Finite Group Databasebeta

← C137×#18266 #46867652

Subgroups — C137×#18266 #46867652

Order 35072.

1000 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3536×C137C211yesmaximal, characteristic
2#3092×C137C211yesmaximal, characteristic, complemented
3C2×#483×C137C211yesmaximal, characteristic, complemented
4#2504×C137C211yesmaximal, characteristic, complemented
5#1612×C137C214yesmaximal, complemented
6#1608×C137C214yesmaximal, complemented
7#1508×C137C211yesmaximal, characteristic, complemented
8#1501×C137C211yesmaximal, characteristic, complemented
9#1471×C137C211yesmaximal, characteristic, complemented
10#562×C137—22no—
11#554×C137—21no—
12#548×C137—21no—
13#547×C137—21no—
14#546×C137—22no—
15#546×C137—21no—
16#543×C137C2212yes—
17#542×C137—21no—
18#539×C137—21no—
19#538×C137C2212yes—
20#537×C137—21no—
21#537×C137—21no—
22#525×C137C2211yescharacteristic
23C2×C22⋊Q8×C137C2211yescharacteristic
24C2×#121×C137C2211yescharacteristic
25C22×C4⋊C4×C137C2211yescharacteristic
26#484×C137—21no—
27#483×C137—22no—
28#483×C137C2214yes—
29#483×C137C2212yes—
30#481×C137—21no—
31#480×C137C2212yes—
32#428×C137—21no—
33C2×#106×C137—21no—
34#417×C137—21no—
35C2×#103×C137—21no—
36#407×C137C2211yescharacteristic
37C2×C22⋊C8×C137C2211yescharacteristic
38#397×C137C2214yes—
39#390×C137—21no—
40#386×C137—22no—
41#382×C137—21no—
42C4×C22⋊C4×C137C2211yescharacteristic
43#328×C137C2212yes—
44#328×C137C2214yes—
45#327×C137C2212yes—
46#323×C137C2214yes—
47C2×#41×C137—22no—
48C2×#41×C137—21no—
49C2×#41×C137D411yescharacteristic
50C22×Q8×C137D411yescharacteristic
51C22×D4×C137D412yes—
52C22×D4×C137D411yescharacteristic
53C23×C4×C137—21no—
54C23×C4×C137C2311yescharacteristic
55C2×M16×C137—21no—
56C22×C8×C137—21no—
57C4⋊Q8×C137—21no—
58#126×C137—22no—
59#125×C137—22no—
60#124×C137—21no—
61#124×C137—22no—
62#124×C137—22no—
63#123×C137—22no—
64#123×C137—22no—
65#123×C137—22no—
66#123×C137—21no—
67#123×C137—22no—
68C22⋊Q8×C137—21no—
69C22⋊Q8×C137—22no—
70C22⋊Q8×C137—22no—
71C22⋊Q8×C137—22no—
72C22⋊Q8×C137—21no—
73C22⋊Q8×C137—21no—
74C22⋊Q8×C137—22no—
75C22⋊Q8×C137—22no—
76C22⋊Q8×C137—22no—
77C22⋊Q8×C137D412yes—
78C22⋊Q8×C137D411yescharacteristic
79C22⋊Q8×C137C2312yes—
80C22⋊Q8×C137C2311yescharacteristic
81#121×C137—21no—
82#121×C137—22no—
83#121×C137—22no—
84#121×C137—22no—
85#121×C137—22no—
86#121×C137—22no—
87#121×C137—21no—
88#121×C137—21no—
89#121×C137—21no—
90#121×C137—21no—
91#121×C137D411yescharacteristic
92#121×C137C2311yescharacteristic
93#120×C137—21no—
94#120×C137—22no—
95C4×Q8×C137—21no—
96C4×D4×C137—22no—
97C4×D4×C137—21no—
98C4×D4×C137—21no—
99C4×D4×C137—21no—
100C4×D4×C137—22no—
101#117×C137D411yescharacteristic
102C2×C4⋊C4×C137—21no—
103C2×C4⋊C4×C137—21no—
104C2×C4⋊C4×C137—21no—
105C2×C4⋊C4×C137—21no—
106C2×C4⋊C4×C137—21no—
107C2×C4⋊C4×C137—21no—
108C2×C4⋊C4×C137—21no—
109C2×C4⋊C4×C137D411yescharacteristic
110C2×C4⋊C4×C137D411yescharacteristic
111C2×C4⋊C4×C137C2312yes—
112C2×C22⋊C4×C137—21no—
113C2×C22⋊C4×C137—22no—
114C2×C22⋊C4×C137—21no—
115C2×C22⋊C4×C137—24no—
116C2×C22⋊C4×C137C2312yes—
117C2×C42×C137—21no—
118#107×C137—22no—
119#106×C137—24no—
120#106×C137—22no—
121#103×C137—24no—
122#103×C137—22no—
123#102×C137—22no—
124C22⋊C8×C137—21no—
125C22⋊C8×C137—21no—
126C22⋊C8×C137C2312yes—
127C22⋊C8×C137C2312yes—
128#95×C137—24no—
129#95×C137C2312yes—
130C24×C137C2×D411yescharacteristic
131C24×C137C2×D412yes—
132#41×C137—42no—
133#41×C137—84no—
134#41×C137—82no—
135C2×Q8×C137—22no—
136C2×Q8×C137—21no—
137C2×Q8×C137—42no—
138C2×Q8×C137C2×D411yescharacteristic
139C2×Q8×C137C2×D412yes—
140C2×D4×C137—21no—
141C2×D4×C137—22no—
142C2×D4×C137—21no—
143C2×D4×C137—22no—
144C2×D4×C137—22no—
145C2×D4×C137—21no—
146C2×D4×C137—22no—
147C2×D4×C137—21no—
148C2×D4×C137—21no—
149C2×D4×C137—42no—
150C2×D4×C137—44no—
151C2×D4×C137C2×D411yescharacteristic
152C22×C4×C137—21no—
153C22×C4×C137—22no—
154C22×C4×C137—22no—
155C22×C4×C137—22no—
156C22×C4×C137—21no—
157C22×C4×C137—21no—
158C22×C4×C137—22no—
159C22×C4×C137—22no—
160C22×C4×C137—42no—
161C22×C4×C137—41no—
162C22×C4×C137—41no—
163C22×C4×C137—82no—
164C22×C4×C137C2×D411yescharacteristic
165C22×C4×C137C2×D411yescharacteristic
166C22×C4×C137C2×D411yescharacteristic
167C22×C4×C137C2×D412yes—
168C22×C4×C137C2×D411yescharacteristic
169C22×C4×C137C2×D411yescharacteristic
170C22×C4×C137C2411yescharacteristic
171M16×C137—42no—
172C2×C8×C137—22no—
173C2×C8×C137—22no—
174C2×C8×C137—42no—
175C4⋊C4×C137—22no—
176C4⋊C4×C137—22no—
177C4⋊C4×C137—22no—
178C4⋊C4×C137—22no—
179C4⋊C4×C137—21no—
180C4⋊C4×C137—21no—
181C4⋊C4×C137—22no—
182C4⋊C4×C137—21no—
183C4⋊C4×C137—21no—
184C4⋊C4×C137—22no—
185C4⋊C4×C137—21no—
186C4⋊C4×C137—22no—
187C4⋊C4×C137—22no—
188C4⋊C4×C137—41no—
189C4⋊C4×C137—41no—
190C4⋊C4×C137C2×D412yes—
191C4⋊C4×C137C2×D411yescharacteristic
192C4⋊C4×C137C2×D411yescharacteristic
193C22⋊C4×C137—22no—
194C22⋊C4×C137—22no—
195C22⋊C4×C137—22no—
196C22⋊C4×C137—22no—
197C22⋊C4×C137—22no—
198C22⋊C4×C137—22no—
199C22⋊C4×C137—21no—
200C22⋊C4×C137—21no—
201C22⋊C4×C137—22no—
202C22⋊C4×C137—21no—
203C22⋊C4×C137—21no—
204C22⋊C4×C137—22no—
205C22⋊C4×C137—22no—
206C22⋊C4×C137—21no—
207C22⋊C4×C137—42no—
208C22⋊C4×C137—84no—
209C42×C137—22no—
210C42×C137—21no—
211C42×C137—42no—
212C23×C137—22no—
213C23×C137—22no—
214C23×C137—22no—
215C23×C137—21no—
216C23×C137—24no—
217C23×C137—41no—
218C23×C137—42no—
219C23×C137—42no—
220C23×C137#12011yescharacteristic
221C23×C137#12011yescharacteristic
222C23×C137C22×D411yescharacteristic
223Q8×C137—42no—
224Q8×C137—41no—
225Q8×C137—42no—
226Q8×C137—82no—
227D4×C137—41no—
228D4×C137—42no—
229D4×C137—44no—
230D4×C137—42no—
231D4×C137—42no—
232D4×C137—81no—
233D4×C137—82no—
234D4×C137—81no—
235D4×C137—82no—
236D4×C137—82no—
237C2×C4×C137—22no—
238C2×C4×C137—22no—
239C2×C4×C137—22no—
240C2×C4×C137—21no—
241C2×C4×C137—22no—
242C2×C4×C137—21no—
243C2×C4×C137—22no—
244C2×C4×C137—21no—
245C2×C4×C137—22no—
246C2×C4×C137—41no—
247C2×C4×C137—41no—
248C2×C4×C137—41no—
249C2×C4×C137—42no—
250C2×C4×C137—42no—
251C2×C4×C137—42no—
252C2×C4×C137—42no—
253C2×C4×C137—42no—
254C2×C4×C137—42no—
255C2×C4×C137—42no—
256C2×C4×C137—44no—
257C2×C4×C137—81no—
258C2×C4×C137—82no—
259C2×C4×C137—82no—
260C2×C4×C137—82no—
261C2×C4×C137—82no—
262C2×C4×C137—82no—
263C2×C4×C137#12011yescharacteristic
264C2×C4×C137#12011yescharacteristic
265C2×C4×C137C22×D411yescharacteristic
266C2×C4×C137C22×D411yescharacteristic
267C8×C137—42no—
268C8×C137—42no—
269C22×C137—21no—
270C22×C137—21no—
271C22×C137—21no—
272C22×C137—21no—
273C22×C137—21no—
274C22×C137—21no—
275C22×C137—21no—
276C22×C137—44no—
277C22×C137—41no—
278C22×C137—42no—
279C22×C137—41no—
280C22×C137—42no—
281C22×C137—41no—
282C22×C137—42no—
283C22×C137—84no—
284C22×C137#45512yes—
285C22×C137C2×#12011yescharacteristic
286C22×C137#57712yes—
287C4×C137—21no—
288C4×C137—21no—
289C4×C137—21no—
290C4×C137—41no—
291C4×C137—42no—
292C4×C137—42no—
293C4×C137—41no—
294C4×C137—41no—
295C4×C137—42no—
296C4×C137—82no—
297C4×C137—82no—
298C4×C137#45812yes—
299C2×C137—22no—
300C2×C137—22no—
301C2×C137—41no—
302C2×C137—81no—
303C2×C137—82no—
304C2×C137#301011yescharacteristic
305C2×C137C2×#45511yescharacteristic
306C2×C137C2×#45811yescharacteristic
307#18266C13711yesmaximal, characteristic, direct product factor, Sylow
308C137#1826611yescharacteristic, direct product factor, Sylow
309#3536C2×C13711yescharacteristic
310#3092C2×C13711yescharacteristic, complemented
311C2×#483C2×C13711yescharacteristic, complemented
312#2504C2×C13711yescharacteristic, complemented
313#1612C2×C13714yescomplemented
314#1608C2×C13714yescomplemented
315#1508C2×C13711yescharacteristic, complemented
316#1501C2×C13711yescharacteristic, complemented
317#1471C2×C13711yescharacteristic, complemented
318#562—22no—
319#554—21no—
320#548—21no—
321#547—21no—
322#546—22no—
323#546—21no—
324#543C22×C13712yes—
325#542—21no—
326#539—21no—
327#538C22×C13712yes—
328#537—21no—
329#537—21no—
330#525C22×C13711yescharacteristic
331C2×C22⋊Q8C22×C13711yescharacteristic
332C2×#121C22×C13711yescharacteristic
333C22×C4⋊C4C22×C13711yescharacteristic
334#484—21no—
335#483—22no—
336#483C22×C13714yes—
337#483C22×C13712yes—
338#481—21no—
339#480C22×C13712yes—
340#428—21no—
341C2×#106—21no—
342#417—21no—
343C2×#103—21no—
344#407C22×C13711yescharacteristic
345C2×C22⋊C8C22×C13711yescharacteristic
346#397C22×C13714yes—
347#390—21no—
348#386—22no—
349#382—21no—
350C4×C22⋊C4C22×C13711yescharacteristic
351#328C22×C13712yes—
352#328C22×C13714yes—
353#327C22×C13712yes—
354#323C22×C13714yes—
355C2×#41—22no—
356C2×#41—21no—
357C2×#41D4×C13711yescharacteristic
358C22×Q8D4×C13711yescharacteristic
359C22×D4D4×C13712yes—
360C22×D4D4×C13711yescharacteristic
361C23×C4—21no—
362C23×C4C23×C13711yescharacteristic
363C2×M16—21no—
364C22×C8—21no—
365C4⋊Q8—21no—
366#126—22no—
367#125—22no—
368#124—21no—
369#124—22no—
370#124—22no—
371#123—22no—
372#123—22no—
373#123—22no—
374#123—21no—
375#123—22no—
376C22⋊Q8—21no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8—21no—
381C22⋊Q8—21no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8D4×C13712yes—
386C22⋊Q8D4×C13711yescharacteristic
387C22⋊Q8C23×C13712yes—
388C22⋊Q8C23×C13711yescharacteristic
389#121—21no—
390#121—22no—
391#121—22no—
392#121—22no—
393#121—22no—
394#121—22no—
395#121—21no—
396#121—21no—
397#121—21no—
398#121—21no—
399#121D4×C13711yescharacteristic
400#121C23×C13711yescharacteristic
401#120—21no—
402#120—22no—
403C4×Q8—21no—
404C4×D4—22no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—22no—
409#117D4×C13711yescharacteristic
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4D4×C13711yescharacteristic
418C2×C4⋊C4D4×C13711yescharacteristic
419C2×C4⋊C4C23×C13712yes—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—22no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—24no—
424C2×C22⋊C4C23×C13712yes—
425C2×C42—21no—
426#107—22no—
427#106—24no—
428#106—22no—
429#103—24no—
430#103—22no—
431#102—22no—
432C22⋊C8—21no—
433C22⋊C8—21no—
434C22⋊C8C23×C13712yes—
435C22⋊C8C23×C13712yes—
436#95—24no—
437#95C23×C13712yes—
438C24C2×D4×C13711yescharacteristic
439C24C2×D4×C13712yes—
440#41—42no—
441#41—84no—
442#41—82no—
443C2×Q8—22no—
444C2×Q8—21no—
445C2×Q8—42no—
446C2×Q8C2×D4×C13711yescharacteristic
447C2×Q8C2×D4×C13712yes—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—42no—
458C2×D4—44no—
459C2×D4C2×D4×C13711yescharacteristic
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—82no—
472C22×C4C2×D4×C13711yescharacteristic
473C22×C4C2×D4×C13711yescharacteristic
474C22×C4C2×D4×C13711yescharacteristic
475C22×C4C2×D4×C13712yes—
476C22×C4C2×D4×C13711yescharacteristic
477C22×C4C2×D4×C13711yescharacteristic
478C22×C4C24×C13711yescharacteristic
479M16—42no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—42no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4C2×D4×C13712yes—
499C4⋊C4C2×D4×C13711yescharacteristic
500C4⋊C4C2×D4×C13711yescharacteristic
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—22no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—21no—
515C22⋊C4—42no—
516C22⋊C4—84no—
517C42—22no—
518C42—21no—
519C42—42no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—21no—
524C23—24no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23#120×C13711yescharacteristic
529C23#120×C13711yescharacteristic
530C23C22×D4×C13711yescharacteristic
531Q8—42no—
532Q8—41no—
533Q8—42no—
534Q8—82no—
535D4—41no—
536D4—44no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—81no—
541D4—82no—
542D4—81no—
543D4—82no—
544D4—82no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—81no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4#120×C13711yescharacteristic
572C2×C4#120×C13711yescharacteristic
573C2×C4C22×D4×C13711yescharacteristic
574C2×C4C22×D4×C13711yescharacteristic
575C8—42no—
576C8—42no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—44no—
585C22—41no—
586C22—42no—
587C22—41no—
588C22—42no—
589C22—41no—
590C22—42no—
591C22—84no—
592C22#455×C13712yes—
593C22C2×#120×C13711yescharacteristic
594C22#577×C13712yes—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—41no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—42no—
604C4—82no—
605C4—82no—
606C4#458×C13712yes—
607C2—22no—
608C2—22no—
609C2—41no—
610C2—82no—
611C2—81no—
612C2#3010×C13711yescharacteristic
613C2C2×#455×C13711yescharacteristic
614C2C2×#458×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC22×C4C24×C137
prime radical#3536×C137C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3092×C137C2semidirect
3. C2×#483×C137C2semidirect
4. #2504×C137C2semidirect
5. #1612×C137C2semidirect
6. #1608×C137C2semidirect
7. #1508×C137C2semidirect
8. #1501×C137C2semidirect
9. #1471×C137C2semidirect
307. #18266C137direct (N × H)
308. C137#18266direct (N × H)
310. #3092C2×C137semidirect
311. C2×#483C2×C137semidirect
312. #2504C2×C137semidirect
313. #1612C2×C137semidirect
314. #1608C2×C137semidirect
315. #1508C2×C137semidirect
316. #1501C2×C137semidirect
317. #1471C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3536×C1372
#3092×C1372
C2×#483×C1372
#2504×C1372
#1612×C1372
#1608×C1372
#1508×C1372
#1501×C1372
#1471×C1372
#18266137