Finite Group Databasebeta

← C137×#25854 #46875240

Subgroups — C137×#25854 #46875240

Order 35072.

1094 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3501×C137C211yesmaximal, characteristic
2#2583×C137C212yesmaximal, complemented
3#2579×C137C211yesmaximal, characteristic
4#2062×C137C212yesmaximal, complemented
5#2062×C137C212yesmaximal, complemented
6#1996×C137C212yesmaximal, complemented
7#1867×C137C212yesmaximal
8#1867×C137C212yesmaximal
9C4×#99×C137C211yesmaximal, characteristic
10#554×C137—24no—
11#554×C137—22no—
12D4×Q8×C137—22no—
13#544×C137—21no—
14#543×C137—21no—
15#543×C137C2212yes—
16#533×C137—21no—
17C2×C22⋊Q8×C137—21no—
18C2×C22⋊Q8×C137C2212yes—
19#518×C137C2212yes—
20C2×C4×D4×C137C2211yescharacteristic
21C22×C4⋊C4×C137C2211yescharacteristic
22C2×#99×C137C2212yes—
23#398×C137—21no—
24#398×C137—22no—
25#396×C137—21no—
26#395×C137—22no—
27#389×C137—24no—
28#387×C137—21no—
29#386×C137C2211yescharacteristic
30#386×C137C2211yescharacteristic
31#385×C137—21no—
32#385×C137C2211yescharacteristic
33#385×C137C2211yescharacteristic
34#385×C137C2214yes—
35#385×C137C2214yes—
36#380×C137C2211yescharacteristic
37C4×C22⋊C4×C137C2212yes—
38#342×C137C2212yes—
39#342×C137C2214yes—
40#342×C137C2214yes—
41C22×Q8×C137—21no—
42C22×Q8×C137D412yes—
43C22×D4×C137C2311yescharacteristic
44C23×C4×C137D411yescharacteristic
45C23×C4×C137C2311yescharacteristic
46C23×C4×C137C2311yescharacteristic
47C23×C4×C137C2311yescharacteristic
48C4⋊Q8×C137—21no—
49C4⋊Q8×C137—21no—
50C4⋊Q8×C137—24no—
51#125×C137—22no—
52#125×C137—24no—
53C22⋊Q8×C137—22no—
54C22⋊Q8×C137—22no—
55C22⋊Q8×C137—24no—
56C22⋊Q8×C137—24no—
57C22⋊Q8×C137—22no—
58C22⋊Q8×C137—22no—
59C22⋊Q8×C137—22no—
60C22⋊Q8×C137—22no—
61C22⋊Q8×C137—22no—
62C22⋊Q8×C137—22no—
63C22⋊Q8×C137—24no—
64C22⋊Q8×C137—24no—
65C4×Q8×C137—22no—
66C4×Q8×C137—22no—
67C4×D4×C137—22no—
68C4×D4×C137—22no—
69C4×D4×C137—24no—
70C4×D4×C137—24no—
71#117×C137—22no—
72C2×C4⋊C4×C137—21no—
73C2×C4⋊C4×C137—22no—
74C2×C4⋊C4×C137—21no—
75C2×C4⋊C4×C137—21no—
76C2×C4⋊C4×C137—24no—
77C2×C4⋊C4×C137—22no—
78C2×C4⋊C4×C137—22no—
79C2×C4⋊C4×C137—21no—
80C2×C4⋊C4×C137—21no—
81C2×C4⋊C4×C137—21no—
82C2×C4⋊C4×C137—24no—
83C2×C4⋊C4×C137D414yes—
84C2×C4⋊C4×C137D411yescharacteristic
85C2×C4⋊C4×C137D412yes—
86C2×C22⋊C4×C137—22no—
87C2×C22⋊C4×C137—21no—
88C2×C22⋊C4×C137—21no—
89C2×C22⋊C4×C137D411yescharacteristic
90C2×C22⋊C4×C137C2311yescharacteristic
91C2×C22⋊C4×C137C2311yescharacteristic
92C2×C22⋊C4×C137C2311yescharacteristic
93C2×C22⋊C4×C137C2314yes—
94C2×C22⋊C4×C137C2314yes—
95C2×C42×C137—22no—
96C2×C42×C137D411yescharacteristic
97#99×C137Q818yescomplemented
98#95×C137—22no—
99#95×C137—21no—
100#95×C137—21no—
101#95×C137—21no—
102#95×C137—24no—
103#95×C137—24no—
104#95×C137—21no—
105#95×C137—21no—
106C24×C137C2×D411yescharacteristic
107C24×C137C2411yescharacteristic
108C2×Q8×C137—22no—
109C2×Q8×C137—22no—
110C2×Q8×C137—22no—
111C2×Q8×C137—22no—
112C2×Q8×C137—82no—
113C2×Q8×C137—82no—
114C2×D4×C137—22no—
115C2×D4×C137—22no—
116C2×D4×C137C2×Q814yes—
117C22×C4×C137—24no—
118C22×C4×C137—21no—
119C22×C4×C137—22no—
120C22×C4×C137—22no—
121C22×C4×C137—22no—
122C22×C4×C137—21no—
123C22×C4×C137—21no—
124C22×C4×C137—21no—
125C22×C4×C137—21no—
126C22×C4×C137—21no—
127C22×C4×C137—22no—
128C22×C4×C137—42no—
129C22×C4×C137—81no—
130C22×C4×C137—81no—
131C22×C4×C137—81no—
132C22×C4×C137C2×D411yescharacteristic
133C22×C4×C137C2×D411yescharacteristic
134C22×C4×C137C2×D414yes—
135C22×C4×C137C2×D414yes—
136C22×C4×C137C2×D411yescharacteristic
137C22×C4×C137C2×D411yescharacteristic
138C22×C4×C137C2×D412yes—
139C22×C4×C137C2×D412yes—
140C22×C4×C137C2×D411yescharacteristic
141C4⋊C4×C137—22no—
142C4⋊C4×C137—24no—
143C4⋊C4×C137—22no—
144C4⋊C4×C137—22no—
145C4⋊C4×C137—24no—
146C4⋊C4×C137—22no—
147C4⋊C4×C137—24no—
148C4⋊C4×C137—22no—
149C4⋊C4×C137—22no—
150C4⋊C4×C137—22no—
151C4⋊C4×C137—24no—
152C4⋊C4×C137—42no—
153C4⋊C4×C137—48no—
154C4⋊C4×C137—41no—
155C4⋊C4×C137—41no—
156C4⋊C4×C137—42no—
157C4⋊C4×C137—81no—
158C4⋊C4×C137—84no—
159C4⋊C4×C137—81no—
160C22⋊C4×C137—24no—
161C22⋊C4×C137—24no—
162C22⋊C4×C137—22no—
163C22⋊C4×C137—24no—
164C22⋊C4×C137—24no—
165C22⋊C4×C137—22no—
166C22⋊C4×C137—22no—
167C22⋊C4×C137—22no—
168C22⋊C4×C137—42no—
169C22⋊C4×C137—81no—
170C22⋊C4×C137C2×Q818yescomplemented
171C42×C137—24no—
172C42×C137—21no—
173C42×C137—21no—
174C42×C137—82no—
175C23×C137—21no—
176C23×C137—21no—
177C23×C137—41no—
178C23×C137—41no—
179C23×C137—41no—
180C23×C137—41no—
181C23×C137C4⋊Q812yes—
182C23×C137C22×D411yescharacteristic
183C23×C137C22×D412yes—
184C23×C137C22×Q812yes—
185C23×C1372−1+412yes—
186Q8×C137—44no—
187Q8×C137—82no—
188Q8×C137—82no—
189Q8×C137—82no—
190D4×C137—28no—
191C2×C4×C137—21no—
192C2×C4×C137—21no—
193C2×C4×C137—22no—
194C2×C4×C137—24no—
195C2×C4×C137—21no—
196C2×C4×C137—24no—
197C2×C4×C137—24no—
198C2×C4×C137—21no—
199C2×C4×C137—28no—
200C2×C4×C137—42no—
201C2×C4×C137—41no—
202C2×C4×C137—41no—
203C2×C4×C137—42no—
204C2×C4×C137—42no—
205C2×C4×C137—44no—
206C2×C4×C137—44no—
207C2×C4×C137—42no—
208C2×C4×C137—41no—
209C2×C4×C137—41no—
210C2×C4×C137—44no—
211C2×C4×C137—41no—
212C2×C4×C137—41no—
213C2×C4×C137—44no—
214C2×C4×C137—41no—
215C2×C4×C137—41no—
216C2×C4×C137—41no—
217C2×C4×C137—41no—
218C2×C4×C137—41no—
219C2×C4×C137—44no—
220C2×C4×C137—41no—
221C2×C4×C137—41no—
222C2×C4×C137—41no—
223C2×C4×C137—41no—
224C2×C4×C137—81no—
225C2×C4×C137—82no—
226C2×C4×C137—82no—
227C2×C4×C137#12011yescharacteristic
228C2×C4×C137#12011yescharacteristic
229C2×C4×C137#12012yes—
230C2×C4×C137C4⋊Q812yes—
231C22×C137—22no—
232C22×C137—21no—
233C22×C137—22no—
234C22×C137—21no—
235C22×C137—22no—
236C22×C137—42no—
237C22×C137—42no—
238C22×C137—42no—
239C22×C137—41no—
240C22×C137—41no—
241C22×C137—81no—
242C22×C137C2×#12011yescharacteristic
243C22×C137C2×C4⋊Q811yescharacteristic
244C22×C137#53611yescharacteristic
245C22×C137D4×Q814yes—
246C4×C137—22no—
247C4×C137—21no—
248C4×C137—21no—
249C4×C137—22no—
250C4×C137—48no—
251C4×C137—42no—
252C4×C137—44no—
253C4×C137—41no—
254C4×C137—41no—
255C4×C137—44no—
256C4×C137—81no—
257C4×C137—81no—
258C4×C137—81no—
259C2×C137—24no—
260C2×C137—21no—
261C2×C137—21no—
262C2×C137—42no—
263C2×C137#243411yescharacteristic
264C2×C137#302511yescharacteristic
265C2×C137#302611yescharacteristic
266#25854C13711yesmaximal, characteristic, direct product factor, Sylow
267C137#2585411yescharacteristic, direct product factor, Sylow
268#3501C2×C13711yescharacteristic
269#2583C2×C13712yescomplemented
270#2579C2×C13711yescharacteristic
271#2062C2×C13712yescomplemented
272#2062C2×C13712yescomplemented
273#1996C2×C13712yescomplemented
274#1867C2×C13712yes—
275#1867C2×C13712yes—
276C4×#99C2×C13711yescharacteristic
277#554—24no—
278#554—22no—
279D4×Q8—22no—
280#544—21no—
281#543—21no—
282#543C22×C13712yes—
283#533—21no—
284C2×C22⋊Q8—21no—
285C2×C22⋊Q8C22×C13712yes—
286#518C22×C13712yes—
287C2×C4×D4C22×C13711yescharacteristic
288C22×C4⋊C4C22×C13711yescharacteristic
289C2×#99C22×C13712yes—
290#398—21no—
291#398—22no—
292#396—21no—
293#395—22no—
294#389—24no—
295#387—21no—
296#386C22×C13711yescharacteristic
297#386C22×C13711yescharacteristic
298#385—21no—
299#385C22×C13711yescharacteristic
300#385C22×C13711yescharacteristic
301#385C22×C13714yes—
302#385C22×C13714yes—
303#380C22×C13711yescharacteristic
304C4×C22⋊C4C22×C13712yes—
305#342C22×C13712yes—
306#342C22×C13714yes—
307#342C22×C13714yes—
308C22×Q8—21no—
309C22×Q8D4×C13712yes—
310C22×D4C23×C13711yescharacteristic
311C23×C4D4×C13711yescharacteristic
312C23×C4C23×C13711yescharacteristic
313C23×C4C23×C13711yescharacteristic
314C23×C4C23×C13711yescharacteristic
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—24no—
318#125—22no—
319#125—24no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322C22⋊Q8—24no—
323C22⋊Q8—24no—
324C22⋊Q8—22no—
325C22⋊Q8—22no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—24no—
331C22⋊Q8—24no—
332C4×Q8—22no—
333C4×Q8—22no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—24no—
337C4×D4—24no—
338#117—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4D4×C13711yescharacteristic
351C2×C4⋊C4D4×C13714yes—
352C2×C4⋊C4D4×C13712yes—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4D4×C13711yescharacteristic
357C2×C22⋊C4C23×C13711yescharacteristic
358C2×C22⋊C4C23×C13711yescharacteristic
359C2×C22⋊C4C23×C13711yescharacteristic
360C2×C22⋊C4C23×C13714yes—
361C2×C22⋊C4C23×C13714yes—
362C2×C42—22no—
363C2×C42D4×C13711yescharacteristic
364#99Q8×C13718yescomplemented
365#95—22no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—24no—
370#95—24no—
371#95—21no—
372#95—21no—
373C24C2×D4×C13711yescharacteristic
374C24C24×C13711yescharacteristic
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—82no—
380C2×Q8—82no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4C2×Q8×C13714yes—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—42no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4—81no—
399C22×C4C2×D4×C13711yescharacteristic
400C22×C4C2×D4×C13711yescharacteristic
401C22×C4C2×D4×C13714yes—
402C22×C4C2×D4×C13714yes—
403C22×C4C2×D4×C13711yescharacteristic
404C22×C4C2×D4×C13711yescharacteristic
405C22×C4C2×D4×C13712yes—
406C22×C4C2×D4×C13712yes—
407C22×C4C2×D4×C13711yescharacteristic
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—42no—
420C4⋊C4—48no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—42no—
424C4⋊C4—81no—
425C4⋊C4—84no—
426C4⋊C4—81no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—22no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—42no—
436C22⋊C4—81no—
437C22⋊C4C2×Q8×C13718yescomplemented
438C42—24no—
439C42—21no—
440C42—21no—
441C42—82no—
442C23—21no—
443C23—21no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23C4⋊Q8×C13712yes—
449C23C22×D4×C13711yescharacteristic
450C23C22×D4×C13712yes—
451C23C22×Q8×C13712yes—
452C232−1+4×C13712yes—
453Q8—44no—
454Q8—82no—
455Q8—82no—
456Q8—82no—
457D4—28no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—21no—
466C2×C4—28no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—44no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4#120×C13711yescharacteristic
495C2×C4#120×C13711yescharacteristic
496C2×C4#120×C13712yes—
497C2×C4C4⋊Q8×C13712yes—
498C22—22no—
499C22—21no—
500C22—22no—
501C22—21no—
502C22—22no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—81no—
509C22C2×#120×C13711yescharacteristic
510C22C2×C4⋊Q8×C13711yescharacteristic
511C22#536×C13711yescharacteristic
512C22D4×Q8×C13714yes—
513C4—22no—
514C4—21no—
515C4—21no—
516C4—22no—
517C4—48no—
518C4—42no—
519C4—44no—
520C4—41no—
521C4—41no—
522C4—44no—
523C4—81no—
524C4—81no—
525C4—81no—
526C2—24no—
527C2—21no—
528C2—21no—
529C2—42no—
530C2#2434×C13711yescharacteristic
531C2#3025×C13711yescharacteristic
532C2#3026×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC24C24×C137
prime radicalC22×D4×C137C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2583×C137C2semidirect
4. #2062×C137C2semidirect
5. #2062×C137C2semidirect
6. #1996×C137C2semidirect
97. #99×C137Q8semidirect
170. C22⋊C4×C137C2×Q8semidirect
266. #25854C137direct (N × H)
267. C137#25854direct (N × H)
269. #2583C2×C137semidirect
271. #2062C2×C137semidirect
272. #2062C2×C137semidirect
273. #1996C2×C137semidirect
364. #99Q8×C137semidirect
437. C22⋊C4C2×Q8×C137semidirect

Maximal subgroups

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

subgroupindex
#3501×C1372
#2583×C1372
#2579×C1372
#2062×C1372
#1996×C1372
#1867×C1372
C4×#99×C1372
#25854137