Finite Group Databasebeta

← C137×#19234 #46868620

Subgroups — C137×#19234 #46868620

Order 35072.

704 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3246×C137C211yesmaximal, characteristic, complemented
2#3205×C137C211yesmaximal, characteristic
3#3138×C137C211yesmaximal, characteristic, complemented
4#2935×C137C211yesmaximal, characteristic, complemented
5#1701×C137C212yesmaximal, complemented
6#1697×C137C212yesmaximal, complemented
7#1681×C137C212yesmaximal, complemented
8#1677×C137C212yesmaximal, complemented
9#1565×C137C211yesmaximal, characteristic, complemented
10#1542×C137C211yesmaximal, characteristic, complemented
11C2×#331×C137C211yesmaximal, characteristic, complemented
12C2×#131×C137—21no—
13#533×C137C2211yescharacteristic
14#532×C137C2211yescharacteristic
15C2×#127×C137C2211yescharacteristic
16#493×C137—21no—
17#490×C137—22no—
18#486×C137—21no—
19#486×C137C2212yes—
20#486×C137C2212yes—
21#476×C137C2212yescomplemented
22#475×C137—21no—
23#465×C137—22no—
24#461×C137C2212yescomplemented
25#460×C137—21no—
26#459×C137C2212yes—
27#459×C137C2212yes—
28#447×C137—22no—
29#436×C137C2212yescomplemented
30#435×C137—22no—
31C8×D4×C137C2212yescomplemented
32#431×C137C2211yescharacteristic
33C2×C4⋊C8×C137C2211yescharacteristic
34C2×C4⋊C8×C137C2211yescharacteristic
35#418×C137—21no—
36#418×C137—21no—
37#408×C137—21no—
38C4×M16×C137C2211yescharacteristic
39#335×C137C2212yescomplemented
40#334×C137C2212yescomplemented
41#333×C137C2211yescharacteristic, complemented
42#333×C137C2211yescharacteristic, complemented
43#332×C137C2211yescharacteristic, complemented
44#332×C137C2211yescharacteristic, complemented
45#331×C137C2212yes—
46#331×C137C2212yes—
47C2×#41×C137—21no—
48C22×D4×C137—21no—
49C22×D4×C137—21no—
50C22×D4×C137—21no—
51C2×SD16×C137—22no—
52C2×D8×C137—22no—
53C2×D8×C137—22no—
54#131×C137—44no—
55C2×M16×C137—22no—
56C2×M16×C137D411yescharacteristic
57C2×M16×C137D411yescharacteristic
58C22×C8×C137—21no—
59C22×C8×C137—21no—
60C22×C8×C137—22no—
61C4⋊Q8×C137D411yescharacteristic
62C4⋊Q8×C137C2311yescharacteristic
63C4⋊Q8×C137C2311yescharacteristic
64#127×C137—21no—
65#127×C137—21no—
66#127×C137—21no—
67#127×C137D411yescharacteristic
68#127×C137C2312yes—
69#125×C137—21no—
70#124×C137—21no—
71C22⋊Q8×C137—21no—
72C22⋊Q8×C137—21no—
73#121×C137—21no—
74#121×C137D412yes—
75C4×Q8×C137—21no—
76C4×Q8×C137—21no—
77C4×D4×C137—21no—
78C4×D4×C137C2312yes—
79#117×C137—21no—
80C2×C42×C137C2311yescharacteristic
81#106×C137—22no—
82C4⋊C8×C137—21no—
83C4⋊C8×C137—21no—
84C4⋊C8×C137C2311yescharacteristic
85C4⋊C8×C137C2311yescharacteristic
86C4⋊C8×C137C2312yes—
87C4⋊C8×C137C2311yescharacteristic
88C4⋊C8×C137C2311yescharacteristic
89#103×C137—22no—
90#102×C137—22no—
91#102×C137—22no—
92#102×C137—22no—
93#102×C137—22no—
94#102×C137—22no—
95#102×C137—22no—
96#102×C137—22no—
97C22⋊C8×C137—21no—
98C22⋊C8×C137—22no—
99C22⋊C8×C137—22no—
100#97×C137D412yes—
101C4×C8×C137—21no—
102C4×C8×C137C2312yes—
103C24×C137—41no—
104#41×C137—42no—
105#41×C137—42no—
106C2×Q8×C137—21no—
107C2×Q8×C137—21no—
108C2×Q8×C137—21no—
109C2×D4×C137—22no—
110C2×D4×C137—21no—
111C2×D4×C137—21no—
112C2×D4×C137—22no—
113C2×D4×C137—22no—
114C2×D4×C137—22no—
115C2×D4×C137—41no—
116C2×D4×C137—41no—
117C2×D4×C137—41no—
118C2×D4×C137—42no—
119C2×D4×C137—42no—
120C2×D4×C137—81no—
121C2×D4×C137—81no—
122C2×D4×C137—82no—
123C2×D4×C137C2×D412yes—
124C22×C4×C137—22no—
125C22×C4×C137C2×D411yescharacteristic
126C22×C4×C137C2×D411yescharacteristic
127C22×C4×C137#4111yescharacteristic
128SD16×C137—82no—
129D8×C137—82no—
130D8×C137—82no—
131M16×C137—22no—
132M16×C137—22no—
133M16×C137—44no—
134C2×C8×C137—21no—
135C2×C8×C137—21no—
136C2×C8×C137—21no—
137C2×C8×C137—21no—
138C2×C8×C137—22no—
139C2×C8×C137—44no—
140C2×C8×C137—42no—
141C2×C8×C137—81no—
142C2×C8×C137C2×D412yes—
143C2×C8×C137C2×D412yes—
144C4⋊C4×C137—21no—
145C4⋊C4×C137—21no—
146C4⋊C4×C137—21no—
147C4⋊C4×C137—21no—
148C4⋊C4×C137—21no—
149C4⋊C4×C137—41no—
150C4⋊C4×C137—41no—
151C4⋊C4×C137C2×D412yes—
152C22⋊C4×C137—22no—
153C22⋊C4×C137—22no—
154C22⋊C4×C137—41no—
155C42×C137—41no—
156C42×C137C2×D411yescharacteristic
157C42×C137C2×D411yescharacteristic
158C42×C137#4111yescharacteristic
159C42×C137C2411yescharacteristic
160C23×C137—22no—
161C23×C137—42no—
162C23×C137—82no—
163C23×C137—82no—
164C23×C137—82no—
165C23×C137#12111yescharacteristic
166Q8×C137—42no—
167Q8×C137—41no—
168Q8×C137—41no—
169Q8×C137—81no—
170D4×C137—22no—
171D4×C137—22no—
172D4×C137—44no—
173D4×C137—42no—
174D4×C137—42no—
175D4×C137—42no—
176D4×C137—42no—
177D4×C137—42no—
178D4×C137—81no—
179D4×C137—82no—
180D4×C137—82no—
181D4×C137—82no—
182D4×C137—82no—
183C2×C4×C137—21no—
184C2×C4×C137—21no—
185C2×C4×C137—22no—
186C2×C4×C137—41no—
187C2×C4×C137—42no—
188C2×C4×C137—42no—
189C2×C4×C137—41no—
190C2×C4×C137—41no—
191C2×C4×C137#12111yescharacteristic
192C2×C4×C137#12111yescharacteristic
193C2×C4×C137#12111yescharacteristic
194C2×C4×C137#13512yes—
195C2×C4×C137AGL(1,ℤ/8)12yes—
196C2×C4×C137C22×D411yescharacteristic
197C2×C4×C137C22×D411yescharacteristic
198C2×C4×C137C2×#4111yescharacteristic
199C8×C137—22no—
200C8×C137—22no—
201C8×C137—41no—
202C8×C137—41no—
203C8×C137—42no—
204C8×C137—41no—
205C8×C137—41no—
206C22×C137—21no—
207C22×C137—21no—
208C22×C137—42no—
209C22×C137—42no—
210C22×C137—42no—
211C22×C137—82no—
212C22×C137—82no—
213C22×C137—82no—
214C22×C137—164no—
215C22×C137#46312yes—
216C22×C137C2×#12111yescharacteristic
217C4×C137—21no—
218C4×C137—21no—
219C4×C137—21no—
220C4×C137—42no—
221C4×C137—81no—
222C4×C137—81no—
223C4×C137#46312yes—
224C4×C137C2×#13511yescharacteristic
225C4×C137C2×AGL(1,ℤ/8)11yescharacteristic
226C4×C137#57611yescharacteristic
227C4×C137#57711yescharacteristic
228C2×C137—22no—
229C2×C137—82no—
230C2×C137—162no—
231C2×C137C2×#46311yescharacteristic
232C2×C137#306211yescharacteristic
233C2×C137#306811yescharacteristic
234#19234C13711yesmaximal, characteristic, direct product factor, Sylow
235C137#1923411yescharacteristic, direct product factor, Sylow
236#3246C2×C13711yescharacteristic, complemented
237#3205C2×C13711yescharacteristic
238#3138C2×C13711yescharacteristic, complemented
239#2935C2×C13711yescharacteristic, complemented
240#1701C2×C13712yescomplemented
241#1697C2×C13712yescomplemented
242#1681C2×C13712yescomplemented
243#1677C2×C13712yescomplemented
244#1565C2×C13711yescharacteristic, complemented
245#1542C2×C13711yescharacteristic, complemented
246C2×#331C2×C13711yescharacteristic, complemented
247C2×#131—21no—
248#533C22×C13711yescharacteristic
249#532C22×C13711yescharacteristic
250C2×#127C22×C13711yescharacteristic
251#493—21no—
252#490—22no—
253#486—21no—
254#486C22×C13712yes—
255#486C22×C13712yes—
256#476C22×C13712yescomplemented
257#475—21no—
258#465—22no—
259#461C22×C13712yescomplemented
260#460—21no—
261#459C22×C13712yes—
262#459C22×C13712yes—
263#447—22no—
264#436C22×C13712yescomplemented
265#435—22no—
266C8×D4C22×C13712yescomplemented
267#431C22×C13711yescharacteristic
268C2×C4⋊C8C22×C13711yescharacteristic
269C2×C4⋊C8C22×C13711yescharacteristic
270#418—21no—
271#418—21no—
272#408—21no—
273C4×M16C22×C13711yescharacteristic
274#335C22×C13712yescomplemented
275#334C22×C13712yescomplemented
276#333C22×C13711yescharacteristic, complemented
277#333C22×C13711yescharacteristic, complemented
278#332C22×C13711yescharacteristic, complemented
279#332C22×C13711yescharacteristic, complemented
280#331C22×C13712yes—
281#331C22×C13712yes—
282C2×#41—21no—
283C22×D4—21no—
284C22×D4—21no—
285C22×D4—21no—
286C2×SD16—22no—
287C2×D8—22no—
288C2×D8—22no—
289#131—44no—
290C2×M16—22no—
291C2×M16D4×C13711yescharacteristic
292C2×M16D4×C13711yescharacteristic
293C22×C8—21no—
294C22×C8—21no—
295C22×C8—22no—
296C4⋊Q8D4×C13711yescharacteristic
297C4⋊Q8C23×C13711yescharacteristic
298C4⋊Q8C23×C13711yescharacteristic
299#127—21no—
300#127—21no—
301#127—21no—
302#127D4×C13711yescharacteristic
303#127C23×C13712yes—
304#125—21no—
305#124—21no—
306C22⋊Q8—21no—
307C22⋊Q8—21no—
308#121—21no—
309#121D4×C13712yes—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×D4—21no—
313C4×D4C23×C13712yes—
314#117—21no—
315C2×C42C23×C13711yescharacteristic
316#106—22no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8C23×C13711yescharacteristic
320C4⋊C8C23×C13711yescharacteristic
321C4⋊C8C23×C13712yes—
322C4⋊C8C23×C13711yescharacteristic
323C4⋊C8C23×C13711yescharacteristic
324#103—22no—
325#102—22no—
326#102—22no—
327#102—22no—
328#102—22no—
329#102—22no—
330#102—22no—
331#102—22no—
332C22⋊C8—21no—
333C22⋊C8—22no—
334C22⋊C8—22no—
335#97D4×C13712yes—
336C4×C8—21no—
337C4×C8C23×C13712yes—
338C24—41no—
339#41—42no—
340#41—42no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×D4—22no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—81no—
356C2×D4—81no—
357C2×D4—82no—
358C2×D4C2×D4×C13712yes—
359C22×C4—22no—
360C22×C4C2×D4×C13711yescharacteristic
361C22×C4C2×D4×C13711yescharacteristic
362C22×C4#41×C13711yescharacteristic
363SD16—82no—
364D8—82no—
365D8—82no—
366M16—22no—
367M16—22no—
368M16—44no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—22no—
374C2×C8—44no—
375C2×C8—42no—
376C2×C8—81no—
377C2×C8C2×D4×C13712yes—
378C2×C8C2×D4×C13712yes—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4C2×D4×C13712yes—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—41no—
390C42—41no—
391C42C2×D4×C13711yescharacteristic
392C42C2×D4×C13711yescharacteristic
393C42#41×C13711yescharacteristic
394C42C24×C13711yescharacteristic
395C23—22no—
396C23—42no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23#121×C13711yescharacteristic
401Q8—42no—
402Q8—41no—
403Q8—41no—
404Q8—81no—
405D4—22no—
406D4—22no—
407D4—44no—
408D4—42no—
409D4—42no—
410D4—42no—
411D4—42no—
412D4—42no—
413D4—81no—
414D4—82no—
415D4—82no—
416D4—82no—
417D4—82no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4#121×C13711yescharacteristic
427C2×C4#121×C13711yescharacteristic
428C2×C4#121×C13711yescharacteristic
429C2×C4#135×C13712yes—
430C2×C4AGL(1,ℤ/8)×C13712yes—
431C2×C4C22×D4×C13711yescharacteristic
432C2×C4C22×D4×C13711yescharacteristic
433C2×C4C2×#41×C13711yescharacteristic
434C8—22no—
435C8—22no—
436C8—41no—
437C8—41no—
438C8—42no—
439C8—41no—
440C8—41no—
441C22—21no—
442C22—21no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22—164no—
450C22#463×C13712yes—
451C22C2×#121×C13711yescharacteristic
452C4—21no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—81no—
457C4—81no—
458C4#463×C13712yes—
459C4C2×#135×C13711yescharacteristic
460C4C2×AGL(1,ℤ/8)×C13711yescharacteristic
461C4#576×C13711yescharacteristic
462C4#577×C13711yescharacteristic
463C2—22no—
464C2—82no—
465C2—162no—
466C2C2×#463×C13711yescharacteristic
467C2#3062×C13711yescharacteristic
468C2#3068×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#121
commutator, Frattini, cosocleC42C24×C137
prime radical#3205×C137C2
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. #3246×C137C2semidirect
3. #3138×C137C2semidirect
4. #2935×C137C2semidirect
5. #1701×C137C2semidirect
6. #1697×C137C2semidirect
7. #1681×C137C2semidirect
8. #1677×C137C2semidirect
9. #1565×C137C2semidirect
10. #1542×C137C2semidirect
11. C2×#331×C137C2semidirect
21. #476×C137C22semidirect
24. #461×C137C22semidirect
29. #436×C137C22semidirect
31. C8×D4×C137C22semidirect
39. #335×C137C22semidirect
40. #334×C137C22semidirect
41. #333×C137C22semidirect
42. #333×C137C22semidirect
43. #332×C137C22semidirect
44. #332×C137C22semidirect
234. #19234C137direct (N × H)
235. C137#19234direct (N × H)
236. #3246C2×C137semidirect
238. #3138C2×C137semidirect
239. #2935C2×C137semidirect
240. #1701C2×C137semidirect
241. #1697C2×C137semidirect
242. #1681C2×C137semidirect
243. #1677C2×C137semidirect
244. #1565C2×C137semidirect
245. #1542C2×C137semidirect
246. C2×#331C2×C137semidirect
256. #476C22×C137semidirect
259. #461C22×C137semidirect
264. #436C22×C137semidirect
266. C8×D4C22×C137semidirect
274. #335C22×C137semidirect
275. #334C22×C137semidirect
276. #333C22×C137semidirect
277. #333C22×C137semidirect
278. #332C22×C137semidirect
279. #332C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3246×C1372
#3205×C1372
#3138×C1372
#2935×C1372
#1701×C1372
#1697×C1372
#1681×C1372
#1677×C1372
#1565×C1372
#1542×C1372
C2×#331×C1372
#19234137