Finite Group Databasebeta

← C137×#18887 #46868273

Subgroups — C137×#18887 #46868273

Order 35072.

854 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3509×C137C211yesmaximal, characteristic
2#3249×C137C211yesmaximal, characteristic, complemented
3#3229×C137C211yesmaximal, characteristic, complemented
4#3076×C137C211yesmaximal, characteristic
5#1653×C137C212yesmaximal, complemented
6#1652×C137C212yesmaximal, complemented
7#1638×C137C212yesmaximal, complemented
8#1636×C137C212yesmaximal, complemented
9C2×#332×C137C211yesmaximal, characteristic, complemented
10#1481×C137C211yesmaximal, characteristic, complemented
11#1477×C137C211yesmaximal, characteristic
12#557×C137—21no—
13#557×C137C2212yes—
14#556×C137—21no—
15#556×C137C2212yes—
16#552×C137—22no—
17#552×C137—22no—
18#541×C137—22no—
19#536×C137—21no—
20#533×C137—21no—
21#533×C137—21no—
22#533×C137C2211yescharacteristic
23#533×C137C2211yescharacteristic
24C2×C4⋊Q8×C137C2211yescharacteristic
25#519×C137—21no—
26C4×#41×C137C2211yescharacteristic
27#498×C137C2212yes—
28#487×C137C2212yes—
29#483×C137—22no—
30#479×C137—21no—
31#478×C137—21no—
32#477×C137—21no—
33#477×C137C2211yescharacteristic
34#477×C137C2212yes—
35#476×C137C2211yescharacteristic
36#475×C137—22no—
37#475×C137C2211yescharacteristic
38#474×C137C2211yescharacteristic
39#467×C137—22no—
40#462×C137—21no—
41#462×C137C2212yes—
42#431×C137C2211yescharacteristic
43#427×C137—21no—
44C2×C4⋊C8×C137C2211yescharacteristic
45C2×C4⋊C8×C137C2211yescharacteristic
46#416×C137—21no—
47#332×C137C2212yes—
48#332×C137C2212yes—
49Q8⋊C8×C137C2212yes—
50Q8⋊C8×C137C2212yes—
51Q8⋊C8×C137C2212yes—
52D4⋊C8×C137C2212yes—
53C2×#41×C137D411yescharacteristic
54C22×Q8×C137—21no—
55C2×Q16×C137—22no—
56C22×C8×C137—21no—
57C22×C8×C137—21no—
58C4⋊Q8×C137—22no—
59C4⋊Q8×C137—21no—
60C4⋊Q8×C137—21no—
61C4⋊Q8×C137D411yescharacteristic
62C4⋊Q8×C137D412yes—
63C4⋊Q8×C137D411yescharacteristic
64C4⋊Q8×C137C2311yescharacteristic
65C4⋊Q8×C137C2311yescharacteristic
66C4⋊Q8×C137C2312yes—
67#126×C137—24no—
68#126×C137—24no—
69#125×C137—22no—
70#125×C137—24no—
71#125×C137—22no—
72#125×C137—24no—
73#125×C137—21no—
74#125×C137—21no—
75#124×C137—22no—
76#123×C137—24no—
77C22⋊Q8×C137—22no—
78C22⋊Q8×C137—22no—
79C22⋊Q8×C137—22no—
80C22⋊Q8×C137—22no—
81C22⋊Q8×C137—21no—
82C22⋊Q8×C137—21no—
83C22⋊Q8×C137—21no—
84C22⋊Q8×C137D412yes—
85C4×Q8×C137—22no—
86C4×Q8×C137—22no—
87C4×Q8×C137—22no—
88C4×Q8×C137—22no—
89C4×Q8×C137—22no—
90C4×Q8×C137—22no—
91C4×Q8×C137—21no—
92C4×Q8×C137—21no—
93C4×Q8×C137—21no—
94C4×Q8×C137D411yescharacteristic
95C4×Q8×C137C2312yes—
96C4×Q8×C137C2311yescharacteristic
97C4×D4×C137—21no—
98C4×D4×C137—21no—
99C4×D4×C137D411yescharacteristic
100C4×D4×C137C2311yescharacteristic
101#117×C137—24no—
102#117×C137—22no—
103#117×C137—21no—
104#117×C137—22no—
105#117×C137—21no—
106#117×C137—21no—
107#117×C137D411yescharacteristic
108C2×C4⋊C4×C137—21no—
109C2×C4⋊C4×C137D412yes—
110C2×C42×C137—21no—
111C2×C42×C137C2311yescharacteristic
112#107×C137—21no—
113#107×C137—21no—
114#107×C137—22no—
115#106×C137—21no—
116#106×C137—21no—
117#106×C137—24no—
118C4⋊C8×C137—21no—
119C4⋊C8×C137—21no—
120C4⋊C8×C137—21no—
121C4⋊C8×C137C2312yes—
122C4⋊C8×C137C2311yescharacteristic
123C4⋊C8×C137C2311yescharacteristic
124#103×C137—21no—
125#103×C137—21no—
126#103×C137—22no—
127#103×C137—22no—
128#103×C137—24no—
129#102×C137—21no—
130#102×C137—21no—
131C22⋊C8×C137—21no—
132C4×C8×C137C2312yes—
133#41×C137—22no—
134#41×C137—22no—
135C2×Q8×C137—21no—
136C2×Q8×C137—22no—
137C2×Q8×C137—22no—
138C2×Q8×C137—21no—
139C2×Q8×C137—21no—
140C2×Q8×C137—42no—
141C2×Q8×C137—42no—
142C2×Q8×C137C2×D411yescharacteristic
143C2×Q8×C137C2×D412yes—
144C2×D4×C137—21no—
145C2×D4×C137C2×D411yescharacteristic
146C22×C4×C137—22no—
147C22×C4×C137—21no—
148C22×C4×C137—21no—
149C22×C4×C137—21no—
150C22×C4×C137C2×D411yescharacteristic
151C22×C4×C137C2×D411yescharacteristic
152C22×C4×C137C2×D411yescharacteristic
153Q16×C137—82no—
154C2×C8×C137—22no—
155C2×C8×C137—21no—
156C2×C8×C137—21no—
157C2×C8×C137—22no—
158C2×C8×C137—42no—
159C2×C8×C137—81no—
160C4⋊C4×C137—24no—
161C4⋊C4×C137—24no—
162C4⋊C4×C137—24no—
163C4⋊C4×C137—24no—
164C4⋊C4×C137—22no—
165C4⋊C4×C137—22no—
166C4⋊C4×C137—21no—
167C4⋊C4×C137—21no—
168C4⋊C4×C137—21no—
169C4⋊C4×C137—21no—
170C4⋊C4×C137—22no—
171C4⋊C4×C137—22no—
172C4⋊C4×C137—21no—
173C4⋊C4×C137—22no—
174C4⋊C4×C137—21no—
175C4⋊C4×C137—21no—
176C4⋊C4×C137—21no—
177C4⋊C4×C137—21no—
178C4⋊C4×C137—41no—
179C4⋊C4×C137—41no—
180C4⋊C4×C137C2×D414yes—
181C4⋊C4×C137C2×D412yes—
182C4⋊C4×C137C2×D411yescharacteristic
183C4⋊C4×C137C2×D411yescharacteristic
184C22⋊C4×C137—24no—
185C22⋊C4×C137—24no—
186C22⋊C4×C137—21no—
187C22⋊C4×C137—22no—
188C22⋊C4×C137—21no—
189C22⋊C4×C137—21no—
190C22⋊C4×C137—41no—
191C42×C137—24no—
192C42×C137—24no—
193C42×C137—21no—
194C42×C137—21no—
195C42×C137—21no—
196C42×C137—22no—
197C42×C137—41no—
198C42×C137C2×D411yescharacteristic
199C42×C137C2×D411yescharacteristic
200C42×C137C2×D411yescharacteristic
201C42×C137C2411yescharacteristic
202C23×C137—21no—
203C23×C137#12011yescharacteristic
204Q8×C137—21no—
205Q8×C137—21no—
206Q8×C137—22no—
207Q8×C137—22no—
208Q8×C137—42no—
209Q8×C137—42no—
210Q8×C137—42no—
211Q8×C137—42no—
212Q8×C137—44no—
213Q8×C137—41no—
214Q8×C137—41no—
215D4×C137—21no—
216D4×C137—21no—
217D4×C137—42no—
218C2×C4×C137—21no—
219C2×C4×C137—21no—
220C2×C4×C137—21no—
221C2×C4×C137—21no—
222C2×C4×C137—22no—
223C2×C4×C137—22no—
224C2×C4×C137—24no—
225C2×C4×C137—21no—
226C2×C4×C137—41no—
227C2×C4×C137—42no—
228C2×C4×C137—44no—
229C2×C4×C137—41no—
230C2×C4×C137—41no—
231C2×C4×C137—42no—
232C2×C4×C137—41no—
233C2×C4×C137—41no—
234C2×C4×C137#12011yescharacteristic
235C2×C4×C137#12011yescharacteristic
236C2×C4×C137#12011yescharacteristic
237C2×C4×C137#13512yes—
238C2×C4×C137#13712yes—
239C2×C4×C137C22×D411yescharacteristic
240C2×C4×C137C22×D411yescharacteristic
241C2×C4×C137C22×D411yescharacteristic
242C8×C137—42no—
243C8×C137—41no—
244C8×C137—41no—
245C8×C137—42no—
246C22×C137—21no—
247C22×C137—21no—
248C22×C137—41no—
249C22×C137—41no—
250C22×C137—41no—
251C22×C137#45212yes—
252C22×C137C2×#12011yescharacteristic
253C4×C137—21no—
254C4×C137—21no—
255C4×C137—21no—
256C4×C137—41no—
257C4×C137—44no—
258C4×C137—41no—
259C4×C137—42no—
260C4×C137—41no—
261C4×C137—42no—
262C4×C137—81no—
263C4×C137—81no—
264C4×C137#45212yes—
265C4×C137C2×#13511yescharacteristic
266C4×C137C2×#13711yescharacteristic
267C4×C137#57711yescharacteristic
268C4×C137#57811yescharacteristic
269C2×C137—22no—
270C2×C137—81no—
271C2×C137C2×#45211yescharacteristic
272C2×C137#301311yescharacteristic
273C2×C137#302011yescharacteristic
274#18887C13711yesmaximal, characteristic, direct product factor, Sylow
275C137#1888711yescharacteristic, direct product factor, Sylow
276#3509C2×C13711yescharacteristic
277#3249C2×C13711yescharacteristic, complemented
278#3229C2×C13711yescharacteristic, complemented
279#3076C2×C13711yescharacteristic
280#1653C2×C13712yescomplemented
281#1652C2×C13712yescomplemented
282#1638C2×C13712yescomplemented
283#1636C2×C13712yescomplemented
284C2×#332C2×C13711yescharacteristic, complemented
285#1481C2×C13711yescharacteristic, complemented
286#1477C2×C13711yescharacteristic
287#557—21no—
288#557C22×C13712yes—
289#556—21no—
290#556C22×C13712yes—
291#552—22no—
292#552—22no—
293#541—22no—
294#536—21no—
295#533—21no—
296#533—21no—
297#533C22×C13711yescharacteristic
298#533C22×C13711yescharacteristic
299C2×C4⋊Q8C22×C13711yescharacteristic
300#519—21no—
301C4×#41C22×C13711yescharacteristic
302#498C22×C13712yes—
303#487C22×C13712yes—
304#483—22no—
305#479—21no—
306#478—21no—
307#477—21no—
308#477C22×C13711yescharacteristic
309#477C22×C13712yes—
310#476C22×C13711yescharacteristic
311#475—22no—
312#475C22×C13711yescharacteristic
313#474C22×C13711yescharacteristic
314#467—22no—
315#462—21no—
316#462C22×C13712yes—
317#431C22×C13711yescharacteristic
318#427—21no—
319C2×C4⋊C8C22×C13711yescharacteristic
320C2×C4⋊C8C22×C13711yescharacteristic
321#416—21no—
322#332C22×C13712yes—
323#332C22×C13712yes—
324Q8⋊C8C22×C13712yes—
325Q8⋊C8C22×C13712yes—
326Q8⋊C8C22×C13712yes—
327D4⋊C8C22×C13712yes—
328C2×#41D4×C13711yescharacteristic
329C22×Q8—21no—
330C2×Q16—22no—
331C22×C8—21no—
332C22×C8—21no—
333C4⋊Q8—22no—
334C4⋊Q8—21no—
335C4⋊Q8—21no—
336C4⋊Q8D4×C13711yescharacteristic
337C4⋊Q8D4×C13712yes—
338C4⋊Q8D4×C13711yescharacteristic
339C4⋊Q8C23×C13711yescharacteristic
340C4⋊Q8C23×C13711yescharacteristic
341C4⋊Q8C23×C13712yes—
342#126—24no—
343#126—24no—
344#125—22no—
345#125—24no—
346#125—22no—
347#125—24no—
348#125—21no—
349#125—21no—
350#124—22no—
351#123—24no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355C22⋊Q8—22no—
356C22⋊Q8—21no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8D4×C13712yes—
360C4×Q8—22no—
361C4×Q8—22no—
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8—22no—
365C4×Q8—22no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8D4×C13711yescharacteristic
370C4×Q8C23×C13711yescharacteristic
371C4×Q8C23×C13712yes—
372C4×D4—21no—
373C4×D4—21no—
374C4×D4D4×C13711yescharacteristic
375C4×D4C23×C13711yescharacteristic
376#117—24no—
377#117—22no—
378#117—21no—
379#117—22no—
380#117—21no—
381#117—21no—
382#117D4×C13711yescharacteristic
383C2×C4⋊C4—21no—
384C2×C4⋊C4D4×C13712yes—
385C2×C42—21no—
386C2×C42C23×C13711yescharacteristic
387#107—21no—
388#107—21no—
389#107—22no—
390#106—21no—
391#106—21no—
392#106—24no—
393C4⋊C8—21no—
394C4⋊C8—21no—
395C4⋊C8—21no—
396C4⋊C8C23×C13712yes—
397C4⋊C8C23×C13711yescharacteristic
398C4⋊C8C23×C13711yescharacteristic
399#103—21no—
400#103—21no—
401#103—22no—
402#103—22no—
403#103—24no—
404#102—21no—
405#102—21no—
406C22⋊C8—21no—
407C4×C8C23×C13712yes—
408#41—22no—
409#41—22no—
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8C2×D4×C13711yescharacteristic
418C2×Q8C2×D4×C13712yes—
419C2×D4—21no—
420C2×D4C2×D4×C13711yescharacteristic
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4C2×D4×C13711yescharacteristic
426C22×C4C2×D4×C13711yescharacteristic
427C22×C4C2×D4×C13711yescharacteristic
428Q16—82no—
429C2×C8—22no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—22no—
433C2×C8—42no—
434C2×C8—81no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4C2×D4×C13714yes—
456C4⋊C4C2×D4×C13712yes—
457C4⋊C4C2×D4×C13711yescharacteristic
458C4⋊C4C2×D4×C13711yescharacteristic
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—21no—
462C22⋊C4—22no—
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—41no—
466C42—24no—
467C42—24no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—22no—
472C42—41no—
473C42C2×D4×C13711yescharacteristic
474C42C2×D4×C13711yescharacteristic
475C42C2×D4×C13711yescharacteristic
476C42C24×C13711yescharacteristic
477C23—21no—
478C23#120×C13711yescharacteristic
479Q8—21no—
480Q8—21no—
481Q8—22no—
482Q8—22no—
483Q8—42no—
484Q8—42no—
485Q8—44no—
486Q8—42no—
487Q8—42no—
488Q8—41no—
489Q8—41no—
490D4—21no—
491D4—21no—
492D4—42no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4#120×C13711yescharacteristic
510C2×C4#120×C13711yescharacteristic
511C2×C4#120×C13711yescharacteristic
512C2×C4#135×C13712yes—
513C2×C4#137×C13712yes—
514C2×C4C22×D4×C13711yescharacteristic
515C2×C4C22×D4×C13711yescharacteristic
516C2×C4C22×D4×C13711yescharacteristic
517C8—42no—
518C8—41no—
519C8—41no—
520C8—42no—
521C22—21no—
522C22—21no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22#452×C13712yes—
527C22C2×#120×C13711yescharacteristic
528C4—21no—
529C4—21no—
530C4—21no—
531C4—41no—
532C4—44no—
533C4—41no—
534C4—42no—
535C4—41no—
536C4—42no—
537C4—81no—
538C4—81no—
539C4#452×C13712yes—
540C4C2×#135×C13711yescharacteristic
541C4C2×#137×C13711yescharacteristic
542C4#577×C13711yescharacteristic
543C4#578×C13711yescharacteristic
544C2—22no—
545C2—81no—
546C2C2×#452×C13711yescharacteristic
547C2#3013×C13711yescharacteristic
548C2#3020×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radicalC2×#41×C137D4
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
2. #3249×C137C2semidirect
3. #3229×C137C2semidirect
5. #1653×C137C2semidirect
6. #1652×C137C2semidirect
7. #1638×C137C2semidirect
8. #1636×C137C2semidirect
9. C2×#332×C137C2semidirect
10. #1481×C137C2semidirect
274. #18887C137direct (N × H)
275. C137#18887direct (N × H)
277. #3249C2×C137semidirect
278. #3229C2×C137semidirect
280. #1653C2×C137semidirect
281. #1652C2×C137semidirect
282. #1638C2×C137semidirect
283. #1636C2×C137semidirect
284. C2×#332C2×C137semidirect
285. #1481C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3509×C1372
#3249×C1372
#3229×C1372
#3076×C1372
#1653×C1372
#1652×C1372
#1638×C1372
#1636×C1372
C2×#332×C1372
#1481×C1372
#1477×C1372
#18887137