Finite Group Databasebeta

← C137×#18222 #46867608

Subgroups — C137×#18222 #46867608

Order 35072.

1078 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3506×C137C211yesmaximal, characteristic, complemented
2#3101×C137C211yesmaximal, characteristic, complemented
3#3100×C137C211yesmaximal, characteristic, complemented
4#2535×C137C211yesmaximal, characteristic, complemented
5#1612×C137C214yesmaximal, complemented
6#1600×C137C214yesmaximal, complemented
7#1518×C137C211yesmaximal, characteristic, complemented
8#1517×C137C211yesmaximal, characteristic, complemented
9#1460×C137C211yesmaximal, characteristic, complemented
10#559×C137—22no—
11#556×C137—21no—
12#555×C137—21no—
13#553×C137—22no—
14#550×C137—21no—
15#547×C137—21no—
16#538×C137C2214yes—
17#535×C137—21no—
18#533×C137C2211yescharacteristic
19#532×C137C2211yescharacteristic
20#525×C137—22no—
21C2×#121×C137—21no—
22#520×C137—21no—
23C2×C4×D4×C137C2211yescharacteristic
24C2×C4×D4×C137C2211yescharacteristic
25#483×C137C2212yes—
26#480×C137C2212yes—
27#476×C137—21no—
28#474×C137—21no—
29#461×C137—21no—
30#459×C137—21no—
31#450×C137C2212yes—
32#447×C137C2212yes—
33C4×SD16×C137—22no—
34C4×D8×C137—22no—
35#431×C137C2211yescharacteristic
36#431×C137C2211yescharacteristic
37#397×C137C2212yes—
38#392×C137C2212yes—
39#386×C137—22no—
40#382×C137C2211yescharacteristic
41#380×C137—21no—
42#328×C137C2214yes—
43#327×C137C2214yes—
44#323×C137C2214yes—
45C2×#41×C137—22no—
46C2×#41×C137—21no—
47C22×D4×C137—22no—
48C22×D4×C137D411yescharacteristic
49C22×D4×C137C2311yescharacteristic
50C23×C4×C137—21no—
51C23×C4×C137D412yes—
52C2×SD16×C137—22no—
53C2×D8×C137—22no—
54C4⋊Q8×C137D411yescharacteristic, complemented
55C4⋊Q8×C137D411yescharacteristic, complemented
56C4⋊Q8×C137D411yescharacteristic, complemented
57#127×C137—22no—
58#127×C137D411yescharacteristic, complemented
59#126×C137—24no—
60#125×C137—21no—
61#124×C137—22no—
62#124×C137—24no—
63#124×C137—21no—
64#123×C137—24no—
65#123×C137—24no—
66C22⋊Q8×C137—22no—
67C22⋊Q8×C137—21no—
68C22⋊Q8×C137C2312yes—
69#121×C137—22no—
70#121×C137—22no—
71#121×C137—22no—
72#121×C137—24no—
73#121×C137—21no—
74#121×C137—44no—
75#121×C137C2312yes—
76#120×C137—24no—
77C4×Q8×C137—22no—
78C4×Q8×C137—21no—
79C4×Q8×C137D412yescomplemented
80C4×D4×C137—22no—
81C4×D4×C137—24no—
82C4×D4×C137—24no—
83C4×D4×C137—21no—
84C4×D4×C137—21no—
85C4×D4×C137—21no—
86C4×D4×C137—21no—
87C4×D4×C137—21no—
88C4×D4×C137—21no—
89C4×D4×C137—21no—
90C4×D4×C137—21no—
91C4×D4×C137—21no—
92C4×D4×C137D412yescomplemented
93#117×C137—22no—
94#117×C137—22no—
95#117×C137—21no—
96C2×C4⋊C4×C137—22no—
97C2×C4⋊C4×C137—21no—
98C2×C4⋊C4×C137D411yescharacteristic
99C2×C4⋊C4×C137C2311yescharacteristic
100C2×C22⋊C4×C137—22no—
101C2×C22⋊C4×C137—21no—
102C2×C22⋊C4×C137—22no—
103C2×C22⋊C4×C137C2312yes—
104C2×C42×C137C2311yescharacteristic
105#107×C137—22no—
106#106×C137—22no—
107C4⋊C8×C137—21no—
108C4⋊C8×C137—21no—
109#103×C137—22no—
110#102×C137—22no—
111#102×C137—22no—
112#102×C137—22no—
113C22⋊C8×C137C2312yes—
114C22⋊C8×C137C2312yes—
115C4×C8×C137—21no—
116C4×C8×C137—21no—
117#95×C137—22no—
118#95×C137—21no—
119#95×C137C2312yes—
120C24×C137—21no—
121C24×C137C2×D412yes—
122#41×C137—42no—
123#41×C137—42no—
124#41×C137—84no—
125C2×Q8×C137—21no—
126C2×Q8×C137—21no—
127C2×Q8×C137C2×D412yes—
128C2×D4×C137—22no—
129C2×D4×C137—24no—
130C2×D4×C137—24no—
131C2×D4×C137—21no—
132C2×D4×C137—21no—
133C2×D4×C137—21no—
134C2×D4×C137—21no—
135C2×D4×C137—22no—
136C2×D4×C137—22no—
137C2×D4×C137—21no—
138C2×D4×C137—21no—
139C2×D4×C137—42no—
140C2×D4×C137—44no—
141C2×D4×C137—44no—
142C2×D4×C137—82no—
143C2×D4×C137C2×D412yes—
144C22×C4×C137—24no—
145C22×C4×C137—22no—
146C22×C4×C137—22no—
147C22×C4×C137—22no—
148C22×C4×C137—22no—
149C22×C4×C137—21no—
150C22×C4×C137—41no—
151C22×C4×C137—41no—
152C22×C4×C137—42no—
153C22×C4×C137—42no—
154C22×C4×C137—82no—
155C22×C4×C137C2×D412yes—
156C22×C4×C137C2×D411yescharacteristic
157C22×C4×C137C2×D411yescharacteristic
158C22×C4×C137C2411yescharacteristic
159SD16×C137—44no—
160D8×C137—44no—
161C2×C8×C137—22no—
162C2×C8×C137—22no—
163C4⋊C4×C137—24no—
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—22no—
171C4⋊C4×C137—22no—
172C4⋊C4×C137—21no—
173C4⋊C4×C137—21no—
174C4⋊C4×C137—42no—
175C4⋊C4×C137C2×D412yes—
176C4⋊C4×C137C2×D412yes—
177C22⋊C4×C137—24no—
178C22⋊C4×C137—24no—
179C22⋊C4×C137—24no—
180C22⋊C4×C137—22no—
181C22⋊C4×C137—22no—
182C22⋊C4×C137—22no—
183C22⋊C4×C137—22no—
184C22⋊C4×C137—22no—
185C22⋊C4×C137—44no—
186C22⋊C4×C137—41no—
187C22⋊C4×C137—41no—
188C22⋊C4×C137—82no—
189C42×C137—24no—
190C42×C137—22no—
191C42×C137C2×D411yescharacteristic
192C42×C137C2×D411yescharacteristic
193C42×C137C2×D411yescharacteristic
194C42×C137C2×D411yescharacteristic
195C23×C137—22no—
196C23×C137—22no—
197C23×C137—22no—
198C23×C137—22no—
199C23×C137—22no—
200C23×C137—42no—
201C23×C137—42no—
202C23×C137—41no—
203C23×C137—41no—
204C23×C137—41no—
205C23×C137—41no—
206C23×C137—41no—
207C23×C137—41no—
208C23×C137C22×D411yescharacteristic
209Q8×C137—42no—
210Q8×C137—42no—
211Q8×C137—42no—
212D4×C137—22no—
213D4×C137—22no—
214D4×C137—42no—
215D4×C137—41no—
216D4×C137—41no—
217D4×C137—42no—
218D4×C137—42no—
219D4×C137—42no—
220D4×C137—42no—
221D4×C137—42no—
222D4×C137—42no—
223D4×C137—84no—
224D4×C137—84no—
225D4×C137—88no—
226C2×C4×C137—22no—
227C2×C4×C137—22no—
228C2×C4×C137—22no—
229C2×C4×C137—22no—
230C2×C4×C137—41no—
231C2×C4×C137—41no—
232C2×C4×C137—44no—
233C2×C4×C137—44no—
234C2×C4×C137—42no—
235C2×C4×C137—42no—
236C2×C4×C137—42no—
237C2×C4×C137—41no—
238C2×C4×C137—41no—
239C2×C4×C137—41no—
240C2×C4×C137—42no—
241C2×C4×C137—82no—
242C2×C4×C137—84no—
243C2×C4×C137—82no—
244C2×C4×C137—82no—
245C2×C4×C137#12011yescharacteristic
246C2×C4×C137#12011yescharacteristic
247C2×C4×C137#12011yescharacteristic
248C2×C4×C137#12011yescharacteristic
249C2×C4×C137#13512yes—
250C2×C4×C137AGL(1,ℤ/8)12yes—
251C2×C4×C137C22×D411yescharacteristic
252C2×C4×C137C22×D411yescharacteristic
253C8×C137—42no—
254C8×C137—42no—
255C22×C137—21no—
256C22×C137—21no—
257C22×C137—42no—
258C22×C137—44no—
259C22×C137—42no—
260C22×C137—42no—
261C22×C137—42no—
262C22×C137—42no—
263C22×C137—42no—
264C22×C137—42no—
265C22×C137—42no—
266C22×C137—42no—
267C22×C137—82no—
268C22×C137—81no—
269C22×C137—81no—
270C22×C137—81no—
271C22×C137—81no—
272C22×C137C2×#12011yescharacteristic
273C22×C137C2×#13511yescharacteristic
274C22×C137C2×AGL(1,ℤ/8)11yescharacteristic
275C4×C137—21no—
276C4×C137—21no—
277C4×C137—21no—
278C4×C137—21no—
279C4×C137—42no—
280C4×C137—42no—
281C4×C137—42no—
282C4×C137—42no—
283C4×C137—82no—
284C4×C137—81no—
285C4×C137#44912yes—
286C4×C137#44912yes—
287C2×C137—21no—
288C2×C137—21no—
289C2×C137—82no—
290C2×C137—82no—
291C2×C137—82no—
292C2×C137C2×#44911yescharacteristic
293C2×C137#301911yescharacteristic
294C2×C137#302511yescharacteristic
295#18222C13711yesmaximal, characteristic, direct product factor, Sylow
296C137#1822211yescharacteristic, direct product factor, Sylow
297#3506C2×C13711yescharacteristic, complemented
298#3101C2×C13711yescharacteristic, complemented
299#3100C2×C13711yescharacteristic, complemented
300#2535C2×C13711yescharacteristic, complemented
301#1612C2×C13714yescomplemented
302#1600C2×C13714yescomplemented
303#1518C2×C13711yescharacteristic, complemented
304#1517C2×C13711yescharacteristic, complemented
305#1460C2×C13711yescharacteristic, complemented
306#559—22no—
307#556—21no—
308#555—21no—
309#553—22no—
310#550—21no—
311#547—21no—
312#538C22×C13714yes—
313#535—21no—
314#533C22×C13711yescharacteristic
315#532C22×C13711yescharacteristic
316#525—22no—
317C2×#121—21no—
318#520—21no—
319C2×C4×D4C22×C13711yescharacteristic
320C2×C4×D4C22×C13711yescharacteristic
321#483C22×C13712yes—
322#480C22×C13712yes—
323#476—21no—
324#474—21no—
325#461—21no—
326#459—21no—
327#450C22×C13712yes—
328#447C22×C13712yes—
329C4×SD16—22no—
330C4×D8—22no—
331#431C22×C13711yescharacteristic
332#431C22×C13711yescharacteristic
333#397C22×C13712yes—
334#392C22×C13712yes—
335#386—22no—
336#382C22×C13711yescharacteristic
337#380—21no—
338#328C22×C13714yes—
339#327C22×C13714yes—
340#323C22×C13714yes—
341C2×#41—22no—
342C2×#41—21no—
343C22×D4—22no—
344C22×D4D4×C13711yescharacteristic
345C22×D4C23×C13711yescharacteristic
346C23×C4—21no—
347C23×C4D4×C13712yes—
348C2×SD16—22no—
349C2×D8—22no—
350C4⋊Q8D4×C13711yescharacteristic, complemented
351C4⋊Q8D4×C13711yescharacteristic, complemented
352C4⋊Q8D4×C13711yescharacteristic, complemented
353#127—22no—
354#127D4×C13711yescharacteristic, complemented
355#126—24no—
356#125—21no—
357#124—22no—
358#124—24no—
359#124—21no—
360#123—24no—
361#123—24no—
362C22⋊Q8—22no—
363C22⋊Q8—21no—
364C22⋊Q8C23×C13712yes—
365#121—22no—
366#121—22no—
367#121—22no—
368#121—24no—
369#121—21no—
370#121—44no—
371#121C23×C13712yes—
372#120—24no—
373C4×Q8—22no—
374C4×Q8—21no—
375C4×Q8D4×C13712yescomplemented
376C4×D4—22no—
377C4×D4—24no—
378C4×D4—24no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—21no—
387C4×D4—21no—
388C4×D4D4×C13712yescomplemented
389#117—22no—
390#117—22no—
391#117—21no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4D4×C13711yescharacteristic
395C2×C4⋊C4C23×C13711yescharacteristic
396C2×C22⋊C4—22no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4C23×C13712yes—
400C2×C42C23×C13711yescharacteristic
401#107—22no—
402#106—22no—
403C4⋊C8—21no—
404C4⋊C8—21no—
405#103—22no—
406#102—22no—
407#102—22no—
408#102—22no—
409C22⋊C8C23×C13712yes—
410C22⋊C8C23×C13712yes—
411C4×C8—21no—
412C4×C8—21no—
413#95—22no—
414#95—21no—
415#95C23×C13712yes—
416C24—21no—
417C24C2×D4×C13712yes—
418#41—42no—
419#41—42no—
420#41—84no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8C2×D4×C13712yes—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—42no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—82no—
439C2×D4C2×D4×C13712yes—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—82no—
451C22×C4C2×D4×C13712yes—
452C22×C4C2×D4×C13711yescharacteristic
453C22×C4C2×D4×C13711yescharacteristic
454C22×C4C24×C13711yescharacteristic
455SD16—44no—
456D8—44no—
457C2×C8—22no—
458C2×C8—22no—
459C4⋊C4—24no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—42no—
471C4⋊C4C2×D4×C13712yes—
472C4⋊C4C2×D4×C13712yes—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—44no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—82no—
485C42—24no—
486C42—22no—
487C42C2×D4×C13711yescharacteristic
488C42C2×D4×C13711yescharacteristic
489C42C2×D4×C13711yescharacteristic
490C42C2×D4×C13711yescharacteristic
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23C22×D4×C13711yescharacteristic
505Q8—42no—
506Q8—42no—
507Q8—42no—
508D4—22no—
509D4—22no—
510D4—42no—
511D4—41no—
512D4—41no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—84no—
520D4—84no—
521D4—88no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—82no—
538C2×C4—84no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4#120×C13711yescharacteristic
542C2×C4#120×C13711yescharacteristic
543C2×C4#120×C13711yescharacteristic
544C2×C4#120×C13711yescharacteristic
545C2×C4#135×C13712yes—
546C2×C4AGL(1,ℤ/8)×C13712yes—
547C2×C4C22×D4×C13711yescharacteristic
548C2×C4C22×D4×C13711yescharacteristic
549C8—42no—
550C8—42no—
551C22—21no—
552C22—21no—
553C22—42no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—82no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22C2×#120×C13711yescharacteristic
569C22C2×#135×C13711yescharacteristic
570C22C2×AGL(1,ℤ/8)×C13711yescharacteristic
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—82no—
580C4—81no—
581C4#449×C13712yes—
582C4#449×C13712yes—
583C2—21no—
584C2—21no—
585C2—82no—
586C2—82no—
587C2—82no—
588C2C2×#449×C13711yescharacteristic
589C2#3019×C13711yescharacteristic
590C2#3025×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC22×C4C24×C137
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3506×C137C2semidirect
2. #3101×C137C2semidirect
3. #3100×C137C2semidirect
4. #2535×C137C2semidirect
5. #1612×C137C2semidirect
6. #1600×C137C2semidirect
7. #1518×C137C2semidirect
8. #1517×C137C2semidirect
9. #1460×C137C2semidirect
54. C4⋊Q8×C137D4semidirect
55. C4⋊Q8×C137D4semidirect
56. C4⋊Q8×C137D4semidirect
58. #127×C137D4semidirect
79. C4×Q8×C137D4semidirect
92. C4×D4×C137D4semidirect
295. #18222C137direct (N × H)
296. C137#18222direct (N × H)
297. #3506C2×C137semidirect
298. #3101C2×C137semidirect
299. #3100C2×C137semidirect
300. #2535C2×C137semidirect
301. #1612C2×C137semidirect
302. #1600C2×C137semidirect
303. #1518C2×C137semidirect
304. #1517C2×C137semidirect
305. #1460C2×C137semidirect
350. C4⋊Q8D4×C137semidirect
351. C4⋊Q8D4×C137semidirect
352. C4⋊Q8D4×C137semidirect
354. #127D4×C137semidirect
375. C4×Q8D4×C137semidirect
388. C4×D4D4×C137semidirect

Maximal subgroups

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

subgroupindex
#3506×C1372
#3101×C1372
#3100×C1372
#2535×C1372
#1612×C1372
#1600×C1372
#1518×C1372
#1517×C1372
#1460×C1372
#18222137