Finite Group Databasebeta

← C29×#23901 #20318801

Subgroups — C29×#23901 #20318801

Order 7424.

2056 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C29×#3484C211yesmaximal, characteristic
2C22×C29×#123C211yesmaximal, characteristic, complemented
3C2×C29×#458C212yesmaximal, complemented
4C29×#2849C212yesmaximal, complemented
5C29×#2022C214yesmaximal, complemented
6C29×#1898C212yesmaximal, complemented
7C29×#1890C212yesmaximal, complemented
8C29×#1781C211yesmaximal, characteristic, complemented
9C2×C29×H(5,𝔽2)C2211yescharacteristic
10C23×D4×C29C2211yescharacteristic
11C24×C4×C29C2211yescharacteristic, complemented
12C29×#546—24no—
13D42×C29—24no—
14C29×#535—21no—
15C29×#534—21no—
16C29×#534C2212yes—
17C29×#525—21no—
18C2×C29×#123—21no—
19C2×C29×#123—21no—
20C2×C29×#123—28no—
21C2×C29×#123C2212yes—
22C2×C29×#123C2212yescomplemented
23C2×C29×#123C2214yescomplemented
24C2×C29×#120—21no—
25C2×C29×#120C2212yes—
26C29×#518C2211yescharacteristic
27C22×C4⋊C4×C29—21no—
28C22×C22⋊C4×C29—21no—
29C22×C22⋊C4×C29C2211yescharacteristic, complemented
30C29×#458—28no—
31C2×C29×#99C2212yes—
32C2×C29×#99C2214yescomplemented
33C2×C29×#99C2212yescomplemented
34C29×#386—24no—
35C29×#385—24no—
36C29×#381C2214yescomplemented
37C29×#379C2212yescomplemented
38C29×#342C2214yescomplemented
39C25×C29D411yescharacteristic
40C25×C29C2311yescharacteristic
41C29×H(5,𝔽2)—24no—
42C29×H(5,𝔽2)—42no—
43C2×#41×C29—21no—
44C2×#41×C29—22no—
45C22×D4×C29—21no—
46C22×D4×C29—22no—
47C22×D4×C29—24no—
48C22×D4×C29—22no—
49C22×D4×C29—24no—
50C22×D4×C29—42no—
51C22×D4×C29D411yescharacteristic
52C22×D4×C29D411yescharacteristic
53C22×D4×C29D414yes—
54C22×D4×C29C2312yes—
55C22×D4×C29C2311yescharacteristic
56C22×D4×C29C2311yescharacteristic
57C23×C4×C29—21no—
58C23×C4×C29—21no—
59C23×C4×C29—21no—
60C23×C4×C29—24no—
61C23×C4×C29—24no—
62C23×C4×C29—42no—
63C23×C4×C29D411yescharacteristic
64C23×C4×C29C2312yes—
65C29×#127—22no—
66C29×#124—22no—
67C29×#123—28no—
68C29×#123—28no—
69C29×#123—22no—
70C29×#123—24no—
71C29×#123—22no—
72C29×#123—22no—
73C29×#123—24no—
74C29×#123—24no—
75C29×#123—28no—
76C29×#123—44no—
77C29×#123D418yescomplemented
78C29×C22⋊Q8—22no—
79C29×#121—22no—
80C29×#121—24no—
81C29×#121—24no—
82C29×#121—24no—
83C29×#120—24no—
84C29×#120—22no—
85C29×#120—22no—
86C29×#120—22no—
87C29×#120—22no—
88C29×#120—24no—
89C29×#120—22no—
90C4×D4×C29—24no—
91C4×D4×C29—24no—
92C29×#117—21no—
93C2×C4⋊C4×C29—28no—
94C2×C4⋊C4×C29—22no—
95C2×C4⋊C4×C29—22no—
96C2×C4⋊C4×C29—24no—
97C2×C22⋊C4×C29—28no—
98C2×C22⋊C4×C29—22no—
99C2×C22⋊C4×C29—22no—
100C2×C22⋊C4×C29—22no—
101C2×C22⋊C4×C29—21no—
102C2×C22⋊C4×C29—21no—
103C2×C22⋊C4×C29—24no—
104C2×C22⋊C4×C29—24no—
105C2×C22⋊C4×C29—24no—
106C2×C22⋊C4×C29D414yes—
107C2×C22⋊C4×C29C2314yes—
108C2×C22⋊C4×C29C2312yes—
109C2×C22⋊C4×C29C2312yes—
110C29×#99—24no—
111C29×#99—28no—
112C29×#99—42no—
113C29×#95—28no—
114C24×C29—22no—
115C24×C29—22no—
116C24×C29—21no—
117C24×C29—22no—
118C24×C29—22no—
119C24×C29—44no—
120C24×C29—41no—
121C24×C29—44no—
122C24×C29C2×D411yescharacteristic
123C24×C29C2×D414yes—
124C24×C29C2×D411yescharacteristic
125C24×C29C2×D412yes—
126C24×C29C2×D411yescharacteristic
127C24×C29C2×D411yescharacteristic
128C24×C29C2411yescharacteristic
129#41×C29—48no—
130#41×C29—42no—
131#41×C29—42no—
132C2×Q8×C29—21no—
133C2×D4×C29—22no—
134C2×D4×C29—22no—
135C2×D4×C29—22no—
136C2×D4×C29—21no—
137C2×D4×C29—24no—
138C2×D4×C29—22no—
139C2×D4×C29—22no—
140C2×D4×C29—22no—
141C2×D4×C29—22no—
142C2×D4×C29—22no—
143C2×D4×C29—22no—
144C2×D4×C29—22no—
145C2×D4×C29—24no—
146C2×D4×C29—24no—
147C2×D4×C29—22no—
148C2×D4×C29—24no—
149C2×D4×C29—24no—
150C2×D4×C29—22no—
151C2×D4×C29—41no—
152C2×D4×C29—41no—
153C2×D4×C29—416no—
154C2×D4×C29—48no—
155C2×D4×C29—42no—
156C2×D4×C29—42no—
157C2×D4×C29—48no—
158C2×D4×C29—84no—
159C2×D4×C29C2×D414yes—
160C22×C4×C29—24no—
161C22×C4×C29—24no—
162C22×C4×C29—21no—
163C22×C4×C29—22no—
164C22×C4×C29—24no—
165C22×C4×C29—22no—
166C22×C4×C29—22no—
167C22×C4×C29—22no—
168C22×C4×C29—24no—
169C22×C4×C29—22no—
170C22×C4×C29—24no—
171C22×C4×C29—22no—
172C22×C4×C29—22no—
173C22×C4×C29—24no—
174C22×C4×C29—41no—
175C22×C4×C29—416no—
176C22×C4×C29—44no—
177C22×C4×C29—42no—
178C22×C4×C29—42no—
179C22×C4×C29—42no—
180C22×C4×C29—42no—
181C22×C4×C29C2×D412yes—
182C22×C4×C29C2×D411yescharacteristic
183C22×C4×C29C2×D411yescharacteristic
184C22×C4×C29C2×D414yes—
185C22×C4×C29C2×D414yescomplemented
186C4⋊C4×C29—28no—
187C4⋊C4×C29—22no—
188C4⋊C4×C29—24no—
189C4⋊C4×C29—28no—
190C4⋊C4×C29—48no—
191C22⋊C4×C29—24no—
192C22⋊C4×C29—28no—
193C22⋊C4×C29—22no—
194C22⋊C4×C29—24no—
195C22⋊C4×C29—24no—
196C22⋊C4×C29—28no—
197C22⋊C4×C29—24no—
198C22⋊C4×C29—24no—
199C22⋊C4×C29—22no—
200C22⋊C4×C29—22no—
201C22⋊C4×C29—22no—
202C22⋊C4×C29—24no—
203C22⋊C4×C29—24no—
204C22⋊C4×C29—48no—
205C22⋊C4×C29—48no—
206C22⋊C4×C29C2×D414yescomplemented
207C42×C29—22no—
208#23901C2911yesmaximal, characteristic, direct product factor, Sylow
209C23×C29—21no—
210C23×C29—21no—
211C23×C29—24no—
212C23×C29—21no—
213C23×C29—21no—
214C23×C29—21no—
215C23×C29—28no—
216C23×C29—28no—
217C23×C29—21no—
218C23×C29—21no—
219C23×C29—21no—
220C23×C29—22no—
221C23×C29—21no—
222C23×C29—24no—
223C23×C29—44no—
224C23×C29—42no—
225C23×C29—42no—
226C23×C29—44no—
227C23×C29—48no—
228C23×C29—48no—
229C23×C29—42no—
230C23×C29—42no—
231C23×C29—48no—
232C23×C29—42no—
233C23×C29—44no—
234C23×C29—44no—
235C23×C29—82no—
236C23×C29—84no—
237C23×C29—82no—
238C23×C29#12012yes—
239C23×C29#12012yes—
240C23×C29#12012yes—
241C23×C29C22×D411yescharacteristic
242C23×C29C22×D412yes—
243C23×C29C22×D412yes—
244C23×C29H(5,𝔽2)12yes—
245Q8×C29—42no—
246D4×C29—28no—
247D4×C29—44no—
248D4×C29—42no—
249D4×C29—44no—
250D4×C29—44no—
251D4×C29—42no—
252D4×C29—42no—
253D4×C29—44no—
254D4×C29—44no—
255D4×C29—48no—
256D4×C29—84no—
257D4×C29—84no—
258C2×C4×C29—21no—
259C2×C4×C29—21no—
260C2×C4×C29—24no—
261C2×C4×C29—24no—
262C2×C4×C29—21no—
263C2×C4×C29—24no—
264C2×C4×C29—22no—
265C2×C4×C29—21no—
266C2×C4×C29—21no—
267C2×C4×C29—28no—
268C2×C4×C29—24no—
269C2×C4×C29—44no—
270C2×C4×C29—44no—
271C2×C4×C29—48no—
272C2×C4×C29—44no—
273C2×C4×C29—48no—
274C2×C4×C29—48no—
275C2×C4×C29—42no—
276C2×C4×C29—42no—
277C2×C4×C29—42no—
278C2×C4×C29—44no—
279C2×C4×C29—44no—
280C2×C4×C29—48no—
281C2×C4×C29—44no—
282C2×C4×C29—82no—
283C2×C4×C29#12012yes—
284#3484C2×C2911yescharacteristic
285C22×#123C2×C2911yescharacteristic, complemented
286C2×#458C2×C2912yescomplemented
287#2849C2×C2912yescomplemented
288#2022C2×C2914yescomplemented
289#1898C2×C2912yescomplemented
290#1890C2×C2912yescomplemented
291#1781C2×C2911yescharacteristic, complemented
292C22×C29—24no—
293C22×C29—24no—
294C22×C29—22no—
295C22×C29—24no—
296C22×C29—22no—
297C22×C29—22no—
298C22×C29—22no—
299C22×C29—22no—
300C22×C29—24no—
301C22×C29—42no—
302C22×C29—42no—
303C22×C29—416no—
304C22×C29—42no—
305C22×C29—42no—
306C22×C29—42no—
307C22×C29—42no—
308C22×C29—42no—
309C22×C29—42no—
310C22×C29—41no—
311C22×C29—44no—
312C22×C29—42no—
313C22×C29—81no—
314C22×C29—88no—
315C22×C29—84no—
316C22×C29—81no—
317C22×C29—162no—
318C22×C29#45814yescomplemented
319C22×C29C2×#12011yescharacteristic
320C22×C29C2×#12011yescharacteristic
321C22×C29#53411yescharacteristic
322C22×C29D4214yes—
323C4×C29—22no—
324C4×C29—44no—
325C4×C29—44no—
326C4×C29—41no—
327C4×C29—42no—
328C4×C29—44no—
329C4×C29—84no—
330C2×H(5,𝔽2)C22×C2911yescharacteristic
331C23×D4C22×C2911yescharacteristic
332C24×C4C22×C2911yescharacteristic, complemented
333#546—24no—
334D42—24no—
335#535—21no—
336#534—21no—
337#534C22×C2912yes—
338#525—21no—
339C2×#123—21no—
340C2×#123—21no—
341C2×#123—28no—
342C2×#123C22×C2912yes—
343C2×#123C22×C2912yescomplemented
344C2×#123C22×C2914yescomplemented
345C2×#120—21no—
346C2×#120C22×C2912yes—
347#518C22×C2911yescharacteristic
348C22×C4⋊C4—21no—
349C22×C22⋊C4—21no—
350C22×C22⋊C4C22×C2911yescharacteristic, complemented
351#458—28no—
352C2×#99C22×C2912yes—
353C2×#99C22×C2914yescomplemented
354C2×#99C22×C2912yescomplemented
355#386—24no—
356#385—24no—
357#381C22×C2914yescomplemented
358#379C22×C2912yescomplemented
359#342C22×C2914yescomplemented
360C2×C29—21no—
361C2×C29—24no—
362C2×C29—21no—
363C2×C29—24no—
364C2×C29—42no—
365C2×C29—42no—
366C2×C29—81no—
367C2×C29—82no—
368C2×C29#240311yescharacteristic
369C2×C29C2×#45812yes—
370C25D4×C2911yescharacteristic
371C25C23×C2911yescharacteristic
372H(5,𝔽2)—24no—
373H(5,𝔽2)—42no—
374C2×#41—21no—
375C2×#41—22no—
376C22×D4—21no—
377C22×D4—22no—
378C22×D4—24no—
379C22×D4—22no—
380C22×D4—24no—
381C22×D4—42no—
382C22×D4D4×C2911yescharacteristic
383C22×D4D4×C2911yescharacteristic
384C22×D4D4×C2914yes—
385C22×D4C23×C2912yes—
386C22×D4C23×C2911yescharacteristic
387C22×D4C23×C2911yescharacteristic
388C23×C4—21no—
389C23×C4—21no—
390C23×C4—21no—
391C23×C4—24no—
392C23×C4—24no—
393C23×C4—42no—
394C23×C4D4×C2911yescharacteristic
395C23×C4C23×C2912yes—
396#127—22no—
397#124—22no—
398#123—28no—
399#123—28no—
400#123—22no—
401#123—24no—
402#123—22no—
403#123—22no—
404#123—24no—
405#123—24no—
406#123—28no—
407#123—44no—
408#123D4×C2918yescomplemented
409C22⋊Q8—22no—
410#121—22no—
411#121—24no—
412#121—24no—
413#121—24no—
414#120—24no—
415#120—22no—
416#120—22no—
417#120—22no—
418#120—22no—
419#120—24no—
420#120—22no—
421C4×D4—24no—
422C4×D4—24no—
423#117—21no—
424C2×C4⋊C4—28no—
425C2×C4⋊C4—22no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—24no—
428C2×C22⋊C4—28no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—24no—
435C2×C22⋊C4—24no—
436C2×C22⋊C4—24no—
437C2×C22⋊C4D4×C2914yes—
438C2×C22⋊C4C23×C2914yes—
439C2×C22⋊C4C23×C2912yes—
440C2×C22⋊C4C23×C2912yes—
441#99—24no—
442#99—28no—
443#99—42no—
444#95—28no—
445C29#2390111yescharacteristic, direct product factor, Sylow
446C24—22no—
447C24—22no—
448C24—21no—
449C24—22no—
450C24—22no—
451C24—44no—
452C24—41no—
453C24—44no—
454C24C2×D4×C2911yescharacteristic
455C24C2×D4×C2914yes—
456C24C2×D4×C2911yescharacteristic
457C24C2×D4×C2912yes—
458C24C2×D4×C2911yescharacteristic
459C24C2×D4×C2911yescharacteristic
460C24C24×C2911yescharacteristic
461#41—48no—
462#41—42no—
463#41—42no—
464C2×Q8—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—24no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—24no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—24no—
482C2×D4—22no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—416no—
486C2×D4—48no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—48no—
490C2×D4—84no—
491C2×D4C2×D4×C2914yes—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—24no—
506C22×C4—41no—
507C22×C4—416no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—44no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4C2×D4×C2912yes—
514C22×C4C2×D4×C2911yescharacteristic
515C22×C4C2×D4×C2911yescharacteristic
516C22×C4C2×D4×C2914yes—
517C22×C4C2×D4×C2914yescomplemented
518C4⋊C4—28no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—28no—
522C4⋊C4—48no—
523C22⋊C4—24no—
524C22⋊C4—28no—
525C22⋊C4—22no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—28no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4C2×D4×C2914yescomplemented
539C42—22no—
540C23—24no—
541C23—21no—
542C23—21no—
543C23—28no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—28no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—21no—
553C23—24no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—48no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—48no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—82no—
567C23—84no—
568C23—82no—
569C23C29×#12012yes—
570C23C29×#12012yes—
571C23C29×#12012yes—
572C23C22×D4×C2911yescharacteristic
573C23C22×D4×C2912yes—
574C23C22×D4×C2912yes—
575C23C29×H(5,𝔽2)12yes—
576Q8—42no—
577D4—28no—
578D4—44no—
579D4—42no—
580D4—44no—
581D4—44no—
582D4—48no—
583D4—42no—
584D4—42no—
585D4—44no—
586D4—44no—
587D4—84no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—82no—
614C2×C4C29×#12012yes—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—24no—
622C22—22no—
623C22—22no—
624C22—42no—
625C22—42no—
626C22—416no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—41no—
636C22—81no—
637C22—88no—
638C22—81no—
639C22—84no—
640C22—162no—
641C22C29×#45814yescomplemented
642C22C2×C29×#12011yescharacteristic
643C22C2×C29×#12011yescharacteristic
644C22C29×#53411yescharacteristic
645C22D42×C2914yes—
646C4—22no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—41no—
651C4—42no—
652C4—84no—
653C2—21no—
654C2—24no—
655C2—24no—
656C2—21no—
657C2—42no—
658C2—42no—
659C2—81no—
660C2—82no—
661C2C29×#240311yescharacteristic
662C2C2×C29×#45812yes—
subgroupG/H
center, socleC22×C29C2×#120
commutator, Frattini, cosocleC24C24×C29
prime radicalC29×#3484C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C29 ◁ C25×C29 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C29×#123C2semidirect
3. C2×C29×#458C2semidirect
4. C29×#2849C2semidirect
5. C29×#2022C2semidirect
6. C29×#1898C2semidirect
7. C29×#1890C2semidirect
8. C29×#1781C2semidirect
11. C24×C4×C29C22semidirect
22. C2×C29×#123C22semidirect
23. C2×C29×#123C22semidirect
29. C22×C22⋊C4×C29C22semidirect
32. C2×C29×#99C22semidirect
33. C2×C29×#99C22semidirect
36. C29×#381C22semidirect
37. C29×#379C22semidirect
38. C29×#342C22semidirect
77. C29×#123D4semidirect
185. C22×C4×C29C2×D4semidirect
206. C22⋊C4×C29C2×D4semidirect
208. #23901C29direct (N × H)
285. C22×#123C2×C29semidirect
286. C2×#458C2×C29semidirect
287. #2849C2×C29semidirect
288. #2022C2×C29semidirect
289. #1898C2×C29semidirect
290. #1890C2×C29semidirect
291. #1781C2×C29semidirect
318. C22×C29#458semidirect
332. C24×C4C22×C29semidirect
343. C2×#123C22×C29semidirect
344. C2×#123C22×C29semidirect
350. C22×C22⋊C4C22×C29semidirect
353. C2×#99C22×C29semidirect
354. C2×#99C22×C29semidirect
357. #381C22×C29semidirect
358. #379C22×C29semidirect
359. #342C22×C29semidirect
408. #123D4×C29semidirect
445. C29#23901direct (N × H)
517. C22×C4C2×D4×C29semidirect
538. C22⋊C4C2×D4×C29semidirect
641. C22C29×#458semidirect

Maximal subgroups

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

subgroupindex
C29×#34842
C22×C29×#1232
C2×C29×#4582
C29×#28492
C29×#20222
C29×#18982
C29×#18902
C29×#17812
#2390129