Finite Group Databasebeta

← C137×#9924 #46859310

Subgroups — C137×#9924 #46859310

Order 35072.

890 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#451×C137C211yesmaximal, characteristic, complemented
2#2626×C137C211yesmaximal, characteristic
3C2×#393×C137C211yesmaximal, characteristic
4#1787×C137C211yesmaximal, characteristic, complemented
5C2×#328×C137C211yesmaximal, characteristic, complemented
6C2×#323×C137C211yesmaximal, characteristic
7#1294×C137C211yesmaximal, characteristic, complemented
8C23×Q8×C137C2211yescharacteristic
9C22×Q16×C137—21no—
10C2×C22⋊Q8×C137C2211yescharacteristic
11C22×C22⋊C4×C137—21no—
12C22×C22⋊C4×C137C2211yescharacteristic
13#451×C137—22no—
14#451×C137—22no—
15C2×#103×C137—21no—
16C2×#103×C137—22no—
17C2×C22⋊C8×C137C2211yescharacteristic
18C2×C22⋊C8×C137C2211yescharacteristic
19#399×C137—21no—
20#396×C137—22no—
21#393×C137—21no—
22#393×C137—22no—
23#393×C137—22no—
24#385×C137—21no—
25#380×C137C2211yescharacteristic
26#379×C137—21no—
27C2×#95×C137—21no—
28C2×#95×C137C2211yescharacteristic
29#328×C137—22no—
30#323×C137—22no—
31C25×C137D411yescharacteristic
32C22×Q8×C137—21no—
33C22×Q8×C137—21no—
34C22×Q8×C137—21no—
35C22×Q8×C137—21no—
36C22×Q8×C137—21no—
37C22×Q8×C137—44no—
38C22×Q8×C137D412yes—
39C22×Q8×C137D411yescharacteristic
40C23×C4×C137—21no—
41C23×C4×C137—21no—
42C23×C4×C137D411yescharacteristic
43C23×C4×C137C2311yescharacteristic
44C2×Q16×C137—24no—
45C2×Q16×C137—44no—
46C22×C8×C137—21no—
47C22×C8×C137—21no—
48C22⋊Q8×C137—21no—
49C22⋊Q8×C137—21no—
50C22⋊Q8×C137—21no—
51C22⋊Q8×C137—21no—
52C2×C4⋊C4×C137—21no—
53C2×C4⋊C4×C137—22no—
54C2×C4⋊C4×C137D411yescharacteristic
55C2×C4⋊C4×C137D412yes—
56C2×C22⋊C4×C137—24no—
57C2×C22⋊C4×C137—21no—
58C2×C22⋊C4×C137—21no—
59C2×C22⋊C4×C137—21no—
60C2×C22⋊C4×C137—21no—
61C2×C22⋊C4×C137—21no—
62C2×C22⋊C4×C137—21no—
63C2×C22⋊C4×C137—21no—
64C2×C22⋊C4×C137—21no—
65C2×C22⋊C4×C137—22no—
66C2×C22⋊C4×C137—22no—
67C2×C22⋊C4×C137—44no—
68#103×C137—24no—
69#103×C137—44no—
70C22⋊C8×C137—21no—
71C22⋊C8×C137—21no—
72C22⋊C8×C137—21no—
73C22⋊C8×C137D412yes—
74#95×C137—24no—
75#95×C137—22no—
76#95×C137—21no—
77#95×C137—21no—
78#95×C137D412yes—
79C24×C137—21no—
80C24×C137—21no—
81C24×C137—21no—
82C24×C137—21no—
83C24×C137—21no—
84C24×C137—21no—
85C24×C137—42no—
86C24×C137—42no—
87C24×C137Q1612yescomplemented
88C24×C137C2×D411yescharacteristic
89C2×Q8×C137—24no—
90C2×Q8×C137—24no—
91C2×Q8×C137—22no—
92C2×Q8×C137—21no—
93C2×Q8×C137—22no—
94C2×Q8×C137—21no—
95C2×Q8×C137—44no—
96C2×Q8×C137—42no—
97C2×Q8×C137—42no—
98C2×Q8×C137—44no—
99C2×Q8×C137—42no—
100C2×Q8×C137—44no—
101C2×Q8×C137—44no—
102C2×Q8×C137—41no—
103C2×Q8×C137—41no—
104C22×C4×C137—24no—
105C22×C4×C137—22no—
106C22×C4×C137—21no—
107C22×C4×C137—22no—
108C22×C4×C137—22no—
109C22×C4×C137—21no—
110C22×C4×C137—21no—
111C22×C4×C137—21no—
112C22×C4×C137—21no—
113C22×C4×C137—42no—
114C22×C4×C137—42no—
115C22×C4×C137—41no—
116C22×C4×C137—41no—
117C22×C4×C137—42no—
118C22×C4×C137—42no—
119C22×C4×C137—42no—
120C22×C4×C137—41no—
121C22×C4×C137—41no—
122C22×C4×C137C2×D411yescharacteristic
123C22×C4×C137C2×D411yescharacteristic
124C22×C4×C137C2×D411yescharacteristic
125C22×C4×C137C2×D411yescharacteristic
126C22×C4×C137C2×D411yescharacteristic
127C22×C4×C137#4111yescharacteristic
128Q16×C137—48no—
129C2×C8×C137—22no—
130C2×C8×C137—42no—
131C2×C8×C137—42no—
132C2×C8×C137—41no—
133C4⋊C4×C137—21no—
134C4⋊C4×C137—21no—
135C4⋊C4×C137—42no—
136C4⋊C4×C137—41no—
137C4⋊C4×C137—41no—
138C4⋊C4×C137—82no—
139C22⋊C4×C137—48no—
140C22⋊C4×C137—41no—
141C22⋊C4×C137—41no—
142C22⋊C4×C137—41no—
143C22⋊C4×C137—41no—
144C22⋊C4×C137—41no—
145C22⋊C4×C137—41no—
146C22⋊C4×C137—41no—
147C22⋊C4×C137—41no—
148C22⋊C4×C137—82no—
149C22⋊C4×C137—82no—
150C23×C137—22no—
151C23×C137—22no—
152C23×C137—22no—
153C23×C137—22no—
154C23×C137—22no—
155C23×C137—44no—
156C23×C137—42no—
157C23×C137—44no—
158C23×C137—42no—
159C23×C137—41no—
160C23×C137—41no—
161C23×C137—41no—
162C23×C137—42no—
163C23×C137—42no—
164C23×C137—41no—
165C23×C137—82no—
166C23×C137—82no—
167C23×C137—81no—
168C23×C137—81no—
169C23×C137#12011yescharacteristic
170C23×C137#12011yescharacteristic
171C23×C137#12111yescharacteristic
172C23×C137C2×Q1611yescharacteristic
173C23×C137#13511yescharacteristic
174C23×C137#13711yescharacteristic
175C23×C137#13711yescharacteristic
176Q8×C137—24no—
177Q8×C137—24no—
178Q8×C137—42no—
179Q8×C137—44no—
180Q8×C137—42no—
181Q8×C137—44no—
182Q8×C137—42no—
183Q8×C137—81no—
184C2×C4×C137—22no—
185C2×C4×C137—22no—
186C2×C4×C137—24no—
187C2×C4×C137—41no—
188C2×C4×C137—41no—
189C2×C4×C137—41no—
190C2×C4×C137—41no—
191C2×C4×C137—42no—
192C2×C4×C137—44no—
193C2×C4×C137—42no—
194C2×C4×C137—44no—
195C2×C4×C137—42no—
196C2×C4×C137—42no—
197C2×C4×C137—42no—
198C2×C4×C137—41no—
199C2×C4×C137—41no—
200C2×C4×C137—41no—
201C2×C4×C137—42no—
202C2×C4×C137—42no—
203C2×C4×C137—42no—
204C2×C4×C137—42no—
205C2×C4×C137—42no—
206C2×C4×C137—42no—
207C2×C4×C137—42no—
208C2×C4×C137—82no—
209C2×C4×C137—82no—
210C2×C4×C137—81no—
211C2×C4×C137#12011yescharacteristic
212C2×C4×C137#12111yescharacteristic
213C2×C4×C137#12111yescharacteristic
214C2×C4×C137#12711yescharacteristic
215C8×C137—42no—
216C8×C137—81no—
217C22×C137—22no—
218C22×C137—22no—
219C22×C137—22no—
220C22×C137—22no—
221C22×C137—21no—
222C22×C137—21no—
223C22×C137—21no—
224C22×C137—21no—
225C22×C137—44no—
226C22×C137—44no—
227C22×C137—42no—
228C22×C137—82no—
229C22×C137—82no—
230C22×C137—82no—
231C22×C137—82no—
232C22×C137—81no—
233C22×C137—81no—
234C22×C137#39211yescharacteristic
235C22×C137#45111yescharacteristic
236C22×C137#45111yescharacteristic
237C22×C137#45211yescharacteristic
238C22×C137#45211yescharacteristic
239C22×C137#45512yes—
240C22×C137B(4,𝔽2)12yes—
241C22×C137#46711yescharacteristic
242C22×C137#47011yescharacteristic
243C4×C137—21no—
244C4×C137—21no—
245C4×C137—21no—
246C4×C137—21no—
247C4×C137—44no—
248C4×C137—82no—
249C4×C137—82no—
250C4×C137—81no—
251C4×C137—81no—
252C4×C137—81no—
253C2×C137—21no—
254C2×C137—21no—
255C2×C137—21no—
256C2×C137—21no—
257C2×C137—82no—
258C2×C137#160212yes—
259C2×C137#160612yes—
260C2×C137#200111yescharacteristic
261C2×C137#200411yescharacteristic
262C2×C137#202111yescharacteristic
263#9924C13711yesmaximal, characteristic, direct product factor, Sylow
264C137#992411yescharacteristic, direct product factor, Sylow
265C2×#451C2×C13711yescharacteristic, complemented
266#2626C2×C13711yescharacteristic
267C2×#393C2×C13711yescharacteristic
268#1787C2×C13711yescharacteristic, complemented
269C2×#328C2×C13711yescharacteristic, complemented
270C2×#323C2×C13711yescharacteristic
271#1294C2×C13711yescharacteristic, complemented
272C23×Q8C22×C13711yescharacteristic
273C22×Q16—21no—
274C2×C22⋊Q8C22×C13711yescharacteristic
275C22×C22⋊C4—21no—
276C22×C22⋊C4C22×C13711yescharacteristic
277#451—22no—
278#451—22no—
279C2×#103—21no—
280C2×#103—22no—
281C2×C22⋊C8C22×C13711yescharacteristic
282C2×C22⋊C8C22×C13711yescharacteristic
283#399—21no—
284#396—22no—
285#393—21no—
286#393—22no—
287#393—22no—
288#385—21no—
289#380C22×C13711yescharacteristic
290#379—21no—
291C2×#95—21no—
292C2×#95C22×C13711yescharacteristic
293#328—22no—
294#323—22no—
295C25D4×C13711yescharacteristic
296C22×Q8—21no—
297C22×Q8—21no—
298C22×Q8—21no—
299C22×Q8—21no—
300C22×Q8—21no—
301C22×Q8—44no—
302C22×Q8D4×C13711yescharacteristic
303C22×Q8D4×C13712yes—
304C23×C4—21no—
305C23×C4—21no—
306C23×C4D4×C13711yescharacteristic
307C23×C4C23×C13711yescharacteristic
308C2×Q16—24no—
309C2×Q16—44no—
310C22×C8—21no—
311C22×C8—21no—
312C22⋊Q8—21no—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315C22⋊Q8—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4D4×C13711yescharacteristic
319C2×C4⋊C4D4×C13712yes—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—44no—
332#103—24no—
333#103—44no—
334C22⋊C8—21no—
335C22⋊C8—21no—
336C22⋊C8—21no—
337C22⋊C8D4×C13712yes—
338#95—24no—
339#95—22no—
340#95—21no—
341#95—21no—
342#95D4×C13712yes—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—21no—
348C24—21no—
349C24—42no—
350C24—42no—
351C24Q16×C13712yescomplemented
352C24C2×D4×C13711yescharacteristic
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—21no—
359C2×Q8—44no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—44no—
363C2×Q8—42no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—41no—
367C2×Q8—41no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4C2×D4×C13711yescharacteristic
387C22×C4C2×D4×C13711yescharacteristic
388C22×C4C2×D4×C13711yescharacteristic
389C22×C4C2×D4×C13711yescharacteristic
390C22×C4C2×D4×C13711yescharacteristic
391C22×C4#41×C13711yescharacteristic
392Q16—48no—
393C2×C8—22no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—41no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—42no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—82no—
403C22⋊C4—48no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—44no—
420C23—42no—
421C23—44no—
422C23—42no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—42no—
428C23—41no—
429C23—82no—
430C23—81no—
431C23—82no—
432C23—81no—
433C23#120×C13711yescharacteristic
434C23#120×C13711yescharacteristic
435C23#121×C13711yescharacteristic
436C23C2×Q16×C13711yescharacteristic
437C23#135×C13711yescharacteristic
438C23#137×C13711yescharacteristic
439C23#137×C13711yescharacteristic
440Q8—24no—
441Q8—24no—
442Q8—42no—
443Q8—44no—
444Q8—42no—
445Q8—44no—
446Q8—42no—
447Q8—81no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4#120×C13711yescharacteristic
476C2×C4#121×C13711yescharacteristic
477C2×C4#121×C13711yescharacteristic
478C2×C4#127×C13711yescharacteristic
479C8—42no—
480C8—81no—
481C22—21no—
482C22—21no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—21no—
488C22—21no—
489C22—44no—
490C22—44no—
491C22—42no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—81no—
496C22—82no—
497C22—81no—
498C22#392×C13711yescharacteristic
499C22#451×C13711yescharacteristic
500C22#451×C13711yescharacteristic
501C22#452×C13711yescharacteristic
502C22#452×C13711yescharacteristic
503C22#455×C13712yes—
504C22B(4,𝔽2)×C13712yes—
505C22#467×C13711yescharacteristic
506C22#470×C13711yescharacteristic
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—44no—
512C4—82no—
513C4—81no—
514C4—82no—
515C4—81no—
516C4—81no—
517C2—21no—
518C2—21no—
519C2—21no—
520C2—21no—
521C2—82no—
522C2#1602×C13712yes—
523C2#1606×C13712yes—
524C2#2001×C13711yescharacteristic
525C2#2004×C13711yescharacteristic
526C2#2021×C13711yescharacteristic
subgroupG/H
center, socleC23×C137#120
commutator, Frattini, cosocleC23×C4C23×C137
prime radicalC25×C137D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C137 ◁ C23×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#451×C137C2semidirect
4. #1787×C137C2semidirect
5. C2×#328×C137C2semidirect
7. #1294×C137C2semidirect
87. C24×C137Q16semidirect
263. #9924C137direct (N × H)
264. C137#9924direct (N × H)
265. C2×#451C2×C137semidirect
268. #1787C2×C137semidirect
269. C2×#328C2×C137semidirect
271. #1294C2×C137semidirect
351. C24Q16×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#451×C1372
#2626×C1372
C2×#393×C1372
#1787×C1372
C2×#328×C1372
C2×#323×C1372
#1294×C1372
#9924137