Finite Group Databasebeta

← C137×#18716 #46868102

Subgroups — C137×#18716 #46868102

Order 35072.

1060 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3548×C137C211yesmaximal, characteristic, complemented
2#3208×C137C211yesmaximal, characteristic, complemented
3#3205×C137C211yesmaximal, characteristic, complemented
4#3151×C137C211yesmaximal, characteristic, complemented
5#1622×C137C214yesmaximal, complemented
6#1621×C137C214yesmaximal, complemented
7#1543×C137C211yesmaximal, characteristic, complemented
8#1485×C137C211yesmaximal, characteristic, complemented
9#1484×C137C211yesmaximal, characteristic, complemented
10C2×AGL(1,ℤ/8)×C137—22no—
11#562×C137—22no—
12#554×C137—21no—
13#554×C137C2212yescomplemented
14#550×C137—21no—
15#550×C137C2212yescomplemented
16#547×C137—21no—
17#547×C137—21no—
18#540×C137—22no—
19#537×C137—22no—
20#537×C137—22no—
21#533×C137C2211yescharacteristic
22#532×C137C2211yescharacteristic
23#532×C137C2211yescharacteristic
24C2×#127×C137C2211yescharacteristic
25#496×C137—22no—
26#493×C137—21no—
27#493×C137C2212yes—
28#492×C137—21no—
29#492×C137C2212yes—
30#486×C137—21no—
31#486×C137—21no—
32#460×C137C2214yes—
33#459×C137C2214yes—
34#450×C137—22no—
35#447×C137—22no—
36#431×C137C2211yescharacteristic
37#431×C137C2211yescharacteristic
38C4×M16×C137C2211yescharacteristic
39#333×C137C2212yescomplemented
40#331×C137C2212yescomplemented
41Q8⋊C8×C137C2214yescomplemented
42D4⋊C8×C137C2214yescomplemented
43C2×#41×C137—22no—
44C2×#41×C137—22no—
45C2×#41×C137—21no—
46C2×#41×C137D412yes—
47C22×D4×C137—21no—
48C22×D4×C137—21no—
49C22×D4×C137—21no—
50AGL(1,ℤ/8)×C137—48no—
51C2×SD16×C137—24no—
52C2×SD16×C137—24no—
53C2×D8×C137—24no—
54C2×D8×C137—24no—
55C2×M16×C137—21no—
56C4⋊Q8×C137D411yescharacteristic
57C4⋊Q8×C137D411yescharacteristic
58C4⋊Q8×C137D411yescharacteristic
59C4⋊Q8×C137C2311yescharacteristic
60#127×C137—22no—
61#127×C137—21no—
62#127×C137—21no—
63#127×C137—21no—
64#127×C137D411yescharacteristic
65#127×C137C2312yes—
66#127×C137C2311yescharacteristic
67#125×C137—22no—
68#125×C137—21no—
69#124×C137—21no—
70#124×C137D412yes—
71#123×C137—24no—
72#123×C137—24no—
73C22⋊Q8×C137—22no—
74C22⋊Q8×C137—24no—
75C22⋊Q8×C137—22no—
76C22⋊Q8×C137—24no—
77C22⋊Q8×C137—21no—
78C22⋊Q8×C137D412yes—
79#121×C137—24no—
80#121×C137—24no—
81#121×C137—22no—
82#121×C137—24no—
83#121×C137—24no—
84#121×C137—21no—
85#121×C137—21no—
86#121×C137—21no—
87#121×C137—22no—
88#121×C137D412yes—
89C4×Q8×C137—21no—
90C4×Q8×C137C2312yes—
91C4×D4×C137—22no—
92C4×D4×C137—22no—
93C4×D4×C137—21no—
94C4×D4×C137—21no—
95C4×D4×C137—21no—
96C4×D4×C137C2312yes—
97#117×C137—21no—
98C2×C4⋊C4×C137—22no—
99C2×C4⋊C4×C137—22no—
100C2×C42×C137C2311yescharacteristic
101C4⋊C8×C137C2312yes—
102C4⋊C8×C137C2312yes—
103#102×C137—24no—
104#102×C137—24no—
105C22⋊C8×C137—21no—
106C22⋊C8×C137—21no—
107#97×C137—21no—
108C4×C8×C137—21no—
109C4×C8×C137—21no—
110C4×C8×C137C2312yes—
111C24×C137—41no—
112#41×C137—44no—
113#41×C137—82no—
114#41×C137—84no—
115#41×C137—84no—
116C2×Q8×C137—21no—
117C2×Q8×C137—21no—
118C2×Q8×C137C2×D412yes—
119C2×Q8×C137C2×D412yes—
120C2×D4×C137—24no—
121C2×D4×C137—24no—
122C2×D4×C137—22no—
123C2×D4×C137—22no—
124C2×D4×C137—21no—
125C2×D4×C137—21no—
126C2×D4×C137—21no—
127C2×D4×C137—21no—
128C2×D4×C137—22no—
129C2×D4×C137—22no—
130C2×D4×C137—22no—
131C2×D4×C137—44no—
132C2×D4×C137—41no—
133C2×D4×C137—41no—
134C2×D4×C137—41no—
135C2×D4×C137—82no—
136C2×D4×C137—82no—
137C2×D4×C137C2×D412yes—
138C2×D4×C137C2×D412yes—
139C22×C4×C137—24no—
140C22×C4×C137—24no—
141C22×C4×C137—22no—
142C22×C4×C137—22no—
143C22×C4×C137C2×D411yescharacteristic
144C22×C4×C137C2×D411yescharacteristic
145C22×C4×C137C2×D411yescharacteristic
146SD16×C137—48no—
147SD16×C137—84no—
148D8×C137—48no—
149D8×C137—84no—
150M16×C137—24no—
151C2×C8×C137—22no—
152C2×C8×C137—22no—
153C2×C8×C137—22no—
154C4⋊C4×C137—24no—
155C4⋊C4×C137—24no—
156C4⋊C4×C137—21no—
157C4⋊C4×C137—21no—
158C4⋊C4×C137—21no—
159C4⋊C4×C137—21no—
160C4⋊C4×C137—22no—
161C4⋊C4×C137—21no—
162C4⋊C4×C137—21no—
163C4⋊C4×C137—22no—
164C4⋊C4×C137C2×D412yes—
165C4⋊C4×C137C2×D412yes—
166C22⋊C4×C137—24no—
167C22⋊C4×C137—24no—
168C22⋊C4×C137—22no—
169C22⋊C4×C137—22no—
170C22⋊C4×C137—22no—
171C22⋊C4×C137—22no—
172C22⋊C4×C137—22no—
173C42×C137—22no—
174C42×C137C2×D411yescharacteristic
175C42×C137C2×D411yescharacteristic
176C42×C137C2×D411yescharacteristic
177C42×C137C2411yescharacteristic
178C23×C137—22no—
179C23×C137—22no—
180C23×C137—42no—
181C23×C137—82no—
182C23×C137—82no—
183C23×C137—82no—
184C23×C137#12011yescharacteristic
185Q8×C137—24no—
186Q8×C137—42no—
187Q8×C137—42no—
188Q8×C137—81no—
189D4×C137—24no—
190D4×C137—42no—
191D4×C137—44no—
192D4×C137—42no—
193D4×C137—44no—
194D4×C137—42no—
195D4×C137—44no—
196D4×C137—42no—
197D4×C137—81no—
198D4×C137—82no—
199D4×C137—82no—
200D4×C137—82no—
201D4×C137—82no—
202D4×C137—84no—
203D4×C137—84no—
204D4×C137—82no—
205C2×C4×C137—22no—
206C2×C4×C137—22no—
207C2×C4×C137—22no—
208C2×C4×C137—22no—
209C2×C4×C137—22no—
210C2×C4×C137—22no—
211C2×C4×C137—22no—
212C2×C4×C137—44no—
213C2×C4×C137—44no—
214C2×C4×C137—84no—
215C2×C4×C137—84no—
216C2×C4×C137#12011yescharacteristic
217C2×C4×C137#12011yescharacteristic
218C2×C4×C137#12011yescharacteristic
219C2×C4×C137C22×D411yescharacteristic
220C2×C4×C137C22×D411yescharacteristic
221C2×C4×C137C22×D411yescharacteristic
222C8×C137—24no—
223C8×C137—42no—
224C8×C137—42no—
225C22×C137—21no—
226C22×C137—21no—
227C22×C137—21no—
228C22×C137—42no—
229C22×C137—42no—
230C22×C137—42no—
231C22×C137—42no—
232C22×C137—42no—
233C22×C137—42no—
234C22×C137—82no—
235C22×C137—82no—
236C22×C137—82no—
237C22×C137—164no—
238C22×C137C2×#12011yescharacteristic
239C4×C137—21no—
240C4×C137—21no—
241C4×C137—21no—
242C4×C137—42no—
243C4×C137—42no—
244C4×C137—42no—
245C4×C137—42no—
246C4×C137#45312yes—
247C4×C137#57612yes—
248C4×C137#57712yes—
249C2×C137—41no—
250C2×C137—82no—
251C2×C137—82no—
252C2×C137—162no—
253C2×C137#300811yescharacteristic
254C2×C137#301011yescharacteristic
255C2×C137C2×#45311yescharacteristic
256#18716C13711yesmaximal, characteristic, direct product factor, Sylow
257C137#1871611yescharacteristic, direct product factor, Sylow
258#3548C2×C13711yescharacteristic, complemented
259#3208C2×C13711yescharacteristic, complemented
260#3205C2×C13711yescharacteristic, complemented
261#3151C2×C13711yescharacteristic, complemented
262#1622C2×C13714yescomplemented
263#1621C2×C13714yescomplemented
264#1543C2×C13711yescharacteristic, complemented
265#1485C2×C13711yescharacteristic, complemented
266#1484C2×C13711yescharacteristic, complemented
267C2×AGL(1,ℤ/8)—22no—
268#562—22no—
269#554—21no—
270#554C22×C13712yescomplemented
271#550—21no—
272#550C22×C13712yescomplemented
273#547—21no—
274#547—21no—
275#540—22no—
276#537—22no—
277#537—22no—
278#533C22×C13711yescharacteristic
279#532C22×C13711yescharacteristic
280#532C22×C13711yescharacteristic
281C2×#127C22×C13711yescharacteristic
282#496—22no—
283#493—21no—
284#493C22×C13712yes—
285#492—21no—
286#492C22×C13712yes—
287#486—21no—
288#486—21no—
289#460C22×C13714yes—
290#459C22×C13714yes—
291#450—22no—
292#447—22no—
293#431C22×C13711yescharacteristic
294#431C22×C13711yescharacteristic
295C4×M16C22×C13711yescharacteristic
296#333C22×C13712yescomplemented
297#331C22×C13712yescomplemented
298Q8⋊C8C22×C13714yescomplemented
299D4⋊C8C22×C13714yescomplemented
300C2×#41—22no—
301C2×#41—22no—
302C2×#41—21no—
303C2×#41D4×C13712yes—
304C22×D4—21no—
305C22×D4—21no—
306C22×D4—21no—
307AGL(1,ℤ/8)—48no—
308C2×SD16—24no—
309C2×SD16—24no—
310C2×D8—24no—
311C2×D8—24no—
312C2×M16—21no—
313C4⋊Q8D4×C13711yescharacteristic
314C4⋊Q8D4×C13711yescharacteristic
315C4⋊Q8D4×C13711yescharacteristic
316C4⋊Q8C23×C13711yescharacteristic
317#127—22no—
318#127—21no—
319#127—21no—
320#127—21no—
321#127D4×C13711yescharacteristic
322#127C23×C13712yes—
323#127C23×C13711yescharacteristic
324#125—22no—
325#125—21no—
326#124—21no—
327#124D4×C13712yes—
328#123—24no—
329#123—24no—
330C22⋊Q8—22no—
331C22⋊Q8—24no—
332C22⋊Q8—22no—
333C22⋊Q8—24no—
334C22⋊Q8—21no—
335C22⋊Q8D4×C13712yes—
336#121—24no—
337#121—24no—
338#121—22no—
339#121—24no—
340#121—24no—
341#121—21no—
342#121—21no—
343#121—21no—
344#121—22no—
345#121D4×C13712yes—
346C4×Q8—21no—
347C4×Q8C23×C13712yes—
348C4×D4—22no—
349C4×D4—22no—
350C4×D4—21no—
351C4×D4—21no—
352C4×D4—21no—
353C4×D4C23×C13712yes—
354#117—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C42C23×C13711yescharacteristic
358C4⋊C8C23×C13712yes—
359C4⋊C8C23×C13712yes—
360#102—24no—
361#102—24no—
362C22⋊C8—21no—
363C22⋊C8—21no—
364#97—21no—
365C4×C8—21no—
366C4×C8—21no—
367C4×C8C23×C13712yes—
368C24—41no—
369#41—44no—
370#41—82no—
371#41—84no—
372#41—84no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8C2×D4×C13712yes—
376C2×Q8C2×D4×C13712yes—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—44no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—82no—
393C2×D4—82no—
394C2×D4C2×D4×C13712yes—
395C2×D4C2×D4×C13712yes—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4C2×D4×C13711yescharacteristic
401C22×C4C2×D4×C13711yescharacteristic
402C22×C4C2×D4×C13711yescharacteristic
403SD16—48no—
404SD16—84no—
405D8—48no—
406D8—84no—
407M16—24no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4C2×D4×C13712yes—
422C4⋊C4C2×D4×C13712yes—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C42—22no—
431C42C2×D4×C13711yescharacteristic
432C42C2×D4×C13711yescharacteristic
433C42C2×D4×C13711yescharacteristic
434C42C24×C13711yescharacteristic
435C23—22no—
436C23—22no—
437C23—42no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23#120×C13711yescharacteristic
442Q8—24no—
443Q8—42no—
444Q8—42no—
445Q8—81no—
446D4—24no—
447D4—42no—
448D4—44no—
449D4—42no—
450D4—42no—
451D4—44no—
452D4—42no—
453D4—44no—
454D4—81no—
455D4—82no—
456D4—82no—
457D4—82no—
458D4—82no—
459D4—84no—
460D4—84no—
461D4—82no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—84no—
472C2×C4—84no—
473C2×C4#120×C13711yescharacteristic
474C2×C4#120×C13711yescharacteristic
475C2×C4#120×C13711yescharacteristic
476C2×C4C22×D4×C13711yescharacteristic
477C2×C4C22×D4×C13711yescharacteristic
478C2×C4C22×D4×C13711yescharacteristic
479C8—24no—
480C8—42no—
481C8—42no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—164no—
495C22C2×#120×C13711yescharacteristic
496C4—21no—
497C4—21no—
498C4—21no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4#453×C13712yes—
504C4#576×C13712yes—
505C4#577×C13712yes—
506C2—41no—
507C2—82no—
508C2—82no—
509C2—162no—
510C2#3008×C13711yescharacteristic
511C2#3010×C13711yescharacteristic
512C2C2×#453×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
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. #3548×C137C2semidirect
2. #3208×C137C2semidirect
3. #3205×C137C2semidirect
4. #3151×C137C2semidirect
5. #1622×C137C2semidirect
6. #1621×C137C2semidirect
7. #1543×C137C2semidirect
8. #1485×C137C2semidirect
9. #1484×C137C2semidirect
13. #554×C137C22semidirect
15. #550×C137C22semidirect
39. #333×C137C22semidirect
40. #331×C137C22semidirect
41. Q8⋊C8×C137C22semidirect
42. D4⋊C8×C137C22semidirect
256. #18716C137direct (N × H)
257. C137#18716direct (N × H)
258. #3548C2×C137semidirect
259. #3208C2×C137semidirect
260. #3205C2×C137semidirect
261. #3151C2×C137semidirect
262. #1622C2×C137semidirect
263. #1621C2×C137semidirect
264. #1543C2×C137semidirect
265. #1485C2×C137semidirect
266. #1484C2×C137semidirect
270. #554C22×C137semidirect
272. #550C22×C137semidirect
296. #333C22×C137semidirect
297. #331C22×C137semidirect
298. Q8⋊C8C22×C137semidirect
299. D4⋊C8C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3548×C1372
#3208×C1372
#3205×C1372
#3151×C1372
#1622×C1372
#1621×C1372
#1543×C1372
#1485×C1372
#1484×C1372
#18716137