Finite Group Databasebeta

← C137×#18847 #46868233

Subgroups — C137×#18847 #46868233

Order 35072.

902 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3501×C137C211yesmaximal, characteristic
2#3255×C137C211yesmaximal, characteristic, complemented
3#3231×C137C211yesmaximal, characteristic, complemented
4#3110×C137C211yesmaximal, characteristic
5#1651×C137C212yesmaximal, complemented
6#1649×C137C212yesmaximal, complemented
7#1644×C137C212yesmaximal, complemented
8#1643×C137C212yesmaximal, complemented
9#1559×C137C211yesmaximal, characteristic, complemented
10#1497×C137C211yesmaximal, characteristic, complemented
11#1486×C137C211yesmaximal, characteristic
12#554×C137—21no—
13#554×C137—22no—
14#554×C137—22no—
15#554×C137C2212yes—
16D4×Q8×C137—21no—
17D4×Q8×C137C2212yes—
18#544×C137—21no—
19#543×C137—22no—
20#533×C137C2211yescharacteristic
21#533×C137C2211yescharacteristic
22#533×C137C2211yescharacteristic
23C2×C22⋊Q8×C137—22no—
24#518×C137—21no—
25C2×C4×D4×C137C2211yescharacteristic
26#500×C137C2211yescharacteristic, complemented
27#498×C137C2211yescharacteristic, complemented
28#487×C137C2211yescharacteristic, complemented
29#487×C137C2211yescharacteristic, complemented
30#485×C137—22no—
31#484×C137—21no—
32#479×C137—22no—
33#475×C137—21no—
34#475×C137C2212yescomplemented
35#474×C137C2212yes—
36#470×C137—22no—
37#462×C137—21no—
38#462×C137C2212yescomplemented
39#461×C137C2212yes—
40#450×C137—21no—
41#440×C137—22no—
42#432×C137C2211yescharacteristic
43#431×C137C2211yescharacteristic
44#423×C137C2211yescharacteristic
45#333×C137C2212yescomplemented
46#332×C137C2212yescomplemented
47Q8⋊C8×C137C2212yescomplemented
48Q8⋊C8×C137C2212yescomplemented
49D4⋊C8×C137C2212yes—
50D4⋊C8×C137C2212yes—
51C22×Q8×C137—22no—
52C22×D4×C137D411yescharacteristic
53C23×C4×C137—21no—
54C2×Q16×C137—22no—
55C2×SD16×C137—22no—
56C2×M16×C137—21no—
57C4⋊Q8×C137—22no—
58C4⋊Q8×C137—22no—
59C4⋊Q8×C137D411yescharacteristic
60C4⋊Q8×C137D411yescharacteristic
61C4⋊Q8×C137C2311yescharacteristic
62C4⋊Q8×C137C2311yescharacteristic
63C4⋊Q8×C137C2311yescharacteristic
64C4⋊Q8×C137C2311yescharacteristic
65#125×C137—22no—
66#125×C137—22no—
67#125×C137—21no—
68#125×C137—21no—
69#125×C137—21no—
70C22⋊Q8×C137—24no—
71C22⋊Q8×C137—24no—
72C22⋊Q8×C137—24no—
73C22⋊Q8×C137—24no—
74C22⋊Q8×C137—22no—
75C22⋊Q8×C137—24no—
76C22⋊Q8×C137—21no—
77C22⋊Q8×C137—21no—
78C22⋊Q8×C137—22no—
79C22⋊Q8×C137—24no—
80C22⋊Q8×C137—21no—
81C22⋊Q8×C137—21no—
82C22⋊Q8×C137D412yes—
83C22⋊Q8×C137D412yes—
84C4×Q8×C137—21no—
85C4×Q8×C137—21no—
86C4×Q8×C137—21no—
87C4×Q8×C137—21no—
88C4×Q8×C137D412yes—
89C4×Q8×C137C2312yes—
90C4×D4×C137—22no—
91C4×D4×C137—22no—
92C4×D4×C137—22no—
93C4×D4×C137—22no—
94C4×D4×C137—21no—
95C4×D4×C137—21no—
96C4×D4×C137D412yes—
97C4×D4×C137C2312yes—
98#117×C137—21no—
99#117×C137—21no—
100#117×C137—21no—
101C2×C4⋊C4×C137—24no—
102C2×C4⋊C4×C137—22no—
103C2×C4⋊C4×C137D411yescharacteristic
104C2×C22⋊C4×C137—24no—
105C2×C22⋊C4×C137—21no—
106C2×C42×C137C2311yescharacteristic
107#107×C137—22no—
108#107×C137—22no—
109#106×C137—22no—
110#106×C137—22no—
111C4⋊C8×C137—21no—
112C4⋊C8×C137—21no—
113C4⋊C8×C137C2312yes—
114C4⋊C8×C137C2312yes—
115#103×C137—22no—
116#103×C137—22no—
117#103×C137—22no—
118#103×C137—22no—
119#103×C137—22no—
120#102×C137—22no—
121C22⋊C8×C137—21no—
122C22⋊C8×C137—21no—
123#97×C137—21no—
124C4×C8×C137C2311yescharacteristic
125C4×C8×C137C2311yescharacteristic
126C24×C137—21no—
127C2×Q8×C137—24no—
128C2×Q8×C137—21no—
129C2×Q8×C137—21no—
130C2×Q8×C137—21no—
131C2×Q8×C137—21no—
132C2×Q8×C137—82no—
133C2×Q8×C137—82no—
134C2×Q8×C137C2×D412yes—
135C2×Q8×C137C2×D412yes—
136C2×D4×C137—24no—
137C2×D4×C137—21no—
138C2×D4×C137C2×D412yes—
139C22×C4×C137—24no—
140C22×C4×C137—24no—
141C22×C4×C137—21no—
142C22×C4×C137—22no—
143C22×C4×C137—22no—
144C22×C4×C137—81no—
145C22×C4×C137—81no—
146C22×C4×C137C2×D411yescharacteristic
147C22×C4×C137C2×D411yescharacteristic
148C22×C4×C137C2×D411yescharacteristic
149Q16×C137—82no—
150SD16×C137—44no—
151M16×C137—81no—
152C2×C8×C137—21no—
153C2×C8×C137—21no—
154C2×C8×C137—22no—
155C2×C8×C137—22no—
156C4⋊C4×C137—24no—
157C4⋊C4×C137—24no—
158C4⋊C4×C137—24no—
159C4⋊C4×C137—22no—
160C4⋊C4×C137—22no—
161C4⋊C4×C137—21no—
162C4⋊C4×C137—21no—
163C4⋊C4×C137—21no—
164C4⋊C4×C137—21no—
165C4⋊C4×C137—22no—
166C4⋊C4×C137—21no—
167C4⋊C4×C137—21no—
168C4⋊C4×C137—21no—
169C4⋊C4×C137—22no—
170C4⋊C4×C137—21no—
171C4⋊C4×C137—21no—
172C4⋊C4×C137—21no—
173C4⋊C4×C137—21no—
174C4⋊C4×C137—41no—
175C4⋊C4×C137—41no—
176C4⋊C4×C137C2×D412yes—
177C4⋊C4×C137C2×D412yes—
178C4⋊C4×C137C2×D412yes—
179C22⋊C4×C137—24no—
180C22⋊C4×C137—24no—
181C22⋊C4×C137—22no—
182C22⋊C4×C137—22no—
183C22⋊C4×C137—22no—
184C22⋊C4×C137—22no—
185C22⋊C4×C137—41no—
186C42×C137—22no—
187C42×C137—22no—
188C42×C137—41no—
189C42×C137C2×D411yescharacteristic
190C42×C137C2×D411yescharacteristic
191C42×C137C2×D411yescharacteristic
192C42×C137C2411yescharacteristic
193C23×C137—24no—
194C23×C137—22no—
195C23×C137—81no—
196C23×C137—81no—
197C23×C137#12011yescharacteristic
198Q8×C137—22no—
199Q8×C137—22no—
200Q8×C137—42no—
201Q8×C137—41no—
202Q8×C137—41no—
203Q8×C137—42no—
204Q8×C137—41no—
205Q8×C137—41no—
206Q8×C137—81no—
207Q8×C137—84no—
208D4×C137—24no—
209D4×C137—22no—
210D4×C137#13714yes—
211C2×C4×C137—22no—
212C2×C4×C137—21no—
213C2×C4×C137—21no—
214C2×C4×C137—22no—
215C2×C4×C137—22no—
216C2×C4×C137—22no—
217C2×C4×C137—22no—
218C2×C4×C137—22no—
219C2×C4×C137—41no—
220C2×C4×C137—42no—
221C2×C4×C137—42no—
222C2×C4×C137—41no—
223C2×C4×C137—41no—
224C2×C4×C137—82no—
225C2×C4×C137—81no—
226C2×C4×C137—84no—
227C2×C4×C137—84no—
228C2×C4×C137#12011yescharacteristic
229C2×C4×C137#12011yescharacteristic
230C2×C4×C137#12011yescharacteristic
231C2×C4×C137C22×D411yescharacteristic
232C2×C4×C137C22×D411yescharacteristic
233C2×C4×C137C22×D411yescharacteristic
234C8×C137—41no—
235C8×C137—41no—
236C8×C137—42no—
237C8×C137—42no—
238C22×C137—21no—
239C22×C137—24no—
240C22×C137—21no—
241C22×C137—21no—
242C22×C137—42no—
243C22×C137—42no—
244C22×C137—84no—
245C22×C137C2×#12011yescharacteristic
246C4×C137—21no—
247C4×C137—21no—
248C4×C137—21no—
249C4×C137—21no—
250C4×C137—42no—
251C4×C137—42no—
252C4×C137—42no—
253C4×C137—42no—
254C4×C137—42no—
255C4×C137—81no—
256C4×C137—81no—
257C4×C137C2×#13712yes—
258C4×C137#57611yescharacteristic
259C4×C137#57711yescharacteristic
260C2×C137—41no—
261C2×C137—44no—
262C2×C137#300411yescharacteristic
263C2×C137#301211yescharacteristic
264C2×C137#302211yescharacteristic
265#18847C13711yesmaximal, characteristic, direct product factor, Sylow
266C137#1884711yescharacteristic, direct product factor, Sylow
267#3501C2×C13711yescharacteristic
268#3255C2×C13711yescharacteristic, complemented
269#3231C2×C13711yescharacteristic, complemented
270#3110C2×C13711yescharacteristic
271#1651C2×C13712yescomplemented
272#1649C2×C13712yescomplemented
273#1644C2×C13712yescomplemented
274#1643C2×C13712yescomplemented
275#1559C2×C13711yescharacteristic, complemented
276#1497C2×C13711yescharacteristic, complemented
277#1486C2×C13711yescharacteristic
278#554—21no—
279#554—22no—
280#554—22no—
281#554C22×C13712yes—
282D4×Q8—21no—
283D4×Q8C22×C13712yes—
284#544—21no—
285#543—22no—
286#533C22×C13711yescharacteristic
287#533C22×C13711yescharacteristic
288#533C22×C13711yescharacteristic
289C2×C22⋊Q8—22no—
290#518—21no—
291C2×C4×D4C22×C13711yescharacteristic
292#500C22×C13711yescharacteristic, complemented
293#498C22×C13711yescharacteristic, complemented
294#487C22×C13711yescharacteristic, complemented
295#487C22×C13711yescharacteristic, complemented
296#485—22no—
297#484—21no—
298#479—22no—
299#475—21no—
300#475C22×C13712yescomplemented
301#474C22×C13712yes—
302#470—22no—
303#462—21no—
304#462C22×C13712yescomplemented
305#461C22×C13712yes—
306#450—21no—
307#440—22no—
308#432C22×C13711yescharacteristic
309#431C22×C13711yescharacteristic
310#423C22×C13711yescharacteristic
311#333C22×C13712yescomplemented
312#332C22×C13712yescomplemented
313Q8⋊C8C22×C13712yescomplemented
314Q8⋊C8C22×C13712yescomplemented
315D4⋊C8C22×C13712yes—
316D4⋊C8C22×C13712yes—
317C22×Q8—22no—
318C22×D4D4×C13711yescharacteristic
319C23×C4—21no—
320C2×Q16—22no—
321C2×SD16—22no—
322C2×M16—21no—
323C4⋊Q8—22no—
324C4⋊Q8—22no—
325C4⋊Q8D4×C13711yescharacteristic
326C4⋊Q8D4×C13711yescharacteristic
327C4⋊Q8C23×C13711yescharacteristic
328C4⋊Q8C23×C13711yescharacteristic
329C4⋊Q8C23×C13711yescharacteristic
330C4⋊Q8C23×C13711yescharacteristic
331#125—22no—
332#125—22no—
333#125—21no—
334#125—21no—
335#125—21no—
336C22⋊Q8—24no—
337C22⋊Q8—24no—
338C22⋊Q8—24no—
339C22⋊Q8—24no—
340C22⋊Q8—22no—
341C22⋊Q8—24no—
342C22⋊Q8—21no—
343C22⋊Q8—21no—
344C22⋊Q8—22no—
345C22⋊Q8—24no—
346C22⋊Q8—21no—
347C22⋊Q8—21no—
348C22⋊Q8D4×C13712yes—
349C22⋊Q8D4×C13712yes—
350C4×Q8—21no—
351C4×Q8—21no—
352C4×Q8—21no—
353C4×Q8—21no—
354C4×Q8D4×C13712yes—
355C4×Q8C23×C13712yes—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4D4×C13712yes—
363C4×D4C23×C13712yes—
364#117—21no—
365#117—21no—
366#117—21no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4D4×C13711yescharacteristic
370C2×C22⋊C4—24no—
371C2×C22⋊C4—21no—
372C2×C42C23×C13711yescharacteristic
373#107—22no—
374#107—22no—
375#106—22no—
376#106—22no—
377C4⋊C8—21no—
378C4⋊C8—21no—
379C4⋊C8C23×C13712yes—
380C4⋊C8C23×C13712yes—
381#103—22no—
382#103—22no—
383#103—22no—
384#103—22no—
385#103—22no—
386#102—22no—
387C22⋊C8—21no—
388C22⋊C8—21no—
389#97—21no—
390C4×C8C23×C13711yescharacteristic
391C4×C8C23×C13711yescharacteristic
392C24—21no—
393C2×Q8—24no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×Q8C2×D4×C13712yes—
401C2×Q8C2×D4×C13712yes—
402C2×D4—24no—
403C2×D4—21no—
404C2×D4C2×D4×C13712yes—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4C2×D4×C13711yescharacteristic
413C22×C4C2×D4×C13711yescharacteristic
414C22×C4C2×D4×C13711yescharacteristic
415Q16—82no—
416SD16—44no—
417M16—81no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4C2×D4×C13712yes—
443C4⋊C4C2×D4×C13712yes—
444C4⋊C4C2×D4×C13712yes—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—41no—
452C42—22no—
453C42—22no—
454C42—41no—
455C42C2×D4×C13711yescharacteristic
456C42C2×D4×C13711yescharacteristic
457C42C2×D4×C13711yescharacteristic
458C42C24×C13711yescharacteristic
459C23—24no—
460C23—22no—
461C23—81no—
462C23—81no—
463C23#120×C13711yescharacteristic
464Q8—22no—
465Q8—22no—
466Q8—42no—
467Q8—41no—
468Q8—41no—
469Q8—42no—
470Q8—41no—
471Q8—41no—
472Q8—81no—
473Q8—84no—
474D4—24no—
475D4—22no—
476D4#137×C13714yes—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—84no—
493C2×C4—84no—
494C2×C4#120×C13711yescharacteristic
495C2×C4#120×C13711yescharacteristic
496C2×C4#120×C13711yescharacteristic
497C2×C4C22×D4×C13711yescharacteristic
498C2×C4C22×D4×C13711yescharacteristic
499C2×C4C22×D4×C13711yescharacteristic
500C8—41no—
501C8—41no—
502C8—42no—
503C8—42no—
504C22—21no—
505C22—24no—
506C22—21no—
507C22—21no—
508C22—42no—
509C22—42no—
510C22—84no—
511C22C2×#120×C13711yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—81no—
522C4—81no—
523C4C2×#137×C13712yes—
524C4#576×C13711yescharacteristic
525C4#577×C13711yescharacteristic
526C2—41no—
527C2—44no—
528C2#3004×C13711yescharacteristic
529C2#3012×C13711yescharacteristic
530C2#3022×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
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
2. #3255×C137C2semidirect
3. #3231×C137C2semidirect
5. #1651×C137C2semidirect
6. #1649×C137C2semidirect
7. #1644×C137C2semidirect
8. #1643×C137C2semidirect
9. #1559×C137C2semidirect
10. #1497×C137C2semidirect
26. #500×C137C22semidirect
27. #498×C137C22semidirect
28. #487×C137C22semidirect
29. #487×C137C22semidirect
34. #475×C137C22semidirect
38. #462×C137C22semidirect
45. #333×C137C22semidirect
46. #332×C137C22semidirect
47. Q8⋊C8×C137C22semidirect
48. Q8⋊C8×C137C22semidirect
265. #18847C137direct (N × H)
266. C137#18847direct (N × H)
268. #3255C2×C137semidirect
269. #3231C2×C137semidirect
271. #1651C2×C137semidirect
272. #1649C2×C137semidirect
273. #1644C2×C137semidirect
274. #1643C2×C137semidirect
275. #1559C2×C137semidirect
276. #1497C2×C137semidirect
292. #500C22×C137semidirect
293. #498C22×C137semidirect
294. #487C22×C137semidirect
295. #487C22×C137semidirect
300. #475C22×C137semidirect
304. #462C22×C137semidirect
311. #333C22×C137semidirect
312. #332C22×C137semidirect
313. Q8⋊C8C22×C137semidirect
314. Q8⋊C8C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3501×C1372
#3255×C1372
#3231×C1372
#3110×C1372
#1651×C1372
#1649×C1372
#1644×C1372
#1643×C1372
#1559×C1372
#1497×C1372
#1486×C1372
#18847137