Finite Group Databasebeta

← C137×#26118 #46875504

Subgroups — C137×#26118 #46875504

Order 35072.

764 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2570×C137C211yesmaximal, characteristic, complemented
2D4×C4⋊C4×C137C211yesmaximal, characteristic
3#2101×C137C211yesmaximal, characteristic, complemented
4#2099×C137C211yesmaximal, characteristic, complemented
5#2088×C137C211yesmaximal, characteristic, complemented
6#2088×C137C211yesmaximal, characteristic, complemented
7#1948×C137C211yesmaximal, characteristic, complemented
8#1940×C137C211yesmaximal, characteristic, complemented
9#1864×C137C211yesmaximal, characteristic
10#1864×C137C211yesmaximal, characteristic
11#1836×C137C211yesmaximal, characteristic, complemented
12#1804×C137C211yesmaximal, characteristic
13#1804×C137C211yesmaximal, characteristic
14#1802×C137C211yesmaximal, characteristic
15#1762×C137C211yesmaximal, characteristic
16C2×C4×D4×C137C2211yescharacteristic
17C2×C4×D4×C137C2211yescharacteristic
18C2×C4×D4×C137C2211yescharacteristic
19C22×C4⋊C4×C137—21no—
20C2×#106×C137C2211yescharacteristic
21C2×#106×C137C2211yescharacteristic
22C2×C4⋊C8×C137C2211yescharacteristic
23C2×C4⋊C8×C137C2211yescharacteristic
24C2×C4⋊C8×C137C2211yescharacteristic
25C2×#102×C137C2211yescharacteristic
26C2×#102×C137C2211yescharacteristic
27C2×#102×C137C2211yescharacteristic
28C2×#102×C137C2211yescharacteristic
29C2×#97×C137C2211yescharacteristic
30#400×C137—21no—
31#398×C137—21no—
32#389×C137C2211yescharacteristic
33#389×C137C2211yescharacteristic
34#389×C137C2211yescharacteristic
35#389×C137C2211yescharacteristic
36#387×C137—21no—
37#387×C137—21no—
38#385×C137—21no—
39#385×C137—21no—
40#384×C137C2211yescharacteristic
41#384×C137C2211yescharacteristic
42#382×C137C2211yescharacteristic
43#382×C137C2211yescharacteristic
44#380×C137—21no—
45C4×C4⋊C4×C137C2211yescharacteristic
46C4×C4⋊C4×C137C2211yescharacteristic
47#340×C137C2211yescharacteristic
48#340×C137C2211yescharacteristic
49#340×C137C2211yescharacteristic
50#340×C137C2211yescharacteristic
51#340×C137C2211yescharacteristic
52#340×C137C2211yescharacteristic
53#340×C137C2211yescharacteristic
54#340×C137C2211yescharacteristic
55#340×C137C2211yescharacteristic
56#340×C137C2211yescharacteristic
57#336×C137C2211yescharacteristic
58#336×C137C2211yescharacteristic
59C22×D4×C137C2311yescharacteristic
60C23×C4×C137—21no—
61C23×C4×C137—21no—
62C23×C4×C137—21no—
63C22×C8×C137C2311yescharacteristic
64C22×C8×C137C2311yescharacteristic
65C22×C8×C137C2311yescharacteristic
66C22×C8×C137C2311yescharacteristic
67C4×D4×C137—22no—
68C4×D4×C137—22no—
69C4×D4×C137—22no—
70C4×D4×C137—41no—
71C4×D4×C137—41no—
72C4×D4×C137Q814yes—
73C2×C4⋊C4×C137—21no—
74C2×C4⋊C4×C137—21no—
75C2×C4⋊C4×C137—21no—
76C2×C4⋊C4×C137—21no—
77C2×C4⋊C4×C137—21no—
78C2×C4⋊C4×C137—21no—
79C2×C4⋊C4×C137—21no—
80C2×C4⋊C4×C137—21no—
81C2×C4⋊C4×C137—44no—
82C2×C4⋊C4×C137D411yescharacteristic
83C2×C4⋊C4×C137D411yescharacteristic
84C2×C4⋊C4×C137C2311yescharacteristic
85C2×C4⋊C4×C137C2311yescharacteristic
86C2×C4⋊C4×C137C2311yescharacteristic
87C2×C4⋊C4×C137C2311yescharacteristic
88C2×C4⋊C4×C137C2311yescharacteristic
89C2×C4⋊C4×C137C2311yescharacteristic
90C2×C4⋊C4×C137C2311yescharacteristic
91C2×C22⋊C4×C137—21no—
92C2×C22⋊C4×C137—21no—
93C2×C22⋊C4×C137—21no—
94C2×C42×C137—21no—
95C2×C42×C137C2311yescharacteristic
96C2×C42×C137C2311yescharacteristic
97C2×C42×C137C2311yescharacteristic
98#106×C137—41no—
99#106×C137—41no—
100C4⋊C8×C137—22no—
101C4⋊C8×C137—41no—
102C4⋊C8×C137—41no—
103#102×C137—22no—
104#102×C137—22no—
105#102×C137—22no—
106#102×C137—22no—
107#97×C137—22no—
108#95×C137—21no—
109#95×C137—21no—
110#95×C137—21no—
111#95×C137—21no—
112#95×C137—21no—
113#95×C137—21no—
114#95×C137—21no—
115#95×C137—21no—
116#95×C137—21no—
117#95×C137D411yescharacteristic
118#95×C137D411yescharacteristic
119C24×C137—21no—
120C2×D4×C137—21no—
121C2×D4×C137—21no—
122C2×D4×C137—21no—
123C2×D4×C137C2×Q812yes—
124C2×D4×C137#4112yes—
125C2×D4×C137#4112yes—
126C22×C4×C137—21no—
127C22×C4×C137—24no—
128C22×C4×C137—21no—
129C22×C4×C137—21no—
130C22×C4×C137—21no—
131C22×C4×C137—21no—
132C22×C4×C137—21no—
133C22×C4×C137—21no—
134C22×C4×C137—42no—
135C22×C4×C137—42no—
136C22×C4×C137—42no—
137C22×C4×C137—42no—
138C22×C4×C137—42no—
139C22×C4×C137—42no—
140C22×C4×C137—81no—
141C22×C4×C137—81no—
142C22×C4×C137C2×D411yescharacteristic
143C22×C4×C137C2×D411yescharacteristic
144C22×C4×C137C2×D411yescharacteristic
145C22×C4×C137C2×D411yescharacteristic
146C22×C4×C137C2×D411yescharacteristic
147C22×C4×C137C2×D411yescharacteristic
148C22×C4×C137C2411yescharacteristic
149C2×C8×C137—21no—
150C2×C8×C137—21no—
151C2×C8×C137—21no—
152C2×C8×C137—21no—
153C2×C8×C137—21no—
154C2×C8×C137—21no—
155C2×C8×C137—21no—
156C2×C8×C137—21no—
157C2×C8×C137—21no—
158C2×C8×C137—21no—
159C2×C8×C137—21no—
160C2×C8×C137—21no—
161C4⋊C4×C137—22no—
162C4⋊C4×C137—21no—
163C4⋊C4×C137—21no—
164C4⋊C4×C137—21no—
165C4⋊C4×C137—21no—
166C4⋊C4×C137—21no—
167C4⋊C4×C137—21no—
168C4⋊C4×C137—21no—
169C4⋊C4×C137—21no—
170C4⋊C4×C137—21no—
171C4⋊C4×C137—21no—
172C4⋊C4×C137—21no—
173C4⋊C4×C137—21no—
174C4⋊C4×C137—21no—
175C4⋊C4×C137—41no—
176C4⋊C4×C137—42no—
177C4⋊C4×C137—42no—
178C4⋊C4×C137—42no—
179C4⋊C4×C137—42no—
180C4⋊C4×C137—42no—
181C4⋊C4×C137—81no—
182C4⋊C4×C137—81no—
183C4⋊C4×C137—81no—
184C4⋊C4×C137C2×Q812yes—
185C22⋊C4×C137—24no—
186C22⋊C4×C137—42no—
187C22⋊C4×C137—42no—
188C42×C137—21no—
189C42×C137—21no—
190C42×C137—21no—
191C42×C137—21no—
192C42×C137—81no—
193C42×C137C2×Q812yes—
194C42×C137#4112yes—
195C23×C137—22no—
196C23×C137—22no—
197C23×C137—22no—
198C23×C137—41no—
199C23×C137—41no—
200C23×C137—41no—
201C23×C137—41no—
202C23×C137C22×D411yescharacteristic
203D4×C137—22no—
204D4×C137—22no—
205D4×C137—22no—
206D4×C137#12514yes—
207C2×C4×C137—21no—
208C2×C4×C137—21no—
209C2×C4×C137—21no—
210C2×C4×C137—21no—
211C2×C4×C137—21no—
212C2×C4×C137—21no—
213C2×C4×C137—22no—
214C2×C4×C137—21no—
215C2×C4×C137—21no—
216C2×C4×C137—21no—
217C2×C4×C137—21no—
218C2×C4×C137—21no—
219C2×C4×C137—21no—
220C2×C4×C137—21no—
221C2×C4×C137—21no—
222C2×C4×C137—21no—
223C2×C4×C137—48no—
224C2×C4×C137—41no—
225C2×C4×C137—41no—
226C2×C4×C137—41no—
227C2×C4×C137—41no—
228C2×C4×C137—41no—
229C2×C4×C137—41no—
230C2×C4×C137—41no—
231C2×C4×C137—41no—
232C2×C4×C137—41no—
233C2×C4×C137—41no—
234C2×C4×C137—41no—
235C2×C4×C137—41no—
236C2×C4×C137—41no—
237C2×C4×C137—41no—
238C2×C4×C137—41no—
239C2×C4×C137—41no—
240C2×C4×C137—41no—
241C2×C4×C137—41no—
242C2×C4×C137—41no—
243C2×C4×C137—41no—
244C2×C4×C137—84no—
245C2×C4×C137—84no—
246C2×C4×C137C22⋊Q812yes—
247C2×C4×C137C22⋊Q812yes—
248C2×C4×C137AGL(1,ℤ/8)12yes—
249C2×C4×C137C22×Q811yescharacteristic
250C2×C4×C137C2×#4111yescharacteristic
251C2×C4×C137C2×#4111yescharacteristic
252C2×C4×C137C2×#4111yescharacteristic
253C2×C4×C137H(5,𝔽2)11yescharacteristic
254C2×C4×C1372−1+411yescharacteristic
255C8×C137—41no—
256C8×C137—41no—
257C8×C137—41no—
258C8×C137—41no—
259C22×C137—24no—
260C22×C137—42no—
261C22×C137—42no—
262C22×C137—42no—
263C22×C137—42no—
264C22×C137—42no—
265C22×C137—42no—
266C22×C137C2×C22⋊Q811yescharacteristic
267C22×C137#54611yescharacteristic
268C22×C137#54811yescharacteristic
269C22×C137C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C137#57511yescharacteristic
271C22×C137#57711yescharacteristic
272C22×C137#57711yescharacteristic
273C4×C137—22no—
274C4×C137—22no—
275C4×C137—22no—
276C4×C137—42no—
277C4×C137—41no—
278C4×C137—41no—
279C4×C137—41no—
280C4×C137—81no—
281C4×C137—81no—
282C4×C137—81no—
283C4×C137—81no—
284C4×C137—81no—
285C4×C137—81no—
286C4×C137C2×#12511yescharacteristic
287C4×C137#54411yescharacteristic
288C4×C137#55211yescharacteristic
289C4×C137#55311yescharacteristic
290C2×C137—44no—
291C2×C137#244211yescharacteristic
292C2×C137#307711yescharacteristic
293C2×C137#308411yescharacteristic
294C2×C137#330711yescharacteristic
295C2×C137#331011yescharacteristic
296C2×C137#337311yescharacteristic
297C2×C137#337611yescharacteristic
298#26118C13711yesmaximal, characteristic, direct product factor, Sylow
299C137#2611811yescharacteristic, direct product factor, Sylow
300#2570C2×C13711yescharacteristic, complemented
301D4×C4⋊C4C2×C13711yescharacteristic
302#2101C2×C13711yescharacteristic, complemented
303#2099C2×C13711yescharacteristic, complemented
304#2088C2×C13711yescharacteristic, complemented
305#2088C2×C13711yescharacteristic, complemented
306#1948C2×C13711yescharacteristic, complemented
307#1940C2×C13711yescharacteristic, complemented
308#1864C2×C13711yescharacteristic
309#1864C2×C13711yescharacteristic
310#1836C2×C13711yescharacteristic, complemented
311#1804C2×C13711yescharacteristic
312#1804C2×C13711yescharacteristic
313#1802C2×C13711yescharacteristic
314#1762C2×C13711yescharacteristic
315C2×C4×D4C22×C13711yescharacteristic
316C2×C4×D4C22×C13711yescharacteristic
317C2×C4×D4C22×C13711yescharacteristic
318C22×C4⋊C4—21no—
319C2×#106C22×C13711yescharacteristic
320C2×#106C22×C13711yescharacteristic
321C2×C4⋊C8C22×C13711yescharacteristic
322C2×C4⋊C8C22×C13711yescharacteristic
323C2×C4⋊C8C22×C13711yescharacteristic
324C2×#102C22×C13711yescharacteristic
325C2×#102C22×C13711yescharacteristic
326C2×#102C22×C13711yescharacteristic
327C2×#102C22×C13711yescharacteristic
328C2×#97C22×C13711yescharacteristic
329#400—21no—
330#398—21no—
331#389C22×C13711yescharacteristic
332#389C22×C13711yescharacteristic
333#389C22×C13711yescharacteristic
334#389C22×C13711yescharacteristic
335#387—21no—
336#387—21no—
337#385—21no—
338#385—21no—
339#384C22×C13711yescharacteristic
340#384C22×C13711yescharacteristic
341#382C22×C13711yescharacteristic
342#382C22×C13711yescharacteristic
343#380—21no—
344C4×C4⋊C4C22×C13711yescharacteristic
345C4×C4⋊C4C22×C13711yescharacteristic
346#340C22×C13711yescharacteristic
347#340C22×C13711yescharacteristic
348#340C22×C13711yescharacteristic
349#340C22×C13711yescharacteristic
350#340C22×C13711yescharacteristic
351#340C22×C13711yescharacteristic
352#340C22×C13711yescharacteristic
353#340C22×C13711yescharacteristic
354#340C22×C13711yescharacteristic
355#340C22×C13711yescharacteristic
356#336C22×C13711yescharacteristic
357#336C22×C13711yescharacteristic
358C22×D4C23×C13711yescharacteristic
359C23×C4—21no—
360C23×C4—21no—
361C23×C4—21no—
362C22×C8C23×C13711yescharacteristic
363C22×C8C23×C13711yescharacteristic
364C22×C8C23×C13711yescharacteristic
365C22×C8C23×C13711yescharacteristic
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—41no—
370C4×D4—41no—
371C4×D4Q8×C13714yes—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—44no—
381C2×C4⋊C4D4×C13711yescharacteristic
382C2×C4⋊C4D4×C13711yescharacteristic
383C2×C4⋊C4C23×C13711yescharacteristic
384C2×C4⋊C4C23×C13711yescharacteristic
385C2×C4⋊C4C23×C13711yescharacteristic
386C2×C4⋊C4C23×C13711yescharacteristic
387C2×C4⋊C4C23×C13711yescharacteristic
388C2×C4⋊C4C23×C13711yescharacteristic
389C2×C4⋊C4C23×C13711yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C42—21no—
394C2×C42C23×C13711yescharacteristic
395C2×C42C23×C13711yescharacteristic
396C2×C42C23×C13711yescharacteristic
397#106—41no—
398#106—41no—
399C4⋊C8—22no—
400C4⋊C8—41no—
401C4⋊C8—41no—
402#102—22no—
403#102—22no—
404#102—22no—
405#102—22no—
406#97—22no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95D4×C13711yescharacteristic
417#95D4×C13711yescharacteristic
418C24—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×Q8×C13712yes—
423C2×D4#41×C13712yes—
424C2×D4#41×C13712yes—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4C2×D4×C13711yescharacteristic
442C22×C4C2×D4×C13711yescharacteristic
443C22×C4C2×D4×C13711yescharacteristic
444C22×C4C2×D4×C13711yescharacteristic
445C22×C4C2×D4×C13711yescharacteristic
446C22×C4C2×D4×C13711yescharacteristic
447C22×C4C24×C13711yescharacteristic
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—41no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4C2×Q8×C13712yes—
484C22⋊C4—24no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—81no—
492C42C2×Q8×C13712yes—
493C42#41×C13712yes—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23C22×D4×C13711yescharacteristic
502D4—22no—
503D4—22no—
504D4—22no—
505D4#125×C13714yes—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—48no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4C22⋊Q8×C13712yes—
546C2×C4C22⋊Q8×C13712yes—
547C2×C4AGL(1,ℤ/8)×C13712yes—
548C2×C4C22×Q8×C13711yescharacteristic
549C2×C4C2×#41×C13711yescharacteristic
550C2×C4C2×#41×C13711yescharacteristic
551C2×C4C2×#41×C13711yescharacteristic
552C2×C4H(5,𝔽2)×C13711yescharacteristic
553C2×C42−1+4×C13711yescharacteristic
554C8—41no—
555C8—41no—
556C8—41no—
557C8—41no—
558C22—24no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22C2×C22⋊Q8×C13711yescharacteristic
566C22#546×C13711yescharacteristic
567C22#548×C13711yescharacteristic
568C22C2×AGL(1,ℤ/8)×C13711yescharacteristic
569C22#575×C13711yescharacteristic
570C22#577×C13711yescharacteristic
571C22#577×C13711yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—42no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4C2×#125×C13711yescharacteristic
586C4#544×C13711yescharacteristic
587C4#552×C13711yescharacteristic
588C4#553×C13711yescharacteristic
589C2—44no—
590C2#2442×C13711yescharacteristic
591C2#3077×C13711yescharacteristic
592C2#3084×C13711yescharacteristic
593C2#3307×C13711yescharacteristic
594C2#3310×C13711yescharacteristic
595C2#3373×C13711yescharacteristic
596C2#3376×C13711yescharacteristic
subgroupG/H
center, socleC23×C137C22×D4
commutator, Frattini, cosocleC22×C4C24×C137
prime radicalC22×D4×C137C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C137 ◁ C4×C4⋊C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #2570×C137C2semidirect
3. #2101×C137C2semidirect
4. #2099×C137C2semidirect
5. #2088×C137C2semidirect
6. #2088×C137C2semidirect
7. #1948×C137C2semidirect
8. #1940×C137C2semidirect
11. #1836×C137C2semidirect
298. #26118C137direct (N × H)
299. C137#26118direct (N × H)
300. #2570C2×C137semidirect
302. #2101C2×C137semidirect
303. #2099C2×C137semidirect
304. #2088C2×C137semidirect
305. #2088C2×C137semidirect
306. #1948C2×C137semidirect
307. #1940C2×C137semidirect
310. #1836C2×C137semidirect

Maximal subgroups

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

subgroupindex
#2570×C1372
D4×C4⋊C4×C1372
#2101×C1372
#2099×C1372
#2088×C1372
#1948×C1372
#1940×C1372
#1864×C1372
#1836×C1372
#1804×C1372
#1802×C1372
#1762×C1372
#26118137