Finite Group Databasebeta

← C137×#18657 #46868043

Subgroups — C137×#18657 #46868043

Order 35072.

940 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#3511×C137C211yesmaximal, characteristic
2#3206×C137C211yesmaximal, characteristic, complemented
3C2×#500×C137C211yesmaximal, characteristic, complemented
4#3079×C137C211yesmaximal, characteristic, complemented
5#1650×C137C214yesmaximal, complemented
6#1647×C137C214yesmaximal, complemented
7#1549×C137C211yesmaximal, characteristic, complemented
8#1492×C137C211yesmaximal, characteristic, complemented
9#1476×C137C211yesmaximal, characteristic, complemented
10#556×C137C2212yes—
11#555×C137—21no—
12#554×C137—21no—
13#553×C137—22no—
14#548×C137—22no—
15#547×C137C2212yes—
16#542×C137—22no—
17#537×C137—21no—
18#533×C137—21no—
19#532×C137—21no—
20#532×C137C2211yescharacteristic
21C2×C4⋊Q8×C137C2211yescharacteristic
22C2×C4⋊Q8×C137C2211yescharacteristic
23#520×C137—21no—
24C4×#41×C137C2211yescharacteristic
25#500×C137—22no—
26#500×C137C2214yes—
27#492×C137—21no—
28#487×C137—21no—
29#477×C137—21no—
30#476×C137—21no—
31#475×C137C2212yes—
32#474×C137C2212yes—
33#461×C137C2214yes—
34#448×C137—22no—
35#431×C137C2211yescharacteristic
36#428×C137—21no—
37C2×#107×C137—22no—
38#423×C137C2211yescharacteristic
39#417×C137—21no—
40C2×C4×C8×C137C2211yescharacteristic
41#332×C137C2214yes—
42Q8⋊C8×C137C2212yes—
43D4⋊C8×C137C2212yes—
44D4⋊C8×C137C2214yes—
45C2×#41×C137—22no—
46C2×#41×C137—21no—
47C2×#41×C137D411yescharacteristic
48C22×Q8×C137—21no—
49C22×Q8×C137—21no—
50C2×SD16×C137—24no—
51C2×M16×C137—21no—
52C22×C8×C137—21no—
53C4⋊Q8×C137—21no—
54C4⋊Q8×C137—21no—
55C4⋊Q8×C137—21no—
56C4⋊Q8×C137—21no—
57C4⋊Q8×C137—21no—
58C4⋊Q8×C137D411yescharacteristic
59C4⋊Q8×C137D412yes—
60C4⋊Q8×C137C2312yes—
61C4⋊Q8×C137C2312yes—
62#127×C137D411yescharacteristic
63#126×C137—24no—
64#125×C137—22no—
65#124×C137—24no—
66#124×C137—22no—
67#124×C137—21no—
68#123×C137—24no—
69C22⋊Q8×C137—22no—
70C22⋊Q8×C137—24no—
71C22⋊Q8×C137—22no—
72C22⋊Q8×C137—22no—
73#121×C137—22no—
74#121×C137—22no—
75#121×C137—24no—
76#121×C137—22no—
77#121×C137—21no—
78#121×C137D412yes—
79C4×Q8×C137—22no—
80C4×Q8×C137—22no—
81C4×Q8×C137D411yescharacteristic
82C4×Q8×C137C2311yescharacteristic
83C4×D4×C137—22no—
84C4×D4×C137—24no—
85C4×D4×C137—22no—
86C4×D4×C137—21no—
87C4×D4×C137—21no—
88C4×D4×C137—21no—
89C4×D4×C137D411yescharacteristic
90C4×D4×C137C2312yes—
91C4×D4×C137C2311yescharacteristic
92#117×C137—22no—
93#117×C137—22no—
94#117×C137—21no—
95#117×C137D411yescharacteristic
96C2×C4⋊C4×C137—21no—
97C2×C4⋊C4×C137—22no—
98C2×C4⋊C4×C137—21no—
99C2×C4⋊C4×C137—21no—
100C2×C4⋊C4×C137D412yes—
101C2×C42×C137—21no—
102C2×C42×C137C2311yescharacteristic
103#107×C137—28no—
104#107×C137—22no—
105#106×C137—22no—
106C4⋊C8×C137—21no—
107C4⋊C8×C137C2312yes—
108C4⋊C8×C137C2312yes—
109#103×C137—22no—
110#103×C137—24no—
111#102×C137—22no—
112C22⋊C8×C137—21no—
113C4×C8×C137—21no—
114C4×C8×C137—21no—
115C4×C8×C137C2312yes—
116#41×C137—22no—
117#41×C137—22no—
118#41×C137—42no—
119#41×C137—42no—
120#41×C137—84no—
121C2×Q8×C137—21no—
122C2×Q8×C137—22no—
123C2×Q8×C137—22no—
124C2×Q8×C137—22no—
125C2×Q8×C137—41no—
126C2×Q8×C137—82no—
127C2×Q8×C137—82no—
128C2×Q8×C137C2×D411yescharacteristic
129C2×D4×C137—21no—
130C2×D4×C137—22no—
131C2×D4×C137—24no—
132C2×D4×C137—21no—
133C2×D4×C137—21no—
134C2×D4×C137C2×D411yescharacteristic
135C2×D4×C137C2×D412yes—
136C22×C4×C137—24no—
137C22×C4×C137—22no—
138C22×C4×C137—21no—
139C22×C4×C137—21no—
140C22×C4×C137—21no—
141C22×C4×C137—22no—
142C22×C4×C137—41no—
143C22×C4×C137C2×D411yescharacteristic
144C22×C4×C137C2×D411yescharacteristic
145C22×C4×C137C2×D411yescharacteristic
146SD16×C137—84no—
147M16×C137—42no—
148C2×C8×C137—24no—
149C2×C8×C137—22no—
150C2×C8×C137—22no—
151C2×C8×C137—22no—
152C4⋊C4×C137—24no—
153C4⋊C4×C137—24no—
154C4⋊C4×C137—22no—
155C4⋊C4×C137—22no—
156C4⋊C4×C137—21no—
157C4⋊C4×C137—21no—
158C4⋊C4×C137—22no—
159C4⋊C4×C137—21no—
160C4⋊C4×C137—22no—
161C4⋊C4×C137—22no—
162C4⋊C4×C137—41no—
163C4⋊C4×C137—41no—
164C4⋊C4×C137C2×D412yes—
165C4⋊C4×C137C2×D411yescharacteristic
166C4⋊C4×C137C2×D411yescharacteristic
167C4⋊C4×C137C2×D414yes—
168C22⋊C4×C137—24no—
169C22⋊C4×C137—24no—
170C22⋊C4×C137—24no—
171C22⋊C4×C137—21no—
172C22⋊C4×C137—21no—
173C22⋊C4×C137—22no—
174C22⋊C4×C137—21no—
175C22⋊C4×C137—22no—
176C42×C137—24no—
177C42×C137—21no—
178C42×C137—21no—
179C42×C137—21no—
180C42×C137C2×D411yescharacteristic
181C42×C137C2×D411yescharacteristic
182C42×C137C2×D411yescharacteristic
183C42×C137C2411yescharacteristic
184C23×C137—21no—
185C23×C137—22no—
186C23×C137#12011yescharacteristic
187Q8×C137—21no—
188Q8×C137—21no—
189Q8×C137—44no—
190Q8×C137—42no—
191Q8×C137—44no—
192Q8×C137—82no—
193Q8×C137—82no—
194D4×C137—21no—
195D4×C137—21no—
196D4×C137—24no—
197D4×C137—42no—
198D4×C137—44no—
199D4×C137—42no—
200D4×C137—42no—
201D4×C137—84no—
202C2×C4×C137—22no—
203C2×C4×C137—22no—
204C2×C4×C137—21no—
205C2×C4×C137—21no—
206C2×C4×C137—22no—
207C2×C4×C137—21no—
208C2×C4×C137—24no—
209C2×C4×C137—42no—
210C2×C4×C137—42no—
211C2×C4×C137—42no—
212C2×C4×C137—42no—
213C2×C4×C137—42no—
214C2×C4×C137—42no—
215C2×C4×C137—82no—
216C2×C4×C137—84no—
217C2×C4×C137—82no—
218C2×C4×C137#12011yescharacteristic
219C2×C4×C137#12011yescharacteristic
220C2×C4×C137#12011yescharacteristic
221C2×C4×C137#13512yes—
222C2×C4×C137#13712yes—
223C2×C4×C137C22×D411yescharacteristic
224C2×C4×C137C22×D411yescharacteristic
225C2×C4×C137C22×D411yescharacteristic
226C8×C137—24no—
227C8×C137—42no—
228C8×C137—42no—
229C22×C137—21no—
230C22×C137—21no—
231C22×C137—41no—
232C22×C137—42no—
233C22×C137—41no—
234C22×C137—42no—
235C22×C137—41no—
236C22×C137—42no—
237C22×C137#45212yes—
238C22×C137C2×#12011yescharacteristic
239C4×C137—21no—
240C4×C137—21no—
241C4×C137—41no—
242C4×C137—42no—
243C4×C137—41no—
244C4×C137—44no—
245C4×C137—41no—
246C4×C137—82no—
247C4×C137#45212yes—
248C4×C137#45612yes—
249C4×C137C2×#13511yescharacteristic
250C4×C137C2×#13711yescharacteristic
251C4×C137#57612yes—
252C2×C137—22no—
253C2×C137—81no—
254C2×C137—82no—
255C2×C137C2×#45211yescharacteristic
256C2×C137#300811yescharacteristic
257C2×C137C2×#45611yescharacteristic
258#18657C13711yesmaximal, characteristic, direct product factor, Sylow
259C137#1865711yescharacteristic, direct product factor, Sylow
260#3511C2×C13711yescharacteristic
261#3206C2×C13711yescharacteristic, complemented
262C2×#500C2×C13711yescharacteristic, complemented
263#3079C2×C13711yescharacteristic, complemented
264#1650C2×C13714yescomplemented
265#1647C2×C13714yescomplemented
266#1549C2×C13711yescharacteristic, complemented
267#1492C2×C13711yescharacteristic, complemented
268#1476C2×C13711yescharacteristic, complemented
269#556C22×C13712yes—
270#555—21no—
271#554—21no—
272#553—22no—
273#548—22no—
274#547C22×C13712yes—
275#542—22no—
276#537—21no—
277#533—21no—
278#532—21no—
279#532C22×C13711yescharacteristic
280C2×C4⋊Q8C22×C13711yescharacteristic
281C2×C4⋊Q8C22×C13711yescharacteristic
282#520—21no—
283C4×#41C22×C13711yescharacteristic
284#500—22no—
285#500C22×C13714yes—
286#492—21no—
287#487—21no—
288#477—21no—
289#476—21no—
290#475C22×C13712yes—
291#474C22×C13712yes—
292#461C22×C13714yes—
293#448—22no—
294#431C22×C13711yescharacteristic
295#428—21no—
296C2×#107—22no—
297#423C22×C13711yescharacteristic
298#417—21no—
299C2×C4×C8C22×C13711yescharacteristic
300#332C22×C13714yes—
301Q8⋊C8C22×C13712yes—
302D4⋊C8C22×C13712yes—
303D4⋊C8C22×C13714yes—
304C2×#41—22no—
305C2×#41—21no—
306C2×#41D4×C13711yescharacteristic
307C22×Q8—21no—
308C22×Q8—21no—
309C2×SD16—24no—
310C2×M16—21no—
311C22×C8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8D4×C13712yes—
318C4⋊Q8D4×C13711yescharacteristic
319C4⋊Q8C23×C13712yes—
320C4⋊Q8C23×C13712yes—
321#127D4×C13711yescharacteristic
322#126—24no—
323#125—22no—
324#124—24no—
325#124—22no—
326#124—21no—
327#123—24no—
328C22⋊Q8—22no—
329C22⋊Q8—24no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332#121—22no—
333#121—22no—
334#121—24no—
335#121—22no—
336#121—21no—
337#121D4×C13712yes—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8D4×C13711yescharacteristic
341C4×Q8C23×C13711yescharacteristic
342C4×D4—22no—
343C4×D4—24no—
344C4×D4—22no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4D4×C13711yescharacteristic
349C4×D4C23×C13711yescharacteristic
350C4×D4C23×C13712yes—
351#117—22no—
352#117—22no—
353#117—21no—
354#117D4×C13711yescharacteristic
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4D4×C13712yes—
360C2×C42—21no—
361C2×C42C23×C13711yescharacteristic
362#107—28no—
363#107—22no—
364#106—22no—
365C4⋊C8—21no—
366C4⋊C8C23×C13712yes—
367C4⋊C8C23×C13712yes—
368#103—22no—
369#103—24no—
370#102—22no—
371C22⋊C8—21no—
372C4×C8—21no—
373C4×C8—21no—
374C4×C8C23×C13712yes—
375#41—22no—
376#41—22no—
377#41—42no—
378#41—42no—
379#41—84no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—41no—
385C2×Q8—82no—
386C2×Q8—82no—
387C2×Q8C2×D4×C13711yescharacteristic
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—24no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4C2×D4×C13711yescharacteristic
394C2×D4C2×D4×C13712yes—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—41no—
402C22×C4C2×D4×C13711yescharacteristic
403C22×C4C2×D4×C13711yescharacteristic
404C22×C4C2×D4×C13711yescharacteristic
405SD16—84no—
406M16—42no—
407C2×C8—24no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4C2×D4×C13712yes—
424C4⋊C4C2×D4×C13711yescharacteristic
425C4⋊C4C2×D4×C13711yescharacteristic
426C4⋊C4C2×D4×C13714yes—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—21no—
431C22⋊C4—21no—
432C22⋊C4—22no—
433C22⋊C4—21no—
434C22⋊C4—22no—
435C42—24no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42C2×D4×C13711yescharacteristic
440C42C2×D4×C13711yescharacteristic
441C42C2×D4×C13711yescharacteristic
442C42C24×C13711yescharacteristic
443C23—21no—
444C23—22no—
445C23#120×C13711yescharacteristic
446Q8—21no—
447Q8—21no—
448Q8—42no—
449Q8—44no—
450Q8—44no—
451Q8—82no—
452Q8—82no—
453D4—21no—
454D4—21no—
455D4—24no—
456D4—42no—
457D4—44no—
458D4—42no—
459D4—42no—
460D4—84no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—24no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—82no—
475C2×C4—84no—
476C2×C4—82no—
477C2×C4#120×C13711yescharacteristic
478C2×C4#120×C13711yescharacteristic
479C2×C4#120×C13711yescharacteristic
480C2×C4#135×C13712yes—
481C2×C4#137×C13712yes—
482C2×C4C22×D4×C13711yescharacteristic
483C2×C4C22×D4×C13711yescharacteristic
484C2×C4C22×D4×C13711yescharacteristic
485C8—24no—
486C8—42no—
487C8—42no—
488C22—21no—
489C22—21no—
490C22—41no—
491C22—42no—
492C22—41no—
493C22—42no—
494C22—41no—
495C22—42no—
496C22#452×C13712yes—
497C22C2×#120×C13711yescharacteristic
498C4—21no—
499C4—21no—
500C4—41no—
501C4—42no—
502C4—41no—
503C4—44no—
504C4—41no—
505C4—82no—
506C4#452×C13712yes—
507C4#456×C13712yes—
508C4C2×#135×C13711yescharacteristic
509C4C2×#137×C13711yescharacteristic
510C4#576×C13712yes—
511C2—22no—
512C2—81no—
513C2—82no—
514C2C2×#452×C13711yescharacteristic
515C2#3008×C13711yescharacteristic
516C2C2×#456×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3511×C137C2
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. #3206×C137C2semidirect
3. C2×#500×C137C2semidirect
4. #3079×C137C2semidirect
5. #1650×C137C2semidirect
6. #1647×C137C2semidirect
7. #1549×C137C2semidirect
8. #1492×C137C2semidirect
9. #1476×C137C2semidirect
258. #18657C137direct (N × H)
259. C137#18657direct (N × H)
261. #3206C2×C137semidirect
262. C2×#500C2×C137semidirect
263. #3079C2×C137semidirect
264. #1650C2×C137semidirect
265. #1647C2×C137semidirect
266. #1549C2×C137semidirect
267. #1492C2×C137semidirect
268. #1476C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3511×C1372
#3206×C1372
C2×#500×C1372
#3079×C1372
#1650×C1372
#1647×C1372
#1549×C1372
#1492×C1372
#1476×C1372
#18657137