Finite Group Databasebeta

← C137×#19468 #46868854

Subgroups — C137×#19468 #46868854

Order 35072.

672 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3208×C137C211yesmaximal, characteristic
2#3113×C137C211yesmaximal, characteristic, complemented
3C2×#460×C137C211yesmaximal, characteristic
4#2984×C137C211yesmaximal, characteristic, complemented
5#1702×C137C212yesmaximal, complemented
6#1686×C137C212yesmaximal, complemented
7#1682×C137C212yesmaximal, complemented
8#1666×C137C212yesmaximal, complemented
9#1573×C137C211yesmaximal, characteristic, complemented
10#1555×C137C211yesmaximal, characteristic
11#1552×C137C211yesmaximal, characteristic, complemented
12C22×SD16×C137—21no—
13#533×C137C2211yescharacteristic
14C2×#127×C137C2211yescharacteristic
15C2×C4×Q8×C137C2211yescharacteristic
16#492×C137C2212yes—
17#486×C137C2212yes—
18#483×C137—21no—
19#475×C137C2211yescharacteristic, complemented
20#475×C137C2211yescharacteristic, complemented
21#462×C137C2211yescharacteristic, complemented
22#462×C137C2211yescharacteristic, complemented
23#460×C137—21no—
24#460×C137—21no—
25#460×C137—22no—
26#460×C137C2212yes—
27#460×C137C2212yes—
28#451×C137—21no—
29#450×C137—21no—
30#450×C137—21no—
31#446×C137—22no—
32#446×C137C2212yescomplemented
33C8×Q8×C137C2212yescomplemented
34#441×C137—21no—
35#441×C137—21no—
36#432×C137—21no—
37#432×C137C2211yescharacteristic
38#432×C137C2211yescharacteristic
39#431×C137—21no—
40#431×C137C2211yescharacteristic
41C2×C4⋊C8×C137C2211yescharacteristic
42C2×#102×C137—21no—
43#335×C137C2212yescomplemented
44#334×C137C2212yescomplemented
45#332×C137C2212yescomplemented
46#332×C137C2212yescomplemented
47#331×C137C2212yes—
48#331×C137C2212yes—
49C22×Q8×C137D411yescharacteristic
50C22×D4×C137—21no—
51C22×D4×C137—21no—
52C22×D4×C137—21no—
53C2×Q16×C137—21no—
54C2×Q16×C137—21no—
55C2×SD16×C137—22no—
56C2×SD16×C137—22no—
57C2×SD16×C137—22no—
58C2×SD16×C137—22no—
59C2×SD16×C137—44no—
60C22×C8×C137—21no—
61C4⋊Q8×C137C2311yescharacteristic
62C4⋊Q8×C137C2311yescharacteristic
63#127×C137—21no—
64#127×C137—21no—
65#127×C137—21no—
66#127×C137C2312yes—
67#125×C137—21no—
68C22⋊Q8×C137—21no—
69C22⋊Q8×C137—21no—
70C4×Q8×C137—21no—
71C4×Q8×C137—21no—
72C4×Q8×C137—21no—
73C4×Q8×C137—21no—
74C4×Q8×C137D411yescharacteristic
75C4×Q8×C137D411yescharacteristic
76C4×Q8×C137C2311yescharacteristic
77C4×Q8×C137C2311yescharacteristic
78#117×C137—21no—
79C2×C4⋊C4×C137—21no—
80C2×C4⋊C4×C137D411yescharacteristic
81C2×C42×C137—21no—
82C2×C42×C137C2311yescharacteristic
83#106×C137—21no—
84#106×C137—21no—
85C4⋊C8×C137—22no—
86C4⋊C8×C137—22no—
87C4⋊C8×C137—21no—
88C4⋊C8×C137D412yes—
89C4⋊C8×C137C2312yes—
90C4⋊C8×C137C2312yes—
91C4⋊C8×C137C2311yescharacteristic
92C4⋊C8×C137C2311yescharacteristic
93#103×C137—21no—
94#103×C137—21no—
95#103×C137—21no—
96#103×C137—21no—
97#102×C137—22no—
98#102×C137—22no—
99#102×C137—22no—
100#102×C137—22no—
101C22⋊C8×C137—21no—
102C22⋊C8×C137—21no—
103C22⋊C8×C137D411yescharacteristic
104C22⋊C8×C137D411yescharacteristic
105#97×C137—22no—
106#97×C137—21no—
107#97×C137—21no—
108C4×C8×C137—22no—
109C4×C8×C137C2312yes—
110C24×C137—41no—
111C2×Q8×C137—21no—
112C2×Q8×C137—22no—
113C2×Q8×C137—21no—
114C2×Q8×C137—21no—
115C2×Q8×C137—41no—
116C2×Q8×C137C2×D411yescharacteristic
117C2×Q8×C137C2×D411yescharacteristic
118C2×D4×C137—22no—
119C2×D4×C137—22no—
120C2×D4×C137—22no—
121C2×D4×C137—42no—
122C2×D4×C137—42no—
123C2×D4×C137—41no—
124C2×D4×C137—41no—
125C2×D4×C137—41no—
126C2×D4×C137—82no—
127C2×D4×C137—82no—
128C22×C4×C137—21no—
129C22×C4×C137—21no—
130C22×C4×C137C2×D411yescharacteristic
131C22×C4×C137C2×D411yescharacteristic
132C22×C4×C137#4111yescharacteristic
133Q16×C137—42no—
134Q16×C137—42no—
135SD16×C137—44no—
136SD16×C137—44no—
137SD16×C137—82no—
138SD16×C137—82no—
139C2×C8×C137—22no—
140C2×C8×C137—21no—
141C2×C8×C137—21no—
142C2×C8×C137—22no—
143C2×C8×C137—42no—
144C2×C8×C137C2×D412yes—
145C2×C8×C137C2×D412yes—
146C4⋊C4×C137—21no—
147C4⋊C4×C137—21no—
148C4⋊C4×C137—21no—
149C4⋊C4×C137—21no—
150C4⋊C4×C137—21no—
151C4⋊C4×C137—21no—
152C4⋊C4×C137—21no—
153C4⋊C4×C137—21no—
154C4⋊C4×C137—21no—
155C4⋊C4×C137—41no—
156C4⋊C4×C137—41no—
157C4⋊C4×C137C2×D411yescharacteristic
158C4⋊C4×C137C2×D411yescharacteristic
159C22⋊C4×C137—41no—
160C42×C137—21no—
161C42×C137—21no—
162C42×C137—21no—
163C42×C137—21no—
164C42×C137—41no—
165C42×C137C2×D411yescharacteristic
166C42×C137C2×D411yescharacteristic
167C42×C137#4111yescharacteristic
168C42×C137C2411yescharacteristic
169C23×C137—42no—
170C23×C137—82no—
171C23×C137—82no—
172C23×C137—82no—
173C23×C137#12111yescharacteristic
174Q8×C137—21no—
175Q8×C137—21no—
176Q8×C137—21no—
177Q8×C137—42no—
178Q8×C137—41no—
179Q8×C137—41no—
180Q8×C137—42no—
181Q8×C137AGL(1,ℤ/8)12yes—
182D4×C137—42no—
183D4×C137—42no—
184D4×C137—42no—
185D4×C137—42no—
186D4×C137—82no—
187D4×C137—82no—
188D4×C137—82no—
189D4×C137—82no—
190C2×C4×C137—21no—
191C2×C4×C137—22no—
192C2×C4×C137—21no—
193C2×C4×C137—21no—
194C2×C4×C137—22no—
195C2×C4×C137—21no—
196C2×C4×C137—21no—
197C2×C4×C137—21no—
198C2×C4×C137—42no—
199C2×C4×C137—42no—
200C2×C4×C137—41no—
201C2×C4×C137—41no—
202C2×C4×C137#12111yescharacteristic
203C2×C4×C137#12111yescharacteristic
204C2×C4×C137#12111yescharacteristic
205C2×C4×C137C22×D411yescharacteristic
206C2×C4×C137C22×D411yescharacteristic
207C2×C4×C137C2×#4111yescharacteristic
208C8×C137—22no—
209C8×C137—22no—
210C8×C137—42no—
211C8×C137—41no—
212C8×C137—41no—
213C8×C137—42no—
214C22×C137—21no—
215C22×C137—21no—
216C22×C137—82no—
217C22×C137—82no—
218C22×C137—82no—
219C22×C137—164no—
220C22×C137#47112yes—
221C22×C137C2×#12111yescharacteristic
222C4×C137—21no—
223C4×C137—21no—
224C4×C137—22no—
225C4×C137—21no—
226C4×C137—21no—
227C4×C137—41no—
228C4×C137—41no—
229C4×C137—41no—
230C4×C137—81no—
231C4×C137—81no—
232C4×C137C2×AGL(1,ℤ/8)11yescharacteristic
233C4×C137#57511yescharacteristic
234C4×C137#57712yes—
235C2×C137—22no—
236C2×C137—162no—
237C2×C137#304311yescharacteristic
238C2×C137#305511yescharacteristic
239C2×C137C2×#47111yescharacteristic
240#19468C13711yesmaximal, characteristic, direct product factor, Sylow
241C137#1946811yescharacteristic, direct product factor, Sylow
242#3208C2×C13711yescharacteristic
243#3113C2×C13711yescharacteristic, complemented
244C2×#460C2×C13711yescharacteristic
245#2984C2×C13711yescharacteristic, complemented
246#1702C2×C13712yescomplemented
247#1686C2×C13712yescomplemented
248#1682C2×C13712yescomplemented
249#1666C2×C13712yescomplemented
250#1573C2×C13711yescharacteristic, complemented
251#1555C2×C13711yescharacteristic
252#1552C2×C13711yescharacteristic, complemented
253C22×SD16—21no—
254#533C22×C13711yescharacteristic
255C2×#127C22×C13711yescharacteristic
256C2×C4×Q8C22×C13711yescharacteristic
257#492C22×C13712yes—
258#486C22×C13712yes—
259#483—21no—
260#475C22×C13711yescharacteristic, complemented
261#475C22×C13711yescharacteristic, complemented
262#462C22×C13711yescharacteristic, complemented
263#462C22×C13711yescharacteristic, complemented
264#460—21no—
265#460—21no—
266#460—22no—
267#460C22×C13712yes—
268#460C22×C13712yes—
269#451—21no—
270#450—21no—
271#450—21no—
272#446—22no—
273#446C22×C13712yescomplemented
274C8×Q8C22×C13712yescomplemented
275#441—21no—
276#441—21no—
277#432—21no—
278#432C22×C13711yescharacteristic
279#432C22×C13711yescharacteristic
280#431—21no—
281#431C22×C13711yescharacteristic
282C2×C4⋊C8C22×C13711yescharacteristic
283C2×#102—21no—
284#335C22×C13712yescomplemented
285#334C22×C13712yescomplemented
286#332C22×C13712yescomplemented
287#332C22×C13712yescomplemented
288#331C22×C13712yes—
289#331C22×C13712yes—
290C22×Q8D4×C13711yescharacteristic
291C22×D4—21no—
292C22×D4—21no—
293C22×D4—21no—
294C2×Q16—21no—
295C2×Q16—21no—
296C2×SD16—22no—
297C2×SD16—22no—
298C2×SD16—22no—
299C2×SD16—22no—
300C2×SD16—44no—
301C22×C8—21no—
302C4⋊Q8C23×C13711yescharacteristic
303C4⋊Q8C23×C13711yescharacteristic
304#127—21no—
305#127—21no—
306#127—21no—
307#127C23×C13712yes—
308#125—21no—
309C22⋊Q8—21no—
310C22⋊Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8D4×C13711yescharacteristic
316C4×Q8D4×C13711yescharacteristic
317C4×Q8C23×C13711yescharacteristic
318C4×Q8C23×C13711yescharacteristic
319#117—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4D4×C13711yescharacteristic
322C2×C42—21no—
323C2×C42C23×C13711yescharacteristic
324#106—21no—
325#106—21no—
326C4⋊C8—22no—
327C4⋊C8—22no—
328C4⋊C8—21no—
329C4⋊C8D4×C13712yes—
330C4⋊C8C23×C13712yes—
331C4⋊C8C23×C13711yescharacteristic
332C4⋊C8C23×C13711yescharacteristic
333C4⋊C8C23×C13712yes—
334#103—21no—
335#103—21no—
336#103—21no—
337#103—21no—
338#102—22no—
339#102—22no—
340#102—22no—
341#102—22no—
342C22⋊C8—21no—
343C22⋊C8—21no—
344C22⋊C8D4×C13711yescharacteristic
345C22⋊C8D4×C13711yescharacteristic
346#97—22no—
347#97—21no—
348#97—21no—
349C4×C8—22no—
350C4×C8C23×C13712yes—
351C24—41no—
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—41no—
357C2×Q8C2×D4×C13711yescharacteristic
358C2×Q8C2×D4×C13711yescharacteristic
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—82no—
368C2×D4—82no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4C2×D4×C13711yescharacteristic
372C22×C4C2×D4×C13711yescharacteristic
373C22×C4#41×C13711yescharacteristic
374Q16—42no—
375Q16—42no—
376SD16—44no—
377SD16—44no—
378SD16—82no—
379SD16—82no—
380C2×C8—22no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—22no—
384C2×C8—42no—
385C2×C8C2×D4×C13712yes—
386C2×C8C2×D4×C13712yes—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4C2×D4×C13711yescharacteristic
399C4⋊C4C2×D4×C13711yescharacteristic
400C22⋊C4—41no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—41no—
406C42C2×D4×C13711yescharacteristic
407C42C2×D4×C13711yescharacteristic
408C42#41×C13711yescharacteristic
409C42C24×C13711yescharacteristic
410C23—42no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23#121×C13711yescharacteristic
415Q8—21no—
416Q8—21no—
417Q8—21no—
418Q8—42no—
419Q8—41no—
420Q8—41no—
421Q8—42no—
422Q8AGL(1,ℤ/8)×C13712yes—
423D4—42no—
424D4—42no—
425D4—42no—
426D4—42no—
427D4—82no—
428D4—82no—
429D4—82no—
430D4—82no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4#121×C13711yescharacteristic
444C2×C4#121×C13711yescharacteristic
445C2×C4#121×C13711yescharacteristic
446C2×C4C22×D4×C13711yescharacteristic
447C2×C4C22×D4×C13711yescharacteristic
448C2×C4C2×#41×C13711yescharacteristic
449C8—22no—
450C8—22no—
451C8—42no—
452C8—41no—
453C8—41no—
454C8—42no—
455C22—21no—
456C22—21no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—164no—
461C22#471×C13712yes—
462C22C2×#121×C13711yescharacteristic
463C4—21no—
464C4—21no—
465C4—22no—
466C4—21no—
467C4—21no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—81no—
472C4—81no—
473C4C2×AGL(1,ℤ/8)×C13711yescharacteristic
474C4#575×C13711yescharacteristic
475C4#577×C13712yes—
476C2—22no—
477C2—162no—
478C2#3043×C13711yescharacteristic
479C2#3055×C13711yescharacteristic
480C2C2×#471×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#121
commutator, Frattini, cosocleC42C24×C137
prime radicalC2×#127×C137C22
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
2. #3113×C137C2semidirect
4. #2984×C137C2semidirect
5. #1702×C137C2semidirect
6. #1686×C137C2semidirect
7. #1682×C137C2semidirect
8. #1666×C137C2semidirect
9. #1573×C137C2semidirect
11. #1552×C137C2semidirect
19. #475×C137C22semidirect
20. #475×C137C22semidirect
21. #462×C137C22semidirect
22. #462×C137C22semidirect
32. #446×C137C22semidirect
33. C8×Q8×C137C22semidirect
43. #335×C137C22semidirect
44. #334×C137C22semidirect
45. #332×C137C22semidirect
46. #332×C137C22semidirect
240. #19468C137direct (N × H)
241. C137#19468direct (N × H)
243. #3113C2×C137semidirect
245. #2984C2×C137semidirect
246. #1702C2×C137semidirect
247. #1686C2×C137semidirect
248. #1682C2×C137semidirect
249. #1666C2×C137semidirect
250. #1573C2×C137semidirect
252. #1552C2×C137semidirect
260. #475C22×C137semidirect
261. #475C22×C137semidirect
262. #462C22×C137semidirect
263. #462C22×C137semidirect
273. #446C22×C137semidirect
274. C8×Q8C22×C137semidirect
284. #335C22×C137semidirect
285. #334C22×C137semidirect
286. #332C22×C137semidirect
287. #332C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3208×C1372
#3113×C1372
C2×#460×C1372
#2984×C1372
#1702×C1372
#1686×C1372
#1682×C1372
#1666×C1372
#1573×C1372
#1555×C1372
#1552×C1372
#19468137