Finite Group Databasebeta

← C137×#15175 #46864561

Subgroups — C137×#15175 #46864561

Order 35072.

1418 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#400×C137C211yesmaximal, characteristic, complemented
2C2×#399×C137C211yesmaximal, characteristic
3C2×#388×C137C211yesmaximal, characteristic
4C2×C4×C4⋊C4×C137C211yesmaximal, characteristic, complemented
5#1442×C137C212yesmaximal, complemented
6#1441×C137C211yesmaximal, characteristic, complemented
7#1440×C137C212yesmaximal
8#1439×C137C211yesmaximal, characteristic
9#1437×C137C212yesmaximal
10#1436×C137C212yesmaximal, complemented
11#1434×C137C211yesmaximal, characteristic, complemented
12C22×C4⋊C4×C137C2211yescharacteristic
13C22×C4⋊C4×C137C2211yescharacteristic
14C22×C4⋊C4×C137C2212yes—
15C22×C4⋊C4×C137C2211yescharacteristic
16C22×C22⋊C4×C137C2211yescharacteristic
17C22×C22⋊C4×C137C2211yescharacteristic
18C22×C22⋊C4×C137C2211yescharacteristic
19C22×C22⋊C4×C137C2212yes—
20C22×C22⋊C4×C137C2212yes—
21C22×C42×C137C2211yescharacteristic
22C22×C42×C137C2212yes—
23#400×C137—24no—
24#399×C137—24no—
25#388×C137—24no—
26C4×C4⋊C4×C137—24no—
27C2×#95×C137C2212yes—
28C2×#95×C137C2211yescharacteristic
29C2×#95×C137C2211yescharacteristic
30C2×#95×C137C2211yescharacteristic
31C2×#95×C137C2211yescharacteristic
32C2×#95×C137C2212yes—
33C2×#95×C137C2212yes—
34C2×#95×C137C2212yes—
35C2×#95×C137C2212yes—
36C2×#95×C137C2212yes—
37C2×#95×C137C2212yes—
38C2×#95×C137C2212yes—
39C25×C137C2311yescharacteristic
40C23×C4×C137C2312yes—
41C23×C4×C137C2311yescharacteristic
42C23×C4×C137C2311yescharacteristic
43C23×C4×C137C2311yescharacteristic
44C23×C4×C137C2312yes—
45C23×C4×C137C2311yescharacteristic
46C23×C4×C137C2312yes—
47C23×C4×C137C2312yes—
48C23×C4×C137C2311yescharacteristic
49C23×C4×C137C2311yescharacteristic
50C2×C4⋊C4×C137—22no—
51C2×C4⋊C4×C137—22no—
52C2×C4⋊C4×C137—24no—
53C2×C4⋊C4×C137—22no—
54C2×C4⋊C4×C137—22no—
55C2×C4⋊C4×C137—24no—
56C2×C4⋊C4×C137—22no—
57C2×C4⋊C4×C137—24no—
58C2×C4⋊C4×C137—22no—
59C2×C4⋊C4×C137—42no—
60C2×C4⋊C4×C137D414yes—
61C2×C22⋊C4×C137—22no—
62C2×C22⋊C4×C137—22no—
63C2×C22⋊C4×C137—22no—
64C2×C22⋊C4×C137—24no—
65C2×C22⋊C4×C137—22no—
66C2×C22⋊C4×C137—22no—
67C2×C22⋊C4×C137—24no—
68C2×C22⋊C4×C137—22no—
69C2×C22⋊C4×C137—24no—
70C2×C22⋊C4×C137—24no—
71C2×C22⋊C4×C137—22no—
72C2×C22⋊C4×C137—24no—
73C2×C22⋊C4×C137—42no—
74C2×C22⋊C4×C137—42no—
75C2×C42×C137—24no—
76C2×C42×C137—24no—
77C2×C42×C137—42no—
78C2×C42×C137—42no—
79C2×C42×C137D414yes—
80#95×C137—24no—
81#95×C137—22no—
82#95×C137—22no—
83#95×C137—22no—
84#95×C137—24no—
85#95×C137—22no—
86#95×C137—24no—
87#95×C137—24no—
88#95×C137—24no—
89#95×C137—24no—
90#95×C137—24no—
91#95×C137—42no—
92C24×C137—21no—
93C24×C137—21no—
94C24×C137—21no—
95C24×C137—22no—
96C24×C137—22no—
97C24×C137—22no—
98C24×C137—21no—
99C24×C137—22no—
100C24×C137—21no—
101C24×C137—21no—
102C24×C137#4112yes—
103C24×C137C2411yescharacteristic
104C22×C4×C137—22no—
105C22×C4×C137—22no—
106C22×C4×C137—22no—
107C22×C4×C137—21no—
108C22×C4×C137—21no—
109C22×C4×C137—21no—
110C22×C4×C137—22no—
111C22×C4×C137—22no—
112C22×C4×C137—21no—
113C22×C4×C137—22no—
114C22×C4×C137—22no—
115C22×C4×C137—22no—
116C22×C4×C137—22no—
117C22×C4×C137—22no—
118C22×C4×C137—22no—
119C22×C4×C137—22no—
120C22×C4×C137—21no—
121C22×C4×C137—21no—
122C22×C4×C137—21no—
123C22×C4×C137—22no—
124C22×C4×C137—22no—
125C22×C4×C137—22no—
126C22×C4×C137—22no—
127C22×C4×C137—22no—
128C22×C4×C137—21no—
129C22×C4×C137—22no—
130C22×C4×C137—22no—
131C22×C4×C137—22no—
132C22×C4×C137—22no—
133C22×C4×C137—22no—
134C22×C4×C137—21no—
135C22×C4×C137—21no—
136C22×C4×C137—22no—
137C22×C4×C137—22no—
138C22×C4×C137—21no—
139C22×C4×C137—21no—
140C22×C4×C137—21no—
141C22×C4×C137—21no—
142C22×C4×C137—22no—
143C22×C4×C137—22no—
144C22×C4×C137—22no—
145C22×C4×C137—21no—
146C22×C4×C137—22no—
147C22×C4×C137—22no—
148C22×C4×C137—22no—
149C22×C4×C137—21no—
150C22×C4×C137—22no—
151C22×C4×C137—22no—
152C22×C4×C137—22no—
153C22×C4×C137—22no—
154C22×C4×C137C2×D412yes—
155C22×C4×C137C2×D414yes—
156C22×C4×C137C2×D412yes—
157C22×C4×C137C2×D412yes—
158C22×C4×C137C2×D412yes—
159C22×C4×C137#4112yes—
160C22×C4×C137#4114yes—
161C22×C4×C137#4114yes—
162C22×C4×C137#4112yes—
163C22×C4×C137#4114yes—
164C4⋊C4×C137—28no—
165C4⋊C4×C137—28no—
166C4⋊C4×C137—48no—
167C4⋊C4×C137—44no—
168C22⋊C4×C137—28no—
169C22⋊C4×C137—48no—
170C22⋊C4×C137—48no—
171C22⋊C4×C137—44no—
172C22⋊C4×C137—44no—
173C42×C137—28no—
174C42×C137—48no—
175C23×C137—22no—
176C23×C137—24no—
177C23×C137—22no—
178C23×C137—24no—
179C23×C137—22no—
180C23×C137—41no—
181C23×C137—41no—
182C23×C137—41no—
183C23×C137—42no—
184C23×C137—42no—
185C23×C137—42no—
186C23×C137—41no—
187C23×C137—41no—
188C23×C137—41no—
189C23×C137—42no—
190C23×C137—42no—
191C23×C137—42no—
192C23×C137—41no—
193C23×C137—42no—
194C23×C137—41no—
195C23×C137—42no—
196C23×C137—42no—
197C23×C137—42no—
198C23×C137C22×D411yescharacteristic
199C23×C137C22×D411yescharacteristic
200C23×C137C2×#4111yescharacteristic
201C23×C137C2×#4112yes—
202C23×C137C2×#4111yescharacteristic
203C23×C137C2×#4111yescharacteristic
204C23×C137C2×#4112yes—
205C23×C137C2×#4112yes—
206C23×C137H(5,𝔽2)12yes—
207C23×C1372−1+411yescharacteristic
208C23×C1372−1+411yescharacteristic
209C2×C4×C137—22no—
210C2×C4×C137—22no—
211C2×C4×C137—24no—
212C2×C4×C137—24no—
213C2×C4×C137—22no—
214C2×C4×C137—22no—
215C2×C4×C137—24no—
216C2×C4×C137—24no—
217C2×C4×C137—24no—
218C2×C4×C137—22no—
219C2×C4×C137—22no—
220C2×C4×C137—22no—
221C2×C4×C137—24no—
222C2×C4×C137—24no—
223C2×C4×C137—22no—
224C2×C4×C137—24no—
225C2×C4×C137—24no—
226C2×C4×C137—24no—
227C2×C4×C137—22no—
228C2×C4×C137—24no—
229C2×C4×C137—22no—
230C2×C4×C137—24no—
231C2×C4×C137—24no—
232C2×C4×C137—41no—
233C2×C4×C137—42no—
234C2×C4×C137—42no—
235C2×C4×C137—42no—
236C2×C4×C137—42no—
237C2×C4×C137—42no—
238C2×C4×C137—42no—
239C2×C4×C137—42no—
240C2×C4×C137—41no—
241C2×C4×C137—42no—
242C2×C4×C137—41no—
243C2×C4×C137—42no—
244C2×C4×C137—42no—
245C2×C4×C137—41no—
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—41no—
253C2×C4×C137—41no—
254C2×C4×C137—42no—
255C2×C4×C137—42no—
256C2×C4×C137—42no—
257C2×C4×C137—42no—
258C2×C4×C137—41no—
259C2×C4×C137—41no—
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#12114yes—
268C2×C4×C137#12314yes—
269C22×C137—44no—
270C22×C137—44no—
271C22×C137—42no—
272C22×C137—42no—
273C22×C137—42no—
274C22×C137—82no—
275C22×C137—81no—
276C22×C137—81no—
277C22×C137—81no—
278C22×C137—81no—
279C22×C137—82no—
280C22×C137C2×#12111yescharacteristic
281C22×C137C2×#12311yescharacteristic
282C22×C137#52511yescharacteristic
283C22×C137#52511yescharacteristic
284C22×C137#52912yes—
285C22×C137#52912yes—
286C22×C137#52912yes—
287C22×C137#53212yes—
288C22×C137#53211yescharacteristic
289C22×C137#53511yescharacteristic
290C22×C137#53611yescharacteristic
291C22×C137#53912yes—
292C22×C137#53912yes—
293C22×C137#54612yes—
294C22×C137#54811yescharacteristic
295C22×C137#54811yescharacteristic
296C22×C137#55112yes—
297C22×C137#55211yescharacteristic
298C22×C137#55212yes—
299C22×C137#55211yescharacteristic
300C22×C137#55312yes—
301C22×C137#55612yes—
302C22×C137#55912yes—
303C4×C137—24no—
304C4×C137—24no—
305C4×C137—44no—
306C4×C137—42no—
307C4×C137—42no—
308C4×C137—44no—
309C4×C137—44no—
310C4×C137—82no—
311C4×C137—81no—
312C4×C137—81no—
313C2×C137—82no—
314C2×C137#239511yescharacteristic
315C2×C137#253611yescharacteristic
316C2×C137#254411yescharacteristic
317C2×C137#255812yes—
318C2×C137#256012yes—
319C2×C137#257512yes—
320C2×C137#259612yes—
321C2×C137#259811yescharacteristic
322C2×C137#260411yescharacteristic
323C2×C137#262311yescharacteristic
324C2×C137#285411yescharacteristic
325#15175C13711yesmaximal, characteristic, direct product factor, Sylow
326C137#1517511yescharacteristic, direct product factor, Sylow
327C2×#400C2×C13711yescharacteristic, complemented
328C2×#399C2×C13711yescharacteristic
329C2×#388C2×C13711yescharacteristic
330C2×C4×C4⋊C4C2×C13711yescharacteristic, complemented
331#1442C2×C13712yescomplemented
332#1441C2×C13711yescharacteristic, complemented
333#1440C2×C13712yes—
334#1439C2×C13711yescharacteristic
335#1437C2×C13712yes—
336#1436C2×C13712yescomplemented
337#1434C2×C13711yescharacteristic, complemented
338C22×C4⋊C4C22×C13711yescharacteristic
339C22×C4⋊C4C22×C13711yescharacteristic
340C22×C4⋊C4C22×C13712yes—
341C22×C4⋊C4C22×C13711yescharacteristic
342C22×C22⋊C4C22×C13711yescharacteristic
343C22×C22⋊C4C22×C13711yescharacteristic
344C22×C22⋊C4C22×C13711yescharacteristic
345C22×C22⋊C4C22×C13712yes—
346C22×C22⋊C4C22×C13712yes—
347C22×C42C22×C13711yescharacteristic
348C22×C42C22×C13712yes—
349#400—24no—
350#399—24no—
351#388—24no—
352C4×C4⋊C4—24no—
353C2×#95C22×C13711yescharacteristic
354C2×#95C22×C13711yescharacteristic
355C2×#95C22×C13712yes—
356C2×#95C22×C13712yes—
357C2×#95C22×C13712yes—
358C2×#95C22×C13712yes—
359C2×#95C22×C13711yescharacteristic
360C2×#95C22×C13712yes—
361C2×#95C22×C13712yes—
362C2×#95C22×C13712yes—
363C2×#95C22×C13711yescharacteristic
364C2×#95C22×C13712yes—
365C25C23×C13711yescharacteristic
366C23×C4C23×C13711yescharacteristic
367C23×C4C23×C13711yescharacteristic
368C23×C4C23×C13712yes—
369C23×C4C23×C13712yes—
370C23×C4C23×C13711yescharacteristic
371C23×C4C23×C13711yescharacteristic
372C23×C4C23×C13712yes—
373C23×C4C23×C13712yes—
374C23×C4C23×C13711yescharacteristic
375C23×C4C23×C13711yescharacteristic
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—24no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—42no—
386C2×C4⋊C4D4×C13714yes—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—42no—
400C2×C22⋊C4—42no—
401C2×C42—24no—
402C2×C42—24no—
403C2×C42—42no—
404C2×C42—42no—
405C2×C42D4×C13714yes—
406#95—24no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—24no—
411#95—22no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—42no—
418C24—21no—
419C24—21no—
420C24—21no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—21no—
425C24—22no—
426C24—21no—
427C24—21no—
428C24#41×C13712yes—
429C24C24×C13711yescharacteristic
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4C2×D4×C13712yes—
481C22×C4C2×D4×C13714yes—
482C22×C4C2×D4×C13712yes—
483C22×C4C2×D4×C13712yes—
484C22×C4C2×D4×C13712yes—
485C22×C4#41×C13712yes—
486C22×C4#41×C13714yes—
487C22×C4#41×C13714yes—
488C22×C4#41×C13712yes—
489C22×C4#41×C13714yes—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C22⋊C4—28no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C42—28no—
500C42—48no—
501C23—22no—
502C23—24no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23C22×D4×C13711yescharacteristic
525C23C22×D4×C13711yescharacteristic
526C23C2×#41×C13711yescharacteristic
527C23C2×#41×C13712yes—
528C23C2×#41×C13711yescharacteristic
529C23C2×#41×C13711yescharacteristic
530C23C2×#41×C13712yes—
531C23C2×#41×C13712yes—
532C23H(5,𝔽2)×C13712yes—
533C232−1+4×C13711yescharacteristic
534C232−1+4×C13711yescharacteristic
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4#121×C13714yes—
594C2×C4#123×C13714yes—
595C22—42no—
596C22—44no—
597C22—44no—
598C22—42no—
599C22—42no—
600C22—81no—
601C22—82no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22C2×#121×C13711yescharacteristic
607C22C2×#123×C13711yescharacteristic
608C22#525×C13711yescharacteristic
609C22#525×C13711yescharacteristic
610C22#529×C13712yes—
611C22#529×C13712yes—
612C22#529×C13712yes—
613C22#532×C13711yescharacteristic
614C22#532×C13712yes—
615C22#535×C13711yescharacteristic
616C22#536×C13711yescharacteristic
617C22#539×C13712yes—
618C22#539×C13712yes—
619C22#546×C13712yes—
620C22#548×C13711yescharacteristic
621C22#548×C13711yescharacteristic
622C22#551×C13712yes—
623C22#552×C13711yescharacteristic
624C22#552×C13711yescharacteristic
625C22#552×C13712yes—
626C22#553×C13712yes—
627C22#556×C13712yes—
628C22#559×C13712yes—
629C4—24no—
630C4—24no—
631C4—42no—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—42no—
636C4—81no—
637C4—82no—
638C4—81no—
639C2—82no—
640C2#2395×C13711yescharacteristic
641C2#2536×C13711yescharacteristic
642C2#2544×C13711yescharacteristic
643C2#2558×C13712yes—
644C2#2560×C13712yes—
645C2#2575×C13712yes—
646C2#2596×C13712yes—
647C2#2598×C13711yescharacteristic
648C2#2604×C13711yescharacteristic
649C2#2623×C13711yescharacteristic
650C2#2854×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
1. C2×#400×C137C2semidirect
4. C2×C4×C4⋊C4×C137C2semidirect
5. #1442×C137C2semidirect
6. #1441×C137C2semidirect
10. #1436×C137C2semidirect
11. #1434×C137C2semidirect
325. #15175C137direct (N × H)
326. C137#15175direct (N × H)
327. C2×#400C2×C137semidirect
330. C2×C4×C4⋊C4C2×C137semidirect
331. #1442C2×C137semidirect
332. #1441C2×C137semidirect
336. #1436C2×C137semidirect
337. #1434C2×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#400×C1372
C2×#399×C1372
C2×#388×C1372
C2×C4×C4⋊C4×C1372
#1442×C1372
#1441×C1372
#1440×C1372
#1439×C1372
#1437×C1372
#1436×C1372
#1434×C1372
#15175137