Finite Group Databasebeta

← C137×#23647 #46873033

Subgroups — C137×#23647 #46873033

Order 35072.

914 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#428×C137C211yesmaximal, characteristic, complemented
2#2896×C137C211yesmaximal, characteristic
3C2×#417×C137C211yesmaximal, characteristic
4#2879×C137C211yesmaximal, characteristic
5#2079×C137C214yesmaximal, complemented
6#2078×C137C214yesmaximal, complemented
7#1742×C137C211yesmaximal, characteristic, complemented
8C2×#343×C137C211yesmaximal, characteristic, complemented
9#1738×C137C211yesmaximal, characteristic, complemented
10C22×#41×C137C2211yescharacteristic
11C22×M16×C137C2211yescharacteristic
12C22×M16×C137C2211yescharacteristic
13C22×M16×C137C2211yescharacteristic
14C22×M16×C137C2211yescharacteristic
15C2×#117×C137C2211yescharacteristic
16C22×C4⋊C4×C137C2211yescharacteristic
17#428×C137—22no—
18#428×C137—22no—
19C2×#107×C137C2212yescomplemented
20C2×#106×C137C2212yescomplemented
21#419×C137—21no—
22#418×C137—21no—
23#417×C137—22no—
24#417×C137C2212yes—
25#417×C137C2214yes—
26#417×C137C2212yes—
27#416×C137—22no—
28C2×#103×C137—21no—
29C2×#102×C137—21no—
30#408×C137—22no—
31#408×C137C2214yes—
32#407×C137—22no—
33#343×C137C2212yescomplemented
34#343×C137C2214yescomplemented
35#343×C137C2212yescomplemented
36#341×C137C2214yescomplemented
37C2×#41×C137—21no—
38C2×#41×C137—21no—
39C2×#41×C137—22no—
40C2×#41×C137—22no—
41C2×#41×C137—22no—
42C2×#41×C137—22no—
43C2×#41×C137D412yes—
44C2×#41×C137C2312yes—
45C22×Q8×C137D411yescharacteristic
46C22×D4×C137—21no—
47C22×D4×C137D411yescharacteristic
48C23×C4×C137—21no—
49C23×C4×C137—21no—
50C23×C4×C137C2311yescharacteristic
51C2×M16×C137—22no—
52C2×M16×C137—22no—
53C2×M16×C137—21no—
54C2×M16×C137—21no—
55C2×M16×C137—21no—
56C2×M16×C137—22no—
57C2×M16×C137—22no—
58C2×M16×C137—22no—
59C2×M16×C137—22no—
60C2×M16×C137—22no—
61C2×M16×C137—22no—
62C2×M16×C137D412yes—
63C2×M16×C137D412yes—
64C2×M16×C137C2312yes—
65C2×M16×C137C2312yes—
66C2×M16×C137C2312yes—
67C22×C8×C137—21no—
68C22×C8×C137—21no—
69C22×C8×C137—21no—
70C22×C8×C137C2312yes—
71#117×C137—21no—
72#117×C137—21no—
73#117×C137—21no—
74#117×C137—21no—
75C2×C4⋊C4×C137—21no—
76C2×C4⋊C4×C137—21no—
77C2×C4⋊C4×C137—21no—
78C2×C4⋊C4×C137—21no—
79C2×C4⋊C4×C137—21no—
80C2×C4⋊C4×C137C2311yescharacteristic
81C2×C4⋊C4×C137C2312yes—
82C2×C4⋊C4×C137C2311yescharacteristic
83C2×C22⋊C4×C137—21no—
84C2×C42×C137—21no—
85#107×C137—24no—
86#106×C137—24no—
87#103×C137—24no—
88#103×C137—22no—
89#103×C137—22no—
90#102×C137—24no—
91#102×C137—22no—
92#102×C137—22no—
93C22⋊C8×C137—28no—
94C24×C137—21no—
95C24×C137#4111yescharacteristic
96#41×C137—24no—
97#41×C137—24no—
98#41×C137—22no—
99#41×C137—22no—
100#41×C137—22no—
101#41×C137—22no—
102#41×C137—42no—
103#41×C137—42no—
104#41×C137—42no—
105#41×C137—42no—
106C2×Q8×C137—21no—
107C2×Q8×C137—22no—
108C2×Q8×C137—22no—
109C2×Q8×C137C2×D412yes—
110C2×D4×C137—22no—
111C2×D4×C137—22no—
112C2×D4×C137—21no—
113C2×D4×C137—22no—
114C2×D4×C137—22no—
115C2×D4×C137—42no—
116C2×D4×C137—42no—
117C2×D4×C137C2×D412yes—
118C22×C4×C137—21no—
119C22×C4×C137—22no—
120C22×C4×C137—22no—
121C22×C4×C137—21no—
122C22×C4×C137—21no—
123C22×C4×C137—22no—
124C22×C4×C137—42no—
125C22×C4×C137—42no—
126C22×C4×C137—41no—
127C22×C4×C137—41no—
128C22×C4×C137—82no—
129C22×C4×C137C2×D411yescharacteristic
130C22×C4×C137C2×D411yescharacteristic
131C22×C4×C137C2×D411yescharacteristic
132C22×C4×C137C2×D411yescharacteristic
133C22×C4×C137#4111yescharacteristic
134C22×C4×C137#4112yes—
135C22×C4×C137#4112yes—
136C22×C4×C137#4112yes—
137C22×C4×C137#4112yes—
138C22×C4×C137C2411yescharacteristic
139M16×C137—24no—
140M16×C137—24no—
141M16×C137—22no—
142M16×C137—22no—
143M16×C137—22no—
144M16×C137—22no—
145M16×C137—22no—
146M16×C137—22no—
147M16×C137—42no—
148M16×C137—42no—
149M16×C137—42no—
150C2×C8×C137—22no—
151C2×C8×C137—22no—
152C2×C8×C137—22no—
153C2×C8×C137—22no—
154C2×C8×C137—22no—
155C2×C8×C137—42no—
156C2×C8×C137—42no—
157C2×C8×C137—42no—
158C2×C8×C137C2×D414yes—
159C4⋊C4×C137—22no—
160C4⋊C4×C137—21no—
161C4⋊C4×C137—21no—
162C4⋊C4×C137—21no—
163C4⋊C4×C137—22no—
164C4⋊C4×C137—21no—
165C4⋊C4×C137—41no—
166C4⋊C4×C137—41no—
167C22⋊C4×C137—41no—
168C22⋊C4×C137—41no—
169C42×C137—41no—
170C42×C137—41no—
171C23×C137—22no—
172C23×C137—41no—
173C23×C137—41no—
174C23×C137—42no—
175C23×C137—81no—
176C23×C137—81no—
177C23×C137#12111yescharacteristic
178C23×C137#12111yescharacteristic
179C23×C137#12612yes—
180C23×C137#12612yes—
181C23×C137C2×#4111yescharacteristic
182Q8×C137—22no—
183Q8×C137—22no—
184Q8×C137—42no—
185D4×C137—22no—
186D4×C137—22no—
187D4×C137—42no—
188D4×C137—42no—
189D4×C137—44no—
190D4×C137—42no—
191C2×C4×C137—22no—
192C2×C4×C137—42no—
193C2×C4×C137—42no—
194C2×C4×C137—42no—
195C2×C4×C137—44no—
196C2×C4×C137—41no—
197C2×C4×C137—41no—
198C2×C4×C137—42no—
199C2×C4×C137—82no—
200C2×C4×C137—82no—
201C2×C4×C137—81no—
202C2×C4×C137—81no—
203C2×C4×C137—82no—
204C2×C4×C137#12111yescharacteristic
205C2×C4×C137#12111yescharacteristic
206C2×C4×C137#12312yes—
207C2×C4×C137#12312yes—
208C2×C4×C137#12312yes—
209C2×C4×C137#12312yes—
210C2×C4×C137#12312yes—
211C2×C4×C137#12312yes—
212C2×C4×C137#12412yes—
213C2×C4×C137#12412yes—
214C2×C4×C137#12612yes—
215C2×C4×C137#12612yes—
216C2×C4×C137C22×D411yescharacteristic
217C2×C4×C137C22×D411yescharacteristic
218C2×C4×C137C2×#4111yescharacteristic
219C2×C4×C137C2×#4111yescharacteristic
220C2×C4×C137C2×#4111yescharacteristic
221C2×C4×C137C2×#4111yescharacteristic
222C8×C137—24no—
223C8×C137—42no—
224C8×C137—42no—
225C8×C137—42no—
226C22×C137—21no—
227C22×C137—21no—
228C22×C137—21no—
229C22×C137—21no—
230C22×C137—21no—
231C22×C137—21no—
232C22×C137—42no—
233C22×C137—42no—
234C22×C137—42no—
235C22×C137—42no—
236C22×C137—42no—
237C22×C137—84no—
238C22×C137#39712yes—
239C22×C137#39712yes—
240C22×C137C2×#12111yescharacteristic
241C22×C137C2×#12611yescharacteristic
242C22×C137C2×#12611yescharacteristic
243C4×C137—42no—
244C4×C137—81no—
245C4×C137—82no—
246C4×C137—81no—
247C4×C137#39712yes—
248C4×C137#39712yes—
249C4×C137C2×#12311yescharacteristic
250C4×C137C2×#12311yescharacteristic
251C4×C137C2×#12311yescharacteristic
252C4×C137C2×#12411yescharacteristic
253C2×C137—22no—
254C2×C137—21no—
255C2×C137—21no—
256C2×C137—22no—
257C2×C137—82no—
258C2×C137C2×#39711yescharacteristic
259C2×C137#306811yescharacteristic
260C2×C137#306911yescharacteristic
261#23647C13711yesmaximal, characteristic, direct product factor, Sylow
262C137#2364711yescharacteristic, direct product factor, Sylow
263C2×#428C2×C13711yescharacteristic, complemented
264#2896C2×C13711yescharacteristic
265C2×#417C2×C13711yescharacteristic
266#2879C2×C13711yescharacteristic
267#2079C2×C13714yescomplemented
268#2078C2×C13714yescomplemented
269#1742C2×C13711yescharacteristic, complemented
270C2×#343C2×C13711yescharacteristic, complemented
271#1738C2×C13711yescharacteristic, complemented
272C22×#41C22×C13711yescharacteristic
273C22×M16C22×C13711yescharacteristic
274C22×M16C22×C13711yescharacteristic
275C22×M16C22×C13711yescharacteristic
276C22×M16C22×C13711yescharacteristic
277C2×#117C22×C13711yescharacteristic
278C22×C4⋊C4C22×C13711yescharacteristic
279#428—22no—
280#428—22no—
281C2×#107C22×C13712yescomplemented
282C2×#106C22×C13712yescomplemented
283#419—21no—
284#418—21no—
285#417—22no—
286#417C22×C13712yes—
287#417C22×C13714yes—
288#417C22×C13712yes—
289#416—22no—
290C2×#103—21no—
291C2×#102—21no—
292#408—22no—
293#408C22×C13714yes—
294#407—22no—
295#343C22×C13712yescomplemented
296#343C22×C13714yescomplemented
297#343C22×C13712yescomplemented
298#341C22×C13714yescomplemented
299C2×#41—21no—
300C2×#41—21no—
301C2×#41—22no—
302C2×#41—22no—
303C2×#41—22no—
304C2×#41—22no—
305C2×#41D4×C13712yes—
306C2×#41C23×C13712yes—
307C22×Q8D4×C13711yescharacteristic
308C22×D4—21no—
309C22×D4D4×C13711yescharacteristic
310C23×C4—21no—
311C23×C4—21no—
312C23×C4C23×C13711yescharacteristic
313C2×M16—22no—
314C2×M16—22no—
315C2×M16—21no—
316C2×M16—21no—
317C2×M16—21no—
318C2×M16—22no—
319C2×M16—22no—
320C2×M16—22no—
321C2×M16—22no—
322C2×M16—22no—
323C2×M16—22no—
324C2×M16D4×C13712yes—
325C2×M16D4×C13712yes—
326C2×M16C23×C13712yes—
327C2×M16C23×C13712yes—
328C2×M16C23×C13712yes—
329C22×C8—21no—
330C22×C8—21no—
331C22×C8—21no—
332C22×C8C23×C13712yes—
333#117—21no—
334#117—21no—
335#117—21no—
336#117—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4C23×C13712yes—
343C2×C4⋊C4C23×C13711yescharacteristic
344C2×C4⋊C4C23×C13711yescharacteristic
345C2×C22⋊C4—21no—
346C2×C42—21no—
347#107—24no—
348#106—24no—
349#103—24no—
350#103—22no—
351#103—22no—
352#102—24no—
353#102—22no—
354#102—22no—
355C22⋊C8—28no—
356C24—21no—
357C24#41×C13711yescharacteristic
358#41—24no—
359#41—24no—
360#41—22no—
361#41—22no—
362#41—22no—
363#41—22no—
364#41—42no—
365#41—42no—
366#41—42no—
367#41—42no—
368C2×Q8—21no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8C2×D4×C13712yes—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4C2×D4×C13712yes—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—82no—
391C22×C4C2×D4×C13711yescharacteristic
392C22×C4C2×D4×C13711yescharacteristic
393C22×C4C2×D4×C13711yescharacteristic
394C22×C4C2×D4×C13711yescharacteristic
395C22×C4#41×C13711yescharacteristic
396C22×C4#41×C13712yes—
397C22×C4#41×C13712yes—
398C22×C4#41×C13712yes—
399C22×C4#41×C13712yes—
400C22×C4C24×C13711yescharacteristic
401M16—24no—
402M16—24no—
403M16—22no—
404M16—22no—
405M16—22no—
406M16—22no—
407M16—22no—
408M16—22no—
409M16—42no—
410M16—42no—
411M16—42no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8C2×D4×C13714yes—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C42—41no—
432C42—41no—
433C23—22no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23—81no—
438C23—81no—
439C23#121×C13711yescharacteristic
440C23#121×C13711yescharacteristic
441C23#126×C13712yes—
442C23#126×C13712yes—
443C23C2×#41×C13711yescharacteristic
444Q8—22no—
445Q8—22no—
446Q8—42no—
447D4—22no—
448D4—22no—
449D4—44no—
450D4—42no—
451D4—42no—
452D4—42no—
453C2×C4—22no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4#121×C13711yescharacteristic
467C2×C4#121×C13711yescharacteristic
468C2×C4#123×C13712yes—
469C2×C4#123×C13712yes—
470C2×C4#123×C13712yes—
471C2×C4#123×C13712yes—
472C2×C4#123×C13712yes—
473C2×C4#123×C13712yes—
474C2×C4#124×C13712yes—
475C2×C4#124×C13712yes—
476C2×C4#126×C13712yes—
477C2×C4#126×C13712yes—
478C2×C4C22×D4×C13711yescharacteristic
479C2×C4C22×D4×C13711yescharacteristic
480C2×C4C2×#41×C13711yescharacteristic
481C2×C4C2×#41×C13711yescharacteristic
482C2×C4C2×#41×C13711yescharacteristic
483C2×C4C2×#41×C13711yescharacteristic
484C8—24no—
485C8—42no—
486C8—42no—
487C8—42no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—84no—
500C22#397×C13712yes—
501C22#397×C13712yes—
502C22C2×#121×C13711yescharacteristic
503C22C2×#126×C13711yescharacteristic
504C22C2×#126×C13711yescharacteristic
505C4—42no—
506C4—82no—
507C4—81no—
508C4—81no—
509C4#397×C13712yes—
510C4#397×C13712yes—
511C4C2×#123×C13711yescharacteristic
512C4C2×#123×C13711yescharacteristic
513C4C2×#123×C13711yescharacteristic
514C4C2×#124×C13711yescharacteristic
515C2—22no—
516C2—22no—
517C2—21no—
518C2—21no—
519C2—82no—
520C2C2×#397×C13711yescharacteristic
521C2#3068×C13711yescharacteristic
522C2#3069×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#121
commutator, Frattini, cosocleC22×C4C24×C137
prime radicalC22×#41×C137C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#428×C137C2semidirect
5. #2079×C137C2semidirect
6. #2078×C137C2semidirect
7. #1742×C137C2semidirect
8. C2×#343×C137C2semidirect
9. #1738×C137C2semidirect
19. C2×#107×C137C22semidirect
20. C2×#106×C137C22semidirect
33. #343×C137C22semidirect
34. #343×C137C22semidirect
35. #343×C137C22semidirect
36. #341×C137C22semidirect
261. #23647C137direct (N × H)
262. C137#23647direct (N × H)
263. C2×#428C2×C137semidirect
267. #2079C2×C137semidirect
268. #2078C2×C137semidirect
269. #1742C2×C137semidirect
270. C2×#343C2×C137semidirect
271. #1738C2×C137semidirect
281. C2×#107C22×C137semidirect
282. C2×#106C22×C137semidirect
295. #343C22×C137semidirect
296. #343C22×C137semidirect
297. #343C22×C137semidirect
298. #341C22×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#428×C1372
#2896×C1372
C2×#417×C1372
#2879×C1372
#2079×C1372
#2078×C1372
#1742×C1372
C2×#343×C1372
#1738×C1372
#23647137