Finite Group Databasebeta

← C137×#18592 #46867978

Subgroups — C137×#18592 #46867978

Order 35072.

860 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3509×C137C211yesmaximal, characteristic
2#3255×C137C211yesmaximal, characteristic, complemented
3#3165×C137C211yesmaximal, characteristic, complemented
4#3107×C137C211yesmaximal, characteristic
5#1653×C137C212yesmaximal, complemented
6#1652×C137C212yesmaximal, complemented
7#1650×C137C212yesmaximal, complemented
8#1648×C137C212yesmaximal, complemented
9#1549×C137C211yesmaximal, characteristic, complemented
10#1489×C137C211yesmaximal, characteristic
11#1485×C137C211yesmaximal, characteristic, complemented
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#501×C137—22no—
28#500×C137C2212yes—
29#498×C137C2212yes—
30#485×C137—21no—
31#475×C137—21no—
32#475×C137C2211yescharacteristic
33#475×C137C2212yes—
34#474×C137C2211yescharacteristic
35#470×C137—22no—
36#462×C137—21no—
37#462×C137C2211yescharacteristic
38#462×C137C2212yes—
39#461×C137C2211yescharacteristic
40#452×C137—21no—
41C4×Q16×C137—21no—
42C4×SD16×C137—21no—
43#431×C137C2211yescharacteristic
44#428×C137—22no—
45#423×C137C2211yescharacteristic
46C4×M16×C137C2211yescharacteristic
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—21no—
56C2×Q16×C137—22no—
57C2×SD16×C137—21no—
58C2×M16×C137—21no—
59C2×M16×C137—21no—
60C4⋊Q8×C137—22no—
61C4⋊Q8×C137—21no—
62C4⋊Q8×C137—21no—
63C4⋊Q8×C137D411yescharacteristic
64C4⋊Q8×C137D411yescharacteristic
65C4⋊Q8×C137D412yes—
66C4⋊Q8×C137C2311yescharacteristic
67C4⋊Q8×C137C2311yescharacteristic
68C4⋊Q8×C137C2312yes—
69#126×C137—24no—
70#126×C137—24no—
71#125×C137—22no—
72#125×C137—24no—
73#125×C137—22no—
74#125×C137—24no—
75#125×C137—21no—
76#125×C137—21no—
77#124×C137—22no—
78#123×C137—24no—
79C22⋊Q8×C137—22no—
80C22⋊Q8×C137—22no—
81C22⋊Q8×C137—22no—
82C22⋊Q8×C137—22no—
83C22⋊Q8×C137—21no—
84C22⋊Q8×C137—21no—
85C22⋊Q8×C137—21no—
86C22⋊Q8×C137D412yes—
87C4×Q8×C137—22no—
88C4×Q8×C137—22no—
89C4×Q8×C137—22no—
90C4×Q8×C137—22no—
91C4×Q8×C137—22no—
92C4×Q8×C137—22no—
93C4×Q8×C137—21no—
94C4×Q8×C137—21no—
95C4×Q8×C137—21no—
96C4×Q8×C137D411yescharacteristic
97C4×Q8×C137C2311yescharacteristic
98C4×Q8×C137C2312yes—
99C4×D4×C137—21no—
100C4×D4×C137—21no—
101C4×D4×C137D411yescharacteristic
102C4×D4×C137C2311yescharacteristic
103#117×C137—24no—
104#117×C137—22no—
105#117×C137—21no—
106#117×C137—22no—
107#117×C137—21no—
108#117×C137—21no—
109#117×C137D411yescharacteristic
110C2×C4⋊C4×C137—21no—
111C2×C4⋊C4×C137D412yes—
112C2×C42×C137—21no—
113C2×C42×C137C2311yescharacteristic
114#107×C137—24no—
115#107×C137—21no—
116#106×C137—24no—
117#106×C137—21no—
118#106×C137—22no—
119C4⋊C8×C137—21no—
120C4⋊C8×C137C2312yes—
121C4⋊C8×C137C2311yescharacteristic
122C4⋊C8×C137C2311yescharacteristic
123#103×C137—21no—
124#103×C137—22no—
125#103×C137—21no—
126#103×C137—21no—
127#103×C137—22no—
128#102×C137—21no—
129C22⋊C8×C137—21no—
130#97×C137—21no—
131C4×C8×C137—21no—
132C4×C8×C137C2312yes—
133#41×C137—41no—
134#41×C137—41no—
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×D412yes—
143C2×Q8×C137C2×D411yescharacteristic
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—42no—
154Q16×C137—82no—
155SD16×C137—42no—
156M16×C137—24no—
157M16×C137—81no—
158C2×C8×C137—22no—
159C2×C8×C137—21no—
160C2×C8×C137—21no—
161C2×C8×C137—22no—
162C4⋊C4×C137—24no—
163C4⋊C4×C137—24no—
164C4⋊C4×C137—24no—
165C4⋊C4×C137—24no—
166C4⋊C4×C137—22no—
167C4⋊C4×C137—22no—
168C4⋊C4×C137—21no—
169C4⋊C4×C137—21no—
170C4⋊C4×C137—21no—
171C4⋊C4×C137—21no—
172C4⋊C4×C137—22no—
173C4⋊C4×C137—22no—
174C4⋊C4×C137—21no—
175C4⋊C4×C137—22no—
176C4⋊C4×C137—21no—
177C4⋊C4×C137—21no—
178C4⋊C4×C137—21no—
179C4⋊C4×C137—21no—
180C4⋊C4×C137—41no—
181C4⋊C4×C137—41no—
182C4⋊C4×C137C2×D412yes—
183C4⋊C4×C137C2×D411yescharacteristic
184C4⋊C4×C137C2×D411yescharacteristic
185C4⋊C4×C137C2×D414yes—
186C22⋊C4×C137—24no—
187C22⋊C4×C137—24no—
188C22⋊C4×C137—21no—
189C22⋊C4×C137—22no—
190C22⋊C4×C137—21no—
191C22⋊C4×C137—21no—
192C22⋊C4×C137—41no—
193C42×C137—24no—
194C42×C137—24no—
195C42×C137—21no—
196C42×C137—21no—
197C42×C137—21no—
198C42×C137—22no—
199C42×C137—41no—
200C42×C137C2×D411yescharacteristic
201C42×C137C2×D411yescharacteristic
202C42×C137C2×D411yescharacteristic
203C42×C137C2411yescharacteristic
204C23×C137—21no—
205C23×C137#12011yescharacteristic
206Q8×C137—21no—
207Q8×C137—21no—
208Q8×C137—22no—
209Q8×C137—22no—
210Q8×C137—44no—
211Q8×C137—42no—
212Q8×C137—42no—
213Q8×C137—42no—
214Q8×C137—41no—
215Q8×C137—41no—
216Q8×C137—41no—
217Q8×C137—41no—
218D4×C137—21no—
219D4×C137—21no—
220D4×C137—42no—
221C2×C4×C137—21no—
222C2×C4×C137—21no—
223C2×C4×C137—21no—
224C2×C4×C137—21no—
225C2×C4×C137—22no—
226C2×C4×C137—21no—
227C2×C4×C137—21no—
228C2×C4×C137—22no—
229C2×C4×C137—22no—
230C2×C4×C137—24no—
231C2×C4×C137—21no—
232C2×C4×C137—42no—
233C2×C4×C137—44no—
234C2×C4×C137—41no—
235C2×C4×C137—41no—
236C2×C4×C137—42no—
237C2×C4×C137—41no—
238C2×C4×C137—41no—
239C2×C4×C137#12011yescharacteristic
240C2×C4×C137#12011yescharacteristic
241C2×C4×C137#12011yescharacteristic
242C2×C4×C137C22×D411yescharacteristic
243C2×C4×C137C22×D411yescharacteristic
244C2×C4×C137C22×D411yescharacteristic
245C8×C137—24no—
246C8×C137—41no—
247C8×C137—41no—
248C8×C137—42no—
249C22×C137—21no—
250C22×C137—21no—
251C22×C137—21no—
252C22×C137—41no—
253C22×C137—41no—
254C22×C137—41no—
255C22×C137C2×#12011yescharacteristic
256C4×C137—21no—
257C4×C137—21no—
258C4×C137—21no—
259C4×C137—41no—
260C4×C137—44no—
261C4×C137—41no—
262C4×C137—42no—
263C4×C137—41no—
264C4×C137—42no—
265C4×C137—81no—
266C4×C137—81no—
267C4×C137#45612yes—
268C4×C137#57511yescharacteristic
269C4×C137#57511yescharacteristic
270C4×C137#57711yescharacteristic
271C4×C137#57811yescharacteristic
272C2×C137—41no—
273C2×C137—81no—
274C2×C137#300511yescharacteristic
275C2×C137#301311yescharacteristic
276C2×C137C2×#45611yescharacteristic
277#18592C13711yesmaximal, characteristic, direct product factor, Sylow
278C137#1859211yescharacteristic, direct product factor, Sylow
279#3509C2×C13711yescharacteristic
280#3255C2×C13711yescharacteristic, complemented
281#3165C2×C13711yescharacteristic, complemented
282#3107C2×C13711yescharacteristic
283#1653C2×C13712yescomplemented
284#1652C2×C13712yescomplemented
285#1650C2×C13712yescomplemented
286#1648C2×C13712yescomplemented
287#1549C2×C13711yescharacteristic, complemented
288#1489C2×C13711yescharacteristic
289#1485C2×C13711yescharacteristic, complemented
290#557—21no—
291#557C22×C13712yes—
292#556—21no—
293#556C22×C13712yes—
294#552—22no—
295#552—22no—
296#541—22no—
297#536—21no—
298#533—21no—
299#533—21no—
300#533C22×C13711yescharacteristic
301#533C22×C13711yescharacteristic
302C2×C4⋊Q8C22×C13711yescharacteristic
303#519—21no—
304C4×#41C22×C13711yescharacteristic
305#501—22no—
306#500C22×C13712yes—
307#498C22×C13712yes—
308#485—21no—
309#475—21no—
310#475C22×C13711yescharacteristic
311#475C22×C13712yes—
312#474C22×C13711yescharacteristic
313#470—22no—
314#462—21no—
315#462C22×C13711yescharacteristic
316#462C22×C13712yes—
317#461C22×C13711yescharacteristic
318#452—21no—
319C4×Q16—21no—
320C4×SD16—21no—
321#431C22×C13711yescharacteristic
322#428—22no—
323#423C22×C13711yescharacteristic
324C4×M16C22×C13711yescharacteristic
325#332C22×C13712yes—
326#332C22×C13712yes—
327Q8⋊C8C22×C13712yes—
328Q8⋊C8C22×C13712yes—
329Q8⋊C8C22×C13712yes—
330D4⋊C8C22×C13712yes—
331C2×#41D4×C13711yescharacteristic
332C22×Q8—21no—
333C2×Q16—21no—
334C2×Q16—22no—
335C2×SD16—21no—
336C2×M16—21no—
337C2×M16—21no—
338C4⋊Q8—22no—
339C4⋊Q8—21no—
340C4⋊Q8—21no—
341C4⋊Q8D4×C13711yescharacteristic
342C4⋊Q8D4×C13712yes—
343C4⋊Q8D4×C13711yescharacteristic
344C4⋊Q8C23×C13711yescharacteristic
345C4⋊Q8C23×C13711yescharacteristic
346C4⋊Q8C23×C13712yes—
347#126—24no—
348#126—24no—
349#125—22no—
350#125—24no—
351#125—22no—
352#125—24no—
353#125—21no—
354#125—21no—
355#124—22no—
356#123—24no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8D4×C13712yes—
365C4×Q8—22no—
366C4×Q8—22no—
367C4×Q8—22no—
368C4×Q8—22no—
369C4×Q8—22no—
370C4×Q8—22no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8D4×C13711yescharacteristic
375C4×Q8C23×C13711yescharacteristic
376C4×Q8C23×C13712yes—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4D4×C13711yescharacteristic
380C4×D4C23×C13711yescharacteristic
381#117—24no—
382#117—22no—
383#117—21no—
384#117—22no—
385#117—21no—
386#117—21no—
387#117D4×C13711yescharacteristic
388C2×C4⋊C4—21no—
389C2×C4⋊C4D4×C13712yes—
390C2×C42—21no—
391C2×C42C23×C13711yescharacteristic
392#107—24no—
393#107—21no—
394#106—24no—
395#106—21no—
396#106—22no—
397C4⋊C8—21no—
398C4⋊C8C23×C13712yes—
399C4⋊C8C23×C13711yescharacteristic
400C4⋊C8C23×C13711yescharacteristic
401#103—21no—
402#103—22no—
403#103—21no—
404#103—21no—
405#103—22no—
406#102—21no—
407C22⋊C8—21no—
408#97—21no—
409C4×C8—21no—
410C4×C8C23×C13712yes—
411#41—41no—
412#41—41no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8C2×D4×C13712yes—
421C2×Q8C2×D4×C13711yescharacteristic
422C2×D4—21no—
423C2×D4C2×D4×C13711yescharacteristic
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4C2×D4×C13711yescharacteristic
429C22×C4C2×D4×C13711yescharacteristic
430C22×C4C2×D4×C13711yescharacteristic
431Q16—42no—
432Q16—82no—
433SD16—42no—
434M16—24no—
435M16—81no—
436C2×C8—22no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4C2×D4×C13712yes—
461C4⋊C4C2×D4×C13711yescharacteristic
462C4⋊C4C2×D4×C13711yescharacteristic
463C4⋊C4C2×D4×C13714yes—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—21no—
467C22⋊C4—22no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—41no—
471C42—24no—
472C42—24no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—22no—
477C42—41no—
478C42C2×D4×C13711yescharacteristic
479C42C2×D4×C13711yescharacteristic
480C42C2×D4×C13711yescharacteristic
481C42C24×C13711yescharacteristic
482C23—21no—
483C23#120×C13711yescharacteristic
484Q8—21no—
485Q8—21no—
486Q8—22no—
487Q8—22no—
488Q8—44no—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—41no—
493Q8—41no—
494Q8—41no—
495Q8—41no—
496D4—21no—
497D4—21no—
498D4—42no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—21no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4#120×C13711yescharacteristic
518C2×C4#120×C13711yescharacteristic
519C2×C4#120×C13711yescharacteristic
520C2×C4C22×D4×C13711yescharacteristic
521C2×C4C22×D4×C13711yescharacteristic
522C2×C4C22×D4×C13711yescharacteristic
523C8—24no—
524C8—41no—
525C8—41no—
526C8—42no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22C2×#120×C13711yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—41no—
538C4—44no—
539C4—41no—
540C4—42no—
541C4—41no—
542C4—42no—
543C4—81no—
544C4—81no—
545C4#456×C13712yes—
546C4#575×C13711yescharacteristic
547C4#575×C13711yescharacteristic
548C4#577×C13711yescharacteristic
549C4#578×C13711yescharacteristic
550C2—41no—
551C2—81no—
552C2#3005×C13711yescharacteristic
553C2#3013×C13711yescharacteristic
554C2C2×#456×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. #3255×C137C2semidirect
3. #3165×C137C2semidirect
5. #1653×C137C2semidirect
6. #1652×C137C2semidirect
7. #1650×C137C2semidirect
8. #1648×C137C2semidirect
9. #1549×C137C2semidirect
11. #1485×C137C2semidirect
277. #18592C137direct (N × H)
278. C137#18592direct (N × H)
280. #3255C2×C137semidirect
281. #3165C2×C137semidirect
283. #1653C2×C137semidirect
284. #1652C2×C137semidirect
285. #1650C2×C137semidirect
286. #1648C2×C137semidirect
287. #1549C2×C137semidirect
289. #1485C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3509×C1372
#3255×C1372
#3165×C1372
#3107×C1372
#1653×C1372
#1652×C1372
#1650×C1372
#1648×C1372
#1549×C1372
#1489×C1372
#1485×C1372
#18592137