Finite Group Databasebeta

← C29×#18337 #20313237

Subgroups — C29×#18337 #20313237

Order 7424.

1016 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×#3461C211yesmaximal, characteristic, complemented
2C29×#3217C212yesmaximal, complemented
3C29×#2548C211yesmaximal, characteristic, complemented
4C29×#1599C212yesmaximal, complemented
5C29×#1598C212yesmaximal, complemented
6C29×#1597C214yesmaximal, complemented
7C29×#1509C212yesmaximal, complemented
8C29×#1467C211yesmaximal, characteristic, complemented
9C29×#544C2211yescharacteristic
10C29×#542—22no—
11C29×#541—22no—
12C29×#539—21no—
13C29×#539C2212yes—
14C29×#537C2211yescharacteristic
15C29×#536—21no—
16C29×#533—22no—
17C2×C29×#127C2211yescharacteristic
18C29×#529—22no—
19C29×#529—21no—
20C29×#529C2212yes—
21C2×C29×#124—21no—
22C4×#41×C29—21no—
23C2×C4×Q8×C29C2211yescharacteristic
24C29×#496—22no—
25C29×#490—22no—
26C29×#460—22no—
27C29×#459—22no—
28C29×#450C2214yes—
29C29×#447C2214yes—
30C29×#432C2212yes—
31C29×#394C2212yes—
32C29×#394C2212yes—
33C29×#390—21no—
34C29×#390—21no—
35C29×#390—21no—
36C4×C4⋊C4×C29C2211yescharacteristic
37C29×#328C2214yes—
38C29×#327C2214yes—
39C29×#324C2212yes—
40C29×#324C2212yes—
41C2×#41×C29—21no—
42C22×Q8×C29D411yescharacteristic
43C22×D4×C29—21no—
44C22×D4×C29—21no—
45C22×D4×C29C2312yes—
46C2×SD16×C29—24no—
47C2×D8×C29—24no—
48C29×C4⋊Q8—22no—
49C29×C4⋊Q8—22no—
50C29×#127—22no—
51C29×#127—22no—
52C29×#126—22no—
53C29×#126—22no—
54C29×#126—22no—
55C29×#126—22no—
56C29×#125—21no—
57C29×#125—22no—
58C29×#125—22no—
59C29×#125—22no—
60C29×#125D412yes—
61C29×#124—22no—
62C29×#124—21no—
63C29×#124—44no—
64C29×#124D412yes—
65C29×#123—22no—
66C29×#123—22no—
67C29×#123—22no—
68C29×#123—22no—
69C29×C22⋊Q8—22no—
70C29×C22⋊Q8—22no—
71C29×C22⋊Q8—22no—
72C29×C22⋊Q8—22no—
73C29×C22⋊Q8—21no—
74C29×C22⋊Q8—22no—
75C29×C22⋊Q8C2312yes—
76C29×#121—22no—
77C29×#121—21no—
78C29×#121C2312yes—
79C4×Q8×C29—21no—
80C4×Q8×C29—21no—
81C4×Q8×C29—21no—
82C4×Q8×C29—22no—
83C4×Q8×C29—22no—
84C4×Q8×C29D412yes—
85C4×D4×C29—21no—
86C4×D4×C29—21no—
87C4×D4×C29—21no—
88C4×D4×C29—22no—
89C4×D4×C29—22no—
90C4×D4×C29D412yes—
91C29×#117—22no—
92C29×#117—22no—
93C29×#117—21no—
94C29×#117—22no—
95C2×C4⋊C4×C29—21no—
96C2×C4⋊C4×C29D411yescharacteristic
97C2×C4⋊C4×C29C2311yescharacteristic
98C2×C4⋊C4×C29C2311yescharacteristic
99C2×C22⋊C4×C29—22no—
100C2×C22⋊C4×C29—22no—
101C2×C22⋊C4×C29—22no—
102C2×C22⋊C4×C29—22no—
103C2×C42×C29—22no—
104C2×C42×C29—21no—
105C2×C42×C29D411yescharacteristic
106C2×C42×C29D411yescharacteristic
107C2×C42×C29C2311yescharacteristic
108C29×C4⋊C8—22no—
109C29×#102—24no—
110C29×#102—24no—
111C29×C22⋊C8C2314yes—
112C29×#97—22no—
113C29×#95C2312yes—
114C24×C29—22no—
115#41×C29—42no—
116#41×C29—42no—
117C2×Q8×C29—21no—
118C2×Q8×C29—21no—
119C2×Q8×C29—22no—
120C2×Q8×C29—22no—
121C2×Q8×C29—41no—
122C2×Q8×C29C2×D412yes—
123C2×D4×C29—21no—
124C2×D4×C29—22no—
125C2×D4×C29—21no—
126C2×D4×C29—24no—
127C2×D4×C29—42no—
128C2×D4×C29—42no—
129C2×D4×C29—42no—
130C2×D4×C29—42no—
131C2×D4×C29—41no—
132C2×D4×C29—42no—
133C2×D4×C29—41no—
134C2×D4×C29—82no—
135C2×D4×C29C2×D412yes—
136C22×C4×C29—22no—
137C22×C4×C29—22no—
138C22×C4×C29—22no—
139C22×C4×C29—22no—
140C22×C4×C29C2×D411yescharacteristic
141C22×C4×C29C2×D411yescharacteristic
142C22×C4×C29C2×D411yescharacteristic
143C22×C4×C29C2×D411yescharacteristic
144C22×C4×C29C2×D411yescharacteristic
145C22×C4×C29C2×D411yescharacteristic
146C22×C4×C29C2411yescharacteristic
147SD16×C29—84no—
148D8×C29—84no—
149C2×C8×C29—24no—
150C4⋊C4×C29—21no—
151C4⋊C4×C29—22no—
152C4⋊C4×C29—21no—
153C4⋊C4×C29—22no—
154C4⋊C4×C29—22no—
155C4⋊C4×C29—21no—
156C4⋊C4×C29—21no—
157C4⋊C4×C29—22no—
158C4⋊C4×C29—22no—
159C4⋊C4×C29—22no—
160C4⋊C4×C29—22no—
161C4⋊C4×C29—22no—
162C4⋊C4×C29—22no—
163C4⋊C4×C29—22no—
164C4⋊C4×C29—22no—
165C4⋊C4×C29—42no—
166C4⋊C4×C29C2×D412yes—
167C4⋊C4×C29C2×D412yes—
168C22⋊C4×C29—22no—
169C22⋊C4×C29—22no—
170C22⋊C4×C29—22no—
171C22⋊C4×C29—22no—
172C22⋊C4×C29—22no—
173C22⋊C4×C29—22no—
174C22⋊C4×C29—22no—
175C22⋊C4×C29—22no—
176C22⋊C4×C29—22no—
177C22⋊C4×C29—44no—
178C22⋊C4×C29—44no—
179C22⋊C4×C29—82no—
180C22⋊C4×C29—82no—
181C42×C29—22no—
182C42×C29—22no—
183C42×C29—21no—
184C42×C29—22no—
185C42×C29—21no—
186C42×C29—21no—
187C42×C29—21no—
188C42×C29—42no—
189C42×C29C2×D412yes—
190C42×C29C2×D412yes—
191#18337C2911yesmaximal, characteristic, direct product factor, Sylow
192C23×C29—22no—
193C23×C29—42no—
194C23×C29—42no—
195C23×C29—42no—
196C23×C29—42no—
197C23×C29—42no—
198C23×C29—42no—
199C23×C29—42no—
200C23×C29C22×D411yescharacteristic
201Q8×C29—44no—
202Q8×C29—42no—
203Q8×C29—42no—
204Q8×C29—81no—
205D4×C29—42no—
206D4×C29—42no—
207D4×C29—44no—
208D4×C29—44no—
209D4×C29—42no—
210D4×C29—42no—
211D4×C29—84no—
212D4×C29—82no—
213D4×C29—81no—
214D4×C29—82no—
215C2×C4×C29—21no—
216C2×C4×C29—21no—
217C2×C4×C29—21no—
218C2×C4×C29—21no—
219C2×C4×C29—22no—
220C2×C4×C29—21no—
221C2×C4×C29—22no—
222C2×C4×C29—22no—
223C2×C4×C29—21no—
224C2×C4×C29—21no—
225C2×C4×C29—22no—
226C2×C4×C29—21no—
227C2×C4×C29—21no—
228C2×C4×C29—41no—
229C2×C4×C29—44no—
230C2×C4×C29—44no—
231C2×C4×C29—41no—
232C2×C4×C29—41no—
233C2×C4×C29—42no—
234C2×C4×C29—42no—
235C2×C4×C29—42no—
236C2×C4×C29—82no—
237C2×C4×C29#12011yescharacteristic
238C2×C4×C29#12012yes—
239C2×C4×C29#12011yescharacteristic
240C2×C4×C29AGL(1,ℤ/8)12yes—
241C2×C4×C29C22×D412yes—
242C8×C29—44no—
243#3461C2×C2911yescharacteristic, complemented
244#3217C2×C2912yescomplemented
245#2548C2×C2911yescharacteristic, complemented
246#1599C2×C2912yescomplemented
247#1598C2×C2912yescomplemented
248#1597C2×C2914yescomplemented
249#1509C2×C2912yescomplemented
250#1467C2×C2911yescharacteristic, complemented
251C22×C29—21no—
252C22×C29—21no—
253C22×C29—42no—
254C22×C29—42no—
255C22×C29—42no—
256C22×C29—82no—
257C22×C29—82no—
258C22×C29—82no—
259C22×C29—82no—
260C22×C29—82no—
261C22×C29—82no—
262C22×C29—82no—
263C22×C29C2×#12011yescharacteristic
264C22×C29C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C29#57511yescharacteristic
266C4×C29—22no—
267C4×C29—22no—
268C4×C29—21no—
269C4×C29—22no—
270C4×C29—41no—
271C4×C29—42no—
272C4×C29—42no—
273C4×C29—41no—
274C4×C29—41no—
275C4×C29—42no—
276C4×C29—82no—
277C4×C29#45412yes—
278#544C22×C2911yescharacteristic
279#542—22no—
280#541—22no—
281#539—21no—
282#539C22×C2912yes—
283#537C22×C2911yescharacteristic
284#536—21no—
285#533—22no—
286C2×#127C22×C2911yescharacteristic
287#529—22no—
288#529—21no—
289#529C22×C2912yes—
290C2×#124—21no—
291C4×#41—21no—
292C2×C4×Q8C22×C2911yescharacteristic
293#496—22no—
294#490—22no—
295#460—22no—
296#459—22no—
297#450C22×C2914yes—
298#447C22×C2914yes—
299#432C22×C2912yes—
300#394C22×C2912yes—
301#394C22×C2912yes—
302#390—21no—
303#390—21no—
304#390—21no—
305C4×C4⋊C4C22×C2911yescharacteristic
306#328C22×C2914yes—
307#327C22×C2914yes—
308#324C22×C2912yes—
309#324C22×C2912yes—
310C2×C29—21no—
311C2×C29—21no—
312C2×C29—82no—
313C2×C29—162no—
314C2×C29#300611yescharacteristic
315C2×C29C2×#45411yescharacteristic
316C2×C29#302711yescharacteristic
317C2×#41—21no—
318C22×Q8D4×C2911yescharacteristic
319C22×D4—21no—
320C22×D4—21no—
321C22×D4C23×C2912yes—
322C2×SD16—24no—
323C2×D8—24no—
324C4⋊Q8—22no—
325C4⋊Q8—22no—
326#127—22no—
327#127—22no—
328#126—22no—
329#126—22no—
330#126—22no—
331#126—22no—
332#125—21no—
333#125—22no—
334#125—22no—
335#125—22no—
336#125D4×C2912yes—
337#124—22no—
338#124—21no—
339#124—44no—
340#124D4×C2912yes—
341#123—22no—
342#123—22no—
343#123—22no—
344#123—22no—
345C22⋊Q8—22no—
346C22⋊Q8—22no—
347C22⋊Q8—22no—
348C22⋊Q8—22no—
349C22⋊Q8—21no—
350C22⋊Q8—22no—
351C22⋊Q8C23×C2912yes—
352#121—22no—
353#121—21no—
354#121C23×C2912yes—
355C4×Q8—21no—
356C4×Q8—21no—
357C4×Q8—21no—
358C4×Q8—22no—
359C4×Q8—22no—
360C4×Q8D4×C2912yes—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4—21no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4D4×C2912yes—
367#117—22no—
368#117—22no—
369#117—21no—
370#117—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4D4×C2911yescharacteristic
373C2×C4⋊C4C23×C2911yescharacteristic
374C2×C4⋊C4C23×C2911yescharacteristic
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42D4×C2911yescharacteristic
382C2×C42D4×C2911yescharacteristic
383C2×C42C23×C2911yescharacteristic
384C4⋊C8—22no—
385#102—24no—
386#102—24no—
387C22⋊C8C23×C2914yes—
388#97—22no—
389#95C23×C2912yes—
390C29#1833711yescharacteristic, direct product factor, Sylow
391C24—22no—
392#41—42no—
393#41—42no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—41no—
399C2×Q8C2×D4×C2912yes—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—24no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—41no—
409C2×D4—42no—
410C2×D4—41no—
411C2×D4—82no—
412C2×D4C2×D4×C2912yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4C2×D4×C2911yescharacteristic
418C22×C4C2×D4×C2911yescharacteristic
419C22×C4C2×D4×C2911yescharacteristic
420C22×C4C2×D4×C2911yescharacteristic
421C22×C4C2×D4×C2911yescharacteristic
422C22×C4C2×D4×C2911yescharacteristic
423C22×C4C24×C2911yescharacteristic
424SD16—84no—
425D8—84no—
426C2×C8—24no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4C2×D4×C2912yes—
444C4⋊C4C2×D4×C2912yes—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C42—22no—
459C42—22no—
460C42—21no—
461C42—22no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—42no—
466C42C2×D4×C2912yes—
467C42C2×D4×C2912yes—
468C23—22no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23C22×D4×C2911yescharacteristic
477Q8—44no—
478Q8—42no—
479Q8—42no—
480Q8—81no—
481D4—42no—
482D4—42no—
483D4—44no—
484D4—44no—
485D4—42no—
486D4—42no—
487D4—84no—
488D4—82no—
489D4—81no—
490D4—82no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—82no—
513C2×C4C29×#12011yescharacteristic
514C2×C4C29×#12012yes—
515C2×C4C29×#12011yescharacteristic
516C2×C4C29×AGL(1,ℤ/8)12yes—
517C2×C4C22×D4×C2912yes—
518C8—44no—
519C22—21no—
520C22—21no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22C2×C29×#12011yescharacteristic
532C22C2×C29×AGL(1,ℤ/8)11yescharacteristic
533C22C29×#57511yescharacteristic
534C4—22no—
535C4—22no—
536C4—21no—
537C4—22no—
538C4—41no—
539C4—42no—
540C4—42no—
541C4—41no—
542C4—41no—
543C4—42no—
544C4—82no—
545C4C29×#45412yes—
546C2—21no—
547C2—21no—
548C2—82no—
549C2—162no—
550C2C29×#300611yescharacteristic
551C2C2×C29×#45411yescharacteristic
552C2C29×#302711yescharacteristic
subgroupG/H
center, socleC22×C29C2×#120
commutator, Frattini, cosocleC22×C4C24×C29
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C29 ◁ C2×C42×C29 ◁ G

Complements

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

NH ≅ G/Ntype
1. C29×#3461C2semidirect
2. C29×#3217C2semidirect
3. C29×#2548C2semidirect
4. C29×#1599C2semidirect
5. C29×#1598C2semidirect
6. C29×#1597C2semidirect
7. C29×#1509C2semidirect
8. C29×#1467C2semidirect
191. #18337C29direct (N × H)
243. #3461C2×C29semidirect
244. #3217C2×C29semidirect
245. #2548C2×C29semidirect
246. #1599C2×C29semidirect
247. #1598C2×C29semidirect
248. #1597C2×C29semidirect
249. #1509C2×C29semidirect
250. #1467C2×C29semidirect
390. C29#18337direct (N × H)

Maximal subgroups

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

subgroupindex
C29×#34612
C29×#32172
C29×#25482
C29×#15992
C29×#15982
C29×#15972
C29×#15092
C29×#14672
#1833729