Finite Group Databasebeta

← C29×#24353 #20319253

Subgroups — C29×#24353 #20319253

Order 7424.

1078 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C29×#3501C211yesmaximal, characteristic
2C29×#2627C212yesmaximal, complemented
3C29×#2322C211yesmaximal, characteristic
4C29×#2062C212yesmaximal, complemented
5C29×#2030C212yesmaximal, complemented
6C29×#1996C212yesmaximal, complemented
7C29×#1890C212yesmaximal
8C29×#1867C212yesmaximal
9C29×#1799C211yesmaximal, characteristic
10C29×#554—24no—
11C29×#554—22no—
12D4×Q8×C29—22no—
13C29×#544—21no—
14C29×#543—21no—
15C29×#543C2212yes—
16C29×#533—21no—
17C2×C29×#123C2211yescharacteristic
18C2×C29×C22⋊Q8—21no—
19C2×C29×C22⋊Q8C2212yes—
20C29×#518C2212yes—
21C2×C4×D4×C29C2211yescharacteristic
22C22×C4⋊C4×C29C2211yescharacteristic
23C2×C29×#99C2212yes—
24C29×#400—22no—
25C29×#398—24no—
26C29×#395—22no—
27C29×#389—21no—
28C29×#388—21no—
29C29×#387—21no—
30C29×#386C2211yescharacteristic
31C29×#385—21no—
32C29×#385C2211yescharacteristic
33C29×#385C2214yes—
34C29×#381C2211yescharacteristic
35C29×#381C2214yes—
36C29×#380C2212yes—
37C4×C22⋊C4×C29C2211yescharacteristic
38C29×#342C2214yes—
39C29×#342C2214yes—
40C29×#342C2212yes—
41C22×Q8×C29—21no—
42C22×Q8×C29D412yes—
43C22×D4×C29C2311yescharacteristic
44C23×C4×C29D411yescharacteristic
45C23×C4×C29C2311yescharacteristic
46C23×C4×C29C2311yescharacteristic
47C23×C4×C29C2311yescharacteristic
48C29×C4⋊Q8—21no—
49C29×C4⋊Q8—21no—
50C29×C4⋊Q8—24no—
51C29×#125—22no—
52C29×#125—24no—
53C29×#123—22no—
54C29×#123Q814yescomplemented
55C29×C22⋊Q8—22no—
56C29×C22⋊Q8—22no—
57C29×C22⋊Q8—24no—
58C29×C22⋊Q8—24no—
59C29×C22⋊Q8—22no—
60C29×C22⋊Q8—22no—
61C29×C22⋊Q8—22no—
62C29×C22⋊Q8—22no—
63C29×C22⋊Q8—22no—
64C29×C22⋊Q8—22no—
65C29×C22⋊Q8—24no—
66C29×C22⋊Q8—24no—
67C4×Q8×C29—22no—
68C4×Q8×C29—22no—
69C4×D4×C29—22no—
70C4×D4×C29—22no—
71C4×D4×C29—24no—
72C4×D4×C29—24no—
73C29×#117—22no—
74C2×C4⋊C4×C29—21no—
75C2×C4⋊C4×C29—22no—
76C2×C4⋊C4×C29—21no—
77C2×C4⋊C4×C29—21no—
78C2×C4⋊C4×C29—21no—
79C2×C4⋊C4×C29—22no—
80C2×C4⋊C4×C29—22no—
81C2×C4⋊C4×C29—22no—
82C2×C4⋊C4×C29—21no—
83C2×C4⋊C4×C29—24no—
84C2×C4⋊C4×C29D411yescharacteristic
85C2×C4⋊C4×C29D414yes—
86C2×C4⋊C4×C29D412yes—
87C2×C22⋊C4×C29—22no—
88C2×C22⋊C4×C29—21no—
89C2×C22⋊C4×C29—21no—
90C2×C22⋊C4×C29D411yescharacteristic
91C2×C22⋊C4×C29C2311yescharacteristic
92C2×C22⋊C4×C29C2314yes—
93C2×C22⋊C4×C29C2311yescharacteristic
94C2×C22⋊C4×C29C2314yes—
95C2×C22⋊C4×C29C2311yescharacteristic
96C2×C42×C29—21no—
97C2×C42×C29D411yescharacteristic
98C29×#99—24no—
99C29×#95—21no—
100C29×#95—22no—
101C29×#95—21no—
102C29×#95—21no—
103C29×#95—24no—
104C29×#95—24no—
105C29×#95—21no—
106C29×#95—24no—
107C29×#95—21no—
108C24×C29C2×D411yescharacteristic
109C24×C29C2411yescharacteristic
110C2×Q8×C29—22no—
111C2×Q8×C29—22no—
112C2×Q8×C29—22no—
113C2×Q8×C29—22no—
114C2×Q8×C29—82no—
115C2×Q8×C29—82no—
116C2×D4×C29—22no—
117C2×D4×C29—22no—
118C2×D4×C29C2×Q812yes—
119C2×D4×C29#4112yes—
120C22×C4×C29—21no—
121C22×C4×C29—21no—
122C22×C4×C29—22no—
123C22×C4×C29—21no—
124C22×C4×C29—22no—
125C22×C4×C29—22no—
126C22×C4×C29—22no—
127C22×C4×C29—21no—
128C22×C4×C29—21no—
129C22×C4×C29—21no—
130C22×C4×C29—21no—
131C22×C4×C29—24no—
132C22×C4×C29—22no—
133C22×C4×C29—81no—
134C22×C4×C29—81no—
135C22×C4×C29—81no—
136C22×C4×C29C2×D411yescharacteristic
137C22×C4×C29C2×D411yescharacteristic
138C22×C4×C29C2×D414yes—
139C22×C4×C29C2×D411yescharacteristic
140C22×C4×C29C2×D414yes—
141C22×C4×C29C2×D411yescharacteristic
142C22×C4×C29C2×D411yescharacteristic
143C22×C4×C29C2×D412yes—
144C22×C4×C29C2×D412yes—
145C22×C4×C29C2×Q812yescomplemented
146C4⋊C4×C29—22no—
147C4⋊C4×C29—24no—
148C4⋊C4×C29—22no—
149C4⋊C4×C29—22no—
150C4⋊C4×C29—24no—
151C4⋊C4×C29—22no—
152C4⋊C4×C29—24no—
153C4⋊C4×C29—22no—
154C4⋊C4×C29—22no—
155C4⋊C4×C29—22no—
156C4⋊C4×C29—24no—
157C4⋊C4×C29—24no—
158C4⋊C4×C29—41no—
159C4⋊C4×C29—41no—
160C4⋊C4×C29—42no—
161C4⋊C4×C29—82no—
162C4⋊C4×C29—81no—
163C4⋊C4×C29—84no—
164C22⋊C4×C29—24no—
165C22⋊C4×C29—24no—
166C22⋊C4×C29—24no—
167C22⋊C4×C29—22no—
168C22⋊C4×C29—24no—
169C22⋊C4×C29—21no—
170C22⋊C4×C29—22no—
171C22⋊C4×C29—24no—
172C22⋊C4×C29—22no—
173C22⋊C4×C29—24no—
174C22⋊C4×C29—81no—
175C22⋊C4×C29C2×Q812yescomplemented
176C42×C29—24no—
177C42×C29—21no—
178C42×C29—21no—
179C42×C29—81no—
180#24353C2911yesmaximal, characteristic, direct product factor, Sylow
181C23×C29—21no—
182C23×C29—21no—
183C23×C29—41no—
184C23×C29—41no—
185C23×C29—41no—
186C23×C29—41no—
187C23×C29C22⋊Q812yes—
188C23×C29C22×D411yescharacteristic
189C23×C29C22×D412yes—
190C23×C29C22×Q811yescharacteristic
191C23×C29C2×#4111yescharacteristic
192C23×C292−1+412yes—
193Q8×C29—44no—
194Q8×C29—82no—
195Q8×C29—82no—
196Q8×C29—82no—
197D4×C29—24no—
198D4×C29—24no—
199C2×C4×C29—21no—
200C2×C4×C29—21no—
201C2×C4×C29—21no—
202C2×C4×C29—24no—
203C2×C4×C29—24no—
204C2×C4×C29—21no—
205C2×C4×C29—22no—
206C2×C4×C29—22no—
207C2×C4×C29—24no—
208C2×C4×C29—41no—
209C2×C4×C29—42no—
210C2×C4×C29—41no—
211C2×C4×C29—42no—
212C2×C4×C29—42no—
213C2×C4×C29—41no—
214C2×C4×C29—41no—
215C2×C4×C29—41no—
216C2×C4×C29—41no—
217C2×C4×C29—42no—
218C2×C4×C29—41no—
219C2×C4×C29—44no—
220C2×C4×C29—44no—
221C2×C4×C29—41no—
222C2×C4×C29—41no—
223C2×C4×C29—41no—
224C2×C4×C29—44no—
225C2×C4×C29—44no—
226C2×C4×C29—41no—
227C2×C4×C29—44no—
228C2×C4×C29—41no—
229C2×C4×C29—41no—
230C2×C4×C29—41no—
231C2×C4×C29—44no—
232C2×C4×C29—81no—
233C2×C4×C29—82no—
234C2×C4×C29—82no—
235C2×C4×C29#12011yescharacteristic
236C2×C4×C29#12011yescharacteristic
237C2×C4×C29#12012yes—
238C2×C4×C29C22⋊Q812yes—
239#3501C2×C2911yescharacteristic
240#2627C2×C2912yescomplemented
241#2322C2×C2911yescharacteristic
242#2062C2×C2912yescomplemented
243#2030C2×C2912yescomplemented
244#1996C2×C2912yescomplemented
245#1890C2×C2912yes—
246#1867C2×C2912yes—
247#1799C2×C2911yescharacteristic
248C22×C29—22no—
249C22×C29—21no—
250C22×C29—22no—
251C22×C29—21no—
252C22×C29—22no—
253C22×C29—41no—
254C22×C29—41no—
255C22×C29—42no—
256C22×C29—42no—
257C22×C29—41no—
258C22×C29—41no—
259C22×C29—81no—
260C22×C29C2×#12011yescharacteristic
261C22×C29C2×C22⋊Q811yescharacteristic
262C22×C29#53611yescharacteristic
263C22×C29#54812yes—
264C22×C29D4×Q812yes—
265C4×C29—22no—
266C4×C29—21no—
267C4×C29—21no—
268C4×C29—22no—
269C4×C29—42no—
270C4×C29—42no—
271C4×C29—41no—
272C4×C29—44no—
273C4×C29—44no—
274C4×C29—41no—
275C4×C29—84no—
276C4×C29—81no—
277C4×C29—81no—
278#554—24no—
279#554—22no—
280D4×Q8—22no—
281#544—21no—
282#543—21no—
283#543C22×C2912yes—
284#533—21no—
285C2×#123C22×C2911yescharacteristic
286C2×C22⋊Q8—21no—
287C2×C22⋊Q8C22×C2912yes—
288#518C22×C2912yes—
289C2×C4×D4C22×C2911yescharacteristic
290C22×C4⋊C4C22×C2911yescharacteristic
291C2×#99C22×C2912yes—
292#400—22no—
293#398—24no—
294#395—22no—
295#389—21no—
296#388—21no—
297#387—21no—
298#386C22×C2911yescharacteristic
299#385—21no—
300#385C22×C2911yescharacteristic
301#385C22×C2914yes—
302#381C22×C2911yescharacteristic
303#381C22×C2914yes—
304#380C22×C2912yes—
305C4×C22⋊C4C22×C2911yescharacteristic
306#342C22×C2912yes—
307#342C22×C2914yes—
308#342C22×C2914yes—
309C2×C29—21no—
310C2×C29—22no—
311C2×C29—21no—
312C2×C29—22no—
313C2×C29—42no—
314C2×C29#240911yescharacteristic
315C2×C29#302611yescharacteristic
316C2×C29#302811yescharacteristic
317C22×Q8—21no—
318C22×Q8D4×C2912yes—
319C22×D4C23×C2911yescharacteristic
320C23×C4D4×C2911yescharacteristic
321C23×C4C23×C2911yescharacteristic
322C23×C4C23×C2911yescharacteristic
323C23×C4C23×C2911yescharacteristic
324C4⋊Q8—21no—
325C4⋊Q8—21no—
326C4⋊Q8—24no—
327#125—22no—
328#125—24no—
329#123—22no—
330#123Q8×C2914yescomplemented
331C22⋊Q8—22no—
332C22⋊Q8—22no—
333C22⋊Q8—24no—
334C22⋊Q8—24no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8—24no—
342C22⋊Q8—24no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×D4—22no—
346C4×D4—22no—
347C4×D4—24no—
348C4×D4—24no—
349#117—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4D4×C2911yescharacteristic
361C2×C4⋊C4D4×C2914yes—
362C2×C4⋊C4D4×C2912yes—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4D4×C2911yescharacteristic
367C2×C22⋊C4C23×C2911yescharacteristic
368C2×C22⋊C4C23×C2914yes—
369C2×C22⋊C4C23×C2911yescharacteristic
370C2×C22⋊C4C23×C2914yes—
371C2×C22⋊C4C23×C2911yescharacteristic
372C2×C42—21no—
373C2×C42D4×C2911yescharacteristic
374#99—24no—
375#95—21no—
376#95—22no—
377#95—21no—
378#95—21no—
379#95—24no—
380#95—24no—
381#95—21no—
382#95—24no—
383#95—21no—
384C29#2435311yescharacteristic, direct product factor, Sylow
385C24C2×D4×C2911yescharacteristic
386C24C24×C2911yescharacteristic
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—82no—
392C2×Q8—82no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4C2×Q8×C2912yes—
396C2×D4#41×C2912yes—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4C2×D4×C2911yescharacteristic
414C22×C4C2×D4×C2911yescharacteristic
415C22×C4C2×D4×C2914yes—
416C22×C4C2×D4×C2914yes—
417C22×C4C2×D4×C2911yescharacteristic
418C22×C4C2×D4×C2911yescharacteristic
419C22×C4C2×D4×C2912yes—
420C22×C4C2×D4×C2911yescharacteristic
421C22×C4C2×D4×C2912yes—
422C22×C4C2×Q8×C2912yescomplemented
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—42no—
438C4⋊C4—82no—
439C4⋊C4—81no—
440C4⋊C4—84no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—24no—
446C22⋊C4—21no—
447C22⋊C4—22no—
448C22⋊C4—24no—
449C22⋊C4—22no—
450C22⋊C4—24no—
451C22⋊C4—81no—
452C22⋊C4C2×Q8×C2912yescomplemented
453C42—24no—
454C42—21no—
455C42—21no—
456C42—81no—
457C23—21no—
458C23—21no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23C29×C22⋊Q812yes—
464C23C22×D4×C2911yescharacteristic
465C23C22×D4×C2912yes—
466C23C22×Q8×C2911yescharacteristic
467C23C2×#41×C2911yescharacteristic
468C23C29×2−1+412yes—
469Q8—44no—
470Q8—82no—
471Q8—82no—
472Q8—82no—
473D4—24no—
474D4—24no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—81no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4C29×#12011yescharacteristic
512C2×C4C29×#12011yescharacteristic
513C2×C4C29×#12012yes—
514C2×C4C29×C22⋊Q812yes—
515C22—22no—
516C22—21no—
517C22—22no—
518C22—21no—
519C22—22no—
520C22—41no—
521C22—41no—
522C22—42no—
523C22—42no—
524C22—41no—
525C22—41no—
526C22—81no—
527C22C2×C29×#12011yescharacteristic
528C22C2×C29×C22⋊Q811yescharacteristic
529C22C29×#53611yescharacteristic
530C22C29×#54812yes—
531C22D4×Q8×C2912yes—
532C4—21no—
533C4—21no—
534C4—22no—
535C4—22no—
536C4—42no—
537C4—41no—
538C4—44no—
539C4—44no—
540C4—41no—
541C4—42no—
542C4—84no—
543C4—81no—
544C4—81no—
545C2—21no—
546C2—22no—
547C2—21no—
548C2—22no—
549C2—42no—
550C2C29×#240911yescharacteristic
551C2C29×#302611yescharacteristic
552C2C29×#302811yescharacteristic
subgroupG/H
center, socleC22×C29C2×#120
commutator, Frattini, cosocleC24C24×C29
prime radicalC22×D4×C29C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C29 ◁ C23×C4×C29 ◁ G

Complements

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

NH ≅ G/Ntype
2. C29×#2627C2semidirect
4. C29×#2062C2semidirect
5. C29×#2030C2semidirect
6. C29×#1996C2semidirect
54. C29×#123Q8semidirect
145. C22×C4×C29C2×Q8semidirect
175. C22⋊C4×C29C2×Q8semidirect
180. #24353C29direct (N × H)
240. #2627C2×C29semidirect
242. #2062C2×C29semidirect
243. #2030C2×C29semidirect
244. #1996C2×C29semidirect
330. #123Q8×C29semidirect
384. C29#24353direct (N × H)
422. C22×C4C2×Q8×C29semidirect
452. C22⋊C4C2×Q8×C29semidirect

Maximal subgroups

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

subgroupindex
C29×#35012
C29×#26272
C29×#23222
C29×#20622
C29×#20302
C29×#19962
C29×#18902
C29×#18672
C29×#17992
#2435329