Finite Group Databasebeta

← C137×#14308 #46863694

Subgroups — C137×#14308 #46863694

Order 35072.

1210 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#398×C137C211yesmaximal, characteristic
2C2×#387×C137C211yesmaximal, characteristic
3C2×C4×C4⋊C4×C137C211yesmaximal, characteristic
4#1445×C137C212yesmaximal
5#1445×C137C211yesmaximal, characteristic
6#1443×C137C212yesmaximal
7#1442×C137C211yesmaximal, characteristic
8#1441×C137C211yesmaximal, characteristic
9#1436×C137C212yesmaximal
10#1436×C137C212yesmaximal
11#1434×C137C211yesmaximal, characteristic
12C22×C4⋊C4×C137C2211yescharacteristic
13C22×C4⋊C4×C137C2211yescharacteristic
14C22×C4⋊C4×C137C2211yescharacteristic
15C22×C4⋊C4×C137C2211yescharacteristic
16C22×C4⋊C4×C137C2212yes—
17C22×C42×C137C2211yescharacteristic
18C22×C42×C137C2212yes—
19#398×C137C418yescomplemented
20#387×C137—24no—
21C4×C4⋊C4×C137—24no—
22C2×#95×C137C2212yes—
23C2×#95×C137C2211yescharacteristic
24C2×#95×C137C2211yescharacteristic
25C2×#95×C137C2211yescharacteristic
26C2×#95×C137C2211yescharacteristic
27C2×#95×C137C2212yes—
28C2×#95×C137C2211yescharacteristic
29C2×#95×C137C2212yes—
30C2×#95×C137C2212yes—
31C2×#95×C137C2212yes—
32C2×#95×C137C2212yes—
33C2×#95×C137C2212yes—
34C2×#95×C137C2212yes—
35C2×#95×C137C2212yes—
36C2×#95×C137C2211yescharacteristic
37C2×#95×C137C2212yes—
38C23×C4×C137C2311yescharacteristic
39C23×C4×C137C2312yes—
40C23×C4×C137C2311yescharacteristic
41C23×C4×C137C2312yes—
42C23×C4×C137C2311yescharacteristic
43C23×C4×C137C2312yes—
44C23×C4×C137C2311yescharacteristic
45C23×C4×C137C2311yescharacteristic
46C23×C4×C137C2312yes—
47C23×C4×C137C2311yescharacteristic
48C23×C4×C137C2311yescharacteristic
49C2×C4⋊C4×C137—22no—
50C2×C4⋊C4×C137—22no—
51C2×C4⋊C4×C137—22no—
52C2×C4⋊C4×C137—24no—
53C2×C4⋊C4×C137—22no—
54C2×C4⋊C4×C137—22no—
55C2×C4⋊C4×C137—22no—
56C2×C4⋊C4×C137—24no—
57C2×C4⋊C4×C137—42no—
58C2×C4⋊C4×C137—42no—
59C2×C4⋊C4×C137C2×C414yes—
60C2×C4⋊C4×C137C2×C414yes—
61C2×C4⋊C4×C137C2×C418yes—
62C2×C42×C137—24no—
63C2×C42×C137—24no—
64C2×C42×C137—42no—
65C2×C42×C137—42no—
66C2×C42×C137D414yes—
67#95×C137—24no—
68#95×C137—22no—
69#95×C137—22no—
70#95×C137—22no—
71#95×C137—22no—
72#95×C137—24no—
73#95×C137—24no—
74#95×C137—24no—
75#95×C137—22no—
76#95×C137—42no—
77#95×C137—42no—
78#95×C137—42no—
79#95×C137—42no—
80#95×C137—42no—
81#95×C137C2×C418yes—
82#95×C137C2×C414yes—
83C24×C137C2411yescharacteristic
84C22×C4×C137—22no—
85C22×C4×C137—22no—
86C22×C4×C137—21no—
87C22×C4×C137—21no—
88C22×C4×C137—22no—
89C22×C4×C137—21no—
90C22×C4×C137—22no—
91C22×C4×C137—22no—
92C22×C4×C137—21no—
93C22×C4×C137—22no—
94C22×C4×C137—22no—
95C22×C4×C137—22no—
96C22×C4×C137—22no—
97C22×C4×C137—21no—
98C22×C4×C137—22no—
99C22×C4×C137—21no—
100C22×C4×C137—22no—
101C22×C4×C137—22no—
102C22×C4×C137—21no—
103C22×C4×C137—21no—
104C22×C4×C137—22no—
105C22×C4×C137—22no—
106C22×C4×C137—22no—
107C22×C4×C137—22no—
108C22×C4×C137—22no—
109C22×C4×C137—22no—
110C22×C4×C137—22no—
111C22×C4×C137—21no—
112C22×C4×C137—21no—
113C22×C4×C137—22no—
114C22×C4×C137—22no—
115C22×C4×C137—22no—
116C22×C4×C137—21no—
117C22×C4×C137—22no—
118C22×C4×C137—21no—
119C22×C4×C137—22no—
120C22×C4×C137—22no—
121C22×C4×C137—21no—
122C22×C4×C137—21no—
123C22×C4×C137—22no—
124C22×C4×C137—22no—
125C22×C4×C137—22no—
126C22×C4×C137—22no—
127C22×C4×C137—22no—
128C22×C4×C137—21no—
129C22×C4×C137—21no—
130C22×C4×C137—21no—
131C22×C4×C137—22no—
132C22×C4×C137—22no—
133C22×C4×C137—22no—
134C22×C4×C137—22no—
135C22×C4×C137—21no—
136C22×C4×C137—22no—
137C22×C4×C137—22no—
138C22×C4×C137C22×C412yes—
139C22×C4×C137C22×C414yes—
140C22×C4×C137C22×C414yes—
141C22×C4×C137C22×C412yes—
142C22×C4×C137C22×C412yes—
143C22×C4×C137C2×D412yes—
144C22×C4×C137C2×D412yes—
145C22×C4×C137C2×D412yes—
146C22×C4×C137#4114yes—
147C22×C4×C137#4112yes—
148C22×C4×C137#4114yes—
149C4⋊C4×C137—28no—
150C4⋊C4×C137—28no—
151C4⋊C4×C137—28no—
152C4⋊C4×C137—48no—
153C4⋊C4×C137—44no—
154C42×C137—28no—
155C42×C137—48no—
156C23×C137C23×C411yescharacteristic
157C23×C137C22×D411yescharacteristic
158C23×C137C2×#4111yescharacteristic
159C23×C137C2×#4112yes—
160C23×C137C2×#4112yes—
161C23×C137H(5,𝔽2)11yescharacteristic
162C23×C137H(5,𝔽2)11yescharacteristic
163C23×C137H(5,𝔽2)11yescharacteristic
164C23×C1372−1+412yes—
165C23×C1372−1+412yes—
166C23×C1372−1+411yescharacteristic
167C2×C4×C137—24no—
168C2×C4×C137—22no—
169C2×C4×C137—22no—
170C2×C4×C137—24no—
171C2×C4×C137—22no—
172C2×C4×C137—22no—
173C2×C4×C137—24no—
174C2×C4×C137—24no—
175C2×C4×C137—22no—
176C2×C4×C137—24no—
177C2×C4×C137—22no—
178C2×C4×C137—24no—
179C2×C4×C137—24no—
180C2×C4×C137—22no—
181C2×C4×C137—24no—
182C2×C4×C137—24no—
183C2×C4×C137—24no—
184C2×C4×C137—22no—
185C2×C4×C137—24no—
186C2×C4×C137—22no—
187C2×C4×C137—24no—
188C2×C4×C137—22no—
189C2×C4×C137—22no—
190C2×C4×C137—24no—
191C2×C4×C137—24no—
192C2×C4×C137—41no—
193C2×C4×C137—41no—
194C2×C4×C137—42no—
195C2×C4×C137—42no—
196C2×C4×C137—42no—
197C2×C4×C137—41no—
198C2×C4×C137—42no—
199C2×C4×C137—41no—
200C2×C4×C137—42no—
201C2×C4×C137—42no—
202C2×C4×C137—42no—
203C2×C4×C137—42no—
204C2×C4×C137—42no—
205C2×C4×C137—41no—
206C2×C4×C137—42no—
207C2×C4×C137—42no—
208C2×C4×C137—42no—
209C2×C4×C137—41no—
210C2×C4×C137—42no—
211C2×C4×C137—42no—
212C2×C4×C137—42no—
213C2×C4×C137—41no—
214C2×C4×C137—41no—
215C2×C4×C137—42no—
216C2×C4×C137—41no—
217C2×C4×C137—41no—
218C2×C4×C137—41no—
219C2×C4×C137—41no—
220C2×C4×C137—42no—
221C2×C4×C137—42no—
222C2×C4×C137—42no—
223C2×C4×C137—42no—
224C2×C4×C137—42no—
225C2×C4×C137—42no—
226C2×C4×C137—42no—
227C2×C4×C137—42no—
228C2×C4×C137—42no—
229C2×C4×C137—42no—
230C2×C4×C137C4×D414yes—
231C2×C4×C137#12314yes—
232C22×C137C2×C4×D411yescharacteristic
233C22×C137C4×#4112yes—
234C22×C137#51811yescharacteristic
235C22×C137#51912yes—
236C22×C137#52011yescharacteristic
237C22×C137C2×#12311yescharacteristic
238C22×C137#52511yescharacteristic
239C22×C137#52912yes—
240C22×C137#53511yescharacteristic
241C22×C137#53611yescharacteristic
242C22×C137#53611yescharacteristic
243C22×C137#53711yescharacteristic
244C22×C137#53912yes—
245C22×C137#53912yes—
246C22×C137#53912yes—
247C22×C137#53912yes—
248C22×C137#54011yescharacteristic
249C22×C137#54112yes—
250C22×C137#54112yes—
251C22×C137#54112yes—
252C22×C137#54211yescharacteristic
253C22×C137#54212yes—
254C22×C137#54212yes—
255C4×C137—24no—
256C4×C137—24no—
257C4×C137—42no—
258C4×C137—44no—
259C4×C137—44no—
260C4×C137—42no—
261C4×C137—44no—
262C4×C137—42no—
263C4×C137—81no—
264C4×C137—81no—
265C4×C137—82no—
266C2×C137C4×#12311yescharacteristic
267C2×C137#232411yescharacteristic
268C2×C137#232711yescharacteristic
269C2×C137#233212yes—
270C2×C137#233612yes—
271C2×C137#262311yescharacteristic
272C2×C137#262511yescharacteristic
273C2×C137#263311yescharacteristic
274C2×C137#263711yescharacteristic
275C2×C137#265412yes—
276C2×C137#265512yes—
277#14308C13711yesmaximal, characteristic, direct product factor, Sylow
278C137#1430811yescharacteristic, direct product factor, Sylow
279C2×#398C2×C13711yescharacteristic
280C2×#387C2×C13711yescharacteristic
281C2×C4×C4⋊C4C2×C13711yescharacteristic
282#1445C2×C13711yescharacteristic
283#1445C2×C13712yes—
284#1443C2×C13712yes—
285#1442C2×C13711yescharacteristic
286#1441C2×C13711yescharacteristic
287#1436C2×C13712yes—
288#1436C2×C13712yes—
289#1434C2×C13711yescharacteristic
290C22×C4⋊C4C22×C13711yescharacteristic
291C22×C4⋊C4C22×C13711yescharacteristic
292C22×C4⋊C4C22×C13711yescharacteristic
293C22×C4⋊C4C22×C13711yescharacteristic
294C22×C4⋊C4C22×C13712yes—
295C22×C42C22×C13711yescharacteristic
296C22×C42C22×C13712yes—
297#398C4×C13718yescomplemented
298#387—24no—
299C4×C4⋊C4—24no—
300C2×#95C22×C13712yes—
301C2×#95C22×C13711yescharacteristic
302C2×#95C22×C13711yescharacteristic
303C2×#95C22×C13712yes—
304C2×#95C22×C13712yes—
305C2×#95C22×C13711yescharacteristic
306C2×#95C22×C13711yescharacteristic
307C2×#95C22×C13712yes—
308C2×#95C22×C13711yescharacteristic
309C2×#95C22×C13712yes—
310C2×#95C22×C13712yes—
311C2×#95C22×C13712yes—
312C2×#95C22×C13712yes—
313C2×#95C22×C13712yes—
314C2×#95C22×C13711yescharacteristic
315C2×#95C22×C13712yes—
316C23×C4C23×C13711yescharacteristic
317C23×C4C23×C13711yescharacteristic
318C23×C4C23×C13712yes—
319C23×C4C23×C13711yescharacteristic
320C23×C4C23×C13711yescharacteristic
321C23×C4C23×C13712yes—
322C23×C4C23×C13712yes—
323C23×C4C23×C13711yescharacteristic
324C23×C4C23×C13712yes—
325C23×C4C23×C13711yescharacteristic
326C23×C4C23×C13711yescharacteristic
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—42no—
336C2×C4⋊C4—42no—
337C2×C4⋊C4C2×C4×C13714yes—
338C2×C4⋊C4C2×C4×C13714yes—
339C2×C4⋊C4C2×C4×C13718yes—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42D4×C13714yes—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—22no—
354#95—42no—
355#95—42no—
356#95—42no—
357#95—42no—
358#95—42no—
359#95C2×C4×C13718yes—
360#95C2×C4×C13714yes—
361C24C24×C13711yescharacteristic
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4C22×C4×C13712yes—
417C22×C4C22×C4×C13714yes—
418C22×C4C22×C4×C13714yes—
419C22×C4C22×C4×C13712yes—
420C22×C4C22×C4×C13712yes—
421C22×C4C2×D4×C13712yes—
422C22×C4C2×D4×C13712yes—
423C22×C4C2×D4×C13712yes—
424C22×C4#41×C13714yes—
425C22×C4#41×C13712yes—
426C22×C4#41×C13714yes—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—48no—
431C4⋊C4—44no—
432C42—28no—
433C42—48no—
434C23C23×C4×C13711yescharacteristic
435C23C22×D4×C13711yescharacteristic
436C23C2×#41×C13711yescharacteristic
437C23C2×#41×C13712yes—
438C23C2×#41×C13712yes—
439C23H(5,𝔽2)×C13711yescharacteristic
440C23H(5,𝔽2)×C13711yescharacteristic
441C23H(5,𝔽2)×C13711yescharacteristic
442C232−1+4×C13712yes—
443C232−1+4×C13712yes—
444C232−1+4×C13711yescharacteristic
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4C4×D4×C13714yes—
509C2×C4#123×C13714yes—
510C22C2×C4×D4×C13711yescharacteristic
511C22C4×#41×C13712yes—
512C22#518×C13711yescharacteristic
513C22#519×C13712yes—
514C22#520×C13711yescharacteristic
515C22C2×#123×C13711yescharacteristic
516C22#525×C13711yescharacteristic
517C22#529×C13712yes—
518C22#535×C13711yescharacteristic
519C22#536×C13711yescharacteristic
520C22#536×C13711yescharacteristic
521C22#537×C13711yescharacteristic
522C22#539×C13712yes—
523C22#539×C13712yes—
524C22#539×C13712yes—
525C22#539×C13712yes—
526C22#540×C13711yescharacteristic
527C22#541×C13712yes—
528C22#541×C13712yes—
529C22#541×C13712yes—
530C22#542×C13711yescharacteristic
531C22#542×C13712yes—
532C22#542×C13712yes—
533C4—24no—
534C4—24no—
535C4—42no—
536C4—42no—
537C4—44no—
538C4—44no—
539C4—42no—
540C4—44no—
541C4—81no—
542C4—81no—
543C4—82no—
544C2C4×#123×C13711yescharacteristic
545C2#2324×C13711yescharacteristic
546C2#2327×C13711yescharacteristic
547C2#2332×C13712yes—
548C2#2336×C13712yes—
549C2#2623×C13711yescharacteristic
550C2#2625×C13711yescharacteristic
551C2#2633×C13711yescharacteristic
552C2#2637×C13711yescharacteristic
553C2#2654×C13712yes—
554C2#2655×C13712yes—
subgroupG/H
center, socle, prime radicalC24×C137C24
commutatorC23C23×C4×C137
Frattini, cosocleC24C24×C137
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C137 ◁ G

Complements

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

NH ≅ G/Ntype
19. #398×C137C4semidirect
277. #14308C137direct (N × H)
278. C137#14308direct (N × H)
297. #398C4×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#398×C1372
C2×#387×C1372
C2×C4×C4⋊C4×C1372
#1445×C1372
#1443×C1372
#1442×C1372
#1441×C1372
#1436×C1372
#1434×C1372
#14308137