Finite Group Databasebeta

← C137×#24721 #46874107

Subgroups — C137×#24721 #46874107

Order 35072.

706 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3079×C137C211yesmaximal, characteristic, complemented
2C2×#476×C137C211yesmaximal, characteristic, complemented
3C4×AGL(1,ℤ/8)×C137C211yesmaximal, characteristic
4#2925×C137C211yesmaximal, characteristic, complemented
5#2079×C137C212yesmaximal, complemented
6#2051×C137C212yesmaximal
7#1997×C137C211yesmaximal, characteristic, complemented
8#1951×C137C212yesmaximal, complemented
9#1950×C137C211yesmaximal, characteristic, complemented
10#1931×C137C212yesmaximal, complemented
11#1800×C137C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C137C2211yescharacteristic
13C2×C4⋊Q8×C137C2211yescharacteristic
14C4×#41×C137C2211yescharacteristic
15C2×C4×D4×C137C2211yescharacteristic
16#477×C137—21no—
17#476×C137—21no—
18#476×C137—21no—
19#476×C137—21no—
20#476×C137—22no—
21#475×C137—21no—
22#474×C137—21no—
23#442×C137—21no—
24#440×C137—21no—
25C4×SD16×C137—21no—
26C4×D8×C137—21no—
27#433×C137—22no—
28#432×C137—21no—
29#431×C137—21no—
30C2×#108×C137C2212yes—
31#428×C137C2212yes—
32#428×C137C2211yescharacteristic
33C2×#106×C137C2212yes—
34#423×C137C2211yescharacteristic
35#423×C137C2211yescharacteristic
36C2×C4⋊C8×C137C2211yescharacteristic
37#418×C137C2211yescharacteristic
38#417×C137C2212yes—
39#417×C137C2211yescharacteristic
40C2×#102×C137C2212yes—
41C2×#100×C137C2212yes—
42#408×C137C2212yes—
43#405×C137C2212yes—
44C4×M16×C137C2211yescharacteristic
45#343×C137C2212yes—
46#343×C137C2212yes—
47#343×C137C2212yes—
48#341×C137C2212yes—
49C2×#41×C137C2311yescharacteristic
50C22×Q8×C137—21no—
51C22×D4×C137C2311yescharacteristic
52C23×C4×C137—21no—
53AGL(1,ℤ/8)×C137—22no—
54AGL(1,ℤ/8)×C137—22no—
55C2×SD16×C137—21no—
56C2×D8×C137—21no—
57C2×M16×C137C2311yescharacteristic
58C2×M16×C137C2312yes—
59C2×M16×C137C2312yes—
60C2×M16×C137C2311yescharacteristic
61C22×C8×C137C2312yes—
62C4⋊Q8×C137—21no—
63C4⋊Q8×C137—21no—
64C4⋊Q8×C137—21no—
65C4⋊Q8×C137—21no—
66C4×Q8×C137D411yescharacteristic
67C4×Q8×C137D411yescharacteristic
68C4×D4×C137—21no—
69C4×D4×C137—21no—
70C4×D4×C137—21no—
71C4×D4×C137—21no—
72C4×D4×C137—21no—
73C4×D4×C137—21no—
74C4×D4×C137D411yescharacteristic
75C4×D4×C137D411yescharacteristic
76#117×C137—21no—
77#117×C137C2311yescharacteristic
78C2×C4⋊C4×C137—21no—
79C2×C4⋊C4×C137C2312yes—
80C2×C4⋊C4×C137C2311yescharacteristic
81C2×C22⋊C4×C137—21no—
82C2×C42×C137—21no—
83C2×C42×C137C2311yescharacteristic
84#108×C137—24no—
85#107×C137—21no—
86#107×C137—22no—
87#106×C137—22no—
88#106×C137—22no—
89#106×C137—21no—
90#106×C137—22no—
91C4⋊C8×C137—21no—
92C4⋊C8×C137—21no—
93C4⋊C8×C137—21no—
94C4⋊C8×C137—21no—
95C4⋊C8×C137D411yescharacteristic
96C4⋊C8×C137D411yescharacteristic
97C4⋊C8×C137D412yes—
98#103×C137—21no—
99#103×C137—22no—
100#102×C137—21no—
101#102×C137—22no—
102#102×C137—22no—
103#102×C137—22no—
104#102×C137—21no—
105#102×C137—21no—
106#100×C137—24no—
107C22⋊C8×C137—22no—
108C22⋊C8×C137—22no—
109#97×C137—22no—
110#97×C137—21no—
111C4×C8×C137—22no—
112C4×C8×C137—21no—
113C24×C137—21no—
114#41×C137—21no—
115#41×C137—21no—
116#41×C137—21no—
117#41×C137—21no—
118C2×Q8×C137—41no—
119C2×Q8×C137—41no—
120C2×Q8×C137—82no—
121C2×Q8×C137C2×D411yescharacteristic
122C2×D4×C137—21no—
123C2×D4×C137—21no—
124C2×D4×C137—21no—
125C2×D4×C137—21no—
126C2×D4×C137—21no—
127C2×D4×C137—21no—
128C2×D4×C137C2×D411yescharacteristic
129C2×D4×C137#4112yes—
130C22×C4×C137—21no—
131C22×C4×C137—21no—
132C22×C4×C137—21no—
133C22×C4×C137—21no—
134C22×C4×C137—22no—
135C22×C4×C137—41no—
136C22×C4×C137—41no—
137C22×C4×C137—82no—
138C22×C4×C137#4111yescharacteristic
139C22×C4×C137#4111yescharacteristic
140C22×C4×C137C2411yescharacteristic
141SD16×C137—24no—
142D8×C137—24no—
143M16×C137—21no—
144M16×C137—22no—
145M16×C137—22no—
146M16×C137—22no—
147M16×C137—21no—
148M16×C137—21no—
149M16×C137—22no—
150M16×C137#4112yes—
151C2×C8×C137—21no—
152C2×C8×C137—22no—
153C2×C8×C137—22no—
154C2×C8×C137—22no—
155C2×C8×C137—21no—
156C2×C8×C137—22no—
157C2×C8×C137C2×D412yes—
158C2×C8×C137C2×D412yes—
159C4⋊C4×C137—21no—
160C4⋊C4×C137—21no—
161C4⋊C4×C137—22no—
162C4⋊C4×C137—21no—
163C4⋊C4×C137—22no—
164C4⋊C4×C137—21no—
165C4⋊C4×C137—41no—
166C4⋊C4×C137—41no—
167C4⋊C4×C137C2×D411yescharacteristic
168C4⋊C4×C137C2×D411yescharacteristic
169C22⋊C4×C137—21no—
170C22⋊C4×C137—21no—
171C22⋊C4×C137—21no—
172C22⋊C4×C137—41no—
173C22⋊C4×C137—41no—
174C42×C137—21no—
175C42×C137—21no—
176C42×C137—21no—
177C42×C137C2×D411yescharacteristic
178C42×C137C2×D411yescharacteristic
179C42×C137C2×D411yescharacteristic
180C42×C137C2×D411yescharacteristic
181C23×C137—21no—
182C23×C137—22no—
183C23×C137—41no—
184C23×C137—41no—
185C23×C137—81no—
186C23×C137—81no—
187C23×C137C2×#4111yescharacteristic
188Q8×C137—21no—
189Q8×C137—21no—
190Q8×C137—82no—
191Q8×C137—82no—
192D4×C137—21no—
193D4×C137—22no—
194D4×C137—21no—
195D4×C137—22no—
196D4×C137—41no—
197D4×C137—41no—
198D4×C137—41no—
199D4×C137—41no—
200C2×C4×C137—22no—
201C2×C4×C137—21no—
202C2×C4×C137—21no—
203C2×C4×C137—21no—
204C2×C4×C137—41no—
205C2×C4×C137—41no—
206C2×C4×C137—41no—
207C2×C4×C137—42no—
208C2×C4×C137—41no—
209C2×C4×C137—42no—
210C2×C4×C137—81no—
211C2×C4×C137—82no—
212C2×C4×C137—82no—
213C2×C4×C137—81no—
214C2×C4×C137—82no—
215C2×C4×C137#12111yescharacteristic
216C2×C4×C137#12111yescharacteristic
217C2×C4×C137#12111yescharacteristic
218C2×C4×C137#12111yescharacteristic
219C2×C4×C137C22×D411yescharacteristic
220C2×C4×C137C22×D411yescharacteristic
221C2×C4×C137C2×#4111yescharacteristic
222C2×C4×C137C2×#4111yescharacteristic
223C2×C4×C1372−1+411yescharacteristic
224C2×C4×C1372−1+411yescharacteristic
225C8×C137—22no—
226C8×C137—22no—
227C8×C137—22no—
228C8×C137—42no—
229C8×C137—41no—
230C8×C137—42no—
231C22×C137—21no—
232C22×C137—21no—
233C22×C137—41no—
234C22×C137—42no—
235C22×C137—41no—
236C22×C137—41no—
237C22×C137—82no—
238C22×C137—82no—
239C22×C137—81no—
240C22×C137—81no—
241C22×C137C2×#12111yescharacteristic
242C22×C137#55211yescharacteristic
243C22×C137#55611yescharacteristic
244C4×C137—21no—
245C4×C137—21no—
246C4×C137—21no—
247C4×C137—41no—
248C4×C137—41no—
249C4×C137—41no—
250C4×C137—81no—
251C4×C137—82no—
252C4×C137#47112yes—
253C4×C137#52511yescharacteristic
254C4×C137#53611yescharacteristic
255C4×C137#54711yescharacteristic
256C4×C137#54811yescharacteristic
257C2×C137—21no—
258C2×C137—21no—
259C2×C137—81no—
260C2×C137—82no—
261C2×C137#241511yescharacteristic
262C2×C137C2×#47111yescharacteristic
263C2×C137#306811yescharacteristic
264#24721C13711yesmaximal, characteristic, direct product factor, Sylow
265C137#2472111yescharacteristic, direct product factor, Sylow
266#3079C2×C13711yescharacteristic, complemented
267C2×#476C2×C13711yescharacteristic, complemented
268C4×AGL(1,ℤ/8)C2×C13711yescharacteristic
269#2925C2×C13711yescharacteristic, complemented
270#2079C2×C13712yescomplemented
271#2051C2×C13712yes—
272#1997C2×C13711yescharacteristic, complemented
273#1951C2×C13712yescomplemented
274#1950C2×C13711yescharacteristic, complemented
275#1931C2×C13712yescomplemented
276#1800C2×C13711yescharacteristic, complemented
277C2×AGL(1,ℤ/8)C22×C13711yescharacteristic
278C2×C4⋊Q8C22×C13711yescharacteristic
279C4×#41C22×C13711yescharacteristic
280C2×C4×D4C22×C13711yescharacteristic
281#477—21no—
282#476—21no—
283#476—21no—
284#476—21no—
285#476—22no—
286#475—21no—
287#474—21no—
288#442—21no—
289#440—21no—
290C4×SD16—21no—
291C4×D8—21no—
292#433—22no—
293#432—21no—
294#431—21no—
295C2×#108C22×C13712yes—
296#428C22×C13712yes—
297#428C22×C13711yescharacteristic
298C2×#106C22×C13712yes—
299#423C22×C13711yescharacteristic
300#423C22×C13711yescharacteristic
301C2×C4⋊C8C22×C13711yescharacteristic
302#418C22×C13711yescharacteristic
303#417C22×C13712yes—
304#417C22×C13711yescharacteristic
305C2×#102C22×C13712yes—
306C2×#100C22×C13712yes—
307#408C22×C13712yes—
308#405C22×C13712yes—
309C4×M16C22×C13711yescharacteristic
310#343C22×C13712yes—
311#343C22×C13712yes—
312#343C22×C13712yes—
313#341C22×C13712yes—
314C2×#41C23×C13711yescharacteristic
315C22×Q8—21no—
316C22×D4C23×C13711yescharacteristic
317C23×C4—21no—
318AGL(1,ℤ/8)—22no—
319AGL(1,ℤ/8)—22no—
320C2×SD16—21no—
321C2×D8—21no—
322C2×M16C23×C13711yescharacteristic
323C2×M16C23×C13712yes—
324C2×M16C23×C13712yes—
325C2×M16C23×C13711yescharacteristic
326C22×C8C23×C13712yes—
327C4⋊Q8—21no—
328C4⋊Q8—21no—
329C4⋊Q8—21no—
330C4⋊Q8—21no—
331C4×Q8D4×C13711yescharacteristic
332C4×Q8D4×C13711yescharacteristic
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4D4×C13711yescharacteristic
340C4×D4D4×C13711yescharacteristic
341#117—21no—
342#117C23×C13711yescharacteristic
343C2×C4⋊C4—21no—
344C2×C4⋊C4C23×C13711yescharacteristic
345C2×C4⋊C4C23×C13712yes—
346C2×C22⋊C4—21no—
347C2×C42—21no—
348C2×C42C23×C13711yescharacteristic
349#108—24no—
350#107—21no—
351#107—22no—
352#106—22no—
353#106—22no—
354#106—21no—
355#106—22no—
356C4⋊C8—21no—
357C4⋊C8—21no—
358C4⋊C8—21no—
359C4⋊C8—21no—
360C4⋊C8D4×C13711yescharacteristic
361C4⋊C8D4×C13711yescharacteristic
362C4⋊C8D4×C13712yes—
363#103—21no—
364#103—22no—
365#102—21no—
366#102—22no—
367#102—22no—
368#102—22no—
369#102—21no—
370#102—21no—
371#100—24no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374#97—22no—
375#97—21no—
376C4×C8—22no—
377C4×C8—21no—
378C24—21no—
379#41—21no—
380#41—21no—
381#41—21no—
382#41—21no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—82no—
386C2×Q8C2×D4×C13711yescharacteristic
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4C2×D4×C13711yescharacteristic
394C2×D4#41×C13712yes—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—82no—
403C22×C4#41×C13711yescharacteristic
404C22×C4#41×C13711yescharacteristic
405C22×C4C24×C13711yescharacteristic
406SD16—24no—
407D8—24no—
408M16—21no—
409M16—22no—
410M16—22no—
411M16—22no—
412M16—21no—
413M16—21no—
414M16—22no—
415M16#41×C13712yes—
416C2×C8—21no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—21no—
421C2×C8—22no—
422C2×C8C2×D4×C13712yes—
423C2×C8C2×D4×C13712yes—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4C2×D4×C13711yescharacteristic
433C4⋊C4C2×D4×C13711yescharacteristic
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42C2×D4×C13711yescharacteristic
443C42C2×D4×C13711yescharacteristic
444C42C2×D4×C13711yescharacteristic
445C42C2×D4×C13711yescharacteristic
446C23—21no—
447C23—22no—
448C23—41no—
449C23—41no—
450C23—81no—
451C23—81no—
452C23C2×#41×C13711yescharacteristic
453Q8—21no—
454Q8—21no—
455Q8—82no—
456Q8—82no—
457D4—21no—
458D4—22no—
459D4—21no—
460D4—22no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—41no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4#121×C13711yescharacteristic
481C2×C4#121×C13711yescharacteristic
482C2×C4#121×C13711yescharacteristic
483C2×C4#121×C13711yescharacteristic
484C2×C4C22×D4×C13711yescharacteristic
485C2×C4C22×D4×C13711yescharacteristic
486C2×C4C2×#41×C13711yescharacteristic
487C2×C4C2×#41×C13711yescharacteristic
488C2×C42−1+4×C13711yescharacteristic
489C2×C42−1+4×C13711yescharacteristic
490C8—22no—
491C8—22no—
492C8—22no—
493C8—42no—
494C8—41no—
495C8—42no—
496C22—21no—
497C22—21no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—42no—
502C22—82no—
503C22—82no—
504C22—81no—
505C22—81no—
506C22C2×#121×C13711yescharacteristic
507C22#552×C13711yescharacteristic
508C22#556×C13711yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—81no—
516C4—82no—
517C4#471×C13712yes—
518C4#525×C13711yescharacteristic
519C4#536×C13711yescharacteristic
520C4#547×C13711yescharacteristic
521C4#548×C13711yescharacteristic
522C2—21no—
523C2—21no—
524C2—81no—
525C2—82no—
526C2#2415×C13711yescharacteristic
527C2C2×#471×C13711yescharacteristic
528C2#3068×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#121
commutator, Frattini, cosocleC22×C4C24×C137
prime radicalC2×AGL(1,ℤ/8)×C137C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3079×C137C2semidirect
2. C2×#476×C137C2semidirect
4. #2925×C137C2semidirect
5. #2079×C137C2semidirect
7. #1997×C137C2semidirect
8. #1951×C137C2semidirect
9. #1950×C137C2semidirect
10. #1931×C137C2semidirect
11. #1800×C137C2semidirect
264. #24721C137direct (N × H)
265. C137#24721direct (N × H)
266. #3079C2×C137semidirect
267. C2×#476C2×C137semidirect
269. #2925C2×C137semidirect
270. #2079C2×C137semidirect
272. #1997C2×C137semidirect
273. #1951C2×C137semidirect
274. #1950C2×C137semidirect
275. #1931C2×C137semidirect
276. #1800C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3079×C1372
C2×#476×C1372
C4×AGL(1,ℤ/8)×C1372
#2925×C1372
#2079×C1372
#2051×C1372
#1997×C1372
#1951×C1372
#1950×C1372
#1931×C1372
#1800×C1372
#24721137