22semidirect
33. #448×C137C22semidirect
34. #448×C137C22semidirect
37. #407×C137C22semidirect
38. C2×C22⋊C8×C137C22semidirect
39. #393×C137C22semidirect
42. C4×C22⋊C4×C137C22semidirect
43. #328×C137C22semidirect
44. #327×C137C22semidirect
45. #327×C137C22semidirect
46. #323×C137C22semidirect
345. #18255C137direct (N × H)
346. C137#18255direct (N × H)
348. #3004C2×C137semidirect
349. C2×#448C2×C137semidirect
350. #2436C2×C137semidirect
351. #1606C2×C137semidirect
352. #1601C2×C137semidirect
353. #1507C2×C137semidirect
354. #1501C2×C137semidirect
355. #1471C2×C137semidirect
356. C23×Q8C22×C137semidirect
376. #451C22×C137semidirect
379. #448C22×C137semidirect
380. #448C22×C137semidirect
383. #407C22×C137semidirect
384. C2×C22⋊C8C22×C137semidirect
385. #393C22×C137semidirect
388. C4×C22⋊C4C22×C137semidirect
389. #328C22×C137semidirect
390. #327C22×C137semidirect
391. #327C22×C137semidirect
392. #323C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3537×C1372
#3004×C1372
C2×#448×C1372
#2436×C1372
#1606×C1372
#1601×C1372
#1507×C1372
#1501×C1372
#1471×C1372
#18255137