Finite Group Databasebeta

← C137×#18849 #46868235

Subgroups — C137×#18849 #46868235

Order 35072.

858 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3507×C137C211yesmaximal, characteristic
2#3254×C137C211yesmaximal, characteristic
3#3231×C137C211yesmaximal, characteristic, complemented
4#3113×C137C211yesmaximal, characteristic, complemented
5#1652×C137C212yesmaximal, complemented
6#1650×C137C212yesmaximal, complemented
7#1644×C137C212yesmaximal, complemented
8#1643×C137C212yesmaximal, complemented
9#1559×C137C211yesmaximal, characteristic, complemented
10#1496×C137C211yesmaximal, characteristic
11#1487×C137C211yesmaximal, characteristic, complemented
12#559×C137—21no—
13#557×C137—21no—
14#556×C137—22no—
15#556×C137C2212yes—
16#554×C137C2212yes—
17#552×C137—22no—
18#539×C137—22no—
19#536×C137—21no—
20#533×C137C2211yescharacteristic
21#533×C137C2211yescharacteristic
22#532×C137C2211yescharacteristic
23#529×C137—22no—
24#520×C137—21no—
25C2×C4×Q8×C137C2211yescharacteristic
26#500×C137C2211yescharacteristic
27#498×C137C2211yescharacteristic
28#487×C137C2211yescharacteristic
29#487×C137C2211yescharacteristic
30#485×C137—22no—
31#483×C137—21no—
32#479×C137—22no—
33#477×C137—21no—
34#475×C137C2212yes—
35#474×C137C2212yes—
36#468×C137—22no—
37#462×C137C2212yes—
38#461×C137C2212yes—
39#460×C137—21no—
40#451×C137—21no—
41#441×C137—22no—
42#432×C137C2211yescharacteristic
43#431×C137C2211yescharacteristic
44#423×C137C2211yescharacteristic
45#333×C137C2212yes—
46#332×C137C2212yes—
47Q8⋊C8×C137C2212yes—
48Q8⋊C8×C137C2212yes—
49D4⋊C8×C137C2212yes—
50D4⋊C8×C137C2212yes—
51C2×#41×C137—21no—
52C22×Q8×C137D411yescharacteristic
53C2×Q16×C137—22no—
54C2×SD16×C137—22no—
55C2×M16×C137—21no—
56C4⋊Q8×C137—22no—
57C4⋊Q8×C137D411yescharacteristic
58C4⋊Q8×C137C2311yescharacteristic
59C4⋊Q8×C137C2311yescharacteristic
60C4⋊Q8×C137C2311yescharacteristic
61C4⋊Q8×C137C2311yescharacteristic
62#127×C137D411yescharacteristic
63#126×C137—24no—
64#126×C137—24no—
65#125×C137—24no—
66#125×C137—22no—
67#125×C137—21no—
68#125×C137—21no—
69#124×C137—22no—
70#124×C137—22no—
71#124×C137—21no—
72#123×C137—24no—
73#123×C137—24no—
74C22⋊Q8×C137—22no—
75C22⋊Q8×C137—22no—
76C22⋊Q8×C137—24no—
77C22⋊Q8×C137—21no—
78C22⋊Q8×C137—21no—
79C22⋊Q8×C137—21no—
80C22⋊Q8×C137D412yes—
81#121×C137—21no—
82#121×C137D412yes—
83C4×Q8×C137—22no—
84C4×Q8×C137—22no—
85C4×Q8×C137—21no—
86C4×Q8×C137—21no—
87C4×Q8×C137—21no—
88C4×Q8×C137—21no—
89C4×Q8×C137—21no—
90C4×Q8×C137D412yes—
91C4×Q8×C137D412yes—
92C4×Q8×C137C2312yes—
93C4×D4×C137—24no—
94C4×D4×C137—22no—
95C4×D4×C137—22no—
96C4×D4×C137—21no—
97C4×D4×C137C2312yes—
98#117×C137—24no—
99#117×C137—21no—
100#117×C137—22no—
101#117×C137—21no—
102C2×C4⋊C4×C137—21no—
103C2×C4⋊C4×C137—22no—
104C2×C4⋊C4×C137D411yescharacteristic
105C2×C42×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
126#41×C137—44no—
127C2×Q8×C137—21no—
128C2×Q8×C137—21no—
129C2×Q8×C137—21no—
130C2×Q8×C137—21no—
131C2×Q8×C137—21no—
132C2×Q8×C137—42no—
133C2×Q8×C137C2×D412yes—
134C2×Q8×C137C2×D412yes—
135C2×D4×C137—22no—
136C2×D4×C137—21no—
137C2×D4×C137—21no—
138C2×D4×C137C2×D412yes—
139C22×C4×C137—21no—
140C22×C4×C137—24no—
141C22×C4×C137—21no—
142C22×C4×C137—22no—
143C22×C4×C137C2×D411yescharacteristic
144C22×C4×C137C2×D411yescharacteristic
145C22×C4×C137C2×D411yescharacteristic
146Q16×C137—44no—
147SD16×C137—82no—
148M16×C137—81no—
149C2×C8×C137—21no—
150C2×C8×C137—21no—
151C2×C8×C137—22no—
152C2×C8×C137—22no—
153C4⋊C4×C137—24no—
154C4⋊C4×C137—24no—
155C4⋊C4×C137—24no—
156C4⋊C4×C137—22no—
157C4⋊C4×C137—21no—
158C4⋊C4×C137—22no—
159C4⋊C4×C137—21no—
160C4⋊C4×C137—22no—
161C4⋊C4×C137—21no—
162C4⋊C4×C137—22no—
163C4⋊C4×C137—21no—
164C4⋊C4×C137—21no—
165C4⋊C4×C137—21no—
166C4⋊C4×C137—21no—
167C4⋊C4×C137—21no—
168C4⋊C4×C137—21no—
169C4⋊C4×C137—41no—
170C4⋊C4×C137—41no—
171C4⋊C4×C137C2×D412yes—
172C4⋊C4×C137C2×D412yes—
173C4⋊C4×C137C2×D412yes—
174C22⋊C4×C137—24no—
175C22⋊C4×C137—24no—
176C22⋊C4×C137—24no—
177C22⋊C4×C137—22no—
178C22⋊C4×C137—22no—
179C22⋊C4×C137—22no—
180C22⋊C4×C137—41no—
181C42×C137—24no—
182C42×C137—22no—
183C42×C137—22no—
184C42×C137—22no—
185C42×C137—41no—
186C42×C137C2×D411yescharacteristic
187C42×C137C2×D411yescharacteristic
188C42×C137C2×D411yescharacteristic
189C42×C137C2411yescharacteristic
190C23×C137—22no—
191C23×C137#12011yescharacteristic
192Q8×C137—22no—
193Q8×C137—22no—
194Q8×C137—42no—
195Q8×C137—42no—
196Q8×C137—42no—
197Q8×C137—41no—
198Q8×C137—41no—
199Q8×C137—42no—
200Q8×C137—41no—
201Q8×C137—41no—
202D4×C137—22no—
203D4×C137—22no—
204D4×C137—42no—
205D4×C137—44no—
206D4×C137—81no—
207C2×C4×C137—22no—
208C2×C4×C137—21no—
209C2×C4×C137—21no—
210C2×C4×C137—22no—
211C2×C4×C137—22no—
212C2×C4×C137—22no—
213C2×C4×C137—22no—
214C2×C4×C137—22no—
215C2×C4×C137—41no—
216C2×C4×C137—42no—
217C2×C4×C137—44no—
218C2×C4×C137—41no—
219C2×C4×C137—41no—
220C2×C4×C137—81no—
221C2×C4×C137—84no—
222C2×C4×C137#12011yescharacteristic
223C2×C4×C137#12011yescharacteristic
224C2×C4×C137#12011yescharacteristic
225C2×C4×C137C22×D411yescharacteristic
226C2×C4×C137C22×D411yescharacteristic
227C2×C4×C137C22×D411yescharacteristic
228C8×C137—41no—
229C8×C137—41no—
230C8×C137—42no—
231C8×C137—42no—
232C22×C137—21no—
233C22×C137—21no—
234C22×C137—21no—
235C22×C137—42no—
236C22×C137—42no—
237C22×C137—42no—
238C22×C137C2×#12011yescharacteristic
239C4×C137—21no—
240C4×C137—21no—
241C4×C137—21no—
242C4×C137—21no—
243C4×C137—42no—
244C4×C137—42no—
245C4×C137—42no—
246C4×C137—42no—
247C4×C137—42no—
248C4×C137—81no—
249C4×C137—81no—
250C4×C137#57512yes—
251C4×C137#57711yescharacteristic
252C4×C137#57811yescharacteristic
253C2×C137—41no—
254C2×C137—82no—
255C2×C137#300511yescharacteristic
256C2×C137#301311yescharacteristic
257C2×C137#302211yescharacteristic
258#18849C13711yesmaximal, characteristic, direct product factor, Sylow
259C137#1884911yescharacteristic, direct product factor, Sylow
260#3507C2×C13711yescharacteristic
261#3254C2×C13711yescharacteristic
262#3231C2×C13711yescharacteristic, complemented
263#3113C2×C13711yescharacteristic, complemented
264#1652C2×C13712yescomplemented
265#1650C2×C13712yescomplemented
266#1644C2×C13712yescomplemented
267#1643C2×C13712yescomplemented
268#1559C2×C13711yescharacteristic, complemented
269#1496C2×C13711yescharacteristic
270#1487C2×C13711yescharacteristic, complemented
271#559—21no—
272#557—21no—
273#556—22no—
274#556C22×C13712yes—
275#554C22×C13712yes—
276#552—22no—
277#539—22no—
278#536—21no—
279#533C22×C13711yescharacteristic
280#533C22×C13711yescharacteristic
281#532C22×C13711yescharacteristic
282#529—22no—
283#520—21no—
284C2×C4×Q8C22×C13711yescharacteristic
285#500C22×C13711yescharacteristic
286#498C22×C13711yescharacteristic
287#487C22×C13711yescharacteristic
288#487C22×C13711yescharacteristic
289#485—22no—
290#483—21no—
291#479—22no—
292#477—21no—
293#475C22×C13712yes—
294#474C22×C13712yes—
295#468—22no—
296#462C22×C13712yes—
297#461C22×C13712yes—
298#460—21no—
299#451—21no—
300#441—22no—
301#432C22×C13711yescharacteristic
302#431C22×C13711yescharacteristic
303#423C22×C13711yescharacteristic
304#333C22×C13712yes—
305#332C22×C13712yes—
306Q8⋊C8C22×C13712yes—
307Q8⋊C8C22×C13712yes—
308D4⋊C8C22×C13712yes—
309D4⋊C8C22×C13712yes—
310C2×#41—21no—
311C22×Q8D4×C13711yescharacteristic
312C2×Q16—22no—
313C2×SD16—22no—
314C2×M16—21no—
315C4⋊Q8—22no—
316C4⋊Q8D4×C13711yescharacteristic
317C4⋊Q8C23×C13711yescharacteristic
318C4⋊Q8C23×C13711yescharacteristic
319C4⋊Q8C23×C13711yescharacteristic
320C4⋊Q8C23×C13711yescharacteristic
321#127D4×C13711yescharacteristic
322#126—24no—
323#126—24no—
324#125—24no—
325#125—22no—
326#125—21no—
327#125—21no—
328#124—22no—
329#124—22no—
330#124—21no—
331#123—24no—
332#123—24no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—24no—
336C22⋊Q8—21no—
337C22⋊Q8—21no—
338C22⋊Q8—21no—
339C22⋊Q8D4×C13712yes—
340#121—21no—
341#121D4×C13712yes—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8—21no—
345C4×Q8—21no—
346C4×Q8—21no—
347C4×Q8—21no—
348C4×Q8—21no—
349C4×Q8D4×C13712yes—
350C4×Q8D4×C13712yes—
351C4×Q8C23×C13712yes—
352C4×D4—24no—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—21no—
356C4×D4C23×C13712yes—
357#117—24no—
358#117—21no—
359#117—22no—
360#117—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4D4×C13711yescharacteristic
364C2×C42—21no—
365C2×C42C23×C13711yescharacteristic
366#107—22no—
367#107—22no—
368#106—22no—
369#106—22no—
370C4⋊C8—21no—
371C4⋊C8—21no—
372C4⋊C8C23×C13712yes—
373C4⋊C8C23×C13712yes—
374#103—22no—
375#103—22no—
376#103—22no—
377#103—22no—
378#103—22no—
379#102—22no—
380C22⋊C8—21no—
381C22⋊C8—21no—
382#97—21no—
383C4×C8C23×C13711yescharacteristic
384C4×C8C23×C13711yescharacteristic
385#41—44no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—42no—
392C2×Q8C2×D4×C13712yes—
393C2×Q8C2×D4×C13712yes—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4C2×D4×C13712yes—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4C2×D4×C13711yescharacteristic
403C22×C4C2×D4×C13711yescharacteristic
404C22×C4C2×D4×C13711yescharacteristic
405Q16—44no—
406SD16—82no—
407M16—81no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—22no—
411C2×C8—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4C2×D4×C13712yes—
431C4⋊C4C2×D4×C13712yes—
432C4⋊C4C2×D4×C13712yes—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—41no—
440C42—24no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—41no—
445C42C2×D4×C13711yescharacteristic
446C42C2×D4×C13711yescharacteristic
447C42C2×D4×C13711yescharacteristic
448C42C24×C13711yescharacteristic
449C23—22no—
450C23#120×C13711yescharacteristic
451Q8—22no—
452Q8—22no—
453Q8—42no—
454Q8—42no—
455Q8—42no—
456Q8—41no—
457Q8—41no—
458Q8—42no—
459Q8—41no—
460Q8—41no—
461D4—22no—
462D4—22no—
463D4—42no—
464D4—44no—
465D4—81no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—81no—
480C2×C4—84no—
481C2×C4#120×C13711yescharacteristic
482C2×C4#120×C13711yescharacteristic
483C2×C4#120×C13711yescharacteristic
484C2×C4C22×D4×C13711yescharacteristic
485C2×C4C22×D4×C13711yescharacteristic
486C2×C4C22×D4×C13711yescharacteristic
487C8—41no—
488C8—41no—
489C8—42no—
490C8—42no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22C2×#120×C13711yescharacteristic
498C4—21no—
499C4—21no—
500C4—21no—
501C4—21no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—81no—
508C4—81no—
509C4#575×C13712yes—
510C4#577×C13711yescharacteristic
511C4#578×C13711yescharacteristic
512C2—41no—
513C2—82no—
514C2#3005×C13711yescharacteristic
515C2#3013×C13711yescharacteristic
516C2#3022×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#532×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
3. #3231×C137C2semidirect
4. #3113×C137C2semidirect
5. #1652×C137C2semidirect
6. #1650×C137C2semidirect
7. #1644×C137C2semidirect
8. #1643×C137C2semidirect
9. #1559×C137C2semidirect
11. #1487×C137C2semidirect
258. #18849C137direct (N × H)
259. C137#18849direct (N × H)
262. #3231C2×C137semidirect
263. #3113C2×C137semidirect
264. #1652C2×C137semidirect
265. #1650C2×C137semidirect
266. #1644C2×C137semidirect
267. #1643C2×C137semidirect
268. #1559C2×C137semidirect
270. #1487C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3507×C1372
#3254×C1372
#3231×C1372
#3113×C1372
#1652×C1372
#1650×C1372
#1644×C1372
#1643×C1372
#1559×C1372
#1496×C1372
#1487×C1372
#18849137