Finite Group Databasebeta

← C137×#18153 #46867539

Subgroups — C137×#18153 #46867539

Order 35072.

1184 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#3449×C137C211yesmaximal, characteristic
2#3112×C137C212yesmaximal, complemented
3#2599×C137C211yesmaximal, characteristic, complemented
4#1610×C137C212yesmaximal, complemented
5#1607×C137C212yesmaximal, complemented
6#1604×C137C212yesmaximal, complemented
7#1601×C137C212yesmaximal, complemented
8#1514×C137C212yesmaximal, complemented
9#1464×C137C211yesmaximal, characteristic, complemented
10C22×#41×C137—21no—
11#537×C137—21no—
12#537×C137C2212yes—
13#536×C137—22no—
14#535×C137—22no—
15#534×C137—21no—
16#534×C137C2212yes—
17#532×C137—21no—
18#532×C137C2212yes—
19#525×C137—24no—
20C2×C4×Q8×C137C2211yescharacteristic, complemented
21C2×#117×C137C2211yescharacteristic
22#480×C137C2214yescomplemented
23#477×C137—22no—
24#460×C137—22no—
25#448×C137C2214yescomplemented
26#440×C137—24no—
27#432×C137C2212yescomplemented
28#398×C137C2211yescharacteristic, complemented
29#397×C137C2211yescharacteristic
30#395×C137C2211yescharacteristic, complemented
31#393×C137C2211yescharacteristic
32#388×C137—21no—
33#387×C137—21no—
34#387×C137—21no—
35#385×C137—21no—
36#383×C137C2211yescharacteristic, complemented
37#327×C137C2214yescomplemented
38#327×C137C2214yescomplemented
39#324×C137C2212yescomplemented
40#323×C137C2212yes—
41C2×#41×C137—22no—
42C2×#41×C137—22no—
43C2×#41×C137—22no—
44C2×#41×C137—22no—
45C2×#41×C137—44no—
46C2×#41×C137—44no—
47C22×Q8×C137—21no—
48C22×Q8×C137C2311yescharacteristic
49C22×D4×C137—21no—
50C22×D4×C137—22no—
51C23×C4×C137—22no—
52C23×C4×C137D411yescharacteristic
53C2×SD16×C137—24no—
54C4⋊Q8×C137—21no—
55C4⋊Q8×C137D412yes—
56#127×C137—21no—
57#127×C137D412yes—
58#124×C137—22no—
59#124×C137—22no—
60#123×C137—24no—
61#123×C137—24no—
62C22⋊Q8×C137—24no—
63C22⋊Q8×C137—24no—
64#121×C137—24no—
65#121×C137—24no—
66#121×C137—22no—
67#121×C137—22no—
68#121×C137—22no—
69#121×C137C2314yes—
70#120×C137—24no—
71#120×C137—24no—
72C4×Q8×C137—22no—
73C4×Q8×C137—22no—
74C4×D4×C137—22no—
75C4×D4×C137—22no—
76C4×D4×C137—22no—
77C4×D4×C137D414yes—
78#117×C137—22no—
79#117×C137—22no—
80C2×C4⋊C4×C137—21no—
81C2×C4⋊C4×C137—21no—
82C2×C4⋊C4×C137—21no—
83C2×C4⋊C4×C137D411yescharacteristic
84C2×C4⋊C4×C137C2311yescharacteristic
85C2×C4⋊C4×C137C2311yescharacteristic
86C2×C22⋊C4×C137—21no—
87C2×C22⋊C4×C137—21no—
88C2×C22⋊C4×C137D411yescharacteristic
89C2×C22⋊C4×C137C2311yescharacteristic
90C2×C42×C137—21no—
91C2×C42×C137D411yescharacteristic
92C2×C42×C137C2311yescharacteristic
93#107×C137—24no—
94C4⋊C8×C137—22no—
95#103×C137—24no—
96#102×C137—24no—
97C22⋊C8×C137C2314yes—
98#97×C137—22no—
99#95×C137—21no—
100#95×C137—21no—
101#95×C137—21no—
102#95×C137—21no—
103#95×C137—21no—
104#95×C137—21no—
105#95×C137C2311yescharacteristic
106#95×C137C2311yescharacteristic
107C24×C137—22no—
108C24×C137C2×D411yescharacteristic
109#41×C137—48no—
110#41×C137—48no—
111#41×C137—44no—
112#41×C137—44no—
113#41×C137—44no—
114#41×C137—44no—
115C2×Q8×C137—22no—
116C2×Q8×C137—22no—
117C2×Q8×C137—22no—
118C2×Q8×C137—42no—
119C2×Q8×C137—44no—
120C2×D4×C137—24no—
121C2×D4×C137—22no—
122C2×D4×C137—24no—
123C2×D4×C137—22no—
124C2×D4×C137—22no—
125C2×D4×C137—22no—
126C2×D4×C137—22no—
127C2×D4×C137—44no—
128C2×D4×C137—44no—
129C2×D4×C137—44no—
130C2×D4×C137C2×D414yes—
131C22×C4×C137—22no—
132C22×C4×C137—21no—
133C22×C4×C137—21no—
134C22×C4×C137—22no—
135C22×C4×C137—24no—
136C22×C4×C137—22no—
137C22×C4×C137—21no—
138C22×C4×C137—24no—
139C22×C4×C137—21no—
140C22×C4×C137—21no—
141C22×C4×C137—21no—
142C22×C4×C137—48no—
143C22×C4×C137C2×D411yescharacteristic
144C22×C4×C137C2×D411yescharacteristic
145C22×C4×C137C2×D411yescharacteristic
146C22×C4×C137C2×D411yescharacteristic
147C22×C4×C137C2×D411yescharacteristic
148C22×C4×C137C2411yescharacteristic
149SD16×C137—48no—
150C2×C8×C137—24no—
151C4⋊C4×C137—22no—
152C4⋊C4×C137—22no—
153C4⋊C4×C137—22no—
154C4⋊C4×C137—22no—
155C4⋊C4×C137—22no—
156C4⋊C4×C137—22no—
157C4⋊C4×C137—42no—
158C4⋊C4×C137—41no—
159C4⋊C4×C137—41no—
160C4⋊C4×C137—42no—
161C4⋊C4×C137C2×D414yes—
162C22⋊C4×C137—24no—
163C22⋊C4×C137—24no—
164C22⋊C4×C137—22no—
165C22⋊C4×C137—24no—
166C22⋊C4×C137—22no—
167C22⋊C4×C137—24no—
168C22⋊C4×C137—42no—
169C22⋊C4×C137—42no—
170C42×C137—21no—
171C42×C137—21no—
172C42×C137—42no—
173C42×C137C2×D412yes—
174C42×C137C2×D412yes—
175C23×C137—21no—
176C23×C137—21no—
177C23×C137—24no—
178C23×C137—41no—
179C23×C137—41no—
180C23×C137—44no—
181C23×C137—82no—
182C23×C137—82no—
183C23×C137AGL(1,ℤ/8)12yes—
184C23×C137C22×D411yescharacteristic
185Q8×C137—24no—
186Q8×C137—42no—
187Q8×C137—44no—
188Q8×C137—44no—
189D4×C137—44no—
190D4×C137—44no—
191D4×C137—44no—
192D4×C137—48no—
193D4×C137—44no—
194D4×C137—44no—
195D4×C137—82no—
196D4×C137—82no—
197D4×C137—82no—
198C2×C4×C137—24no—
199C2×C4×C137—24no—
200C2×C4×C137—21no—
201C2×C4×C137—21no—
202C2×C4×C137—21no—
203C2×C4×C137—21no—
204C2×C4×C137—22no—
205C2×C4×C137—21no—
206C2×C4×C137—21no—
207C2×C4×C137—24no—
208C2×C4×C137—41no—
209C2×C4×C137—41no—
210C2×C4×C137—48no—
211C2×C4×C137—48no—
212C2×C4×C137—44no—
213C2×C4×C137—41no—
214C2×C4×C137—41no—
215C2×C4×C137—41no—
216C2×C4×C137—41no—
217C2×C4×C137—41no—
218C2×C4×C137—41no—
219C2×C4×C137—41no—
220C2×C4×C137—41no—
221C2×C4×C137—41no—
222C2×C4×C137—41no—
223C2×C4×C137—41no—
224C2×C4×C137—41no—
225C2×C4×C137—41no—
226C2×C4×C137#12011yescharacteristic
227C2×C4×C137#12011yescharacteristic
228C2×C4×C137#12012yes—
229C2×C4×C137AGL(1,ℤ/8)12yes—
230C2×C4×C137C22×D412yes—
231C8×C137—44no—
232C22×C137—21no—
233C22×C137—21no—
234C22×C137—22no—
235C22×C137—42no—
236C22×C137—41no—
237C22×C137—41no—
238C22×C137—44no—
239C22×C137—44no—
240C22×C137—44no—
241C22×C137—81no—
242C22×C137—88no—
243C22×C137C2×#12011yescharacteristic
244C22×C137C2×AGL(1,ℤ/8)11yescharacteristic
245C22×C137C2×AGL(1,ℤ/8)11yescharacteristic
246C4×C137—21no—
247C4×C137—22no—
248C4×C137—21no—
249C4×C137—22no—
250C4×C137—22no—
251C4×C137—44no—
252C4×C137—41no—
253C4×C137—42no—
254C4×C137—41no—
255C4×C137—81no—
256C4×C137—81no—
257C4×C137—81no—
258C2×C137—21no—
259C2×C137—21no—
260C2×C137—42no—
261C2×C137—84no—
262C2×C137#300311yescharacteristic
263C2×C137#302211yescharacteristic
264C2×C137#302511yescharacteristic
265#18153C13711yesmaximal, characteristic, direct product factor, Sylow
266C137#1815311yescharacteristic, direct product factor, Sylow
267#3449C2×C13711yescharacteristic
268#3112C2×C13712yescomplemented
269#2599C2×C13711yescharacteristic, complemented
270#1610C2×C13712yescomplemented
271#1607C2×C13712yescomplemented
272#1604C2×C13712yescomplemented
273#1601C2×C13712yescomplemented
274#1514C2×C13712yescomplemented
275#1464C2×C13711yescharacteristic, complemented
276C22×#41—21no—
277#537—21no—
278#537C22×C13712yes—
279#536—22no—
280#535—22no—
281#534—21no—
282#534C22×C13712yes—
283#532—21no—
284#532C22×C13712yes—
285#525—24no—
286C2×C4×Q8C22×C13711yescharacteristic, complemented
287C2×#117C22×C13711yescharacteristic
288#480C22×C13714yescomplemented
289#477—22no—
290#460—22no—
291#448C22×C13714yescomplemented
292#440—24no—
293#432C22×C13712yescomplemented
294#398C22×C13711yescharacteristic, complemented
295#397C22×C13711yescharacteristic
296#395C22×C13711yescharacteristic, complemented
297#393C22×C13711yescharacteristic
298#388—21no—
299#387—21no—
300#387—21no—
301#385—21no—
302#383C22×C13711yescharacteristic, complemented
303#327C22×C13714yescomplemented
304#327C22×C13714yescomplemented
305#324C22×C13712yescomplemented
306#323C22×C13712yes—
307C2×#41—22no—
308C2×#41—22no—
309C2×#41—22no—
310C2×#41—22no—
311C2×#41—44no—
312C2×#41—44no—
313C22×Q8—21no—
314C22×Q8C23×C13711yescharacteristic
315C22×D4—21no—
316C22×D4—22no—
317C23×C4—22no—
318C23×C4D4×C13711yescharacteristic
319C2×SD16—24no—
320C4⋊Q8—21no—
321C4⋊Q8D4×C13712yes—
322#127—21no—
323#127D4×C13712yes—
324#124—22no—
325#124—22no—
326#123—24no—
327#123—24no—
328C22⋊Q8—24no—
329C22⋊Q8—24no—
330#121—24no—
331#121—24no—
332#121—22no—
333#121—22no—
334#121—22no—
335#121C23×C13714yes—
336#120—24no—
337#120—24no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×D4—22no—
341C4×D4—22no—
342C4×D4—22no—
343C4×D4D4×C13714yes—
344#117—22no—
345#117—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4D4×C13711yescharacteristic
350C2×C4⋊C4C23×C13711yescharacteristic
351C2×C4⋊C4C23×C13711yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4D4×C13711yescharacteristic
355C2×C22⋊C4C23×C13711yescharacteristic
356C2×C42—21no—
357C2×C42D4×C13711yescharacteristic
358C2×C42C23×C13711yescharacteristic
359#107—24no—
360C4⋊C8—22no—
361#103—24no—
362#102—24no—
363C22⋊C8C23×C13714yes—
364#97—22no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95C23×C13711yescharacteristic
372#95C23×C13711yescharacteristic
373C24—22no—
374C24C2×D4×C13711yescharacteristic
375#41—48no—
376#41—48no—
377#41—44no—
378#41—44no—
379#41—44no—
380#41—44no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—42no—
385C2×Q8—44no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4C2×D4×C13714yes—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—48no—
409C22×C4C2×D4×C13711yescharacteristic
410C22×C4C2×D4×C13711yescharacteristic
411C22×C4C2×D4×C13711yescharacteristic
412C22×C4C2×D4×C13711yescharacteristic
413C22×C4C2×D4×C13711yescharacteristic
414C22×C4C24×C13711yescharacteristic
415SD16—48no—
416C2×C8—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—42no—
427C4⋊C4C2×D4×C13714yes—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—22no—
431C22⋊C4—24no—
432C22⋊C4—22no—
433C22⋊C4—24no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C42—21no—
437C42—21no—
438C42—42no—
439C42C2×D4×C13712yes—
440C42C2×D4×C13712yes—
441C23—21no—
442C23—21no—
443C23—24no—
444C23—41no—
445C23—41no—
446C23—44no—
447C23—82no—
448C23—82no—
449C23AGL(1,ℤ/8)×C13712yes—
450C23C22×D4×C13711yescharacteristic
451Q8—24no—
452Q8—44no—
453Q8—44no—
454Q8—42no—
455D4—44no—
456D4—44no—
457D4—44no—
458D4—48no—
459D4—44no—
460D4—44no—
461D4—82no—
462D4—82no—
463D4—82no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—48no—
477C2×C4—48no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4#120×C13711yescharacteristic
493C2×C4#120×C13711yescharacteristic
494C2×C4#120×C13712yes—
495C2×C4AGL(1,ℤ/8)×C13712yes—
496C2×C4C22×D4×C13712yes—
497C8—44no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—42no—
502C22—41no—
503C22—41no—
504C22—44no—
505C22—44no—
506C22—44no—
507C22—81no—
508C22—88no—
509C22C2×#120×C13711yescharacteristic
510C22C2×AGL(1,ℤ/8)×C13711yescharacteristic
511C22C2×AGL(1,ℤ/8)×C13711yescharacteristic
512C4—21no—
513C4—22no—
514C4—21no—
515C4—22no—
516C4—22no—
517C4—44no—
518C4—41no—
519C4—42no—
520C4—41no—
521C4—81no—
522C4—81no—
523C4—81no—
524C2—21no—
525C2—21no—
526C2—42no—
527C2—84no—
528C2#3003×C13711yescharacteristic
529C2#3022×C13711yescharacteristic
530C2#3025×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC22×C4C24×C137
prime radical#3449×C137C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3112×C137C2semidirect
3. #2599×C137C2semidirect
4. #1610×C137C2semidirect
5. #1607×C137C2semidirect
6. #1604×C137C2semidirect
7. #1601×C137C2semidirect
8. #1514×C137C2semidirect
9. #1464×C137C2semidirect
20. C2×C4×Q8×C137C22semidirect
22. #480×C137C22semidirect
25. #448×C137C22semidirect
27. #432×C137C22semidirect
28. #398×C137C22semidirect
30. #395×C137C22semidirect
36. #383×C137C22semidirect
37. #327×C137C22semidirect
38. #327×C137C22semidirect
39. #324×C137C22semidirect
265. #18153C137direct (N × H)
266. C137#18153direct (N × H)
268. #3112C2×C137semidirect
269. #2599C2×C137semidirect
270. #1610C2×C137semidirect
271. #1607C2×C137semidirect
272. #1604C2×C137semidirect
273. #1601C2×C137semidirect
274. #1514C2×C137semidirect
275. #1464C2×C137semidirect
286. C2×C4×Q8C22×C137semidirect
288. #480C22×C137semidirect
291. #448C22×C137semidirect
293. #432C22×C137semidirect
294. #398C22×C137semidirect
296. #395C22×C137semidirect
302. #383C22×C137semidirect
303. #327C22×C137semidirect
304. #327C22×C137semidirect
305. #324C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3449×C1372
#3112×C1372
#2599×C1372
#1610×C1372
#1607×C1372
#1604×C1372
#1601×C1372
#1514×C1372
#1464×C1372
#18153137