Finite Group Databasebeta

← C137×#13344 #46862730

Subgroups — C137×#13344 #46862730

Order 35072.

892 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 892 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#458×C137C211yesmaximal, characteristic, complemented
2C2×B(4,𝔽2)×C137C211yesmaximal, characteristic
3#2697×C137C211yesmaximal, characteristic, complemented
4C2×#352×C137C211yesmaximal, characteristic, complemented
5C2×AΣL(1,𝔽16)×C137C211yesmaximal, characteristic, complemented
6#1792×C137C211yesmaximal, characteristic, complemented
7#1393×C137C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)×C137C2211yescharacteristic
9C2×#123×C137—21no—
10C2×#123×C137C2211yescharacteristic
11C2×#120×C137—21no—
12C2×#120×C137C2211yescharacteristic
13#458×C137—22no—
14#458×C137—22no—
15B(4,𝔽2)×C137—22no—
16B(4,𝔽2)×C137—22no—
17C2×#104×C137—21no—
18#417×C137—21no—
19C2×#100×C137C2211yescharacteristic
20C2×#99×C137—21no—
21C2×#99×C137—21no—
22C2×#99×C137C2211yescharacteristic
23C2×#99×C137C2211yescharacteristic
24#397×C137—21no—
25#396×C137—21no—
26#394×C137—21no—
27#390×C137C2211yescharacteristic
28#388×C137—21no—
29#386×C137—21no—
30#379×C137—21no—
31#352×C137—22no—
32AΣL(1,𝔽16)×C137—22no—
33C25×C137D411yescharacteristic
34H(5,𝔽2)×C137—21no—
35H(5,𝔽2)×C137—21no—
36H(5,𝔽2)×C137—21no—
37H(5,𝔽2)×C137—42no—
38H(5,𝔽2)×C137D412yes—
39C2×#41×C137—21no—
40C2×#41×C137—41no—
41C22×D4×C137—21no—
42C22×D4×C137—21no—
43C22×D4×C137—21no—
44C22×D4×C137—41no—
45C22×D4×C137—41no—
46C22×D4×C137C2311yescharacteristic
47C23×C4×C137D411yescharacteristic, complemented
48C2×M16×C137—21no—
49#123×C137—24no—
50#123×C137—21no—
51#123×C137—21no—
52#123×C137—21no—
53#123×C137—21no—
54#123×C137—42no—
55#120×C137—24no—
56#120×C137—21no—
57#120×C137—21no—
58#120×C137—21no—
59#120×C137—21no—
60#120×C137—42no—
61C2×C4⋊C4×C137—21no—
62C2×C4⋊C4×C137—21no—
63C2×C4⋊C4×C137—41no—
64C2×C4⋊C4×C137D411yescharacteristic, complemented
65C2×C22⋊C4×C137—21no—
66C2×C22⋊C4×C137—21no—
67C2×C22⋊C4×C137—21no—
68C2×C22⋊C4×C137—21no—
69C2×C22⋊C4×C137—21no—
70C2×C22⋊C4×C137—21no—
71C2×C22⋊C4×C137—21no—
72C2×C22⋊C4×C137—21no—
73C2×C22⋊C4×C137—21no—
74C2×C22⋊C4×C137—21no—
75C2×C22⋊C4×C137—41no—
76C2×C22⋊C4×C137—41no—
77C2×C22⋊C4×C137D411yescharacteristic
78C2×C22⋊C4×C137D411yescharacteristic, complemented
79C2×C42×C137D411yescharacteristic, complemented
80#104×C137—42no—
81#103×C137—41no—
82#102×C137—41no—
83#100×C137—21no—
84#100×C137—21no—
85#99×C137—24no—
86#99×C137—24no—
87#99×C137—21no—
88#99×C137—21no—
89#99×C137D412yes—
90#99×C137D412yescomplemented
91#95×C137—21no—
92#95×C137—21no—
93#95×C137—21no—
94C24×C137—21no—
95C24×C137—21no—
96C24×C137—21no—
97C24×C137—21no—
98C24×C137—21no—
99C24×C137—41no—
100C24×C137—41no—
101C24×C137—42no—
102C24×C137—81no—
103C24×C137C2×D411yescharacteristic
104C24×C137C2×D411yescharacteristic
105#41×C137—22no—
106#41×C137—22no—
107#41×C137—42no—
108#41×C137—41no—
109#41×C137—42no—
110#41×C137—42no—
111#41×C137—42no—
112#41×C137—41no—
113C2×Q8×C137—21no—
114C2×D4×C137—21no—
115C2×D4×C137—22no—
116C2×D4×C137—22no—
117C2×D4×C137—22no—
118C2×D4×C137—22no—
119C2×D4×C137—41no—
120C2×D4×C137—41no—
121C2×D4×C137—42no—
122C2×D4×C137—42no—
123C2×D4×C137—41no—
124C2×D4×C137—41no—
125C2×D4×C137—41no—
126C2×D4×C137—41no—
127C2×D4×C137—42no—
128C2×D4×C137—42no—
129C2×D4×C137—41no—
130C2×D4×C137—41no—
131C2×D4×C137—41no—
132C2×D4×C137—44no—
133C2×D4×C137—42no—
134C2×D4×C137—41no—
135C2×D4×C137—41no—
136C2×D4×C137—81no—
137C2×D4×C137—82no—
138C2×D4×C137—82no—
139C2×D4×C137C2×D411yescharacteristic
140C2×D4×C137C2×D411yescharacteristic
141C2×D4×C137C2×D411yescharacteristic
142C2×D4×C137#4111yescharacteristic
143C22×C4×C137—21no—
144C22×C4×C137—21no—
145C22×C4×C137—21no—
146C22×C4×C137—21no—
147C22×C4×C137—21no—
148C22×C4×C137—21no—
149C22×C4×C137—21no—
150C22×C4×C137—21no—
151C22×C4×C137—21no—
152C22×C4×C137—41no—
153C22×C4×C137—41no—
154C22×C4×C137—41no—
155C22×C4×C137—41no—
156C22×C4×C137—41no—
157C22×C4×C137C2×D411yescharacteristic
158M16×C137—41no—
159M16×C137—41no—
160C2×C8×C137—41no—
161C4⋊C4×C137—41no—
162C4⋊C4×C137—44no—
163C4⋊C4×C137—41no—
164C4⋊C4×C137—41no—
165C4⋊C4×C137—81no—
166C22⋊C4×C137—21no—
167C22⋊C4×C137—21no—
168C22⋊C4×C137—22no—
169C22⋊C4×C137—22no—
170C22⋊C4×C137—21no—
171C22⋊C4×C137—21no—
172C22⋊C4×C137—44no—
173C22⋊C4×C137—42no—
174C22⋊C4×C137—41no—
175C22⋊C4×C137—41no—
176C22⋊C4×C137—44no—
177C22⋊C4×C137—41no—
178C22⋊C4×C137—41no—
179C22⋊C4×C137—41no—
180C22⋊C4×C137—41no—
181C22⋊C4×C137—81no—
182C22⋊C4×C137—81no—
183C22⋊C4×C137—81no—
184C22⋊C4×C137—81no—
185C22⋊C4×C137—81no—
186C42×C137—41no—
187C23×C137—21no—
188C23×C137—21no—
189C23×C137—21no—
190C23×C137—21no—
191C23×C137—22no—
192C23×C137—22no—
193C23×C137—22no—
194C23×C137—41no—
195C23×C137—41no—
196C23×C137—41no—
197C23×C137—41no—
198C23×C137—42no—
199C23×C137—41no—
200C23×C137—41no—
201C23×C137—41no—
202C23×C137—41no—
203C23×C137—41no—
204C23×C137—42no—
205C23×C137—42no—
206C23×C137—41no—
207C23×C137—41no—
208C23×C137—41no—
209C23×C137—41no—
210C23×C137—81no—
211C23×C137—82no—
212C23×C137—81no—
213C23×C137—82no—
214C23×C137—84no—
215C23×C137—81no—
216C23×C137—81no—
217C23×C137—81no—
218C23×C137—81no—
219C23×C137—81no—
220C23×C137#12011yescharacteristic
221C23×C137#12011yescharacteristic
222C23×C137#12011yescharacteristic
223C23×C137#12111yescharacteristic
224C23×C137#12111yescharacteristic
225Q8×C137—22no—
226Q8×C137—41no—
227D4×C137—22no—
228D4×C137—41no—
229D4×C137—41no—
230D4×C137—41no—
231D4×C137—41no—
232D4×C137—41no—
233D4×C137—42no—
234D4×C137—42no—
235D4×C137—42no—
236D4×C137—42no—
237D4×C137—42no—
238D4×C137—42no—
239D4×C137—82no—
240D4×C137—82no—
241D4×C137—88no—
242C2×C4×C137—21no—
243C2×C4×C137—21no—
244C2×C4×C137—22no—
245C2×C4×C137—22no—
246C2×C4×C137—42no—
247C2×C4×C137—42no—
248C2×C4×C137—41no—
249C2×C4×C137—41no—
250C2×C4×C137—41no—
251C2×C4×C137—42no—
252C2×C4×C137—42no—
253C2×C4×C137—42no—
254C2×C4×C137—41no—
255C2×C4×C137—41no—
256C2×C4×C137—41no—
257C2×C4×C137—41no—
258C2×C4×C137—41no—
259C2×C4×C137—41no—
260C2×C4×C137—41no—
261C2×C4×C137—41no—
262C2×C4×C137—41no—
263C2×C4×C137—41no—
264C2×C4×C137—41no—
265C2×C4×C137—41no—
266C2×C4×C137—41no—
267C2×C4×C137—41no—
268C2×C4×C137—41no—
269C2×C4×C137—81no—
270C2×C4×C137—81no—
271C2×C4×C137—81no—
272C2×C4×C137—81no—
273C2×C4×C137—81no—
274C2×C4×C137#12111yescharacteristic
275C2×C4×C137#12711yescharacteristic
276C8×C137—81no—
277C22×C137—21no—
278C22×C137—21no—
279C22×C137—21no—
280C22×C137—21no—
281C22×C137—21no—
282C22×C137—21no—
283C22×C137—41no—
284C22×C137—41no—
285C22×C137—41no—
286C22×C137—41no—
287C22×C137—42no—
288C22×C137—42no—
289C22×C137—41no—
290C22×C137—41no—
291C22×C137—41no—
292C22×C137—41no—
293C22×C137—44no—
294C22×C137—42no—
295C22×C137—42no—
296C22×C137—81no—
297C22×C137—81no—
298C22×C137—81no—
299C22×C137—81no—
300C22×C137—82no—
301C22×C137—82no—
302C22×C137—82no—
303C22×C137—82no—
304C22×C137—82no—
305C22×C137—81no—
306C22×C137—81no—
307C22×C137—81no—
308C22×C137—81no—
309C22×C137—161no—
310C22×C137—161no—
311C22×C137#39211yescharacteristic
312C22×C137B(4,𝔽2)11yescharacteristic
313C22×C137B(4,𝔽2)11yescharacteristic
314C4×C137—21no—
315C4×C137—21no—
316C4×C137—42no—
317C4×C137—81no—
318C4×C137—81no—
319C4×C137—81no—
320C4×C137—82no—
321C4×C137—82no—
322C4×C137—81no—
323C4×C137—81no—
324C2×C137—21no—
325C2×C137—21no—
326C2×C137—41no—
327C2×C137—41no—
328C2×C137—41no—
329C2×C137—41no—
330C2×C137—82no—
331C2×C137—82no—
332C2×C137#202111yescharacteristic
333C2×C137#219711yescharacteristic
334C2×C137#220011yescharacteristic
335#13344C13711yesmaximal, characteristic, direct product factor, Sylow
336C137#1334411yescharacteristic, direct product factor, Sylow
337C2×#458C2×C13711yescharacteristic, complemented
338C2×B(4,𝔽2)C2×C13711yescharacteristic
339#2697C2×C13711yescharacteristic, complemented
340C2×#352C2×C13711yescharacteristic, complemented
341C2×AΣL(1,𝔽16)C2×C13711yescharacteristic, complemented
342#1792C2×C13711yescharacteristic, complemented
343#1393C2×C13711yescharacteristic, complemented
344C2×H(5,𝔽2)C22×C13711yescharacteristic
345C2×#123—21no—
346C2×#123C22×C13711yescharacteristic
347C2×#120—21no—
348C2×#120C22×C13711yescharacteristic
349#458—22no—
350#458—22no—
351B(4,𝔽2)—22no—
352B(4,𝔽2)—22no—
353C2×#104—21no—
354#417—21no—
355C2×#100C22×C13711yescharacteristic
356C2×#99—21no—
357C2×#99—21no—
358C2×#99C22×C13711yescharacteristic
359C2×#99C22×C13711yescharacteristic
360#397—21no—
361#396—21no—
362#394—21no—
363#390C22×C13711yescharacteristic
364#388—21no—
365#386—21no—
366#379—21no—
367#352—22no—
368AΣL(1,𝔽16)—22no—
369C25D4×C13711yescharacteristic
370H(5,𝔽2)—21no—
371H(5,𝔽2)—21no—
372H(5,𝔽2)—21no—
373H(5,𝔽2)—42no—
374H(5,𝔽2)D4×C13712yes—
375C2×#41—21no—
376C2×#41—41no—
377C22×D4—21no—
378C22×D4—21no—
379C22×D4—21no—
380C22×D4—41no—
381C22×D4—41no—
382C22×D4C23×C13711yescharacteristic
383C23×C4D4×C13711yescharacteristic, complemented
384C2×M16—21no—
385#123—24no—
386#123—21no—
387#123—21no—
388#123—21no—
389#123—21no—
390#123—42no—
391#120—24no—
392#120—21no—
393#120—21no—
394#120—21no—
395#120—21no—
396#120—42no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—41no—
400C2×C4⋊C4D4×C13711yescharacteristic, complemented
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—41no—
412C2×C22⋊C4—41no—
413C2×C22⋊C4D4×C13711yescharacteristic
414C2×C22⋊C4D4×C13711yescharacteristic, complemented
415C2×C42D4×C13711yescharacteristic, complemented
416#104—42no—
417#103—41no—
418#102—41no—
419#100—21no—
420#100—21no—
421#99—24no—
422#99—24no—
423#99—21no—
424#99—21no—
425#99D4×C13712yes—
426#99D4×C13712yescomplemented
427#95—21no—
428#95—21no—
429#95—21no—
430C24—21no—
431C24—21no—
432C24—21no—
433C24—21no—
434C24—21no—
435C24—41no—
436C24—41no—
437C24—42no—
438C24—81no—
439C24C2×D4×C13711yescharacteristic
440C24C2×D4×C13711yescharacteristic
441#41—22no—
442#41—22no—
443#41—42no—
444#41—41no—
445#41—42no—
446#41—42no—
447#41—42no—
448#41—41no—
449C2×Q8—21no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—41no—
456C2×D4—41no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—44no—
469C2×D4—42no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—81no—
473C2×D4—82no—
474C2×D4—82no—
475C2×D4C2×D4×C13711yescharacteristic
476C2×D4C2×D4×C13711yescharacteristic
477C2×D4C2×D4×C13711yescharacteristic
478C2×D4#41×C13711yescharacteristic
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4C2×D4×C13711yescharacteristic
494M16—41no—
495M16—41no—
496C2×C8—41no—
497C4⋊C4—41no—
498C4⋊C4—44no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—81no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—44no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C22⋊C4—81no—
521C22⋊C4—81no—
522C42—41no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—81no—
547C23—82no—
548C23—81no—
549C23—82no—
550C23—84no—
551C23—81no—
552C23—81no—
553C23—81no—
554C23—81no—
555C23—81no—
556C23#120×C13711yescharacteristic
557C23#120×C13711yescharacteristic
558C23#120×C13711yescharacteristic
559C23#121×C13711yescharacteristic
560C23#121×C13711yescharacteristic
561Q8—22no—
562Q8—41no—
563D4—22no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—82no—
576D4—82no—
577D4—88no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4#121×C13711yescharacteristic
611C2×C4#127×C13711yescharacteristic
612C8—81no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—42no—
624C22—42no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—44no—
630C22—42no—
631C22—42no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—161no—
646C22—161no—
647C22#392×C13711yescharacteristic
648C22B(4,𝔽2)×C13711yescharacteristic
649C22B(4,𝔽2)×C13711yescharacteristic
650C4—21no—
651C4—21no—
652C4—42no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—82no—
657C4—82no—
658C4—81no—
659C4—81no—
660C2—21no—
661C2—21no—
662C2—41no—
663C2—41no—
664C2—41no—
665C2—41no—
666C2—82no—
667C2—82no—
668C2#2021×C13711yescharacteristic
669C2#2197×C13711yescharacteristic
670C2#2200×C13711yescharacteristic
subgroupG/H
center, socleC22×C137B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C137
prime radicalC2×B(4,𝔽2)×C137C2
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C137 ◁ C22×D4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#458×C137C2semidirect
3. #2697×C137C2semidirect
4. C2×#352×C137C2semidirect
5. C2×AΣL(1,𝔽16)×C137C2semidirect
6. #1792×C137C2semidirect
7. #1393×C137C2semidirect
47. C23×C4×C137D4semidirect
64. C2×C4⋊C4×C137D4semidirect
78. C2×C22⋊C4×C137D4semidirect
79. C2×C42×C137D4semidirect
90. #99×C137D4semidirect
335. #13344C137direct (N × H)
336. C137#13344direct (N × H)
337. C2×#458C2×C137semidirect
339. #2697C2×C137semidirect
340. C2×#352C2×C137semidirect
341. C2×AΣL(1,𝔽16)C2×C137semidirect
342. #1792C2×C137semidirect
343. #1393C2×C137semidirect
383. C23×C4D4×C137semidirect
400. C2×C4⋊C4D4×C137semidirect
414. C2×C22⋊C4D4×C137semidirect
415. C2×C42D4×C137semidirect
426. #99D4×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#458×C1372
C2×B(4,𝔽2)×C1372
#2697×C1372
C2×#352×C1372
C2×AΣL(1,𝔽16)×C1372
#1792×C1372
#1393×C1372
#13344137