Finite Group Databasebeta

← C137×#16566 #46865952

Subgroups — C137×#16566 #46865952

Order 35072.

1454 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#393×C137C211yesmaximal, characteristic
2C2×#391×C137C211yesmaximal, characteristic, complemented
3C2×#388×C137C211yesmaximal, characteristic
4C2×#382×C137C211yesmaximal, characteristic, complemented
5#1446×C137C211yesmaximal, characteristic, complemented
6#1444×C137C211yesmaximal, characteristic, complemented
7#1442×C137C212yesmaximal, complemented
8#1441×C137C212yesmaximal, complemented
9#1439×C137C212yesmaximal
10#1439×C137C211yesmaximal, characteristic
11#1439×C137C212yesmaximal
12C23×Q8×C137C2211yescharacteristic
13C22×C4⋊C4×C137C2212yes—
14C22×C4⋊C4×C137C2211yescharacteristic
15C22×C4⋊C4×C137C2211yescharacteristic
16C22×C4⋊C4×C137C2212yes—
17C22×C22⋊C4×C137C2211yescharacteristic
18C22×C22⋊C4×C137C2212yes—
19C22×C22⋊C4×C137C2211yescharacteristic
20C22×C22⋊C4×C137C2211yescharacteristic
21C22×C22⋊C4×C137C2212yes—
22C22×C42×C137C2211yescharacteristic
23#393×C137—24no—
24#391×C137—24no—
25#388×C137—24no—
26#382×C137—24no—
27C2×#95×C137C2212yes—
28C2×#95×C137C2211yescharacteristic
29C2×#95×C137C2211yescharacteristic
30C2×#95×C137C2211yescharacteristic
31C2×#95×C137C2212yes—
32C2×#95×C137C2212yes—
33C2×#95×C137C2212yes—
34C2×#95×C137C2212yes—
35C2×#95×C137C2212yes—
36C2×#95×C137C2212yes—
37C2×#95×C137C2212yes—
38C2×#95×C137C2211yescharacteristic
39C25×C137C2311yescharacteristic
40C22×Q8×C137—22no—
41C22×Q8×C137—22no—
42C22×Q8×C137—24no—
43C22×Q8×C137—22no—
44C22×Q8×C137—42no—
45C23×C4×C137C2311yescharacteristic
46C23×C4×C137C2312yes—
47C23×C4×C137C2311yescharacteristic
48C23×C4×C137C2312yes—
49C23×C4×C137C2311yescharacteristic
50C23×C4×C137C2312yes—
51C23×C4×C137C2311yescharacteristic
52C23×C4×C137C2312yes—
53C23×C4×C137C2311yescharacteristic
54C23×C4×C137C2311yescharacteristic
55C2×C4⋊C4×C137—24no—
56C2×C4⋊C4×C137—22no—
57C2×C4⋊C4×C137—22no—
58C2×C4⋊C4×C137—24no—
59C2×C4⋊C4×C137—42no—
60C2×C4⋊C4×C137—42no—
61C2×C4⋊C4×C137—42no—
62C2×C4⋊C4×C137—42no—
63C2×C4⋊C4×C137—42no—
64C2×C4⋊C4×C137—42no—
65C2×C22⋊C4×C137—22no—
66C2×C22⋊C4×C137—22no—
67C2×C22⋊C4×C137—22no—
68C2×C22⋊C4×C137—22no—
69C2×C22⋊C4×C137—24no—
70C2×C22⋊C4×C137—22no—
71C2×C22⋊C4×C137—24no—
72C2×C22⋊C4×C137—22no—
73C2×C22⋊C4×C137—42no—
74C2×C22⋊C4×C137—42no—
75C2×C22⋊C4×C137—42no—
76C2×C22⋊C4×C137—42no—
77C2×C22⋊C4×C137—42no—
78C2×C22⋊C4×C137Q814yescomplemented
79C2×C42×C137—22no—
80C2×C42×C137—22no—
81C2×C42×C137D414yes—
82#95×C137—24no—
83#95×C137—22no—
84#95×C137—22no—
85#95×C137—22no—
86#95×C137—24no—
87#95×C137—24no—
88#95×C137—24no—
89#95×C137—24no—
90#95×C137—42no—
91#95×C137—42no—
92#95×C137—42no—
93#95×C137D414yes—
94C24×C137—21no—
95C24×C137—21no—
96C24×C137—21no—
97C24×C137—21no—
98C24×C137—22no—
99C24×C137—22no—
100C24×C137—22no—
101C24×C137—21no—
102C24×C137—22no—
103C24×C137—21no—
104C24×C137C2×Q812yes—
105C24×C137C2411yescharacteristic
106C2×Q8×C137—28no—
107C2×Q8×C137—48no—
108C2×Q8×C137—48no—
109C2×Q8×C137—44no—
110C2×Q8×C137—44no—
111C22×C4×C137—21no—
112C22×C4×C137—21no—
113C22×C4×C137—22no—
114C22×C4×C137—21no—
115C22×C4×C137—22no—
116C22×C4×C137—21no—
117C22×C4×C137—21no—
118C22×C4×C137—22no—
119C22×C4×C137—21no—
120C22×C4×C137—22no—
121C22×C4×C137—22no—
122C22×C4×C137—21no—
123C22×C4×C137—21no—
124C22×C4×C137—22no—
125C22×C4×C137—22no—
126C22×C4×C137—22no—
127C22×C4×C137—22no—
128C22×C4×C137—22no—
129C22×C4×C137—22no—
130C22×C4×C137—22no—
131C22×C4×C137—22no—
132C22×C4×C137—22no—
133C22×C4×C137—22no—
134C22×C4×C137—22no—
135C22×C4×C137—22no—
136C22×C4×C137—22no—
137C22×C4×C137—21no—
138C22×C4×C137—21no—
139C22×C4×C137—22no—
140C22×C4×C137—22no—
141C22×C4×C137—21no—
142C22×C4×C137—22no—
143C22×C4×C137—21no—
144C22×C4×C137—22no—
145C22×C4×C137—22no—
146C22×C4×C137—22no—
147C22×C4×C137—22no—
148C22×C4×C137—22no—
149C22×C4×C137—22no—
150C22×C4×C137—22no—
151C22×C4×C137—22no—
152C22×C4×C137—22no—
153C22×C4×C137—22no—
154C22×C4×C137—22no—
155C22×C4×C137—22no—
156C22×C4×C137—21no—
157C22×C4×C137—22no—
158C22×C4×C137—22no—
159C22×C4×C137—21no—
160C22×C4×C137—22no—
161C22×C4×C137C2×D414yes—
162C22×C4×C137C2×D412yes—
163C22×C4×C137C2×D412yes—
164C22×C4×C137C2×D412yes—
165C22×C4×C137C2×D412yes—
166C22×C4×C137C2×Q812yes—
167C22×C4×C137C2×Q812yes—
168C22×C4×C137#4114yes—
169C22×C4×C137#4112yes—
170C22×C4×C137#4112yes—
171C4⋊C4×C137—48no—
172C4⋊C4×C137—44no—
173C4⋊C4×C137—44no—
174C4⋊C4×C137—48no—
175C22⋊C4×C137—28no—
176C22⋊C4×C137—48no—
177C22⋊C4×C137—44no—
178C22⋊C4×C137—48no—
179C22⋊C4×C137—44no—
180C42×C137—28no—
181C23×C137—22no—
182C23×C137—24no—
183C23×C137—22no—
184C23×C137—24no—
185C23×C137—22no—
186C23×C137—41no—
187C23×C137—41no—
188C23×C137—41no—
189C23×C137—41no—
190C23×C137—41no—
191C23×C137—41no—
192C23×C137—42no—
193C23×C137—42no—
194C23×C137—42no—
195C23×C137—42no—
196C23×C137—42no—
197C23×C137—42no—
198C23×C137—41no—
199C23×C137—42no—
200C23×C137—41no—
201C23×C137—42no—
202C23×C137—42no—
203C23×C137—42no—
204C23×C137C22×D411yescharacteristic
205C23×C137C22×D411yescharacteristic
206C23×C137C22×Q811yescharacteristic
207C23×C137C2×#4111yescharacteristic
208C23×C137C2×#4111yescharacteristic
209C23×C137C2×#4112yes—
210C23×C137H(5,𝔽2)12yes—
211C23×C137H(5,𝔽2)12yes—
212C23×C137H(5,𝔽2)11yescharacteristic
213C23×C1372−1+412yes—
214C23×C1372−1+411yescharacteristic
215Q8×C137—416no—
216C2×C4×C137—22no—
217C2×C4×C137—22no—
218C2×C4×C137—22no—
219C2×C4×C137—22no—
220C2×C4×C137—22no—
221C2×C4×C137—24no—
222C2×C4×C137—22no—
223C2×C4×C137—24no—
224C2×C4×C137—22no—
225C2×C4×C137—22no—
226C2×C4×C137—22no—
227C2×C4×C137—22no—
228C2×C4×C137—24no—
229C2×C4×C137—22no—
230C2×C4×C137—22no—
231C2×C4×C137—24no—
232C2×C4×C137—24no—
233C2×C4×C137—24no—
234C2×C4×C137—24no—
235C2×C4×C137—24no—
236C2×C4×C137—24no—
237C2×C4×C137—42no—
238C2×C4×C137—41no—
239C2×C4×C137—41no—
240C2×C4×C137—42no—
241C2×C4×C137—42no—
242C2×C4×C137—42no—
243C2×C4×C137—42no—
244C2×C4×C137—42no—
245C2×C4×C137—42no—
246C2×C4×C137—42no—
247C2×C4×C137—42no—
248C2×C4×C137—42no—
249C2×C4×C137—42no—
250C2×C4×C137—41no—
251C2×C4×C137—41no—
252C2×C4×C137—42no—
253C2×C4×C137—41no—
254C2×C4×C137—42no—
255C2×C4×C137—41no—
256C2×C4×C137—41no—
257C2×C4×C137—41no—
258C2×C4×C137—42no—
259C2×C4×C137—42no—
260C2×C4×C137—42no—
261C2×C4×C137—42no—
262C2×C4×C137—42no—
263C2×C4×C137—42no—
264C2×C4×C137—42no—
265C2×C4×C137—42no—
266C2×C4×C137—42no—
267C2×C4×C137—42no—
268C2×C4×C137—42no—
269C2×C4×C137—42no—
270C2×C4×C137—42no—
271C2×C4×C137—42no—
272C2×C4×C137—42no—
273C2×C4×C137—42no—
274C2×C4×C137#12114yes—
275C2×C4×C137C22⋊Q814yes—
276C22×C137—42no—
277C22×C137—42no—
278C22×C137—42no—
279C22×C137—44no—
280C22×C137—44no—
281C22×C137—81no—
282C22×C137—81no—
283C22×C137—81no—
284C22×C137—81no—
285C22×C137—82no—
286C22×C137—82no—
287C22×C137C2×#12111yescharacteristic
288C22×C137C2×C22⋊Q811yescharacteristic
289C22×C137#52511yescharacteristic
290C22×C137#53511yescharacteristic
291C22×C137#53511yescharacteristic
292C22×C137#53611yescharacteristic
293C22×C137#53711yescharacteristic
294C22×C137#53712yes—
295C22×C137#54112yes—
296C22×C137#54212yes—
297C22×C137#54212yes—
298C22×C137#54212yes—
299C22×C137#54412yes—
300C22×C137#54612yes—
301C22×C137#54611yescharacteristic
302C22×C137D4×Q811yescharacteristic
303C22×C137#55112yes—
304C22×C137#55211yescharacteristic
305C22×C137#55412yes—
306C22×C137#55411yescharacteristic
307C22×C137#55912yes—
308C22×C137H(3,𝔽4)12yes—
309C22×C137#56212yes—
310C4×C137—24no—
311C4×C137—24no—
312C4×C137—42no—
313C4×C137—42no—
314C4×C137—44no—
315C4×C137—44no—
316C4×C137—81no—
317C4×C137—82no—
318C4×C137—81no—
319C4×C137—82no—
320C2×C137—82no—
321C2×C137#241111yescharacteristic
322C2×C137#241711yescharacteristic
323C2×C137#258011yescharacteristic
324C2×C137#262011yescharacteristic
325C2×C137#262811yescharacteristic
326C2×C137#263711yescharacteristic
327C2×C137#270412yes—
328C2×C137#271212yes—
329C2×C137#275412yes—
330C2×C137#276312yes—
331C2×C137#281911yescharacteristic
332#16566C13711yesmaximal, characteristic, direct product factor, Sylow
333C137#1656611yescharacteristic, direct product factor, Sylow
334C2×#393C2×C13711yescharacteristic
335C2×#391C2×C13711yescharacteristic, complemented
336C2×#388C2×C13711yescharacteristic
337C2×#382C2×C13711yescharacteristic, complemented
338#1446C2×C13711yescharacteristic, complemented
339#1444C2×C13711yescharacteristic, complemented
340#1442C2×C13712yescomplemented
341#1441C2×C13712yescomplemented
342#1439C2×C13711yescharacteristic
343#1439C2×C13712yes—
344#1439C2×C13712yes—
345C23×Q8C22×C13711yescharacteristic
346C22×C4⋊C4C22×C13712yes—
347C22×C4⋊C4C22×C13711yescharacteristic
348C22×C4⋊C4C22×C13711yescharacteristic
349C22×C4⋊C4C22×C13712yes—
350C22×C22⋊C4C22×C13711yescharacteristic
351C22×C22⋊C4C22×C13712yes—
352C22×C22⋊C4C22×C13711yescharacteristic
353C22×C22⋊C4C22×C13711yescharacteristic
354C22×C22⋊C4C22×C13712yes—
355C22×C42C22×C13711yescharacteristic
356#393—24no—
357#391—24no—
358#388—24no—
359#382—24no—
360C2×#95C22×C13712yes—
361C2×#95C22×C13712yes—
362C2×#95C22×C13712yes—
363C2×#95C22×C13712yes—
364C2×#95C22×C13712yes—
365C2×#95C22×C13711yescharacteristic
366C2×#95C22×C13711yescharacteristic
367C2×#95C22×C13711yescharacteristic
368C2×#95C22×C13712yes—
369C2×#95C22×C13712yes—
370C2×#95C22×C13712yes—
371C2×#95C22×C13711yescharacteristic
372C25C23×C13711yescharacteristic
373C22×Q8—22no—
374C22×Q8—22no—
375C22×Q8—24no—
376C22×Q8—22no—
377C22×Q8—42no—
378C23×C4C23×C13711yescharacteristic
379C23×C4C23×C13712yes—
380C23×C4C23×C13711yescharacteristic
381C23×C4C23×C13712yes—
382C23×C4C23×C13712yes—
383C23×C4C23×C13711yescharacteristic
384C23×C4C23×C13711yescharacteristic
385C23×C4C23×C13712yes—
386C23×C4C23×C13711yescharacteristic
387C23×C4C23×C13711yescharacteristic
388C2×C4⋊C4—24no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—42no—
393C2×C4⋊C4—42no—
394C2×C4⋊C4—42no—
395C2×C4⋊C4—42no—
396C2×C4⋊C4—42no—
397C2×C4⋊C4—42no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—42no—
407C2×C22⋊C4—42no—
408C2×C22⋊C4—42no—
409C2×C22⋊C4—42no—
410C2×C22⋊C4—42no—
411C2×C22⋊C4Q8×C13714yescomplemented
412C2×C42—22no—
413C2×C42—22no—
414C2×C42D4×C13714yes—
415#95—24no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—42no—
424#95—42no—
425#95—42no—
426#95D4×C13714yes—
427C24—21no—
428C24—21no—
429C24—21no—
430C24—21no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—21no—
435C24—22no—
436C24—21no—
437C24C2×Q8×C13712yes—
438C24C24×C13711yescharacteristic
439C2×Q8—28no—
440C2×Q8—48no—
441C2×Q8—48no—
442C2×Q8—44no—
443C2×Q8—44no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4C2×D4×C13714yes—
495C22×C4C2×D4×C13712yes—
496C22×C4C2×D4×C13712yes—
497C22×C4C2×D4×C13712yes—
498C22×C4C2×D4×C13712yes—
499C22×C4C2×Q8×C13712yes—
500C22×C4C2×Q8×C13712yes—
501C22×C4#41×C13714yes—
502C22×C4#41×C13712yes—
503C22×C4#41×C13712yes—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C22⋊C4—28no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C42—28no—
514C23—22no—
515C23—24no—
516C23—22no—
517C23—24no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23C22×D4×C13711yescharacteristic
538C23C22×D4×C13711yescharacteristic
539C23C22×Q8×C13711yescharacteristic
540C23C2×#41×C13711yescharacteristic
541C23C2×#41×C13711yescharacteristic
542C23C2×#41×C13712yes—
543C23H(5,𝔽2)×C13712yes—
544C23H(5,𝔽2)×C13712yes—
545C23H(5,𝔽2)×C13711yescharacteristic
546C232−1+4×C13712yes—
547C232−1+4×C13711yescharacteristic
548Q8—416no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4#121×C13714yes—
608C2×C4C22⋊Q8×C13714yes—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—82no—
619C22—82no—
620C22C2×#121×C13711yescharacteristic
621C22C2×C22⋊Q8×C13711yescharacteristic
622C22#525×C13711yescharacteristic
623C22#535×C13711yescharacteristic
624C22#535×C13711yescharacteristic
625C22#536×C13711yescharacteristic
626C22#537×C13711yescharacteristic
627C22#537×C13712yes—
628C22#541×C13712yes—
629C22#542×C13712yes—
630C22#542×C13712yes—
631C22#542×C13712yes—
632C22#544×C13712yes—
633C22#546×C13712yes—
634C22#546×C13711yescharacteristic
635C22D4×Q8×C13711yescharacteristic
636C22#551×C13712yes—
637C22#552×C13711yescharacteristic
638C22#554×C13712yes—
639C22#554×C13711yescharacteristic
640C22#559×C13712yes—
641C22H(3,𝔽4)×C13712yes—
642C22#562×C13712yes—
643C4—24no—
644C4—24no—
645C4—42no—
646C4—42no—
647C4—44no—
648C4—44no—
649C4—81no—
650C4—82no—
651C4—81no—
652C4—82no—
653C2—82no—
654C2#2411×C13711yescharacteristic
655C2#2417×C13711yescharacteristic
656C2#2580×C13711yescharacteristic
657C2#2620×C13711yescharacteristic
658C2#2628×C13711yescharacteristic
659C2#2637×C13711yescharacteristic
660C2#2704×C13712yes—
661C2#2712×C13712yes—
662C2#2754×C13712yes—
663C2#2763×C13712yes—
664C2#2819×C13711yescharacteristic
subgroupG/H
center, socleC24×C137C24
commutator, Frattini, cosocleC24C24×C137
prime radicalC25×C137C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#391×C137C2semidirect
4. C2×#382×C137C2semidirect
5. #1446×C137C2semidirect
6. #1444×C137C2semidirect
7. #1442×C137C2semidirect
8. #1441×C137C2semidirect
78. C2×C22⋊C4×C137Q8semidirect
332. #16566C137direct (N × H)
333. C137#16566direct (N × H)
335. C2×#391C2×C137semidirect
337. C2×#382C2×C137semidirect
338. #1446C2×C137semidirect
339. #1444C2×C137semidirect
340. #1442C2×C137semidirect
341. #1441C2×C137semidirect
411. C2×C22⋊C4Q8×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#393×C1372
C2×#391×C1372
C2×#388×C1372
C2×#382×C1372
#1446×C1372
#1444×C1372
#1442×C1372
#1441×C1372
#1439×C1372
#16566137