Finite Group Databasebeta

← C137×#19464 #46868850

Subgroups — C137×#19464 #46868850

Order 35072.

658 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3230×C137C211yesmaximal, characteristic, complemented
2#3110×C137C211yesmaximal, characteristic
3C2×#461×C137C211yesmaximal, characteristic
4#2974×C137C211yesmaximal, characteristic
5#1699×C137C212yesmaximal, complemented
6#1693×C137C212yesmaximal, complemented
7#1679×C137C212yesmaximal, complemented
8#1673×C137C212yesmaximal, complemented
9#1573×C137C211yesmaximal, characteristic, complemented
10#1556×C137C211yesmaximal, characteristic, complemented
11#1552×C137C211yesmaximal, characteristic, complemented
12C22×SD16×C137—21no—
13#533×C137C2211yescharacteristic
14C2×C4⋊Q8×C137C2211yescharacteristic
15C2×C4×D4×C137C2211yescharacteristic
16#500×C137C2212yescomplemented
17#487×C137C2212yescomplemented
18#484×C137—21no—
19#484×C137—21no—
20#484×C137—21no—
21#477×C137—22no—
22#474×C137C2211yescharacteristic
23#474×C137C2211yescharacteristic
24#461×C137—21no—
25#461×C137—21no—
26#461×C137C2211yescharacteristic
27#461×C137C2211yescharacteristic
28#461×C137C2212yes—
29#461×C137C2212yes—
30#450×C137—21no—
31#440×C137—21no—
32#440×C137—21no—
33#436×C137C2212yes—
34#435×C137—22no—
35C8×D4×C137C2212yes—
36#432×C137C2211yescharacteristic
37#432×C137C2211yescharacteristic
38#431×C137C2211yescharacteristic
39C2×C4⋊C8×C137C2211yescharacteristic
40C2×#103×C137—21no—
41#407×C137—21no—
42C2×C22⋊C8×C137—21no—
43#335×C137C2212yescomplemented
44#334×C137C2212yescomplemented
45#332×C137C2212yescomplemented
46#332×C137C2212yescomplemented
47#332×C137C2212yescomplemented
48#332×C137C2212yescomplemented
49C22×Q8×C137—21no—
50C22×D4×C137D411yescharacteristic
51C23×C4×C137—21no—
52C2×SD16×C137—21no—
53C2×SD16×C137—21no—
54C2×SD16×C137—22no—
55C2×SD16×C137—22no—
56C2×SD16×C137—44no—
57C2×M16×C137—22no—
58C22×C8×C137—21no—
59C22×C8×C137—22no—
60C4⋊Q8×C137—21no—
61C4⋊Q8×C137—21no—
62C4⋊Q8×C137—21no—
63C4⋊Q8×C137C2312yes—
64C4⋊Q8×C137C2311yescharacteristic
65C4⋊Q8×C137C2311yescharacteristic
66#125×C137—21no—
67C22⋊Q8×C137—21no—
68C22⋊Q8×C137—21no—
69C4×Q8×C137—21no—
70C4×Q8×C137—21no—
71C4×D4×C137—21no—
72C4×D4×C137—21no—
73C4×D4×C137D411yescharacteristic
74C4×D4×C137D411yescharacteristic
75C4×D4×C137C2311yescharacteristic
76C4×D4×C137C2311yescharacteristic
77#117×C137—21no—
78C2×C4⋊C4×C137—21no—
79C2×C4⋊C4×C137—21no—
80C2×C4⋊C4×C137D411yescharacteristic
81C2×C22⋊C4×C137—21no—
82C2×C42×C137C2311yescharacteristic
83#107×C137—22no—
84#107×C137—22no—
85#107×C137—21no—
86#107×C137—21no—
87C4⋊C8×C137—21no—
88C4⋊C8×C137D412yes—
89C4⋊C8×C137C2312yes—
90C4⋊C8×C137C2312yes—
91C4⋊C8×C137C2311yescharacteristic
92C4⋊C8×C137C2311yescharacteristic
93#103×C137—22no—
94#103×C137—22no—
95#103×C137—22no—
96#103×C137—22no—
97#103×C137—21no—
98#103×C137—21no—
99#102×C137—21no—
100#102×C137—21no—
101C22⋊C8×C137—21no—
102C22⋊C8×C137—21no—
103C22⋊C8×C137—22no—
104C22⋊C8×C137—22no—
105C22⋊C8×C137D411yescharacteristic
106C22⋊C8×C137D411yescharacteristic
107#97×C137—21no—
108#97×C137—21no—
109C4×C8×C137C2312yes—
110C24×C137—21no—
111C2×Q8×C137—21no—
112C2×Q8×C137—21no—
113C2×Q8×C137—22no—
114C2×Q8×C137—42no—
115C2×Q8×C137—42no—
116C2×Q8×C137—41no—
117C2×D4×C137—22no—
118C2×D4×C137—21no—
119C2×D4×C137—41no—
120C2×D4×C137C2×D411yescharacteristic
121C2×D4×C137C2×D411yescharacteristic
122C22×C4×C137—21no—
123C22×C4×C137—21no—
124C22×C4×C137—21no—
125C22×C4×C137—21no—
126C22×C4×C137—21no—
127C22×C4×C137—41no—
128C22×C4×C137—82no—
129C22×C4×C137C2×D411yescharacteristic
130C22×C4×C137C2×D411yescharacteristic
131C22×C4×C137#4111yescharacteristic
132SD16×C137—42no—
133SD16×C137—44no—
134SD16×C137—42no—
135SD16×C137—44no—
136M16×C137—82no—
137C2×C8×C137—22no—
138C2×C8×C137—21no—
139C2×C8×C137—21no—
140C2×C8×C137—22no—
141C2×C8×C137—42no—
142C2×C8×C137—82no—
143C2×C8×C137C2×D412yes—
144C2×C8×C137C2×D412yes—
145C4⋊C4×C137—21no—
146C4⋊C4×C137—21no—
147C4⋊C4×C137—22no—
148C4⋊C4×C137—21no—
149C4⋊C4×C137—22no—
150C4⋊C4×C137—21no—
151C4⋊C4×C137—21no—
152C4⋊C4×C137—41no—
153C4⋊C4×C137—41no—
154C4⋊C4×C137—41no—
155C4⋊C4×C137—41no—
156C4⋊C4×C137C2×D411yescharacteristic
157C4⋊C4×C137C2×D411yescharacteristic
158C22⋊C4×C137—21no—
159C22⋊C4×C137—21no—
160C22⋊C4×C137—21no—
161C22⋊C4×C137—21no—
162C22⋊C4×C137—41no—
163C42×C137—41no—
164C42×C137C2×D411yescharacteristic
165C42×C137C2×D411yescharacteristic
166C42×C137#4111yescharacteristic
167C42×C137C2411yescharacteristic
168C23×C137—21no—
169C23×C137—21no—
170C23×C137—42no—
171C23×C137—42no—
172C23×C137—81no—
173C23×C137#12111yescharacteristic
174Q8×C137—42no—
175Q8×C137—41no—
176Q8×C137—41no—
177Q8×C137—42no—
178Q8×C137—42no—
179Q8×C137—82no—
180D4×C137—21no—
181D4×C137—21no—
182D4×C137—21no—
183D4×C137—42no—
184D4×C137AGL(1,ℤ/8)12yes—
185C2×C4×C137—22no—
186C2×C4×C137—21no—
187C2×C4×C137—21no—
188C2×C4×C137—22no—
189C2×C4×C137—21no—
190C2×C4×C137—21no—
191C2×C4×C137—42no—
192C2×C4×C137—42no—
193C2×C4×C137—41no—
194C2×C4×C137—41no—
195C2×C4×C137—42no—
196C2×C4×C137—81no—
197C2×C4×C137—81no—
198C2×C4×C137—81no—
199C2×C4×C137—82no—
200C2×C4×C137#12111yescharacteristic
201C2×C4×C137#12111yescharacteristic
202C2×C4×C137#12111yescharacteristic
203C2×C4×C137C22×D411yescharacteristic
204C2×C4×C137C22×D411yescharacteristic
205C2×C4×C137C2×#4111yescharacteristic
206C8×C137—22no—
207C8×C137—22no—
208C8×C137—42no—
209C8×C137—41no—
210C8×C137—41no—
211C8×C137—42no—
212C22×C137—21no—
213C22×C137—21no—
214C22×C137—22no—
215C22×C137—41no—
216C22×C137—41no—
217C22×C137—41no—
218C22×C137—41no—
219C22×C137—41no—
220C22×C137—84no—
221C22×C137#47312yes—
222C22×C137C2×#12111yescharacteristic
223C4×C137—21no—
224C4×C137—21no—
225C4×C137—21no—
226C4×C137—21no—
227C4×C137—41no—
228C4×C137—41no—
229C4×C137—81no—
230C4×C137—82no—
231C4×C137—81no—
232C4×C137C2×AGL(1,ℤ/8)11yescharacteristic
233C4×C137#57511yescharacteristic
234C4×C137#57712yes—
235C2×C137—22no—
236C2×C137—42no—
237C2×C137—81no—
238C2×C137#304211yescharacteristic
239C2×C137#305511yescharacteristic
240C2×C137C2×#47311yescharacteristic
241#19464C13711yesmaximal, characteristic, direct product factor, Sylow
242C137#1946411yescharacteristic, direct product factor, Sylow
243#3230C2×C13711yescharacteristic, complemented
244#3110C2×C13711yescharacteristic
245C2×#461C2×C13711yescharacteristic
246#2974C2×C13711yescharacteristic
247#1699C2×C13712yescomplemented
248#1693C2×C13712yescomplemented
249#1679C2×C13712yescomplemented
250#1673C2×C13712yescomplemented
251#1573C2×C13711yescharacteristic, complemented
252#1556C2×C13711yescharacteristic, complemented
253#1552C2×C13711yescharacteristic, complemented
254C22×SD16—21no—
255#533C22×C13711yescharacteristic
256C2×C4⋊Q8C22×C13711yescharacteristic
257C2×C4×D4C22×C13711yescharacteristic
258#500C22×C13712yescomplemented
259#487C22×C13712yescomplemented
260#484—21no—
261#484—21no—
262#484—21no—
263#477—22no—
264#474C22×C13711yescharacteristic
265#474C22×C13711yescharacteristic
266#461—21no—
267#461—21no—
268#461C22×C13711yescharacteristic
269#461C22×C13711yescharacteristic
270#461C22×C13712yes—
271#461C22×C13712yes—
272#450—21no—
273#440—21no—
274#440—21no—
275#436C22×C13712yes—
276#435—22no—
277C8×D4C22×C13712yes—
278#432C22×C13711yescharacteristic
279#432C22×C13711yescharacteristic
280#431C22×C13711yescharacteristic
281C2×C4⋊C8C22×C13711yescharacteristic
282C2×#103—21no—
283#407—21no—
284C2×C22⋊C8—21no—
285#335C22×C13712yescomplemented
286#334C22×C13712yescomplemented
287#332C22×C13712yescomplemented
288#332C22×C13712yescomplemented
289#332C22×C13712yescomplemented
290#332C22×C13712yescomplemented
291C22×Q8—21no—
292C22×D4D4×C13711yescharacteristic
293C23×C4—21no—
294C2×SD16—22no—
295C2×SD16—22no—
296C2×SD16—21no—
297C2×SD16—21no—
298C2×SD16—44no—
299C2×M16—22no—
300C22×C8—21no—
301C22×C8—22no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8C23×C13712yes—
306C4⋊Q8C23×C13711yescharacteristic
307C4⋊Q8C23×C13711yescharacteristic
308#125—21no—
309C22⋊Q8—21no—
310C22⋊Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4D4×C13711yescharacteristic
316C4×D4D4×C13711yescharacteristic
317C4×D4C23×C13711yescharacteristic
318C4×D4C23×C13711yescharacteristic
319#117—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4D4×C13711yescharacteristic
323C2×C22⋊C4—21no—
324C2×C42C23×C13711yescharacteristic
325#107—22no—
326#107—22no—
327#107—21no—
328#107—21no—
329C4⋊C8—21no—
330C4⋊C8D4×C13712yes—
331C4⋊C8C23×C13712yes—
332C4⋊C8C23×C13711yescharacteristic
333C4⋊C8C23×C13712yes—
334C4⋊C8C23×C13711yescharacteristic
335#103—22no—
336#103—22no—
337#103—22no—
338#103—22no—
339#103—21no—
340#103—21no—
341#102—21no—
342#102—21no—
343C22⋊C8—21no—
344C22⋊C8—21no—
345C22⋊C8—22no—
346C22⋊C8—22no—
347C22⋊C8D4×C13711yescharacteristic
348C22⋊C8D4×C13711yescharacteristic
349#97—21no—
350#97—21no—
351C4×C8C23×C13712yes—
352C24—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—41no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—41no—
362C2×D4C2×D4×C13711yescharacteristic
363C2×D4C2×D4×C13711yescharacteristic
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—41no—
370C22×C4—82no—
371C22×C4C2×D4×C13711yescharacteristic
372C22×C4C2×D4×C13711yescharacteristic
373C22×C4#41×C13711yescharacteristic
374SD16—42no—
375SD16—44no—
376SD16—42no—
377SD16—44no—
378M16—82no—
379C2×C8—22no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—42no—
384C2×C8—82no—
385C2×C8C2×D4×C13712yes—
386C2×C8C2×D4×C13712yes—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—21no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4C2×D4×C13711yescharacteristic
399C4⋊C4C2×D4×C13711yescharacteristic
400C22⋊C4—21no—
401C22⋊C4—21no—
402C22⋊C4—21no—
403C22⋊C4—21no—
404C22⋊C4—41no—
405C42—41no—
406C42C2×D4×C13711yescharacteristic
407C42C2×D4×C13711yescharacteristic
408C42#41×C13711yescharacteristic
409C42C24×C13711yescharacteristic
410C23—21no—
411C23—21no—
412C23—42no—
413C23—42no—
414C23—81no—
415C23#121×C13711yescharacteristic
416Q8—42no—
417Q8—41no—
418Q8—41no—
419Q8—42no—
420Q8—42no—
421Q8—82no—
422D4—21no—
423D4—21no—
424D4—21no—
425D4—42no—
426D4AGL(1,ℤ/8)×C13712yes—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—82no—
442C2×C4#121×C13711yescharacteristic
443C2×C4#121×C13711yescharacteristic
444C2×C4#121×C13711yescharacteristic
445C2×C4C22×D4×C13711yescharacteristic
446C2×C4C22×D4×C13711yescharacteristic
447C2×C4C2×#41×C13711yescharacteristic
448C8—22no—
449C8—22no—
450C8—42no—
451C8—41no—
452C8—41no—
453C8—42no—
454C22—21no—
455C22—21no—
456C22—22no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—84no—
463C22#473×C13712yes—
464C22C2×#121×C13711yescharacteristic
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—41no—
470C4—41no—
471C4—81no—
472C4—82no—
473C4—81no—
474C4C2×AGL(1,ℤ/8)×C13711yescharacteristic
475C4#575×C13711yescharacteristic
476C4#577×C13712yes—
477C2—22no—
478C2—42no—
479C2—81no—
480C2#3042×C13711yescharacteristic
481C2#3055×C13711yescharacteristic
482C2C2×#473×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#121
commutator, Frattini, cosocleC42C24×C137
prime radicalC22×D4×C137D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3230×C137C2semidirect
5. #1699×C137C2semidirect
6. #1693×C137C2semidirect
7. #1679×C137C2semidirect
8. #1673×C137C2semidirect
9. #1573×C137C2semidirect
10. #1556×C137C2semidirect
11. #1552×C137C2semidirect
16. #500×C137C22semidirect
17. #487×C137C22semidirect
43. #335×C137C22semidirect
44. #334×C137C22semidirect
45. #332×C137C22semidirect
46. #332×C137C22semidirect
47. #332×C137C22semidirect
48. #332×C137C22semidirect
241. #19464C137direct (N × H)
242. C137#19464direct (N × H)
243. #3230C2×C137semidirect
247. #1699C2×C137semidirect
248. #1693C2×C137semidirect
249. #1679C2×C137semidirect
250. #1673C2×C137semidirect
251. #1573C2×C137semidirect
252. #1556C2×C137semidirect
253. #1552C2×C137semidirect
258. #500C22×C137semidirect
259. #487C22×C137semidirect
285. #335C22×C137semidirect
286. #334C22×C137semidirect
287. #332C22×C137semidirect
288. #332C22×C137semidirect
289. #332C22×C137semidirect
290. #332C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3230×C1372
#3110×C1372
C2×#461×C1372
#2974×C1372
#1699×C1372
#1693×C1372
#1679×C1372
#1673×C1372
#1573×C1372
#1556×C1372
#1552×C1372
#19464137