Finite Group Databasebeta

← C137×#18257 #46867643

Subgroups — C137×#18257 #46867643

Order 35072.

1192 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3536×C137C211yesmaximal, characteristic
2#3004×C137C211yesmaximal, characteristic, complemented
3C2×#451×C137C211yesmaximal, characteristic, complemented
4#2436×C137C211yesmaximal, characteristic, complemented
5#1606×C137C214yesmaximal, complemented
6#1602×C137C214yesmaximal, complemented
7#1508×C137C211yesmaximal, characteristic, complemented
8#1501×C137C211yesmaximal, characteristic, complemented
9#1471×C137C211yesmaximal, characteristic, complemented
10C23×Q8×C137C2211yescharacteristic
11C2×#137×C137—21no—
12C22×Q16×C137—21no—
13#562×C137—22no—
14#554×C137—21no—
15#548×C137—21no—
16#547×C137—21no—
17#546×C137—22no—
18#546×C137—21no—
19#543×C137C2212yes—
20#542×C137—21no—
21#539×C137—21no—
22#538×C137C2212yes—
23#537×C137—21no—
24#537×C137—21no—
25C2×#124×C137—21no—
26#525×C137C2211yescharacteristic
27C2×C22⋊Q8×C137C2211yescharacteristic
28C2×#121×C137C2211yescharacteristic
29#451×C137—22no—
30#451×C137—21no—
31#451×C137C2212yes—
32#451×C137C2214yes—
33#448×C137—21no—
34#448×C137C2212yes—
35#419×C137—21no—
36C2×#103×C137—21no—
37#407×C137C2211yescharacteristic
38C2×C22⋊C8×C137C2211yescharacteristic
39#393×C137C2214yes—
40#391×C137—21no—
41#386×C137—22no—
42C4×C22⋊C4×C137C2211yescharacteristic
43#328×C137C2212yes—
44#328×C137C2214yes—
45#327×C137C2212yes—
46#323×C137C2214yes—
47C2×#41×C137—22no—
48C2×#41×C137—21no—
49C2×#41×C137D411yescharacteristic
50C22×Q8×C137—21no—
51C22×Q8×C137—21no—
52C22×Q8×C137—21no—
53C22×Q8×C137—21no—
54C22×Q8×C137—21no—
55C22×Q8×C137—44no—
56C22×Q8×C137D411yescharacteristic
57C22×Q8×C137C2312yes—
58C22×D4×C137D412yes—
59C22×D4×C137D411yescharacteristic
60C23×C4×C137—21no—
61C23×C4×C137C2311yescharacteristic
62#137×C137—44no—
63C2×Q16×C137—24no—
64C2×Q16×C137—22no—
65C2×Q16×C137—44no—
66C2×SD16×C137—22no—
67C2×M16×C137—21no—
68C22×C8×C137—21no—
69C4⋊Q8×C137—21no—
70#126×C137—22no—
71#125×C137—22no—
72#124×C137—21no—
73#124×C137—22no—
74#124×C137—22no—
75#124×C137—44no—
76#123×C137—22no—
77#123×C137—22no—
78#123×C137—22no—
79#123×C137—21no—
80#123×C137—22no—
81C22⋊Q8×C137—21no—
82C22⋊Q8×C137—22no—
83C22⋊Q8×C137—22no—
84C22⋊Q8×C137—22no—
85C22⋊Q8×C137—21no—
86C22⋊Q8×C137—21no—
87C22⋊Q8×C137—22no—
88C22⋊Q8×C137—22no—
89C22⋊Q8×C137—22no—
90C22⋊Q8×C137D412yes—
91C22⋊Q8×C137D411yescharacteristic
92C22⋊Q8×C137C2312yes—
93C22⋊Q8×C137C2311yescharacteristic
94#121×C137—21no—
95#121×C137—22no—
96#121×C137—22no—
97#121×C137—22no—
98#121×C137—22no—
99#121×C137—22no—
100#121×C137—21no—
101#121×C137—21no—
102#121×C137—21no—
103#121×C137—21no—
104#121×C137D411yescharacteristic
105#121×C137C2311yescharacteristic
106#120×C137—21no—
107#120×C137—22no—
108C4×Q8×C137—21no—
109C4×D4×C137—22no—
110C4×D4×C137—21no—
111C4×D4×C137—21no—
112C4×D4×C137—21no—
113C4×D4×C137—22no—
114#117×C137D411yescharacteristic
115C2×C4⋊C4×C137—21no—
116C2×C4⋊C4×C137—21no—
117C2×C4⋊C4×C137D411yescharacteristic
118C2×C4⋊C4×C137D411yescharacteristic
119C2×C22⋊C4×C137—21no—
120C2×C22⋊C4×C137—22no—
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×C137C2×D411yescharacteristic
135C24×C137C2×D412yes—
136#41×C137—42no—
137#41×C137—84no—
138#41×C137—82no—
139C2×Q8×C137—24no—
140C2×Q8×C137—22no—
141C2×Q8×C137—21no—
142C2×Q8×C137—24no—
143C2×Q8×C137—24no—
144C2×Q8×C137—22no—
145C2×Q8×C137—22no—
146C2×Q8×C137—44no—
147C2×Q8×C137—42no—
148C2×Q8×C137—42no—
149C2×Q8×C137—42no—
150C2×Q8×C137—42no—
151C2×Q8×C137—44no—
152C2×Q8×C137—44no—
153C2×Q8×C137—41no—
154C2×Q8×C137—41no—
155C2×Q8×C137C2×D411yescharacteristic
156C2×Q8×C137C2×D412yes—
157C2×D4×C137—21no—
158C2×D4×C137—22no—
159C2×D4×C137—21no—
160C2×D4×C137—22no—
161C2×D4×C137—22no—
162C2×D4×C137—21no—
163C2×D4×C137—22no—
164C2×D4×C137—21no—
165C2×D4×C137—21no—
166C2×D4×C137—42no—
167C2×D4×C137—44no—
168C2×D4×C137C2×D411yescharacteristic
169C22×C4×C137—21no—
170C22×C4×C137—22no—
171C22×C4×C137—22no—
172C22×C4×C137—22no—
173C22×C4×C137—22no—
174C22×C4×C137—21no—
175C22×C4×C137—21no—
176C22×C4×C137—22no—
177C22×C4×C137—42no—
178C22×C4×C137—44no—
179C22×C4×C137—41no—
180C22×C4×C137—41no—
181C22×C4×C137C2×D411yescharacteristic
182C22×C4×C137C2×D411yescharacteristic
183C22×C4×C137C2×D411yescharacteristic
184C22×C4×C137C2×D412yes—
185C22×C4×C137C2×D411yescharacteristic
186C22×C4×C137C2×D411yescharacteristic
187C22×C4×C137C2411yescharacteristic
188Q16×C137—48no—
189Q16×C137—44no—
190SD16×C137—44no—
191M16×C137—42no—
192C2×C8×C137—22no—
193C2×C8×C137—22no—
194C2×C8×C137—42no—
195C4⋊C4×C137—22no—
196C4⋊C4×C137—22no—
197C4⋊C4×C137—22no—
198C4⋊C4×C137—22no—
199C4⋊C4×C137—21no—
200C4⋊C4×C137—21no—
201C4⋊C4×C137—22no—
202C4⋊C4×C137—21no—
203C4⋊C4×C137—21no—
204C4⋊C4×C137—22no—
205C4⋊C4×C137—21no—
206C4⋊C4×C137C2×D412yes—
207C4⋊C4×C137C2×D411yescharacteristic
208C4⋊C4×C137C2×D411yescharacteristic
209C22⋊C4×C137—22no—
210C22⋊C4×C137—22no—
211C22⋊C4×C137—22no—
212C22⋊C4×C137—22no—
213C22⋊C4×C137—22no—
214C22⋊C4×C137—22no—
215C22⋊C4×C137—21no—
216C22⋊C4×C137—21no—
217C22⋊C4×C137—22no—
218C22⋊C4×C137—21no—
219C22⋊C4×C137—21no—
220C22⋊C4×C137—22no—
221C22⋊C4×C137—22no—
222C22⋊C4×C137—21no—
223C22⋊C4×C137—48no—
224C22⋊C4×C137—42no—
225C42×C137—22no—
226C42×C137—21no—
227C42×C137—42no—
228C23×C137—22no—
229C23×C137—22no—
230C23×C137—22no—
231C23×C137—21no—
232C23×C137—24no—
233C23×C137—41no—
234C23×C137—42no—
235C23×C137—42no—
236C23×C137#12011yescharacteristic
237C23×C137#12011yescharacteristic
238C23×C137C22×D411yescharacteristic
239Q8×C137—24no—
240Q8×C137—24no—
241Q8×C137—42no—
242Q8×C137—41no—
243Q8×C137—42no—
244Q8×C137—42no—
245Q8×C137—44no—
246Q8×C137—44no—
247Q8×C137—42no—
248Q8×C137—82no—
249D4×C137—41no—
250D4×C137—42no—
251D4×C137—44no—
252D4×C137—42no—
253D4×C137—42no—
254D4×C137—81no—
255D4×C137—82no—
256D4×C137—81no—
257D4×C137—82no—
258D4×C137—82no—
259C2×C4×C137—22no—
260C2×C4×C137—22no—
261C2×C4×C137—22no—
262C2×C4×C137—24no—
263C2×C4×C137—21no—
264C2×C4×C137—22no—
265C2×C4×C137—21no—
266C2×C4×C137—22no—
267C2×C4×C137—21no—
268C2×C4×C137—22no—
269C2×C4×C137—41no—
270C2×C4×C137—41no—
271C2×C4×C137—41no—
272C2×C4×C137—42no—
273C2×C4×C137—44no—
274C2×C4×C137—44no—
275C2×C4×C137—42no—
276C2×C4×C137—42no—
277C2×C4×C137—42no—
278C2×C4×C137—42no—
279C2×C4×C137—42no—
280C2×C4×C137—42no—
281C2×C4×C137—42no—
282C2×C4×C137—81no—
283C2×C4×C137—82no—
284C2×C4×C137—82no—
285C2×C4×C137—82no—
286C2×C4×C137#12011yescharacteristic
287C2×C4×C137#12011yescharacteristic
288C2×C4×C137C22×D411yescharacteristic
289C2×C4×C137C22×D411yescharacteristic
290C8×C137—42no—
291C8×C137—42no—
292C22×C137—21no—
293C22×C137—21no—
294C22×C137—21no—
295C22×C137—21no—
296C22×C137—21no—
297C22×C137—21no—
298C22×C137—21no—
299C22×C137—44no—
300C22×C137—41no—
301C22×C137—42no—
302C22×C137—41no—
303C22×C137—42no—
304C22×C137—41no—
305C22×C137—42no—
306C22×C137—84no—
307C22×C137#45512yes—
308C22×C137C2×#12011yescharacteristic
309C22×C137#57812yes—
310C4×C137—21no—
311C4×C137—21no—
312C4×C137—21no—
313C4×C137—41no—
314C4×C137—42no—
315C4×C137—42no—
316C4×C137—41no—
317C4×C137—44no—
318C4×C137—41no—
319C4×C137—42no—
320C4×C137—82no—
321C4×C137B(4,𝔽2)12yes—
322C2×C137—22no—
323C2×C137—22no—
324C2×C137—41no—
325C2×C137—81no—
326C2×C137—82no—
327C2×C137#301111yescharacteristic
328C2×C137C2×#45511yescharacteristic
329C2×C137C2×B(4,𝔽2)11yescharacteristic
330#18257C13711yesmaximal, characteristic, direct product factor, Sylow
331C137#1825711yescharacteristic, direct product factor, Sylow
332#3536C2×C13711yescharacteristic
333#3004C2×C13711yescharacteristic, complemented
334C2×#451C2×C13711yescharacteristic, complemented
335#2436C2×C13711yescharacteristic, complemented
336#1606C2×C13714yescomplemented
337#1602C2×C13714yescomplemented
338#1508C2×C13711yescharacteristic, complemented
339#1501C2×C13711yescharacteristic, complemented
340#1471C2×C13711yescharacteristic, complemented
341C23×Q8C22×C13711yescharacteristic
342C2×#137—21no—
343C22×Q16—21no—
344#562—22no—
345#554—21no—
346#548—21no—
347#547—21no—
348#546—22no—
349#546—21no—
350#543C22×C13712yes—
351#542—21no—
352#539—21no—
353#538C22×C13712yes—
354#537—21no—
355#537—21no—
356C2×#124—21no—
357#525C22×C13711yescharacteristic
358C2×C22⋊Q8C22×C13711yescharacteristic
359C2×#121C22×C13711yescharacteristic
360#451—22no—
361#451—21no—
362#451C22×C13712yes—
363#451C22×C13714yes—
364#448—21no—
365#448C22×C13712yes—
366#419—21no—
367C2×#103—21no—
368#407C22×C13711yescharacteristic
369C2×C22⋊C8C22×C13711yescharacteristic
370#393C22×C13714yes—
371#391—21no—
372#386—22no—
373C4×C22⋊C4C22×C13711yescharacteristic
374#328C22×C13712yes—
375#328C22×C13714yes—
376#327C22×C13712yes—
377#323C22×C13714yes—
378C2×#41—22no—
379C2×#41—21no—
380C2×#41D4×C13711yescharacteristic
381C22×Q8—21no—
382C22×Q8—21no—
383C22×Q8—21no—
384C22×Q8—21no—
385C22×Q8—21no—
386C22×Q8—44no—
387C22×Q8D4×C13711yescharacteristic
388C22×Q8C23×C13712yes—
389C22×D4D4×C13711yescharacteristic
390C22×D4D4×C13712yes—
391C23×C4—21no—
392C23×C4C23×C13711yescharacteristic
393#137—44no—
394C2×Q16—24no—
395C2×Q16—22no—
396C2×Q16—44no—
397C2×SD16—22no—
398C2×M16—21no—
399C22×C8—21no—
400C4⋊Q8—21no—
401#126—22no—
402#125—22no—
403#124—21no—
404#124—22no—
405#124—22no—
406#124—44no—
407#123—22no—
408#123—22no—
409#123—22no—
410#123—21no—
411#123—22no—
412C22⋊Q8—21no—
413C22⋊Q8—22no—
414C22⋊Q8—22no—
415C22⋊Q8—22no—
416C22⋊Q8—21no—
417C22⋊Q8—21no—
418C22⋊Q8—22no—
419C22⋊Q8—22no—
420C22⋊Q8—22no—
421C22⋊Q8D4×C13712yes—
422C22⋊Q8D4×C13711yescharacteristic
423C22⋊Q8C23×C13712yes—
424C22⋊Q8C23×C13711yescharacteristic
425#121—21no—
426#121—22no—
427#121—22no—
428#121—22no—
429#121—22no—
430#121—22no—
431#121—21no—
432#121—21no—
433#121—21no—
434#121—21no—
435#121D4×C13711yescharacteristic
436#121C23×C13711yescharacteristic
437#120—21no—
438#120—22no—
439C4×Q8—21no—
440C4×D4—22no—
441C4×D4—21no—
442C4×D4—21no—
443C4×D4—21no—
444C4×D4—22no—
445#117D4×C13711yescharacteristic
446C2×C4⋊C4—21no—
447C2×C4⋊C4—21no—
448C2×C4⋊C4D4×C13711yescharacteristic
449C2×C4⋊C4D4×C13711yescharacteristic
450C2×C22⋊C4—21no—
451C2×C22⋊C4—22no—
452C2×C22⋊C4—24no—
453C2×C22⋊C4—21no—
454C2×C22⋊C4C23×C13712yes—
455C2×C42—21no—
456#103—24no—
457#103—22no—
458#103—22no—
459C22⋊C8—21no—
460C22⋊C8—21no—
461C22⋊C8C23×C13712yes—
462C22⋊C8C23×C13712yes—
463#95—24no—
464#95C23×C13712yes—
465C24C2×D4×C13711yescharacteristic
466C24C2×D4×C13712yes—
467#41—42no—
468#41—84no—
469#41—82no—
470C2×Q8—24no—
471C2×Q8—22no—
472C2×Q8—21no—
473C2×Q8—24no—
474C2×Q8—24no—
475C2×Q8—22no—
476C2×Q8—22no—
477C2×Q8—44no—
478C2×Q8—42no—
479C2×Q8—42no—
480C2×Q8—42no—
481C2×Q8—42no—
482C2×Q8—44no—
483C2×Q8—44no—
484C2×Q8—41no—
485C2×Q8—41no—
486C2×Q8C2×D4×C13711yescharacteristic
487C2×Q8C2×D4×C13712yes—
488C2×D4—21no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—21no—
494C2×D4—22no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—42no—
498C2×D4—44no—
499C2×D4C2×D4×C13711yescharacteristic
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—42no—
509C22×C4—44no—
510C22×C4—41no—
511C22×C4—41no—
512C22×C4C2×D4×C13711yescharacteristic
513C22×C4C2×D4×C13711yescharacteristic
514C22×C4C2×D4×C13711yescharacteristic
515C22×C4C2×D4×C13712yes—
516C22×C4C2×D4×C13711yescharacteristic
517C22×C4C2×D4×C13711yescharacteristic
518C22×C4C24×C13711yescharacteristic
519Q16—48no—
520Q16—44no—
521SD16—44no—
522M16—42no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—42no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4C2×D4×C13712yes—
538C4⋊C4C2×D4×C13711yescharacteristic
539C4⋊C4C2×D4×C13711yescharacteristic
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—21no—
547C22⋊C4—21no—
548C22⋊C4—22no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—21no—
554C22⋊C4—48no—
555C22⋊C4—42no—
556C42—22no—
557C42—21no—
558C42—42no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—21no—
563C23—24no—
564C23—41no—
565C23—42no—
566C23—42no—
567C23#120×C13711yescharacteristic
568C23#120×C13711yescharacteristic
569C23C22×D4×C13711yescharacteristic
570Q8—24no—
571Q8—24no—
572Q8—42no—
573Q8—41no—
574Q8—42no—
575Q8—42no—
576Q8—44no—
577Q8—44no—
578Q8—42no—
579Q8—82no—
580D4—41no—
581D4—44no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—81no—
586D4—82no—
587D4—81no—
588D4—82no—
589D4—82no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—81no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4#120×C13711yescharacteristic
618C2×C4#120×C13711yescharacteristic
619C2×C4C22×D4×C13711yescharacteristic
620C2×C4C22×D4×C13711yescharacteristic
621C8—42no—
622C8—42no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—44no—
631C22—41no—
632C22—42no—
633C22—41no—
634C22—42no—
635C22—41no—
636C22—42no—
637C22—84no—
638C22#455×C13712yes—
639C22C2×#120×C13711yescharacteristic
640C22#578×C13712yes—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—41no—
645C4—42no—
646C4—42no—
647C4—41no—
648C4—44no—
649C4—41no—
650C4—42no—
651C4—82no—
652C4B(4,𝔽2)×C13712yes—
653C2—22no—
654C2—22no—
655C2—41no—
656C2—82no—
657C2—81no—
658C2#3011×C13711yescharacteristic
659C2C2×#455×C13711yescharacteristic
660C2C2×B(4,𝔽2)×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC22×C4C24×C137
prime radical#3536×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×#451×C137C2semidirect
4. #2436×C137C2semidirect
5. #1606×C137C2semidirect
6. #1602×C137C2semidirect
7. #1508×C137C2semidirect
8. #1501×C137C2semidirect
9. #1471×C137C2semidirect
330. #18257C137direct (N × H)
331. C137#18257direct (N × H)
333. #3004C2×C137semidirect
334. C2×#451C2×C137semidirect
335. #2436C2×C137semidirect
336. #1606C2×C137semidirect
337. #1602C2×C137semidirect
338. #1508C2×C137semidirect
339. #1501C2×C137semidirect
340. #1471C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3536×C1372
#3004×C1372
C2×#451×C1372
#2436×C1372
#1606×C1372
#1602×C1372
#1508×C1372
#1501×C1372
#1471×C1372
#18257137