Finite Group Databasebeta

← C29×#12756 #20307656

Subgroups — C29×#12756 #20307656

Order 7424.

758 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C29×#3540C211yesmaximal, characteristic
2C29×#2135C211yesmaximal, characteristic, complemented
3C29×#2122C211yesmaximal, characteristic, complemented
4C29×#2120C211yesmaximal, characteristic, complemented
5C29×#1914C211yesmaximal, characteristic, complemented
6C29×#1909C211yesmaximal, characteristic, complemented
7C29×#1891C211yesmaximal, characteristic, complemented
8C29×#568C2211yescharacteristic
9C29×H(3,𝔽4)C412yescomplemented
10C29×#560—21no—
11C29×#559—21no—
12C29×#556—21no—
13C29×#555—21no—
14C29×#551C412yescomplemented
15C29×#548—21no—
16C29×#546—21no—
17C29×#542—22no—
18C29×#542—21no—
19C29×#538—22no—
20C29×#536C2211yescharacteristic
21C29×#535C2211yescharacteristic
22C29×#525—21no—
23C29×#525C2211yescharacteristic
24C29×#419—21no—
25C29×#413C2211yescharacteristic
26C29×#413C2211yescharacteristic
27C2×C29×#100—21no—
28C29×#410C2211yescharacteristic
29C29×#407—21no—
30C29×#356—22no—
31C29×#352—21no—
32C29×AΣL(1,𝔽16)—21no—
33C29×AΣL(1,𝔽16)—22no—
34C29×#344—21no—
35C29×#344—21no—
36C29×H(3,ℤ/4)—21no—
37C2×#41×C29—21no—
38C2×#41×C29C2311yescharacteristic
39C22×Q8×C29D411yescharacteristic
40C22×D4×C29D411yescharacteristic
41C23×C4×C29D411yescharacteristic
42C29×#131—42no—
43C2×M16×C29—21no—
44C2×M16×C29—21no—
45C2×M16×C29—21no—
46C2×M16×C29—21no—
47C2×M16×C29—21no—
48C29×C4⋊Q8—21no—
49C29×C4⋊Q8D412yes—
50C29×#127—21no—
51C29×#126—22no—
52C29×#126—22no—
53C29×#126—22no—
54C29×#124—22no—
55C29×#124—21no—
56C29×#124—22no—
57C29×#124—21no—
58C29×#124—22no—
59C29×#124—22no—
60C29×#124—22no—
61C29×#124C2×C412yes—
62C29×#123—22no—
63C29×#123—22no—
64C29×#123—21no—
65C29×#123—21no—
66C29×#123—22no—
67C29×#123C2×C411yescharacteristic
68C29×#123D411yescharacteristic
69C29×C22⋊Q8—21no—
70C29×C22⋊Q8—22no—
71C29×C22⋊Q8—22no—
72C29×C22⋊Q8—21no—
73C29×C22⋊Q8—21no—
74C29×C22⋊Q8—21no—
75C29×#121—21no—
76C29×#121—22no—
77C29×#121—22no—
78C29×#121—22no—
79C29×#121—21no—
80C29×#121—21no—
81C29×#121—21no—
82C29×#120—22no—
83C29×#120—22no—
84C29×#120C2×C411yescharacteristic
85C29×#120C2×C412yes—
86C29×#120D411yescharacteristic
87C29×#120D412yes—
88C4×Q8×C29—21no—
89C4×Q8×C29—22no—
90C4×D4×C29—22no—
91C4×D4×C29—22no—
92C4×D4×C29—21no—
93C4×D4×C29—22no—
94C4×D4×C29—21no—
95C4×D4×C29—21no—
96C29×#117—21no—
97C29×#117—22no—
98C29×#117—21no—
99C29×#117D411yescharacteristic
100C29×#117D411yescharacteristic
101C29×#117D411yescharacteristic
102C2×C22⋊C4×C29—22no—
103C29×#103—42no—
104C29×#101—21no—
105C29×#101—21no—
106C29×#100—21no—
107C29×#100—21no—
108C29×#100—42no—
109C29×#99—21no—
110C29×#99—21no—
111C29×C22⋊C8—42no—
112C24×C29C22⋊C412yes—
113C24×C29C22⋊C411yescharacteristic
114#41×C29—41no—
115#41×C29—41no—
116#41×C29—81no—
117#41×C29—81no—
118C2×Q8×C29—22no—
119C2×Q8×C29—21no—
120C2×Q8×C29—41no—
121C2×Q8×C29—41no—
122C2×Q8×C29C22⋊C412yes—
123C2×Q8×C29C2×D411yescharacteristic
124C2×D4×C29—22no—
125C2×D4×C29—22no—
126C2×D4×C29—21no—
127C2×D4×C29—22no—
128C2×D4×C29—21no—
129C2×D4×C29—22no—
130C2×D4×C29—21no—
131C2×D4×C29—21no—
132C2×D4×C29—41no—
133C2×D4×C29—41no—
134C2×D4×C29C22×C411yescharacteristic
135C2×D4×C29C2×D411yescharacteristic
136C2×D4×C29C2×D411yescharacteristic
137C22×C4×C29—22no—
138C22×C4×C29—22no—
139C22×C4×C29—21no—
140C22×C4×C29—21no—
141C22×C4×C29—21no—
142C22×C4×C29—41no—
143C22×C4×C29—41no—
144C22×C4×C29C22⋊C411yescharacteristic
145C22×C4×C29C2×D411yescharacteristic
146C22×C4×C29C2×D411yescharacteristic
147C22×C4×C29C2×D411yescharacteristic
148M16×C29—22no—
149M16×C29—41no—
150M16×C29—41no—
151M16×C29—41no—
152M16×C29—41no—
153M16×C29—41no—
154M16×C29—41no—
155M16×C29—41no—
156C2×C8×C29—22no—
157C2×C8×C29—22no—
158C2×C8×C29—41no—
159C2×C8×C29—41no—
160C2×C8×C29—41no—
161C4⋊C4×C29—22no—
162C4⋊C4×C29—22no—
163C4⋊C4×C29—22no—
164C4⋊C4×C29—21no—
165C4⋊C4×C29—21no—
166C4⋊C4×C29—21no—
167C4⋊C4×C29—22no—
168C4⋊C4×C29—22no—
169C4⋊C4×C29—21no—
170C4⋊C4×C29—21no—
171C4⋊C4×C29—41no—
172C22⋊C4×C29—22no—
173C22⋊C4×C29—22no—
174C22⋊C4×C29—22no—
175C22⋊C4×C29—22no—
176C22⋊C4×C29—22no—
177C22⋊C4×C29—22no—
178C22⋊C4×C29—21no—
179C22⋊C4×C29—22no—
180C22⋊C4×C29—22no—
181C22⋊C4×C29—21no—
182C22⋊C4×C29—21no—
183C22⋊C4×C29—22no—
184C22⋊C4×C29—21no—
185C22⋊C4×C29—21no—
186C22⋊C4×C29—22no—
187C22⋊C4×C29—41no—
188C22⋊C4×C29C22⋊C412yes—
189C22⋊C4×C29C22⋊C411yescharacteristic
190C22⋊C4×C29C22⋊C411yescharacteristic
191C42×C29—22no—
192C42×C29—22no—
193C42×C29—21no—
194C42×C29—21no—
195C42×C29—41no—
196C42×C29C22⋊C412yes—
197#12756C2911yesmaximal, characteristic, direct product factor, Sylow
198C23×C29—22no—
199C23×C29—21no—
200C23×C29—41no—
201C23×C29—42no—
202C23×C29—81no—
203C23×C29—82no—
204C23×C29C2×C22⋊C411yescharacteristic
205C23×C29C2×C22⋊C411yescharacteristic
206C23×C29#12011yescharacteristic
207Q8×C29—41no—
208Q8×C29—42no—
209Q8×C29—41no—
210Q8×C29—41no—
211Q8×C29—82no—
212D4×C29—41no—
213D4×C29—41no—
214D4×C29—41no—
215D4×C29—41no—
216D4×C29—41no—
217D4×C29—81no—
218D4×C29—81no—
219D4×C29—82no—
220D4×C29—82no—
221D4×C29—82no—
222D4×C29—81no—
223D4×C29—82no—
224C2×C4×C29—21no—
225C2×C4×C29—21no—
226C2×C4×C29—21no—
227C2×C4×C29—21no—
228C2×C4×C29—21no—
229C2×C4×C29—21no—
230C2×C4×C29—21no—
231C2×C4×C29—22no—
232C2×C4×C29—21no—
233C2×C4×C29—22no—
234C2×C4×C29—22no—
235C2×C4×C29—21no—
236C2×C4×C29—41no—
237C2×C4×C29—41no—
238C2×C4×C29—41no—
239C2×C4×C29—81no—
240C2×C4×C29—81no—
241C2×C4×C29—82no—
242C2×C4×C29—82no—
243C2×C4×C29—81no—
244C2×C4×C29C2×C22⋊C411yescharacteristic
245C2×C4×C29#12011yescharacteristic
246C2×C4×C29#12011yescharacteristic
247C2×C4×C29#12011yescharacteristic
248C8×C29—42no—
249C8×C29—81no—
250C8×C29—81no—
251#3540C2×C2911yescharacteristic
252#2135C2×C2911yescharacteristic, complemented
253#2122C2×C2911yescharacteristic, complemented
254#2120C2×C2911yescharacteristic, complemented
255#1914C2×C2911yescharacteristic, complemented
256#1909C2×C2911yescharacteristic, complemented
257#1891C2×C2911yescharacteristic, complemented
258C22×C29—21no—
259C22×C29—21no—
260C22×C29—21no—
261C22×C29—41no—
262C22×C29—41no—
263C22×C29—41no—
264C22×C29—42no—
265C22×C29—41no—
266C22×C29—41no—
267C22×C29—42no—
268C22×C29—42no—
269C22×C29—41no—
270C22×C29—42no—
271C22×C29—81no—
272C22×C29—162no—
273C22×C29#37911yescharacteristic
274C4×C29—21no—
275C4×C29—21no—
276C4×C29—21no—
277C4×C29—41no—
278C4×C29—42no—
279C4×C29—42no—
280C4×C29—41no—
281C4×C29—41no—
282C4×C29—42no—
283C4×C29—161no—
284C4×C29B(4,𝔽2)11yescharacteristic
285C4×C29#45811yescharacteristic
286#568C22×C2911yescharacteristic
287H(3,𝔽4)C4×C2912yescomplemented
288#560—21no—
289#559—21no—
290#556—21no—
291#555—21no—
292#551C4×C2912yescomplemented
293#548—21no—
294#546—21no—
295#542—22no—
296#542—21no—
297#538—22no—
298#536C22×C2911yescharacteristic
299#535C22×C2911yescharacteristic
300#525—21no—
301#525C22×C2911yescharacteristic
302#419—21no—
303#413C22×C2911yescharacteristic
304#413C22×C2911yescharacteristic
305C2×#100—21no—
306#410C22×C2911yescharacteristic
307#407—21no—
308#356—22no—
309#352—21no—
310AΣL(1,𝔽16)—21no—
311AΣL(1,𝔽16)—22no—
312#344—21no—
313#344—21no—
314H(3,ℤ/4)—21no—
315C2×C29—21no—
316C2×C29—41no—
317C2×C29—41no—
318C2×C29—41no—
319C2×C29—81no—
320C2×C29—82no—
321C2×C29#179011yescharacteristic
322C2×#41—21no—
323C2×#41C23×C2911yescharacteristic
324C22×Q8D4×C2911yescharacteristic
325C22×D4D4×C2911yescharacteristic
326C23×C4D4×C2911yescharacteristic
327#131—42no—
328C2×M16—21no—
329C2×M16—21no—
330C2×M16—21no—
331C2×M16—21no—
332C2×M16—21no—
333C4⋊Q8—21no—
334C4⋊Q8D4×C2912yes—
335#127—21no—
336#126—22no—
337#126—22no—
338#126—22no—
339#124—22no—
340#124—21no—
341#124—22no—
342#124—21no—
343#124—22no—
344#124—22no—
345#124—22no—
346#124C2×C4×C2912yes—
347#123—22no—
348#123—22no—
349#123—21no—
350#123—21no—
351#123—22no—
352#123C2×C4×C2911yescharacteristic
353#123D4×C2911yescharacteristic
354C22⋊Q8—21no—
355C22⋊Q8—22no—
356C22⋊Q8—22no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360#121—21no—
361#121—22no—
362#121—22no—
363#121—22no—
364#121—21no—
365#121—21no—
366#121—21no—
367#120—22no—
368#120—22no—
369#120C2×C4×C2911yescharacteristic
370#120C2×C4×C2912yes—
371#120D4×C2911yescharacteristic
372#120D4×C2912yes—
373C4×Q8—21no—
374C4×Q8—22no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—21no—
378C4×D4—22no—
379C4×D4—21no—
380C4×D4—21no—
381#117—21no—
382#117—22no—
383#117—21no—
384#117D4×C2911yescharacteristic
385#117D4×C2911yescharacteristic
386#117D4×C2911yescharacteristic
387C2×C22⋊C4—22no—
388#103—42no—
389#101—21no—
390#101—21no—
391#100—21no—
392#100—21no—
393#100—42no—
394#99—21no—
395#99—21no—
396C22⋊C8—42no—
397C29#1275611yescharacteristic, direct product factor, Sylow
398C24C22⋊C4×C2912yes—
399C24C22⋊C4×C2911yescharacteristic
400#41—41no—
401#41—41no—
402#41—81no—
403#41—81no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—41no—
407C2×Q8—41no—
408C2×Q8C22⋊C4×C2912yes—
409C2×Q8C2×D4×C2911yescharacteristic
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4C22×C4×C2911yescharacteristic
421C2×D4C2×D4×C2911yescharacteristic
422C2×D4C2×D4×C2911yescharacteristic
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4C22⋊C4×C2911yescharacteristic
431C22×C4C2×D4×C2911yescharacteristic
432C22×C4C2×D4×C2911yescharacteristic
433C22×C4C2×D4×C2911yescharacteristic
434M16—22no—
435M16—41no—
436M16—41no—
437M16—41no—
438M16—41no—
439M16—41no—
440M16—41no—
441M16—41no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—41no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—21no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—22no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—22no—
473C22⋊C4—41no—
474C22⋊C4C22⋊C4×C2912yes—
475C22⋊C4C22⋊C4×C2911yescharacteristic
476C22⋊C4C22⋊C4×C2911yescharacteristic
477C42—22no—
478C42—22no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42C22⋊C4×C2912yes—
483C23—22no—
484C23—21no—
485C23—41no—
486C23—42no—
487C23—81no—
488C23—82no—
489C23C2×C22⋊C4×C2911yescharacteristic
490C23C2×C22⋊C4×C2911yescharacteristic
491C23C29×#12011yescharacteristic
492Q8—41no—
493Q8—42no—
494Q8—41no—
495Q8—41no—
496Q8—82no—
497D4—41no—
498D4—41no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—81no—
503D4—82no—
504D4—81no—
505D4—82no—
506D4—82no—
507D4—82no—
508D4—81no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4C2×C22⋊C4×C2911yescharacteristic
530C2×C4C29×#12011yescharacteristic
531C2×C4C29×#12011yescharacteristic
532C2×C4C29×#12011yescharacteristic
533C8—42no—
534C8—81no—
535C8—81no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—41no—
544C22—41no—
545C22—42no—
546C22—42no—
547C22—41no—
548C22—42no—
549C22—81no—
550C22—162no—
551C22C29×#37911yescharacteristic
552C4—21no—
553C4—21no—
554C4—21no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—41no—
559C4—41no—
560C4—42no—
561C4—161no—
562C4C29×B(4,𝔽2)11yescharacteristic
563C4C29×#45811yescharacteristic
564C2—21no—
565C2—41no—
566C2—41no—
567C2—41no—
568C2—81no—
569C2—82no—
570C2C29×#179011yescharacteristic
subgroupG/H
center, socleC2×C29#1790
commutatorC2×D4C22×C4×C29
Frattini, cosocleC2×#41C23×C29
prime radicalC29×#3540C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C29 ◁ C2×C4×C29 ◁ C2×#41×C29 ◁ G

Complements

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

NH ≅ G/Ntype
2. C29×#2135C2semidirect
3. C29×#2122C2semidirect
4. C29×#2120C2semidirect
5. C29×#1914C2semidirect
6. C29×#1909C2semidirect
7. C29×#1891C2semidirect
9. C29×H(3,𝔽4)C4semidirect
14. C29×#551C4semidirect
197. #12756C29direct (N × H)
252. #2135C2×C29semidirect
253. #2122C2×C29semidirect
254. #2120C2×C29semidirect
255. #1914C2×C29semidirect
256. #1909C2×C29semidirect
257. #1891C2×C29semidirect
287. H(3,𝔽4)C4×C29semidirect
292. #551C4×C29semidirect
397. C29#12756direct (N × H)

Maximal subgroups

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

subgroupindex
C29×#35402
C29×#21352
C29×#21222
C29×#21202
C29×#19142
C29×#19092
C29×#18912
#1275629