Finite Group Databasebeta

← C2×#1597×C137 #46867484

Subgroups — C2×#1597×C137 #46867484

Order 35072.

1404 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#539×C137C211yesmaximal, characteristic, complemented
2C2×#450×C137C211yesmaximal, characteristic, complemented
3C2×#447×C137C211yesmaximal, characteristic
4C2×#394×C137C211yesmaximal, characteristic, complemented
5#1597×C137C218yesmaximal, direct product factor
6C2×#328×C137C211yesmaximal, characteristic, complemented
7C2×#327×C137C211yesmaximal, characteristic, complemented
8C2×#324×C137C211yesmaximal, characteristic, complemented
9C23×D4×C137C2211yescharacteristic
10C22×SD16×C137—21no—
11C22×D8×C137—21no—
12#539×C137—22no—
13#539×C137—22no—
14#539×C137—22no—
15#539×C137C2214yescomplemented
16C2×#126×C137—21no—
17C2×#125×C137—21no—
18C2×#123×C137—21no—
19C2×#123×C137—21no—
20C2×C22⋊Q8×C137—21no—
21C2×C22⋊Q8×C137C2211yescharacteristic
22C2×#121×C137C2211yescharacteristic
23C2×C4×D4×C137—21no—
24C22×C4⋊C4×C137C2211yescharacteristic
25C22×C22⋊C4×C137—21no—
26C22×C22⋊C4×C137—21no—
27#450×C137—22no—
28#450×C137C2214yescomplemented
29#447×C137—22no—
30#447×C137C2214yes—
31C2×#102×C137—21no—
32C2×#102×C137—21no—
33C2×C22⋊C8×C137C2211yescharacteristic
34C2×C22⋊C8×C137C2211yescharacteristic
35#394×C137—22no—
36#394×C137C2214yescomplemented
37C2×#95×C137C2211yescharacteristic
38#328×C137C2214yescomplemented
39#327×C137C2214yescomplemented
40#324×C137C2214yescomplemented
41C25×C137—21no—
42C22×Q8×C137D411yescharacteristic
43C22×D4×C137—21no—
44C22×D4×C137—21no—
45C22×D4×C137—21no—
46C22×D4×C137—21no—
47C22×D4×C137—21no—
48C22×D4×C137—21no—
49C22×D4×C137—44no—
50C22×D4×C137D411yescharacteristic
51C22×D4×C137C2312yes—
52C23×C4×C137—21no—
53C23×C4×C137—21no—
54C23×C4×C137D411yescharacteristic
55C23×C4×C137D411yescharacteristic
56C23×C4×C137C2311yescharacteristic
57C2×SD16×C137—24no—
58C2×SD16×C137—44no—
59C2×D8×C137—24no—
60C2×D8×C137—44no—
61C22×C8×C137—21no—
62C22×C8×C137—21no—
63#126×C137—24no—
64#126×C137—24no—
65#125×C137—24no—
66#125×C137—24no—
67#123×C137—24no—
68#123×C137—24no—
69#123×C137—24no—
70#123×C137—24no—
71C22⋊Q8×C137—24no—
72C22⋊Q8×C137—24no—
73C22⋊Q8×C137—21no—
74C22⋊Q8×C137—21no—
75C22⋊Q8×C137D412yes—
76C22⋊Q8×C137C2312yes—
77#121×C137—21no—
78#121×C137—21no—
79#121×C137D412yes—
80#121×C137C2312yes—
81C4×D4×C137—24no—
82C4×D4×C137—24no—
83C2×C4⋊C4×C137—21no—
84C2×C4⋊C4×C137—21no—
85C2×C4⋊C4×C137—21no—
86C2×C4⋊C4×C137—21no—
87C2×C4⋊C4×C137—21no—
88C2×C4⋊C4×C137—21no—
89C2×C4⋊C4×C137—21no—
90C2×C4⋊C4×C137—21no—
91C2×C4⋊C4×C137D412yes—
92C2×C4⋊C4×C137D411yescharacteristic
93C2×C4⋊C4×C137D411yescharacteristic
94C2×C4⋊C4×C137C2312yes—
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—24no—
101C2×C22⋊C4×C137—24no—
102C2×C22⋊C4×C137—44no—
103C2×C22⋊C4×C137—44no—
104C2×C42×C137—21no—
105#102×C137—24no—
106#102×C137—24no—
107C22⋊C8×C137—21no—
108C22⋊C8×C137—21no—
109C22⋊C8×C137C2312yes—
110C22⋊C8×C137C2312yes—
111#95×C137—21no—
112#95×C137C2312yes—
113C24×C137—24no—
114C24×C137—21no—
115C24×C137—22no—
116C24×C137—42no—
117C24×C137—42no—
118C24×C137—42no—
119C24×C137—41no—
120C24×C137—41no—
121C24×C137—42no—
122C24×C137—42no—
123C24×C137C2×D411yescharacteristic
124C2×Q8×C137—21no—
125C2×Q8×C137—42no—
126C2×Q8×C137C2×D412yes—
127C2×D4×C137—24no—
128C2×D4×C137—24no—
129C2×D4×C137—22no—
130C2×D4×C137—22no—
131C2×D4×C137—21no—
132C2×D4×C137—22no—
133C2×D4×C137—22no—
134C2×D4×C137—42no—
135C2×D4×C137—42no—
136C2×D4×C137—42no—
137C2×D4×C137—44no—
138C2×D4×C137—44no—
139C2×D4×C137—44no—
140C2×D4×C137—44no—
141C2×D4×C137—42no—
142C2×D4×C137—42no—
143C2×D4×C137—41no—
144C2×D4×C137—41no—
145C2×D4×C137C2×D412yes—
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—22no—
153C22×C4×C137—22no—
154C22×C4×C137—24no—
155C22×C4×C137—21no—
156C22×C4×C137—22no—
157C22×C4×C137—42no—
158C22×C4×C137—42no—
159C22×C4×C137—42no—
160C22×C4×C137—42no—
161C22×C4×C137—42no—
162C22×C4×C137—82no—
163C22×C4×C137C2×D412yes—
164C22×C4×C137C2×D411yescharacteristic
165C22×C4×C137C2×D411yescharacteristic
166C22×C4×C137C2×D412yes—
167C22×C4×C137C2×D411yescharacteristic
168C22×C4×C137C2×D411yescharacteristic
169C22×C4×C137C2×D411yescharacteristic
170C22×C4×C137C2411yescharacteristic
171SD16×C137—48no—
172D8×C137—48no—
173C2×C8×C137—22no—
174C2×C8×C137—22no—
175C2×C8×C137—42no—
176C2×C8×C137—42no—
177C4⋊C4×C137—24no—
178C4⋊C4×C137—24no—
179C4⋊C4×C137—24no—
180C4⋊C4×C137—22no—
181C4⋊C4×C137—22no—
182C4⋊C4×C137—22no—
183C4⋊C4×C137—21no—
184C4⋊C4×C137—21no—
185C4⋊C4×C137—22no—
186C4⋊C4×C137—22no—
187C4⋊C4×C137—22no—
188C4⋊C4×C137C2×D412yes—
189C4⋊C4×C137C2×D412yes—
190C22⋊C4×C137—24no—
191C22⋊C4×C137—24no—
192C22⋊C4×C137—24no—
193C22⋊C4×C137—22no—
194C22⋊C4×C137—22no—
195C22⋊C4×C137—22no—
196C22⋊C4×C137—22no—
197C22⋊C4×C137—48no—
198C22⋊C4×C137—48no—
199C42×C137—24no—
200C23×C137—22no—
201C23×C137—22no—
202C23×C137—24no—
203C23×C137—22no—
204C23×C137—42no—
205C23×C137—44no—
206C23×C137—48no—
207C23×C137—48no—
208C23×C137—42no—
209C23×C137—42no—
210C23×C137—42no—
211C23×C137—42no—
212C23×C137—42no—
213C23×C137—42no—
214C23×C137—42no—
215C23×C137—42no—
216C23×C137—44no—
217C23×C137—84no—
218C23×C137—84no—
219C23×C137—82no—
220C23×C137—82no—
221C23×C137—82no—
222C23×C137—82no—
223C23×C137#12011yescharacteristic
224C23×C137#12011yescharacteristic
225C23×C137#13512yes—
226C23×C137AGL(1,ℤ/8)12yes—
227C23×C137C22×D411yescharacteristic
228Q8×C137—42no—
229Q8×C137—81no—
230D4×C137—24no—
231D4×C137—24no—
232D4×C137—42no—
233D4×C137—42no—
234D4×C137—42no—
235D4×C137—44no—
236D4×C137—44no—
237D4×C137—42no—
238D4×C137—42no—
239D4×C137—44no—
240D4×C137—81no—
241C2×C4×C137—22no—
242C2×C4×C137—22no—
243C2×C4×C137—22no—
244C2×C4×C137—22no—
245C2×C4×C137—22no—
246C2×C4×C137—22no—
247C2×C4×C137—22no—
248C2×C4×C137—22no—
249C2×C4×C137—22no—
250C2×C4×C137—41no—
251C2×C4×C137—41no—
252C2×C4×C137—41no—
253C2×C4×C137—41no—
254C2×C4×C137—42no—
255C2×C4×C137—42no—
256C2×C4×C137—42no—
257C2×C4×C137—42no—
258C2×C4×C137—42no—
259C2×C4×C137—42no—
260C2×C4×C137—42no—
261C2×C4×C137—42no—
262C2×C4×C137—42no—
263C2×C4×C137—42no—
264C2×C4×C137—81no—
265C2×C4×C137—84no—
266C2×C4×C137—84no—
267C2×C4×C137#12011yescharacteristic
268C2×C4×C137#12011yescharacteristic
269C2×C4×C137C22×D411yescharacteristic
270C2×C4×C137C22×D411yescharacteristic
271C8×C137—42no—
272C8×C137—42no—
273C22×C137—24no—
274C22×C137—24no—
275C22×C137—21no—
276C22×C137—21no—
277C22×C137—21no—
278C22×C137—21no—
279C22×C137—44no—
280C22×C137—44no—
281C22×C137—42no—
282C22×C137—42no—
283C22×C137—42no—
284C22×C137—44no—
285C22×C137—82no—
286C22×C137—82no—
287C22×C137—88no—
288C22×C137—84no—
289C22×C137—88no—
290C22×C137—82no—
291C22×C137—82no—
292C22×C137#44912yes—
293C22×C137#44912yes—
294C22×C137#45412yes—
295C22×C137B(4,𝔽2)12yes—
296C22×C137C2×#12011yescharacteristic
297C22×C137C2×#13511yescharacteristic
298C22×C137C2×AGL(1,ℤ/8)11yescharacteristic
299C4×C137—21no—
300C4×C137—21no—
301C4×C137—21no—
302C4×C137—21no—
303C4×C137—42no—
304C4×C137—42no—
305C4×C137—42no—
306C4×C137—42no—
307C4×C137—42no—
308C4×C137—82no—
309C2×C137—21no—
310C2×C137—21no—
311C2×C137—22no—
312C2×C137—82no—
313C2×C137—84no—
314C2×C137#159714yesdirect product factor
315C2×C137C2×#44911yescharacteristic
316C2×C137C2×#45411yescharacteristic
317C2×C137C2×B(4,𝔽2)11yescharacteristic
318C2×#1597C13711yesmaximal, characteristic, direct product factor, Sylow
319C137C2×#159711yescharacteristic, direct product factor, Sylow
320C2×#539C2×C13711yescharacteristic, complemented
321C2×#450C2×C13711yescharacteristic, complemented
322C2×#447C2×C13711yescharacteristic
323C2×#394C2×C13711yescharacteristic, complemented
324#1597C2×C13718yesdirect product factor
325C2×#328C2×C13711yescharacteristic, complemented
326C2×#327C2×C13711yescharacteristic, complemented
327C2×#324C2×C13711yescharacteristic, complemented
328C23×D4C22×C13711yescharacteristic
329C22×SD16—21no—
330C22×D8—21no—
331#539—22no—
332#539—22no—
333#539—22no—
334#539C22×C13714yescomplemented
335C2×#126—21no—
336C2×#125—21no—
337C2×#123—21no—
338C2×#123—21no—
339C2×C22⋊Q8—21no—
340C2×C22⋊Q8C22×C13711yescharacteristic
341C2×#121C22×C13711yescharacteristic
342C2×C4×D4—21no—
343C22×C4⋊C4C22×C13711yescharacteristic
344C22×C22⋊C4—21no—
345C22×C22⋊C4—21no—
346#450—22no—
347#450C22×C13714yescomplemented
348#447—22no—
349#447C22×C13714yes—
350C2×#102—21no—
351C2×#102—21no—
352C2×C22⋊C8C22×C13711yescharacteristic
353C2×C22⋊C8C22×C13711yescharacteristic
354#394—22no—
355#394C22×C13714yescomplemented
356C2×#95C22×C13711yescharacteristic
357#328C22×C13714yescomplemented
358#327C22×C13714yescomplemented
359#324C22×C13714yescomplemented
360C25—21no—
361C22×Q8D4×C13711yescharacteristic
362C22×D4—21no—
363C22×D4—21no—
364C22×D4—21no—
365C22×D4—21no—
366C22×D4—21no—
367C22×D4—21no—
368C22×D4—44no—
369C22×D4D4×C13711yescharacteristic
370C22×D4C23×C13712yes—
371C23×C4—21no—
372C23×C4—21no—
373C23×C4D4×C13711yescharacteristic
374C23×C4D4×C13711yescharacteristic
375C23×C4C23×C13711yescharacteristic
376C2×SD16—24no—
377C2×SD16—44no—
378C2×D8—24no—
379C2×D8—44no—
380C22×C8—21no—
381C22×C8—21no—
382#126—24no—
383#126—24no—
384#125—24no—
385#125—24no—
386#123—24no—
387#123—24no—
388#123—24no—
389#123—24no—
390C22⋊Q8—24no—
391C22⋊Q8—24no—
392C22⋊Q8—21no—
393C22⋊Q8—21no—
394C22⋊Q8D4×C13712yes—
395C22⋊Q8C23×C13712yes—
396#121—21no—
397#121—21no—
398#121D4×C13712yes—
399#121C23×C13712yes—
400C4×D4—24no—
401C4×D4—24no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4D4×C13712yes—
411C2×C4⋊C4D4×C13711yescharacteristic
412C2×C4⋊C4D4×C13711yescharacteristic
413C2×C4⋊C4C23×C13712yes—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—24no—
420C2×C22⋊C4—24no—
421C2×C22⋊C4—44no—
422C2×C22⋊C4—44no—
423C2×C42—21no—
424#102—24no—
425#102—24no—
426C22⋊C8—21no—
427C22⋊C8—21no—
428C22⋊C8C23×C13712yes—
429C22⋊C8C23×C13712yes—
430#95—21no—
431#95C23×C13712yes—
432C24—24no—
433C24—21no—
434C24—22no—
435C24—42no—
436C24—42no—
437C24—42no—
438C24—41no—
439C24—41no—
440C24—42no—
441C24—42no—
442C24C2×D4×C13711yescharacteristic
443C2×Q8—21no—
444C2×Q8—42no—
445C2×Q8C2×D4×C13712yes—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4C2×D4×C13712yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—82no—
482C22×C4C2×D4×C13712yes—
483C22×C4C2×D4×C13711yescharacteristic
484C22×C4C2×D4×C13711yescharacteristic
485C22×C4C2×D4×C13712yes—
486C22×C4C2×D4×C13711yescharacteristic
487C22×C4C2×D4×C13711yescharacteristic
488C22×C4C2×D4×C13711yescharacteristic
489C22×C4C24×C13711yescharacteristic
490SD16—48no—
491D8—48no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—42no—
495C2×C8—42no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4C2×D4×C13712yes—
508C4⋊C4C2×D4×C13712yes—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C42—24no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—42no—
524C23—44no—
525C23—48no—
526C23—48no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—84no—
537C23—84no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23#120×C13711yescharacteristic
543C23#120×C13711yescharacteristic
544C23#135×C13712yes—
545C23AGL(1,ℤ/8)×C13712yes—
546C23C22×D4×C13711yescharacteristic
547Q8—42no—
548Q8—81no—
549D4—24no—
550D4—24no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—44no—
555D4—44no—
556D4—44no—
557D4—42no—
558D4—42no—
559D4—81no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—81no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4#120×C13711yescharacteristic
587C2×C4#120×C13711yescharacteristic
588C2×C4C22×D4×C13711yescharacteristic
589C2×C4C22×D4×C13711yescharacteristic
590C8—42no—
591C8—42no—
592C22—24no—
593C22—24no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—44no—
604C22—82no—
605C22—82no—
606C22—88no—
607C22—88no—
608C22—84no—
609C22—82no—
610C22—82no—
611C22#449×C13712yes—
612C22#449×C13712yes—
613C22#454×C13712yes—
614C22B(4,𝔽2)×C13712yes—
615C22C2×#120×C13711yescharacteristic
616C22C2×#135×C13711yescharacteristic
617C22C2×AGL(1,ℤ/8)×C13711yescharacteristic
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—82no—
628C2—21no—
629C2—21no—
630C2—22no—
631C2—84no—
632C2—82no—
633C2#1597×C13714yesdirect product factor
634C2C2×#449×C13711yescharacteristic
635C2C2×#454×C13711yescharacteristic
636C2C2×B(4,𝔽2)×C13711yescharacteristic
subgroupG/H
center, socleC23×C137#120
commutator, Frattini, cosocleC22×C4C24×C137
prime radicalC2×#447×C137C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C137 ◁ C23×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#539×C137C2semidirect
2. C2×#450×C137C2semidirect
4. C2×#394×C137C2semidirect
5. #1597×C137C2direct (N × H)
6. C2×#328×C137C2semidirect
7. C2×#327×C137C2semidirect
8. C2×#324×C137C2semidirect
15. #539×C137C22semidirect
28. #450×C137C22semidirect
36. #394×C137C22semidirect
38. #328×C137C22semidirect
39. #327×C137C22semidirect
40. #324×C137C22semidirect
314. C2×C137#1597direct (N × H)
318. C2×#1597C137direct (N × H)
319. C137C2×#1597direct (N × H)
320. C2×#539C2×C137semidirect
321. C2×#450C2×C137semidirect
323. C2×#394C2×C137semidirect
324. #1597C2×C137direct (N × H)
325. C2×#328C2×C137semidirect
326. C2×#327C2×C137semidirect
327. C2×#324C2×C137semidirect
334. #539C22×C137semidirect
347. #450C22×C137semidirect
355. #394C22×C137semidirect
357. #328C22×C137semidirect
358. #327C22×C137semidirect
359. #324C22×C137semidirect
633. C2#1597×C137direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#539×C1372
C2×#450×C1372
C2×#447×C1372
C2×#394×C1372
#1597×C1372
C2×#328×C1372
C2×#327×C1372
C2×#324×C1372
C2×#1597137