Finite Group Databasebeta

← C137×#23946 #46873332

Subgroups — C137×#23946 #46873332

Order 35072.

1222 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#542×C137C211yesmaximal, characteristic
2C2×#451×C137C211yesmaximal, characteristic, complemented
3C2×#448×C137C211yesmaximal, characteristic
4#2310×C137C211yesmaximal, characteristic, complemented
5#2024×C137C212yesmaximal, complemented
6#2015×C137C212yesmaximal, complemented
7#1878×C137C212yesmaximal, complemented
8#1877×C137C212yesmaximal, complemented
9#1875×C137C211yesmaximal, characteristic
10#1873×C137C211yesmaximal, characteristic, complemented
11#1818×C137C211yesmaximal, characteristic, complemented
12C23×Q8×C137C2211yescharacteristic
13C22×Q16×C137C2212yes—
14C22×SD16×C137C2212yes—
15#542×C137—24no—
16#542×C137—24no—
17C2×C4⋊Q8×C137C2211yescharacteristic
18C2×#126×C137—21no—
19C2×#124×C137—21no—
20C2×#124×C137C2211yescharacteristic
21C2×#123×C137—21no—
22C2×C22⋊Q8×C137—21no—
23C2×C22⋊Q8×C137C2211yescharacteristic
24C2×#121×C137C2211yescharacteristic
25C2×C4×Q8×C137C2211yescharacteristic
26C2×C4×D4×C137C2211yescharacteristic
27C2×#117×C137C2211yescharacteristic
28C2×#117×C137C2211yescharacteristic
29#451×C137—24no—
30#448×C137—24no—
31C2×#103×C137C2212yes—
32C2×#103×C137C2212yes—
33C2×#103×C137C2212yes—
34C2×#102×C137C2212yes—
35C2×C22⋊C8×C137C2211yescharacteristic
36C2×C22⋊C8×C137C2211yescharacteristic
37#391×C137—21no—
38#391×C137—21no—
39#391×C137C2212yes—
40#391×C137C2212yes—
41#381×C137—21no—
42#381×C137—21no—
43#340×C137C2212yes—
44#340×C137C2212yes—
45#336×C137C2212yes—
46#336×C137C2212yes—
47C22×Q8×C137—24no—
48C22×Q8×C137—21no—
49C22×Q8×C137—21no—
50C22×Q8×C137—22no—
51C22×Q8×C137—22no—
52C22×Q8×C137—42no—
53C22×Q8×C137C2311yescharacteristic
54C22×Q8×C137C2312yes—
55C22×D4×C137—21no—
56C22×D4×C137C2311yescharacteristic
57C23×C4×C137—21no—
58C23×C4×C137—21no—
59C23×C4×C137C2311yescharacteristic
60C2×Q16×C137—24no—
61C2×Q16×C137—24no—
62C2×Q16×C137—42no—
63C2×SD16×C137—24no—
64C2×SD16×C137—24no—
65C2×SD16×C137—42no—
66C22×C8×C137C2312yes—
67C22×C8×C137C2312yes—
68C4⋊Q8×C137—22no—
69C4⋊Q8×C137—22no—
70#126×C137—28no—
71#124×C137—28no—
72#124×C137—22no—
73#124×C137—22no—
74#123×C137—28no—
75C22⋊Q8×C137—28no—
76C22⋊Q8×C137—22no—
77C22⋊Q8×C137D414yes—
78#121×C137—22no—
79#121×C137D414yes—
80C4×Q8×C137—22no—
81C4×Q8×C137—22no—
82C4×D4×C137—22no—
83C4×D4×C137—22no—
84#117×C137—22no—
85#117×C137—22no—
86#117×C137—42no—
87C2×C4⋊C4×C137—21no—
88C2×C4⋊C4×C137—21no—
89C2×C4⋊C4×C137—21no—
90C2×C4⋊C4×C137D411yescharacteristic
91C2×C4⋊C4×C137D411yescharacteristic
92C2×C4⋊C4×C137C2312yes—
93C2×C4⋊C4×C137C2311yescharacteristic
94C2×C4⋊C4×C137C2311yescharacteristic
95C2×C22⋊C4×C137—21no—
96C2×C22⋊C4×C137—21no—
97C2×C22⋊C4×C137—21no—
98C2×C22⋊C4×C137—21no—
99C2×C22⋊C4×C137—21no—
100C2×C22⋊C4×C137—21no—
101C2×C22⋊C4×C137D411yescharacteristic
102C2×C22⋊C4×C137D411yescharacteristic
103C2×C42×C137—21no—
104C2×C42×C137—21no—
105C2×C42×C137C2311yescharacteristic
106C2×C42×C137C2311yescharacteristic
107#103×C137—24no—
108#103×C137—24no—
109#103×C137—42no—
110#102×C137—42no—
111C22⋊C8×C137—22no—
112C22⋊C8×C137—22no—
113#95×C137—22no—
114#95×C137—22no—
115#95×C137—22no—
116#95×C137—22no—
117C24×C137—21no—
118C24×C137C2×D411yescharacteristic
119C2×Q8×C137—24no—
120C2×Q8×C137—24no—
121C2×Q8×C137—21no—
122C2×Q8×C137—22no—
123C2×Q8×C137—22no—
124C2×Q8×C137—22no—
125C2×Q8×C137—22no—
126C2×Q8×C137—22no—
127C2×Q8×C137—22no—
128C2×Q8×C137—22no—
129C2×Q8×C137—48no—
130C2×Q8×C137—48no—
131C2×Q8×C137—42no—
132C2×Q8×C137—42no—
133C2×Q8×C137—82no—
134C2×Q8×C137C2×D412yes—
135C2×Q8×C137#4114yes—
136C2×D4×C137—24no—
137C2×D4×C137—21no—
138C2×D4×C137—22no—
139C2×D4×C137—22no—
140C2×D4×C137—82no—
141C2×D4×C137C2×D412yes—
142C22×C4×C137—22no—
143C22×C4×C137—22no—
144C22×C4×C137—24no—
145C22×C4×C137—22no—
146C22×C4×C137—21no—
147C22×C4×C137—21no—
148C22×C4×C137—21no—
149C22×C4×C137—24no—
150C22×C4×C137—22no—
151C22×C4×C137—42no—
152C22×C4×C137—42no—
153C22×C4×C137—82no—
154C22×C4×C137C2×D411yescharacteristic
155C22×C4×C137C2×D411yescharacteristic
156C22×C4×C137C2×D411yescharacteristic
157C22×C4×C137C2×D411yescharacteristic
158C22×C4×C137C2×D411yescharacteristic
159C22×C4×C137C2×D412yes—
160C22×C4×C137C2×D412yes—
161C22×C4×C137C2411yescharacteristic
162Q16×C137—48no—
163SD16×C137—48no—
164C2×C8×C137—22no—
165C2×C8×C137—22no—
166C2×C8×C137—22no—
167C2×C8×C137—22no—
168C2×C8×C137—22no—
169C2×C8×C137—22no—
170C4⋊C4×C137—24no—
171C4⋊C4×C137—24no—
172C4⋊C4×C137—24no—
173C4⋊C4×C137—22no—
174C4⋊C4×C137—22no—
175C4⋊C4×C137—21no—
176C4⋊C4×C137—21no—
177C4⋊C4×C137—22no—
178C4⋊C4×C137—22no—
179C4⋊C4×C137C2×D412yes—
180C4⋊C4×C137C2×D412yes—
181C22⋊C4×C137—24no—
182C22⋊C4×C137—24no—
183C22⋊C4×C137—24no—
184C22⋊C4×C137—24no—
185C22⋊C4×C137—22no—
186C22⋊C4×C137—22no—
187C22⋊C4×C137—24no—
188C22⋊C4×C137—42no—
189C42×C137—24no—
190C42×C137—21no—
191C42×C137—21no—
192C42×C137—21no—
193C42×C137—21no—
194C42×C137—42no—
195C23×C137—21no—
196C23×C137—21no—
197C23×C137—22no—
198C23×C137—22no—
199C23×C137—22no—
200C23×C137—41no—
201C23×C137—41no—
202C23×C137—42no—
203C23×C137—42no—
204C23×C137#12012yes—
205C23×C137C22×D411yescharacteristic
206Q8×C137—24no—
207Q8×C137—24no—
208Q8×C137—24no—
209Q8×C137—48no—
210Q8×C137—42no—
211Q8×C137—42no—
212Q8×C137—42no—
213Q8×C137—84no—
214D4×C137—24no—
215D4×C137—42no—
216D4×C137—84no—
217C2×C4×C137—21no—
218C2×C4×C137—21no—
219C2×C4×C137—24no—
220C2×C4×C137—24no—
221C2×C4×C137—21no—
222C2×C4×C137—21no—
223C2×C4×C137—22no—
224C2×C4×C137—21no—
225C2×C4×C137—21no—
226C2×C4×C137—22no—
227C2×C4×C137—22no—
228C2×C4×C137—24no—
229C2×C4×C137—22no—
230C2×C4×C137—22no—
231C2×C4×C137—22no—
232C2×C4×C137—22no—
233C2×C4×C137—48no—
234C2×C4×C137—42no—
235C2×C4×C137—41no—
236C2×C4×C137—42no—
237C2×C4×C137—41no—
238C2×C4×C137—42no—
239C2×C4×C137—42no—
240C2×C4×C137—42no—
241C2×C4×C137—42no—
242C2×C4×C137—42no—
243C2×C4×C137—84no—
244C2×C4×C137#12012yes—
245C2×C4×C137C22×D411yescharacteristic
246C2×C4×C137C22×D411yescharacteristic
247C2×C4×C137C2×#4112yes—
248C2×C4×C137H(5,𝔽2)12yes—
249C8×C137—42no—
250C8×C137—42no—
251C22×C137—22no—
252C22×C137—22no—
253C22×C137—22no—
254C22×C137—42no—
255C22×C137—42no—
256C22×C137—42no—
257C22×C137—42no—
258C22×C137—42no—
259C22×C137—82no—
260C22×C137—82no—
261C22×C137C2×#12011yescharacteristic
262C22×C137#52511yescharacteristic
263C22×C137#53511yescharacteristic
264C22×C137#57712yes—
265C22×C137#57812yes—
266C4×C137—22no—
267C4×C137—44no—
268C4×C137—42no—
269C4×C137—41no—
270C4×C137—41no—
271C4×C137—42no—
272C4×C137—42no—
273C4×C137—41no—
274C4×C137—41no—
275C4×C137—82no—
276C4×C137#54612yes—
277C4×C137#54612yes—
278C2×C137—42no—
279C2×C137—82no—
280C2×C137#240411yescharacteristic
281C2×C137#300911yescharacteristic
282C2×C137#301111yescharacteristic
283C2×C137#312112yes—
284C2×C137#322512yes—
285#23946C13711yesmaximal, characteristic, direct product factor, Sylow
286C137#2394611yescharacteristic, direct product factor, Sylow
287C2×#542C2×C13711yescharacteristic
288C2×#451C2×C13711yescharacteristic, complemented
289C2×#448C2×C13711yescharacteristic
290#2310C2×C13711yescharacteristic, complemented
291#2024C2×C13712yescomplemented
292#2015C2×C13712yescomplemented
293#1878C2×C13712yescomplemented
294#1877C2×C13712yescomplemented
295#1875C2×C13711yescharacteristic
296#1873C2×C13711yescharacteristic, complemented
297#1818C2×C13711yescharacteristic, complemented
298C23×Q8C22×C13711yescharacteristic
299C22×Q16C22×C13712yes—
300C22×SD16C22×C13712yes—
301#542—24no—
302#542—24no—
303C2×C4⋊Q8C22×C13711yescharacteristic
304C2×#126—21no—
305C2×#124—21no—
306C2×#124C22×C13711yescharacteristic
307C2×#123—21no—
308C2×C22⋊Q8—21no—
309C2×C22⋊Q8C22×C13711yescharacteristic
310C2×#121C22×C13711yescharacteristic
311C2×C4×Q8C22×C13711yescharacteristic
312C2×C4×D4C22×C13711yescharacteristic
313C2×#117C22×C13711yescharacteristic
314C2×#117C22×C13711yescharacteristic
315#451—24no—
316#448—24no—
317C2×#103C22×C13712yes—
318C2×#103C22×C13712yes—
319C2×#103C22×C13712yes—
320C2×#102C22×C13712yes—
321C2×C22⋊C8C22×C13711yescharacteristic
322C2×C22⋊C8C22×C13711yescharacteristic
323#391—21no—
324#391—21no—
325#391C22×C13712yes—
326#391C22×C13712yes—
327#381—21no—
328#381—21no—
329#340C22×C13712yes—
330#340C22×C13712yes—
331#336C22×C13712yes—
332#336C22×C13712yes—
333C22×Q8—24no—
334C22×Q8—21no—
335C22×Q8—21no—
336C22×Q8—22no—
337C22×Q8—22no—
338C22×Q8—42no—
339C22×Q8C23×C13711yescharacteristic
340C22×Q8C23×C13712yes—
341C22×D4—21no—
342C22×D4C23×C13711yescharacteristic
343C23×C4—21no—
344C23×C4—21no—
345C23×C4C23×C13711yescharacteristic
346C2×Q16—24no—
347C2×Q16—24no—
348C2×Q16—42no—
349C2×SD16—24no—
350C2×SD16—24no—
351C2×SD16—42no—
352C22×C8C23×C13712yes—
353C22×C8C23×C13712yes—
354C4⋊Q8—22no—
355C4⋊Q8—22no—
356#126—28no—
357#124—28no—
358#124—22no—
359#124—22no—
360#123—28no—
361C22⋊Q8—28no—
362C22⋊Q8—22no—
363C22⋊Q8D4×C13714yes—
364#121—22no—
365#121D4×C13714yes—
366C4×Q8—22no—
367C4×Q8—22no—
368C4×D4—22no—
369C4×D4—22no—
370#117—22no—
371#117—22no—
372#117—42no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4D4×C13711yescharacteristic
377C2×C4⋊C4D4×C13711yescharacteristic
378C2×C4⋊C4C23×C13712yes—
379C2×C4⋊C4C23×C13711yescharacteristic
380C2×C4⋊C4C23×C13711yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4D4×C13711yescharacteristic
388C2×C22⋊C4D4×C13711yescharacteristic
389C2×C42—21no—
390C2×C42—21no—
391C2×C42C23×C13711yescharacteristic
392C2×C42C23×C13711yescharacteristic
393#103—24no—
394#103—24no—
395#103—42no—
396#102—42no—
397C22⋊C8—22no—
398C22⋊C8—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403C24—21no—
404C24C2×D4×C13711yescharacteristic
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—48no—
416C2×Q8—48no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—82no—
420C2×Q8C2×D4×C13712yes—
421C2×Q8#41×C13714yes—
422C2×D4—24no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—82no—
427C2×D4C2×D4×C13712yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—82no—
440C22×C4C2×D4×C13711yescharacteristic
441C22×C4C2×D4×C13711yescharacteristic
442C22×C4C2×D4×C13711yescharacteristic
443C22×C4C2×D4×C13711yescharacteristic
444C22×C4C2×D4×C13711yescharacteristic
445C22×C4C2×D4×C13712yes—
446C22×C4C2×D4×C13712yes—
447C22×C4C24×C13711yescharacteristic
448Q16—48no—
449SD16—48no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4C2×D4×C13712yes—
466C4⋊C4C2×D4×C13712yes—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—24no—
474C22⋊C4—42no—
475C42—24no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—42no—
481C23—21no—
482C23—21no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—41no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23#120×C13712yes—
491C23C22×D4×C13711yescharacteristic
492Q8—24no—
493Q8—24no—
494Q8—24no—
495Q8—48no—
496Q8—42no—
497Q8—42no—
498Q8—42no—
499Q8—84no—
500D4—24no—
501D4—42no—
502D4—84no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—84no—
530C2×C4#120×C13712yes—
531C2×C4C22×D4×C13711yescharacteristic
532C2×C4C22×D4×C13711yescharacteristic
533C2×C4C2×#41×C13712yes—
534C2×C4H(5,𝔽2)×C13712yes—
535C8—42no—
536C8—42no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—82no—
546C22—82no—
547C22C2×#120×C13711yescharacteristic
548C22#525×C13711yescharacteristic
549C22#535×C13711yescharacteristic
550C22#577×C13712yes—
551C22#578×C13712yes—
552C4—22no—
553C4—44no—
554C4—42no—
555C4—41no—
556C4—41no—
557C4—42no—
558C4—42no—
559C4—41no—
560C4—41no—
561C4—82no—
562C4#546×C13712yes—
563C4#546×C13712yes—
564C2—42no—
565C2—82no—
566C2#2404×C13711yescharacteristic
567C2#3009×C13711yescharacteristic
568C2#3011×C13711yescharacteristic
569C2#3121×C13712yes—
570C2#3225×C13712yes—
subgroupG/H
center, socleC23×C137C22×D4
commutator, Frattini, cosocleC22×C4C24×C137
prime radicalC2×#121×C137C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C137 ◁ C2×#117×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#451×C137C2semidirect
4. #2310×C137C2semidirect
5. #2024×C137C2semidirect
6. #2015×C137C2semidirect
7. #1878×C137C2semidirect
8. #1877×C137C2semidirect
10. #1873×C137C2semidirect
11. #1818×C137C2semidirect
285. #23946C137direct (N × H)
286. C137#23946direct (N × H)
288. C2×#451C2×C137semidirect
290. #2310C2×C137semidirect
291. #2024C2×C137semidirect
292. #2015C2×C137semidirect
293. #1878C2×C137semidirect
294. #1877C2×C137semidirect
296. #1873C2×C137semidirect
297. #1818C2×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#542×C1372
C2×#451×C1372
C2×#448×C1372
#2310×C1372
#2024×C1372
#2015×C1372
#1878×C1372
#1877×C1372
#1875×C1372
#1873×C1372
#1818×C1372
#23946137