Finite Group Databasebeta

← C137×#23991 #46873377

Subgroups — C137×#23991 #46873377

Order 35072.

1164 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3510×C137C211yesmaximal, characteristic, complemented
2#3027×C137C211yesmaximal, characteristic
3#3022×C137C211yesmaximal, characteristic, complemented
4#2901×C137C211yesmaximal, characteristic, complemented
5#2031×C137C212yesmaximal, complemented
6#2004×C137C212yesmaximal, complemented
7#1914×C137C211yesmaximal, characteristic, complemented
8#1911×C137C212yesmaximal, complemented
9#1909×C137C211yesmaximal, characteristic, complemented
10#1905×C137C212yesmaximal, complemented
11#1827×C137C211yesmaximal, characteristic, complemented
12#585×C137C2211yescharacteristic
13C2×#137×C137C2212yes—
14#568×C137C2211yescharacteristic
15#559×C137—22no—
16#556×C137—22no—
17#551×C137—24no—
18#542×C137—22no—
19#536×C137C2211yescharacteristic
20#535×C137C2212yes—
21#535×C137C2211yescharacteristic
22#533×C137C2211yescharacteristic, complemented
23#529×C137—22no—
24C2×#124×C137C2211yescharacteristic, complemented
25#519×C137C2211yescharacteristic, complemented
26#518×C137C2211yescharacteristic, complemented
27C4×#41×C137C2211yescharacteristic
28B(4,𝔽2)×C137—24no—
29#456×C137—22no—
30#455×C137—22no—
31#421×C137—22no—
32#421×C137C2212yes—
33#421×C137C2212yes—
34C2×#104×C137—22no—
35C2×#104×C137C2212yes—
36#419×C137C2212yescomplemented
37#413×C137C2211yescharacteristic
38C2×#100×C137C2212yescomplemented
39#410×C137C2212yes—
40#410×C137C2211yescharacteristic
41#344×C137C2212yescomplemented
42#339×C137C2212yescomplemented
43H(3,ℤ/4)×C137C2212yescomplemented
44H(3,ℤ/4)×C137C2212yescomplemented
452−1+4×C137—22no—
462−1+4×C137—21no—
47H(5,𝔽2)×C137—22no—
48H(5,𝔽2)×C137—22no—
49H(5,𝔽2)×C137—21no—
50C2×#41×C137—22no—
51C2×#41×C137—42no—
52C2×#41×C137C2312yes—
53C2×#41×C137C2311yescharacteristic
54C22×Q8×C137C2311yescharacteristic
55C22×D4×C137C2311yescharacteristic
56#137×C137—24no—
57#137×C137—42no—
58C2×Q16×C137—22no—
59C2×SD16×C137—22no—
60#131×C137—42no—
61C2×M16×C137—22no—
62C2×M16×C137C2312yes—
63C2×M16×C137C2312yes—
64C4⋊Q8×C137—21no—
65C4⋊Q8×C137D412yescomplemented
66#127×C137—21no—
67#127×C137—22no—
68#126×C137—24no—
69#126×C137—24no—
70#125×C137—21no—
71#124×C137—24no—
72#124×C137—22no—
73#124×C137—21no—
74#124×C137—22no—
75#124×C137—21no—
76#124×C137—21no—
77#124×C137D412yescomplemented
78#123×C137—24no—
79#123×C137—24no—
80#123×C137—22no—
81C22⋊Q8×C137—24no—
82C22⋊Q8×C137—22no—
83C22⋊Q8×C137—22no—
84#121×C137—22no—
85#121×C137—42no—
86#120×C137—24no—
87#120×C137D414yes—
88C4×Q8×C137—21no—
89C4×Q8×C137—22no—
90C4×Q8×C137—22no—
91C4×Q8×C137—22no—
92C4×D4×C137—24no—
93C4×D4×C137—22no—
94C4×D4×C137—21no—
95C4×D4×C137—22no—
96C4×D4×C137—22no—
97#117×C137—24no—
98#117×C137—22no—
99#117×C137D412yescomplemented
100#117×C137C2312yes—
101#117×C137C2311yescharacteristic
102#117×C137C2311yescharacteristic
103#117×C137C2311yescharacteristic
104C2×C22⋊C4×C137—22no—
105C2×C22⋊C4×C137—22no—
106C2×C42×C137D412yescomplemented
107C2×C42×C137C2311yescharacteristic
108#104×C137—22no—
109#104×C137—24no—
110#104×C137—22no—
111#104×C137—24no—
112#103×C137—42no—
113#101×C137—21no—
114#100×C137—21no—
115#100×C137—42no—
116#99×C137—22no—
117#99×C137—24no—
118C24×C137C2×D412yes—
119#41×C137—22no—
120#41×C137—22no—
121#41×C137—24no—
122#41×C137—22no—
123#41×C137—22no—
124#41×C137—22no—
125#41×C137—22no—
126#41×C137—44no—
127#41×C137—42no—
128#41×C137—42no—
129#41×C137—42no—
130#41×C137—42no—
131C2×Q8×C137—22no—
132C2×Q8×C137—21no—
133C2×Q8×C137—22no—
134C2×Q8×C137—22no—
135C2×Q8×C137—22no—
136C2×Q8×C137—42no—
137C2×Q8×C137C2×D412yes—
138C2×Q8×C137C2×D411yescharacteristic
139C2×Q8×C137#4112yes—
140C2×D4×C137—22no—
141C2×D4×C137—22no—
142C2×D4×C137—24no—
143C2×D4×C137—21no—
144C2×D4×C137—21no—
145C2×D4×C137—22no—
146C2×D4×C137—22no—
147C2×D4×C137—42no—
148C2×D4×C137—44no—
149C2×D4×C137—42no—
150C2×D4×C137—44no—
151C2×D4×C137C2×D412yes—
152C2×D4×C137C2×D411yescharacteristic
153C2×D4×C137#4112yes—
154C22×C4×C137—21no—
155C22×C4×C137—24no—
156C22×C4×C137—22no—
157C22×C4×C137—21no—
158C22×C4×C137—22no—
159C22×C4×C137—42no—
160C22×C4×C137C2×D412yes—
161C22×C4×C137C2411yescharacteristic
162Q16×C137—44no—
163SD16×C137—44no—
164M16×C137—22no—
165M16×C137—22no—
166M16×C137—22no—
167M16×C137—22no—
168M16×C137—42no—
169C2×C8×C137—22no—
170C2×C8×C137—22no—
171C2×C8×C137—42no—
172C4⋊C4×C137—22no—
173C4⋊C4×C137—24no—
174C4⋊C4×C137—22no—
175C4⋊C4×C137—22no—
176C4⋊C4×C137—24no—
177C4⋊C4×C137—22no—
178C4⋊C4×C137—21no—
179C4⋊C4×C137—21no—
180C4⋊C4×C137—21no—
181C4⋊C4×C137—22no—
182C22⋊C4×C137—24no—
183C22⋊C4×C137—24no—
184C22⋊C4×C137—24no—
185C22⋊C4×C137—22no—
186C22⋊C4×C137—24no—
187C22⋊C4×C137—21no—
188C22⋊C4×C137—21no—
189C22⋊C4×C137—24no—
190C22⋊C4×C137—22no—
191C22⋊C4×C137—44no—
192C22⋊C4×C137C2×D412yes—
193C42×C137—24no—
194C42×C137—22no—
195C42×C137—21no—
196C42×C137—21no—
197C42×C137—22no—
198C42×C137C2×D411yescharacteristic
199C42×C137C2×D412yes—
200C42×C137C2×D411yescharacteristic
201C42×C137C2×D412yes—
202C23×C137—22no—
203C23×C137—24no—
204C23×C137—22no—
205C23×C137—42no—
206C23×C137—82no—
207C23×C137#12012yes—
208C23×C137C22×D411yescharacteristic
209Q8×C137—22no—
210Q8×C137—22no—
211Q8×C137—24no—
212Q8×C137—22no—
213Q8×C137—42no—
214Q8×C137—41no—
215Q8×C137—41no—
216Q8×C137—82no—
217D4×C137—24no—
218D4×C137—22no—
219D4×C137—22no—
220D4×C137—22no—
221D4×C137—42no—
222D4×C137—44no—
223D4×C137—42no—
224D4×C137—42no—
225D4×C137—41no—
226D4×C137—41no—
227D4×C137—42no—
228D4×C137—84no—
229D4×C137—164no—
230C2×C4×C137—22no—
231C2×C4×C137—22no—
232C2×C4×C137—22no—
233C2×C4×C137—21no—
234C2×C4×C137—22no—
235C2×C4×C137—22no—
236C2×C4×C137—21no—
237C2×C4×C137—22no—
238C2×C4×C137—22no—
239C2×C4×C137—44no—
240C2×C4×C137—42no—
241C2×C4×C137—42no—
242C2×C4×C137—42no—
243C2×C4×C137—42no—
244C2×C4×C137—42no—
245C2×C4×C137—82no—
246C2×C4×C137—84no—
247C2×C4×C137—81no—
248C2×C4×C137#12012yes—
249C2×C4×C137C22×D411yescharacteristic
250C2×C4×C137C22×D411yescharacteristic
251C2×C4×C137C2×#4112yes—
252C2×C4×C137H(5,𝔽2)12yes—
253C8×C137—42no—
254C8×C137—42no—
255C22×C137—22no—
256C22×C137—41no—
257C22×C137—42no—
258C22×C137—44no—
259C22×C137—42no—
260C22×C137—42no—
261C22×C137—82no—
262C22×C137—84no—
263C22×C137—162no—
264C22×C137C2×#12011yescharacteristic
265C22×C137#54612yes—
266C4×C137—22no—
267C4×C137—42no—
268C4×C137—41no—
269C4×C137—42no—
270C4×C137—41no—
271C4×C137—42no—
272C4×C137—42no—
273C4×C137—42no—
274C4×C137—82no—
275C4×C137#52511yescharacteristic
276C4×C137#53511yescharacteristic
277C4×C137#54612yes—
278C2×C137—21no—
279C2×C137—22no—
280C2×C137—42no—
281C2×C137—82no—
282C2×C137—82no—
283C2×C137#240411yescharacteristic
284#23991C13711yesmaximal, characteristic, direct product factor, Sylow
285C137#2399111yescharacteristic, direct product factor, Sylow
286#3510C2×C13711yescharacteristic, complemented
287#3027C2×C13711yescharacteristic
288#3022C2×C13711yescharacteristic, complemented
289#2901C2×C13711yescharacteristic, complemented
290#2031C2×C13712yescomplemented
291#2004C2×C13712yescomplemented
292#1914C2×C13711yescharacteristic, complemented
293#1911C2×C13712yescomplemented
294#1909C2×C13711yescharacteristic, complemented
295#1905C2×C13712yescomplemented
296#1827C2×C13711yescharacteristic, complemented
297#585C22×C13711yescharacteristic
298C2×#137C22×C13712yes—
299#568C22×C13711yescharacteristic
300#559—22no—
301#556—22no—
302#551—24no—
303#542—22no—
304#536C22×C13711yescharacteristic
305#535C22×C13712yes—
306#535C22×C13711yescharacteristic
307#533C22×C13711yescharacteristic, complemented
308#529—22no—
309C2×#124C22×C13711yescharacteristic, complemented
310#519C22×C13711yescharacteristic, complemented
311#518C22×C13711yescharacteristic, complemented
312C4×#41C22×C13711yescharacteristic
313B(4,𝔽2)—24no—
314#456—22no—
315#455—22no—
316#421—22no—
317#421C22×C13712yes—
318#421C22×C13712yes—
319C2×#104—22no—
320C2×#104C22×C13712yes—
321#419C22×C13712yescomplemented
322#413C22×C13711yescharacteristic
323C2×#100C22×C13712yescomplemented
324#410C22×C13711yescharacteristic
325#410C22×C13712yes—
326#344C22×C13712yescomplemented
327#339C22×C13712yescomplemented
328H(3,ℤ/4)C22×C13712yescomplemented
329H(3,ℤ/4)C22×C13712yescomplemented
3302−1+4—22no—
3312−1+4—21no—
332H(5,𝔽2)—22no—
333H(5,𝔽2)—22no—
334H(5,𝔽2)—21no—
335C2×#41—22no—
336C2×#41—42no—
337C2×#41C23×C13712yes—
338C2×#41C23×C13711yescharacteristic
339C22×Q8C23×C13711yescharacteristic
340C22×D4C23×C13711yescharacteristic
341#137—24no—
342#137—42no—
343C2×Q16—22no—
344C2×SD16—22no—
345#131—42no—
346C2×M16—22no—
347C2×M16C23×C13712yes—
348C2×M16C23×C13712yes—
349C4⋊Q8—21no—
350C4⋊Q8D4×C13712yescomplemented
351#127—21no—
352#127—22no—
353#126—24no—
354#126—24no—
355#125—21no—
356#124—24no—
357#124—22no—
358#124—21no—
359#124—22no—
360#124—21no—
361#124—21no—
362#124D4×C13712yescomplemented
363#123—24no—
364#123—24no—
365#123—22no—
366C22⋊Q8—24no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369#121—22no—
370#121—42no—
371#120—24no—
372#120D4×C13714yes—
373C4×Q8—21no—
374C4×Q8—22no—
375C4×Q8—22no—
376C4×Q8—22no—
377C4×D4—24no—
378C4×D4—22no—
379C4×D4—21no—
380C4×D4—22no—
381C4×D4—22no—
382#117—24no—
383#117—22no—
384#117D4×C13712yescomplemented
385#117C23×C13712yes—
386#117C23×C13711yescharacteristic
387#117C23×C13711yescharacteristic
388#117C23×C13711yescharacteristic
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C42D4×C13712yescomplemented
392C2×C42C23×C13711yescharacteristic
393#104—22no—
394#104—24no—
395#104—22no—
396#104—24no—
397#103—42no—
398#101—21no—
399#100—21no—
400#100—42no—
401#99—22no—
402#99—24no—
403C24C2×D4×C13712yes—
404#41—22no—
405#41—22no—
406#41—24no—
407#41—22no—
408#41—22no—
409#41—22no—
410#41—22no—
411#41—44no—
412#41—42no—
413#41—42no—
414#41—42no—
415#41—42no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—42no—
422C2×Q8C2×D4×C13712yes—
423C2×Q8C2×D4×C13711yescharacteristic
424C2×Q8#41×C13712yes—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—24no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—42no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4C2×D4×C13712yes—
437C2×D4C2×D4×C13711yescharacteristic
438C2×D4#41×C13712yes—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—42no—
445C22×C4C2×D4×C13712yes—
446C22×C4C24×C13711yescharacteristic
447Q16—44no—
448SD16—44no—
449M16—22no—
450M16—22no—
451M16—22no—
452M16—22no—
453M16—42no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—42no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—22no—
471C22⋊C4—24no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—24no—
475C22⋊C4—22no—
476C22⋊C4—44no—
477C22⋊C4C2×D4×C13712yes—
478C42—24no—
479C42—22no—
480C42—21no—
481C42—21no—
482C42—22no—
483C42C2×D4×C13711yescharacteristic
484C42C2×D4×C13712yes—
485C42C2×D4×C13711yescharacteristic
486C42C2×D4×C13712yes—
487C23—22no—
488C23—24no—
489C23—22no—
490C23—42no—
491C23—82no—
492C23#120×C13712yes—
493C23C22×D4×C13711yescharacteristic
494Q8—22no—
495Q8—22no—
496Q8—24no—
497Q8—22no—
498Q8—42no—
499Q8—41no—
500Q8—41no—
501Q8—82no—
502D4—24no—
503D4—22no—
504D4—22no—
505D4—22no—
506D4—42no—
507D4—44no—
508D4—42no—
509D4—42no—
510D4—41no—
511D4—41no—
512D4—42no—
513D4—84no—
514D4—164no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—82no—
531C2×C4—84no—
532C2×C4—81no—
533C2×C4#120×C13712yes—
534C2×C4C22×D4×C13711yescharacteristic
535C2×C4C22×D4×C13711yescharacteristic
536C2×C4C2×#41×C13712yes—
537C2×C4H(5,𝔽2)×C13712yes—
538C8—42no—
539C8—42no—
540C22—22no—
541C22—41no—
542C22—42no—
543C22—44no—
544C22—42no—
545C22—42no—
546C22—82no—
547C22—84no—
548C22—162no—
549C22C2×#120×C13711yescharacteristic
550C22#546×C13712yes—
551C4—22no—
552C4—42no—
553C4—41no—
554C4—42no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4#525×C13711yescharacteristic
561C4#535×C13711yescharacteristic
562C4#546×C13712yes—
563C2—21no—
564C2—22no—
565C2—42no—
566C2—82no—
567C2—82no—
568C2#2404×C13711yescharacteristic
subgroupG/H
center, socleC2×C137#2404
commutator, Frattini, cosocleC22×C4C24×C137
prime radical#3027×C137C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C137 ◁ C22×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3510×C137C2semidirect
3. #3022×C137C2semidirect
4. #2901×C137C2semidirect
5. #2031×C137C2semidirect
6. #2004×C137C2semidirect
7. #1914×C137C2semidirect
8. #1911×C137C2semidirect
9. #1909×C137C2semidirect
10. #1905×C137C2semidirect
11. #1827×C137C2semidirect
22. #533×C137C22semidirect
24. C2×#124×C137C22semidirect
25. #519×C137C22semidirect
26. #518×C137C22semidirect
36. #419×C137C22semidirect
38. C2×#100×C137C22semidirect
41. #344×C137C22semidirect
42. #339×C137C22semidirect
43. H(3,ℤ/4)×C137C22semidirect
44. H(3,ℤ/4)×C137C22semidirect
65. C4⋊Q8×C137D4semidirect
77. #124×C137D4semidirect
99. #117×C137D4semidirect
106. C2×C42×C137D4semidirect
284. #23991C137direct (N × H)
285. C137#23991direct (N × H)
286. #3510C2×C137semidirect
288. #3022C2×C137semidirect
289. #2901C2×C137semidirect
290. #2031C2×C137semidirect
291. #2004C2×C137semidirect
292. #1914C2×C137semidirect
293. #1911C2×C137semidirect
294. #1909C2×C137semidirect
295. #1905C2×C137semidirect
296. #1827C2×C137semidirect
307. #533C22×C137semidirect
309. C2×#124C22×C137semidirect
310. #519C22×C137semidirect
311. #518C22×C137semidirect
321. #419C22×C137semidirect
323. C2×#100C22×C137semidirect
326. #344C22×C137semidirect
327. #339C22×C137semidirect
328. H(3,ℤ/4)C22×C137semidirect
329. H(3,ℤ/4)C22×C137semidirect
350. C4⋊Q8D4×C137semidirect
362. #124D4×C137semidirect
384. #117D4×C137semidirect
391. C2×C42D4×C137semidirect

Maximal subgroups

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

subgroupindex
#3510×C1372
#3027×C1372
#3022×C1372
#2901×C1372
#2031×C1372
#2004×C1372
#1914×C1372
#1911×C1372
#1909×C1372
#1905×C1372
#1827×C1372
#23991137