Finite Group Databasebeta

← C137×#19478 #46868864

Subgroups — C137×#19478 #46868864

Order 35072.

558 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3231×C137C211yesmaximal, characteristic
2#3104×C137C211yesmaximal, characteristic
3#3078×C137C211yesmaximal, characteristic
4#2984×C137C211yesmaximal, characteristic
5#1704×C137C211yesmaximal, characteristic, complemented
6#1700×C137C211yesmaximal, characteristic, complemented
7#1696×C137C211yesmaximal, characteristic, complemented
8#1692×C137C211yesmaximal, characteristic, complemented
9#1684×C137C211yesmaximal, characteristic, complemented
10#1680×C137C211yesmaximal, characteristic, complemented
11#1676×C137C211yesmaximal, characteristic, complemented
12#1672×C137C211yesmaximal, characteristic, complemented
13#1574×C137C211yesmaximal, characteristic
14#1559×C137C211yesmaximal, characteristic
15#1554×C137C211yesmaximal, characteristic
16#533×C137C2211yescharacteristic
17#533×C137C2211yescharacteristic
18C2×C4×Q8×C137C2211yescharacteristic
19#500×C137C2211yescharacteristic
20#498×C137C2211yescharacteristic
21#487×C137C2211yescharacteristic
22#487×C137C2211yescharacteristic
23#485×C137—21no—
24#485×C137—21no—
25#484×C137—21no—
26#479×C137—21no—
27#479×C137—21no—
28#479×C137—21no—
29#479×C137—21no—
30#477×C137C2211yescharacteristic
31#477×C137C2211yescharacteristic
32#477×C137C2211yescharacteristic
33#477×C137C2211yescharacteristic
34#475×C137C2211yescharacteristic
35#475×C137C2211yescharacteristic
36#462×C137C2211yescharacteristic
37#462×C137C2211yescharacteristic
38#451×C137—21no—
39#446×C137—21no—
40#446×C137—21no—
41#446×C137C2211yescharacteristic
42#446×C137C2211yescharacteristic
43C8×Q8×C137C2211yescharacteristic
44C8×Q8×C137C2211yescharacteristic
45C4×Q16×C137—21no—
46C4×Q16×C137—21no—
47#432×C137—21no—
48#432×C137C2211yescharacteristic
49#431×C137—21no—
50#431×C137C2211yescharacteristic
51#431×C137C2211yescharacteristic
52#428×C137—21no—
53#423×C137C2211yescharacteristic
54#419×C137—21no—
55#335×C137C2211yescharacteristic
56#335×C137C2211yescharacteristic
57#334×C137C2211yescharacteristic
58#334×C137C2211yescharacteristic
59#333×C137C2211yescharacteristic
60#333×C137C2211yescharacteristic
61#333×C137C2211yescharacteristic
62#333×C137C2211yescharacteristic
63#332×C137C2211yescharacteristic
64#332×C137C2211yescharacteristic
65#332×C137C2211yescharacteristic
66#332×C137C2211yescharacteristic
67C22×Q8×C137D411yescharacteristic
68C2×Q16×C137—21no—
69C2×Q16×C137—21no—
70C2×M16×C137—21no—
71C4⋊Q8×C137C2311yescharacteristic
72C4⋊Q8×C137C2311yescharacteristic
73C4⋊Q8×C137C2311yescharacteristic
74C4⋊Q8×C137C2311yescharacteristic
75#125×C137—21no—
76#125×C137—21no—
77C22⋊Q8×C137—21no—
78C22⋊Q8×C137—21no—
79C22⋊Q8×C137—21no—
80C22⋊Q8×C137—21no—
81C4×Q8×C137—21no—
82C4×Q8×C137—21no—
83C4×Q8×C137—21no—
84C4×Q8×C137—21no—
85C4×Q8×C137—21no—
86C4×Q8×C137—21no—
87C4×Q8×C137D411yescharacteristic
88C4×Q8×C137D411yescharacteristic
89C4×Q8×C137C2311yescharacteristic
90C4×Q8×C137C2311yescharacteristic
91#117×C137—21no—
92#117×C137—21no—
93C2×C4⋊C4×C137—21no—
94C2×C4⋊C4×C137D411yescharacteristic
95C2×C42×C137—21no—
96C2×C42×C137C2311yescharacteristic
97#107×C137—21no—
98#107×C137—21no—
99#107×C137—21no—
100#107×C137—21no—
101#107×C137—21no—
102#107×C137—21no—
103#106×C137—21no—
104#106×C137—21no—
105#106×C137—21no—
106#106×C137—21no—
107C4⋊C8×C137—21no—
108C4⋊C8×C137—21no—
109C4⋊C8×C137—21no—
110C4⋊C8×C137—21no—
111C4⋊C8×C137—21no—
112C4⋊C8×C137D411yescharacteristic
113C4⋊C8×C137D411yescharacteristic
114C4⋊C8×C137C2311yescharacteristic
115C4⋊C8×C137C2311yescharacteristic
116C4⋊C8×C137C2311yescharacteristic
117C4⋊C8×C137C2311yescharacteristic
118C4⋊C8×C137C2311yescharacteristic
119C4⋊C8×C137C2311yescharacteristic
120#103×C137—21no—
121#103×C137—21no—
122#103×C137—21no—
123#103×C137—21no—
124#103×C137—21no—
125#103×C137—21no—
126#103×C137—21no—
127#103×C137—21no—
128#103×C137—21no—
129#103×C137—21no—
130#103×C137—21no—
131#103×C137—21no—
132C22⋊C8×C137—21no—
133C22⋊C8×C137—21no—
134C22⋊C8×C137D411yescharacteristic
135C22⋊C8×C137D411yescharacteristic
136#97×C137—21no—
137#97×C137—21no—
138#97×C137—21no—
139C4×C8×C137—21no—
140C4×C8×C137—21no—
141C4×C8×C137—21no—
142C4×C8×C137C2311yescharacteristic
143C4×C8×C137C2311yescharacteristic
144C2×Q8×C137—21no—
145C2×Q8×C137—22no—
146C2×Q8×C137—21no—
147C2×Q8×C137—21no—
148C2×Q8×C137—21no—
149C2×Q8×C137—21no—
150C2×Q8×C137—41no—
151C2×Q8×C137C2×D411yescharacteristic
152C2×Q8×C137C2×D411yescharacteristic
153C22×C4×C137—21no—
154C22×C4×C137—21no—
155C22×C4×C137C2×D411yescharacteristic
156C22×C4×C137C2×D411yescharacteristic
157C22×C4×C137#4111yescharacteristic
158Q16×C137—42no—
159Q16×C137—42no—
160M16×C137—42no—
161C2×C8×C137—21no—
162C2×C8×C137—21no—
163C2×C8×C137—21no—
164C2×C8×C137—21no—
165C2×C8×C137—21no—
166C2×C8×C137—21no—
167C2×C8×C137C2×D411yescharacteristic
168C2×C8×C137C2×D411yescharacteristic
169C2×C8×C137C2×D411yescharacteristic
170C2×C8×C137C2×D411yescharacteristic
171C4⋊C4×C137—21no—
172C4⋊C4×C137—21no—
173C4⋊C4×C137—21no—
174C4⋊C4×C137—21no—
175C4⋊C4×C137—21no—
176C4⋊C4×C137—21no—
177C4⋊C4×C137—21no—
178C4⋊C4×C137—21no—
179C4⋊C4×C137—21no—
180C4⋊C4×C137—21no—
181C4⋊C4×C137—21no—
182C4⋊C4×C137—21no—
183C4⋊C4×C137—21no—
184C4⋊C4×C137—41no—
185C4⋊C4×C137—41no—
186C4⋊C4×C137—41no—
187C4⋊C4×C137—41no—
188C4⋊C4×C137C2×D411yescharacteristic
189C4⋊C4×C137C2×D411yescharacteristic
190C22⋊C4×C137—41no—
191C22⋊C4×C137—41no—
192C42×C137—21no—
193C42×C137—21no—
194C42×C137—21no—
195C42×C137—21no—
196C42×C137—41no—
197C42×C137—41no—
198C42×C137C2×D411yescharacteristic
199C42×C137C2×D411yescharacteristic
200C42×C137#4111yescharacteristic
201C42×C137C2411yescharacteristic
202C23×C137#12111yescharacteristic
203Q8×C137—21no—
204Q8×C137—21no—
205Q8×C137—21no—
206Q8×C137—42no—
207Q8×C137—41no—
208Q8×C137—41no—
209Q8×C137—41no—
210Q8×C137—41no—
211Q8×C137—81no—
212Q8×C137—81no—
213Q8×C137#13712yes—
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—41no—
223C2×C4×C137—42no—
224C2×C4×C137—42no—
225C2×C4×C137—41no—
226C2×C4×C137—41no—
227C2×C4×C137—41no—
228C2×C4×C137—41no—
229C2×C4×C137#12111yescharacteristic
230C2×C4×C137#12111yescharacteristic
231C2×C4×C137#12111yescharacteristic
232C2×C4×C137C22×D411yescharacteristic
233C2×C4×C137C22×D411yescharacteristic
234C2×C4×C137C2×#4111yescharacteristic
235C8×C137—21no—
236C8×C137—21no—
237C8×C137—21no—
238C8×C137—21no—
239C8×C137—41no—
240C8×C137—41no—
241C8×C137—41no—
242C8×C137—41no—
243C8×C137—41no—
244C8×C137—41no—
245C22×C137—21no—
246C22×C137—21no—
247C22×C137—21no—
248C22×C137C2×#12111yescharacteristic
249C4×C137—21no—
250C4×C137—21no—
251C4×C137—22no—
252C4×C137—21no—
253C4×C137—21no—
254C4×C137—41no—
255C4×C137—41no—
256C4×C137—41no—
257C4×C137—81no—
258C4×C137—81no—
259C4×C137—81no—
260C4×C137—81no—
261C4×C137C2×#13711yescharacteristic
262C4×C137#57511yescharacteristic
263C4×C137#57811yescharacteristic
264C4×C137#57811yescharacteristic
265C2×C137—41no—
266C2×C137#304411yescharacteristic
267C2×C137#305611yescharacteristic
268C2×C137#306711yescharacteristic
269#19478C13711yesmaximal, characteristic, direct product factor, Sylow
270C137#1947811yescharacteristic, direct product factor, Sylow
271#3231C2×C13711yescharacteristic
272#3104C2×C13711yescharacteristic
273#3078C2×C13711yescharacteristic
274#2984C2×C13711yescharacteristic
275#1704C2×C13711yescharacteristic, complemented
276#1700C2×C13711yescharacteristic, complemented
277#1696C2×C13711yescharacteristic, complemented
278#1692C2×C13711yescharacteristic, complemented
279#1684C2×C13711yescharacteristic, complemented
280#1680C2×C13711yescharacteristic, complemented
281#1676C2×C13711yescharacteristic, complemented
282#1672C2×C13711yescharacteristic, complemented
283#1574C2×C13711yescharacteristic
284#1559C2×C13711yescharacteristic
285#1554C2×C13711yescharacteristic
286#533C22×C13711yescharacteristic
287#533C22×C13711yescharacteristic
288C2×C4×Q8C22×C13711yescharacteristic
289#500C22×C13711yescharacteristic
290#498C22×C13711yescharacteristic
291#487C22×C13711yescharacteristic
292#487C22×C13711yescharacteristic
293#485—21no—
294#485—21no—
295#484—21no—
296#479—21no—
297#479—21no—
298#479—21no—
299#479—21no—
300#477C22×C13711yescharacteristic
301#477C22×C13711yescharacteristic
302#477C22×C13711yescharacteristic
303#477C22×C13711yescharacteristic
304#475C22×C13711yescharacteristic
305#475C22×C13711yescharacteristic
306#462C22×C13711yescharacteristic
307#462C22×C13711yescharacteristic
308#451—21no—
309#446—21no—
310#446—21no—
311#446C22×C13711yescharacteristic
312#446C22×C13711yescharacteristic
313C8×Q8C22×C13711yescharacteristic
314C8×Q8C22×C13711yescharacteristic
315C4×Q16—21no—
316C4×Q16—21no—
317#432—21no—
318#432C22×C13711yescharacteristic
319#431—21no—
320#431C22×C13711yescharacteristic
321#431C22×C13711yescharacteristic
322#428—21no—
323#423C22×C13711yescharacteristic
324#419—21no—
325#335C22×C13711yescharacteristic
326#335C22×C13711yescharacteristic
327#334C22×C13711yescharacteristic
328#334C22×C13711yescharacteristic
329#333C22×C13711yescharacteristic
330#333C22×C13711yescharacteristic
331#333C22×C13711yescharacteristic
332#333C22×C13711yescharacteristic
333#332C22×C13711yescharacteristic
334#332C22×C13711yescharacteristic
335#332C22×C13711yescharacteristic
336#332C22×C13711yescharacteristic
337C22×Q8D4×C13711yescharacteristic
338C2×Q16—21no—
339C2×Q16—21no—
340C2×M16—21no—
341C4⋊Q8C23×C13711yescharacteristic
342C4⋊Q8C23×C13711yescharacteristic
343C4⋊Q8C23×C13711yescharacteristic
344C4⋊Q8C23×C13711yescharacteristic
345#125—21no—
346#125—21no—
347C22⋊Q8—21no—
348C22⋊Q8—21no—
349C22⋊Q8—21no—
350C22⋊Q8—21no—
351C4×Q8—21no—
352C4×Q8—21no—
353C4×Q8—21no—
354C4×Q8—21no—
355C4×Q8—21no—
356C4×Q8—21no—
357C4×Q8D4×C13711yescharacteristic
358C4×Q8D4×C13711yescharacteristic
359C4×Q8C23×C13711yescharacteristic
360C4×Q8C23×C13711yescharacteristic
361#117—21no—
362#117—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4D4×C13711yescharacteristic
365C2×C42—21no—
366C2×C42C23×C13711yescharacteristic
367#107—21no—
368#107—21no—
369#107—21no—
370#107—21no—
371#107—21no—
372#107—21no—
373#106—21no—
374#106—21no—
375#106—21no—
376#106—21no—
377C4⋊C8—21no—
378C4⋊C8—21no—
379C4⋊C8—21no—
380C4⋊C8—21no—
381C4⋊C8—21no—
382C4⋊C8D4×C13711yescharacteristic
383C4⋊C8D4×C13711yescharacteristic
384C4⋊C8C23×C13711yescharacteristic
385C4⋊C8C23×C13711yescharacteristic
386C4⋊C8C23×C13711yescharacteristic
387C4⋊C8C23×C13711yescharacteristic
388C4⋊C8C23×C13711yescharacteristic
389C4⋊C8C23×C13711yescharacteristic
390#103—21no—
391#103—21no—
392#103—21no—
393#103—21no—
394#103—21no—
395#103—21no—
396#103—21no—
397#103—21no—
398#103—21no—
399#103—21no—
400#103—21no—
401#103—21no—
402C22⋊C8—21no—
403C22⋊C8—21no—
404C22⋊C8D4×C13711yescharacteristic
405C22⋊C8D4×C13711yescharacteristic
406#97—21no—
407#97—21no—
408#97—21no—
409C4×C8—21no—
410C4×C8—21no—
411C4×C8—21no—
412C4×C8C23×C13711yescharacteristic
413C4×C8C23×C13711yescharacteristic
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—41no—
421C2×Q8C2×D4×C13711yescharacteristic
422C2×Q8C2×D4×C13711yescharacteristic
423C22×C4—21no—
424C22×C4—21no—
425C22×C4C2×D4×C13711yescharacteristic
426C22×C4C2×D4×C13711yescharacteristic
427C22×C4#41×C13711yescharacteristic
428Q16—42no—
429Q16—42no—
430M16—42no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8C2×D4×C13711yescharacteristic
438C2×C8C2×D4×C13711yescharacteristic
439C2×C8C2×D4×C13711yescharacteristic
440C2×C8C2×D4×C13711yescharacteristic
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4C2×D4×C13711yescharacteristic
459C4⋊C4C2×D4×C13711yescharacteristic
460C22⋊C4—41no—
461C22⋊C4—41no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—41no—
467C42—41no—
468C42C2×D4×C13711yescharacteristic
469C42C2×D4×C13711yescharacteristic
470C42#41×C13711yescharacteristic
471C42C24×C13711yescharacteristic
472C23#121×C13711yescharacteristic
473Q8—21no—
474Q8—21no—
475Q8—21no—
476Q8—42no—
477Q8—41no—
478Q8—41no—
479Q8—41no—
480Q8—41no—
481Q8—81no—
482Q8—81no—
483Q8#137×C13712yes—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4#121×C13711yescharacteristic
500C2×C4#121×C13711yescharacteristic
501C2×C4#121×C13711yescharacteristic
502C2×C4C22×D4×C13711yescharacteristic
503C2×C4C22×D4×C13711yescharacteristic
504C2×C4C2×#41×C13711yescharacteristic
505C8—21no—
506C8—21no—
507C8—21no—
508C8—21no—
509C8—41no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—41no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22C2×#121×C13711yescharacteristic
519C4—21no—
520C4—21no—
521C4—22no—
522C4—21no—
523C4—21no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—81no—
531C4C2×#137×C13711yescharacteristic
532C4#575×C13711yescharacteristic
533C4#578×C13711yescharacteristic
534C4#578×C13711yescharacteristic
535C2—41no—
536C2#3044×C13711yescharacteristic
537C2#3056×C13711yescharacteristic
538C2#3067×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#121
commutator, Frattini, cosocleC42C24×C137
prime radicalC23×C137#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
5. #1704×C137C2semidirect
6. #1700×C137C2semidirect
7. #1696×C137C2semidirect
8. #1692×C137C2semidirect
9. #1684×C137C2semidirect
10. #1680×C137C2semidirect
11. #1676×C137C2semidirect
12. #1672×C137C2semidirect
269. #19478C137direct (N × H)
270. C137#19478direct (N × H)
275. #1704C2×C137semidirect
276. #1700C2×C137semidirect
277. #1696C2×C137semidirect
278. #1692C2×C137semidirect
279. #1684C2×C137semidirect
280. #1680C2×C137semidirect
281. #1676C2×C137semidirect
282. #1672C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3231×C1372
#3104×C1372
#3078×C1372
#2984×C1372
#1704×C1372
#1700×C1372
#1696×C1372
#1692×C1372
#1684×C1372
#1680×C1372
#1676×C1372
#1672×C1372
#1574×C1372
#1559×C1372
#1554×C1372
#19478137