Finite Group Databasebeta

← C137×#11043 #46860429

Subgroups — C137×#11043 #46860429

Order 35072.

1454 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#99×C137C212yesmaximal
2C2×#386×C137C211yesmaximal, characteristic
3C2×#380×C137C211yesmaximal, characteristic, complemented
4C2×#342×C137C212yesmaximal, complemented
5C22⋊#95×C137C211yesmaximal, characteristic, complemented
6C23×D4×C137C2211yescharacteristic
7C24×C4×C137C2211yescharacteristic
8C22×C4⋊C4×C137—21no—
9C22×C22⋊C4×C137C412yescomplemented
10C22×C22⋊C4×C137C2211yescharacteristic
11C22×C22⋊C4×C137C2212yes—
12C22×C22⋊C4×C137C2212yes—
13C2×#99×C137—24no—
14C2×#99×C137—24no—
15C2×#99×C137—24no—
16#386×C137—21no—
17#386×C137—22no—
18#386×C137—21no—
19#380×C137—24no—
20C2×#95×C137—21no—
21C2×#95×C137—22no—
22C2×#95×C137C412yescomplemented
23#342×C137—24no—
24C25×C137C2×C411yescharacteristic
25C25×C137C2311yescharacteristic
26C22×D4×C137—21no—
27C22×D4×C137—21no—
28C22×D4×C137—22no—
29C22×D4×C137—22no—
30C22×D4×C137—21no—
31C22×D4×C137—21no—
32C22×D4×C137—22no—
33C22×D4×C137—22no—
34C22×D4×C137D412yes—
35C22×D4×C137Q812yes—
36C23×C4×C137—22no—
37C23×C4×C137—22no—
38C23×C4×C137—22no—
39C23×C4×C137—22no—
40C23×C4×C137—44no—
41C23×C4×C137C2×C411yescharacteristic
42C23×C4×C137C2×C412yes—
43C23×C4×C137C2×C412yes—
44C23×C4×C137D411yescharacteristic
45C23×C4×C137D412yes—
46C23×C4×C137D411yescharacteristic
47C2×C4⋊C4×C137—24no—
48C2×C4⋊C4×C137—44no—
49C2×C22⋊C4×C137—22no—
50C2×C22⋊C4×C137—22no—
51C2×C22⋊C4×C137—22no—
52C2×C22⋊C4×C137—22no—
53C2×C22⋊C4×C137—22no—
54C2×C22⋊C4×C137—22no—
55C2×C22⋊C4×C137—22no—
56C2×C22⋊C4×C137—22no—
57C2×C22⋊C4×C137—22no—
58C2×C22⋊C4×C137—21no—
59C2×C22⋊C4×C137—21no—
60C2×C22⋊C4×C137—22no—
61C2×C22⋊C4×C137—22no—
62C2×C22⋊C4×C137—22no—
63C2×C22⋊C4×C137—22no—
64C2×C22⋊C4×C137—22no—
65C2×C22⋊C4×C137—44no—
66C2×C22⋊C4×C137D414yes—
67C2×C22⋊C4×C137D414yes—
68#99×C137—216no—
69#95×C137—24no—
70#95×C137—22no—
71#95×C137—22no—
72#95×C137—44no—
73C24×C137—21no—
74C24×C137—21no—
75C24×C137—22no—
76C24×C137—22no—
77C24×C137—42no—
78C24×C137—42no—
79C24×C137—42no—
80C24×C137—42no—
81C24×C137C4⋊C412yescomplemented
82C24×C137C22×C411yescharacteristic
83C24×C137C2×D411yescharacteristic
84C24×C137C2×D411yescharacteristic
85C24×C137C2×D411yescharacteristic
86C24×C137C2×D412yes—
87C24×C137C2×D412yes—
88C24×C137C2×Q811yescharacteristic
89C24×C137#4111yescharacteristic
90C24×C137#4111yescharacteristic
91C24×C137#4112yes—
92C24×C137#4112yes—
93C2×D4×C137—24no—
94C2×D4×C137—22no—
95C2×D4×C137—22no—
96C2×D4×C137—22no—
97C2×D4×C137—22no—
98C2×D4×C137—24no—
99C2×D4×C137—24no—
100C2×D4×C137—24no—
101C2×D4×C137—41no—
102C2×D4×C137—42no—
103C2×D4×C137—42no—
104C2×D4×C137—42no—
105C2×D4×C137—41no—
106C2×D4×C137—42no—
107C2×D4×C137—42no—
108C2×D4×C137—41no—
109C2×D4×C137—42no—
110C2×D4×C137—41no—
111C22×C4×C137—21no—
112C22×C4×C137—21no—
113C22×C4×C137—24no—
114C22×C4×C137—21no—
115C22×C4×C137—21no—
116C22×C4×C137—24no—
117C22×C4×C137—22no—
118C22×C4×C137—22no—
119C22×C4×C137—21no—
120C22×C4×C137—24no—
121C22×C4×C137—21no—
122C22×C4×C137—24no—
123C22×C4×C137—24no—
124C22×C4×C137—22no—
125C22×C4×C137—24no—
126C22×C4×C137—24no—
127C22×C4×C137—24no—
128C22×C4×C137—42no—
129C22×C4×C137—42no—
130C22×C4×C137—42no—
131C22×C4×C137—48no—
132C22×C4×C137—48no—
133C22×C4×C137—44no—
134C22×C4×C137—44no—
135C22×C4×C137—44no—
136C22×C4×C137—44no—
137C22×C4×C137—42no—
138C22×C4×C137—42no—
139C22×C4×C137—42no—
140C22×C4×C137—42no—
141C22×C4×C137—42no—
142C22×C4×C137—42no—
143C22×C4×C137C22⋊C412yes—
144C22×C4×C137C22⋊C412yes—
145C22×C4×C137C4⋊C412yescomplemented
146C4⋊C4×C137—48no—
147C22⋊C4×C137—24no—
148C22⋊C4×C137—24no—
149C22⋊C4×C137—24no—
150C22⋊C4×C137—24no—
151C22⋊C4×C137—44no—
152C22⋊C4×C137—42no—
153C22⋊C4×C137—42no—
154C22⋊C4×C137—42no—
155C22⋊C4×C137—42no—
156C22⋊C4×C137—41no—
157C22⋊C4×C137—42no—
158C22⋊C4×C137—41no—
159C22⋊C4×C137—44no—
160C23×C137—22no—
161C23×C137—22no—
162C23×C137—22no—
163C23×C137—22no—
164C23×C137—22no—
165C23×C137—22no—
166C23×C137—22no—
167C23×C137—24no—
168C23×C137—42no—
169C23×C137—42no—
170C23×C137—42no—
171C23×C137—42no—
172C23×C137—44no—
173C23×C137—42no—
174C23×C137—42no—
175C23×C137—42no—
176C23×C137—42no—
177C23×C137—44no—
178C23×C137—44no—
179C23×C137—44no—
180C23×C137—41no—
181C23×C137—42no—
182C23×C137—41no—
183C23×C137—42no—
184C23×C137—42no—
185C23×C137—82no—
186C23×C137—82no—
187C23×C137—84no—
188C23×C137C2×C22⋊C411yescharacteristic
189C23×C137C2×C4⋊C411yescharacteristic
190C23×C137#11711yescharacteristic
191C23×C137C4×D412yes—
192C23×C137C4×D412yes—
193C23×C137#12011yescharacteristic
194C23×C137#12012yes—
195C23×C137#12111yescharacteristic
196C23×C137#12111yescharacteristic
197C23×C137#12112yes—
198C23×C137#12112yes—
199C23×C137#12112yes—
200C23×C137C22⋊Q811yescharacteristic
201C23×C137C22⋊Q811yescharacteristic
202C23×C137C22⋊Q812yes—
203C23×C137C22⋊Q812yes—
204C23×C137#12311yescharacteristic
205C23×C137#12312yes—
206C23×C137#12312yes—
207C23×C137#12411yescharacteristic
208C23×C137#12412yes—
209C23×C137#12612yes—
210C23×C137#12711yescharacteristic
211D4×C137—42no—
212D4×C137—42no—
213D4×C137—44no—
214D4×C137—44no—
215D4×C137—42no—
216D4×C137—42no—
217C2×C4×C137—24no—
218C2×C4×C137—24no—
219C2×C4×C137—22no—
220C2×C4×C137—22no—
221C2×C4×C137—22no—
222C2×C4×C137—24no—
223C2×C4×C137—22no—
224C2×C4×C137—22no—
225C2×C4×C137—22no—
226C2×C4×C137—48no—
227C2×C4×C137—48no—
228C2×C4×C137—42no—
229C2×C4×C137—42no—
230C2×C4×C137—41no—
231C2×C4×C137—42no—
232C2×C4×C137—41no—
233C2×C4×C137—44no—
234C2×C4×C137—44no—
235C2×C4×C137—48no—
236C2×C4×C137—42no—
237C2×C4×C137—44no—
238C2×C4×C137—44no—
239C2×C4×C137—42no—
240C2×C4×C137—42no—
241C2×C4×C137—84no—
242C2×C4×C137—84no—
243C2×C4×C137—82no—
244C2×C4×C137—82no—
245C2×C4×C137—84no—
246C2×C4×C137—84no—
247C2×C4×C137#9914yescomplemented
248C22×C137—22no—
249C22×C137—22no—
250C22×C137—22no—
251C22×C137—22no—
252C22×C137—22no—
253C22×C137—22no—
254C22×C137—21no—
255C22×C137—21no—
256C22×C137—22no—
257C22×C137—22no—
258C22×C137—21no—
259C22×C137—22no—
260C22×C137—22no—
261C22×C137—21no—
262C22×C137—24no—
263C22×C137—22no—
264C22×C137—24no—
265C22×C137—22no—
266C22×C137—42no—
267C22×C137—42no—
268C22×C137—42no—
269C22×C137—42no—
270C22×C137—42no—
271C22×C137—42no—
272C22×C137—42no—
273C22×C137—42no—
274C22×C137—44no—
275C22×C137—42no—
276C22×C137—84no—
277C22×C137—84no—
278C22×C137—81no—
279C22×C137—81no—
280C22×C137—82no—
281C22×C137#38011yescharacteristic
282C22×C137#38512yes—
283C22×C137#38611yescharacteristic
284C22×C137#38812yes—
285C22×C137#39011yescharacteristic
286C22×C137#39211yescharacteristic
287C22×C137#39311yescharacteristic
288C22×C137#39412yes—
289C22×C137#39612yes—
290C22×C137#39711yescharacteristic
291C22×C137#39911yescharacteristic
292C22×C137C2×#9912yes—
293C22×C137#41012yes—
294C4×C137—24no—
295C4×C137—42no—
296C4×C137—42no—
297C4×C137—84no—
298C4×C137—84no—
299C4×C137—82no—
300C2×C137—21no—
301C2×C137—22no—
302C2×C137—22no—
303C2×C137—22no—
304C2×C137—21no—
305C2×C137—22no—
306C2×C137—21no—
307C2×C137—21no—
308C2×C137—82no—
309C2×C137#143911yescharacteristic
310C2×C137#179912yes—
311C2×C137#189811yescharacteristic
312C2×C137#189911yescharacteristic
313C2×C137#196412yes—
314#11043C13711yesmaximal, characteristic, direct product factor, Sylow
315C137#1104311yescharacteristic, direct product factor, Sylow
316C22×#99C2×C13712yes—
317C2×#386C2×C13711yescharacteristic
318C2×#380C2×C13711yescharacteristic, complemented
319C2×#342C2×C13712yescomplemented
320C22⋊#95C2×C13711yescharacteristic, complemented
321C23×D4C22×C13711yescharacteristic
322C24×C4C22×C13711yescharacteristic
323C22×C4⋊C4—21no—
324C22×C22⋊C4C4×C13712yescomplemented
325C22×C22⋊C4C22×C13711yescharacteristic
326C22×C22⋊C4C22×C13712yes—
327C22×C22⋊C4C22×C13712yes—
328C2×#99—24no—
329C2×#99—24no—
330C2×#99—24no—
331#386—21no—
332#386—22no—
333#386—21no—
334#380—24no—
335C2×#95—21no—
336C2×#95—22no—
337C2×#95C4×C13712yescomplemented
338#342—24no—
339C25C2×C4×C13711yescharacteristic
340C25C23×C13711yescharacteristic
341C22×D4—21no—
342C22×D4—21no—
343C22×D4—22no—
344C22×D4—22no—
345C22×D4—21no—
346C22×D4—21no—
347C22×D4—22no—
348C22×D4—22no—
349C22×D4D4×C13712yes—
350C22×D4Q8×C13712yes—
351C23×C4—22no—
352C23×C4—22no—
353C23×C4—22no—
354C23×C4—22no—
355C23×C4—44no—
356C23×C4C2×C4×C13711yescharacteristic
357C23×C4C2×C4×C13712yes—
358C23×C4C2×C4×C13712yes—
359C23×C4D4×C13711yescharacteristic
360C23×C4D4×C13712yes—
361C23×C4D4×C13711yescharacteristic
362C2×C4⋊C4—24no—
363C2×C4⋊C4—44no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—44no—
381C2×C22⋊C4D4×C13714yes—
382C2×C22⋊C4D4×C13714yes—
383#99—216no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—44no—
388C24—21no—
389C24—21no—
390C24—22no—
391C24—22no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24C4⋊C4×C13712yescomplemented
397C24C22×C4×C13711yescharacteristic
398C24C2×D4×C13711yescharacteristic
399C24C2×D4×C13711yescharacteristic
400C24C2×D4×C13711yescharacteristic
401C24C2×D4×C13712yes—
402C24C2×D4×C13712yes—
403C24C2×Q8×C13711yescharacteristic
404C24#41×C13711yescharacteristic
405C24#41×C13711yescharacteristic
406C24#41×C13712yes—
407C24#41×C13712yes—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—41no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—41no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—41no—
424C2×D4—42no—
425C2×D4—41no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4C22⋊C4×C13712yes—
459C22×C4C22⋊C4×C13712yes—
460C22×C4C4⋊C4×C13712yescomplemented
461C4⋊C4—48no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—44no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—41no—
472C22⋊C4—42no—
473C22⋊C4—41no—
474C22⋊C4—44no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—41no—
496C23—42no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—82no—
501C23—82no—
502C23—84no—
503C23C2×C22⋊C4×C13711yescharacteristic
504C23C2×C4⋊C4×C13711yescharacteristic
505C23#117×C13711yescharacteristic
506C23C4×D4×C13712yes—
507C23C4×D4×C13712yes—
508C23#120×C13711yescharacteristic
509C23#120×C13712yes—
510C23#121×C13711yescharacteristic
511C23#121×C13711yescharacteristic
512C23#121×C13712yes—
513C23#121×C13712yes—
514C23#121×C13712yes—
515C23C22⋊Q8×C13711yescharacteristic
516C23C22⋊Q8×C13711yescharacteristic
517C23C22⋊Q8×C13712yes—
518C23C22⋊Q8×C13712yes—
519C23#123×C13711yescharacteristic
520C23#123×C13712yes—
521C23#123×C13712yes—
522C23#124×C13711yescharacteristic
523C23#124×C13712yes—
524C23#126×C13712yes—
525C23#127×C13711yescharacteristic
526D4—42no—
527D4—42no—
528D4—44no—
529D4—44no—
530D4—42no—
531D4—42no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4#99×C13714yescomplemented
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—21no—
570C22—21no—
571C22—22no—
572C22—22no—
573C22—21no—
574C22—22no—
575C22—22no—
576C22—21no—
577C22—24no—
578C22—22no—
579C22—24no—
580C22—22no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—44no—
590C22—42no—
591C22—84no—
592C22—84no—
593C22—81no—
594C22—81no—
595C22—82no—
596C22#380×C13711yescharacteristic
597C22#385×C13712yes—
598C22#386×C13711yescharacteristic
599C22#388×C13712yes—
600C22#390×C13711yescharacteristic
601C22#392×C13711yescharacteristic
602C22#393×C13711yescharacteristic
603C22#394×C13712yes—
604C22#396×C13712yes—
605C22#397×C13711yescharacteristic
606C22#399×C13711yescharacteristic
607C22C2×#99×C13712yes—
608C22#410×C13712yes—
609C4—24no—
610C4—42no—
611C4—42no—
612C4—84no—
613C4—84no—
614C4—82no—
615C2—21no—
616C2—22no—
617C2—22no—
618C2—22no—
619C2—21no—
620C2—22no—
621C2—21no—
622C2—21no—
623C2—82no—
624C2#1439×C13711yescharacteristic
625C2#1799×C13712yes—
626C2#1898×C13711yescharacteristic
627C2#1899×C13711yescharacteristic
628C2#1964×C13712yes—
subgroupG/H
center, socleC23×C137C2×C22⋊C4
commutatorC24C22×C4×C137
Frattini, cosocleC25C23×C137
prime radicalC23×D4×C137C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23×C137 ◁ C24×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#380×C137C2semidirect
4. C2×#342×C137C2semidirect
5. C22⋊#95×C137C2semidirect
9. C22×C22⋊C4×C137C4semidirect
22. C2×#95×C137C4semidirect
81. C24×C137C4⋊C4semidirect
145. C22×C4×C137C4⋊C4semidirect
247. C2×C4×C137#99semidirect
314. #11043C137direct (N × H)
315. C137#11043direct (N × H)
318. C2×#380C2×C137semidirect
319. C2×#342C2×C137semidirect
320. C22⋊#95C2×C137semidirect
324. C22×C22⋊C4C4×C137semidirect
337. C2×#95C4×C137semidirect
396. C24C4⋊C4×C137semidirect
460. C22×C4C4⋊C4×C137semidirect
562. C2×C4#99×C137semidirect

Maximal subgroups

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

subgroupindex
C22×#99×C1372
C2×#386×C1372
C2×#380×C1372
C2×#342×C1372
C22⋊#95×C1372
#11043137