Finite Group Databasebeta

← C29×#61404 #20356304

Subgroups — C29×#61404 #20356304

Order 7424.

1900 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C29×#3480C211yesmaximal, characteristic
2C2×C29×#557C211yesmaximal, characteristic, complemented
3C29×#3400C216yesmaximal, complemented
4C29×#3398C216yesmaximal, complemented
5C29×#3395C212yesmaximal, complemented
6C29×#3392C212yesmaximal, complemented
7C29×#3152C213yesmaximal, complemented
8C29×#3039C213yesmaximal, complemented
9C29×#3037C213yesmaximal, complemented
10Q8×M16×C29C211yesmaximal, characteristic, complemented
11C4×C29×#137C213yesmaximal, complemented
12C2×C29×#137C2213yes—
13C2×C29×#137C2216yes—
14C29×#559C2213yes—
15C29×#557—21no—
16C29×#557—23no—
17C29×#557C2212yes—
18C29×#557C2216yes—
19C29×#557C2211yescharacteristic
20C29×#556C2213yes—
21C29×#553—23no—
22C29×#550C2211yescharacteristic
23C29×#548—21no—
24C29×#532C2213yes—
25C2×C29×C4⋊Q8C2213yes—
26C29×#529—23no—
27C2×C29×#125—23no—
28C4×#41×C29—21no—
29C4×#41×C29C2213yes—
30C2×C4×Q8×C29—21no—
31C2×C4×Q8×C29C2213yes—
32C2×C4×Q8×C29C2211yescharacteristic
33C29×#497C22112yes—
34C29×#494C2216yes—
35C29×#492C2216yes—
36C29×#464C22112yes—
37C29×#462C2216yes—
38C29×#462C2216yes—
39C29×#462C2216yes—
40C29×#461C2216yes—
41C29×#461C2216yes—
42C29×#460C2216yes—
43C29×#446C2216yes—
44C8×Q8×C29C2212yes—
45C29×#441C2216yes—
46C29×#440C2216yes—
47C4×Q16×C29C2216yes—
48C4×SD16×C29C2216yes—
49C29×#428C2211yescharacteristic
50C29×#423C2213yes—
51C29×#419C2213yes—
52C29×#417C2213yes—
53C4×M16×C29C2213yes—
54C2×#41×C29—21no—
55C2×#41×C29C2313yes—
56C22×Q8×C29D411yescharacteristic
57C22×Q8×C29C2313yes—
58C29×#137—212no—
59C29×#137—412no—
60C2×Q16×C29C2316yes—
61C2×Q16×C29C23112yes—
62C2×SD16×C29C2316yes—
63C2×SD16×C29C23112yes—
64C2×M16×C29C2313yes—
65C2×M16×C29C2311yescharacteristic
66C29×C4⋊Q8—23no—
67C29×C4⋊Q8—23no—
68C29×C4⋊Q8C2316yes—
69C29×C4⋊Q8C2316yes—
70C29×C4⋊Q8C2313yes—
71C29×#127C2313yes—
72C29×#126—26no—
73C29×#125—26no—
74C29×#125—26no—
75C29×#125—212no—
76C29×#125—23no—
77C29×#124—23no—
78C29×#124C2316yes—
79C29×#123—26no—
80C29×C22⋊Q8—23no—
81C29×#121—23no—
82C29×#121—23no—
83C29×#121—23no—
84C4×Q8×C29—26no—
85C4×Q8×C29—22no—
86C4×Q8×C29—21no—
87C4×Q8×C29—23no—
88C4×Q8×C29—23no—
89C4×Q8×C29D411yescharacteristic
90C4×Q8×C29D413yes—
91C4×Q8×C29C2316yes—
92C4×Q8×C29C2313yes—
93C4×Q8×C29C2316yes—
94C4×Q8×C29C2313yes—
95C4×Q8×C29C2311yescharacteristic
96C4×Q8×C29C2313yes—
97C4×D4×C29—23no—
98C4×D4×C29—23no—
99C4×D4×C29—23no—
100C4×D4×C29—23no—
101C4×D4×C29—23no—
102C4×D4×C29C2313yes—
103C4×D4×C29C2313yes—
104C29×#117—23no—
105C29×#117—23no—
106C29×#117C2311yescharacteristic
107C2×C4⋊C4×C29—23no—
108C2×C4⋊C4×C29—23no—
109C2×C4⋊C4×C29—23no—
110C2×C4⋊C4×C29D413yes—
111C2×C4⋊C4×C29C2311yescharacteristic
112C2×C42×C29—23no—
113C2×C42×C29—23no—
114C2×C42×C29C2313yes—
115C29×#107C2312yes—
116C29×#106C2312yes—
117C29×C4⋊C8C2316yes—
118C29×C4⋊C8C2316yes—
119C29×#103C2316yes—
120C29×#103C2316yes—
121C29×#103C2316yes—
122C29×#102C2316yes—
123C29×#97C2316yes—
124C4×C8×C29C2316yes—
125#41×C29—26no—
126#41×C29—26no—
127#41×C29—44no—
128C2×Q8×C29—24no—
129C2×Q8×C29—23no—
130C2×Q8×C29—26no—
131C2×Q8×C29—26no—
132C2×Q8×C29C2×D413yes—
133C2×Q8×C29C2×D411yescharacteristic
134C2×Q8×C29C2413yes—
135C2×Q8×C29C2416yes—
136C2×D4×C29—23no—
137C2×D4×C29—23no—
138C2×D4×C29—23no—
139C2×D4×C29C2413yes—
140C22×C4×C29—21no—
141C22×C4×C29—23no—
142C22×C4×C29—23no—
143C22×C4×C29—23no—
144C22×C4×C29—23no—
145C22×C4×C29C2×D413yes—
146C22×C4×C29C2×D413yes—
147C22×C4×C29C2411yescharacteristic
148Q16×C29—212no—
149Q16×C29—412no—
150SD16×C29—212no—
151SD16×C29—412no—
152M16×C29—26no—
153M16×C29#4114yescomplemented
154C2×C8×C29C2416yes—
155C2×C8×C29C2412yes—
156C4⋊C4×C29—26no—
157C4⋊C4×C29—26no—
158C4⋊C4×C29—26no—
159C4⋊C4×C29—26no—
160C4⋊C4×C29—26no—
161C4⋊C4×C29—26no—
162C4⋊C4×C29—21no—
163C4⋊C4×C29—23no—
164C4⋊C4×C29—23no—
165C4⋊C4×C29—23no—
166C4⋊C4×C29—23no—
167C4⋊C4×C29C2×D413yes—
168C4⋊C4×C29C2×D413yes—
169C4⋊C4×C29C2×D416yes—
170C4⋊C4×C29C2412yes—
171C4⋊C4×C29C2411yescharacteristic
172C4⋊C4×C29C2411yescharacteristic
173C22⋊C4×C29—21no—
174C22⋊C4×C29—26no—
175C22⋊C4×C29—23no—
176C22⋊C4×C29—23no—
177C22⋊C4×C29—26no—
178C42×C29—21no—
179C42×C29—23no—
180C42×C29—23no—
181C42×C29—26no—
182C42×C29—26no—
183C42×C29C2×D413yes—
184C42×C29C2×D413yes—
185C42×C29C2413yes—
186C42×C29C2413yes—
187#61404C2911yesmaximal, characteristic, direct product factor, Sylow
188C23×C29—23no—
189C23×C29C22×D411yescharacteristic
190Q8×C29—26no—
191Q8×C29—26no—
192Q8×C29—23no—
193Q8×C29—26no—
194Q8×C29—23no—
195Q8×C29—46no—
196Q8×C29#13714yescomplemented
197D4×C29—23no—
198D4×C29—23no—
199D4×C29—46no—
200D4×C29—46no—
201D4×C29—83no—
202C2×C4×C29—26no—
203C2×C4×C29—22no—
204C2×C4×C29—21no—
205C2×C4×C29—21no—
206C2×C4×C29—23no—
207C2×C4×C29—26no—
208C2×C4×C29—43no—
209C2×C4×C29—42no—
210C2×C4×C29—46no—
211C2×C4×C29—46no—
212C2×C4×C29—46no—
213C2×C4×C29—83no—
214C2×C4×C29C22×D413yes—
215C2×C4×C29C22×D413yes—
216C2×C4×C29C22×D413yes—
217C2×C4×C29C22×D411yescharacteristic
218C2×C4×C29C22×D413yes—
219C2×C4×C29C2×#4112yes—
220C2×C4×C29H(5,𝔽2)12yes—
221C2×C4×C29C2511yescharacteristic
222C8×C29—26no—
223C8×C29C2×#4114yes—
224#3480C2×C2911yescharacteristic
225C2×#557C2×C2911yescharacteristic, complemented
226#3400C2×C2916yescomplemented
227#3398C2×C2916yescomplemented
228#3395C2×C2912yescomplemented
229#3392C2×C2912yescomplemented
230#3152C2×C2913yescomplemented
231#3039C2×C2913yescomplemented
232#3037C2×C2913yescomplemented
233Q8×M16C2×C2911yescharacteristic, complemented
234C4×#137C2×C2913yescomplemented
235C22×C29—21no—
236C22×C29—21no—
237C22×C29—43no—
238C22×C29—43no—
239C22×C29—43no—
240C22×C29#55012yes—
241C22×C29C23×D411yescharacteristic
242C4×C29—23no—
243C4×C29—23no—
244C4×C29—23no—
245C4×C29—42no—
246C4×C29—41no—
247C4×C29—43no—
248C4×C29—46no—
249C4×C29—41no—
250C4×C29#55012yes—
251C4×C29C2×#13716yes—
252C4×C29C22×#4111yescharacteristic
253C4×C29C2×H(5,𝔽2)11yescharacteristic
254C2×#137C22×C2913yes—
255C2×#137C22×C2916yes—
256#559C22×C2913yes—
257#557—21no—
258#557—23no—
259#557C22×C2912yes—
260#557C22×C2911yescharacteristic
261#557C22×C2916yes—
262#556C22×C2913yes—
263#553—23no—
264#550C22×C2911yescharacteristic
265#548—21no—
266#532C22×C2913yes—
267C2×C4⋊Q8C22×C2913yes—
268#529—23no—
269C2×#125—23no—
270C4×#41—21no—
271C4×#41C22×C2913yes—
272C2×C4×Q8—21no—
273C2×C4×Q8C22×C2913yes—
274C2×C4×Q8C22×C2911yescharacteristic
275#497C22×C29112yes—
276#494C22×C2916yes—
277#492C22×C2916yes—
278#464C22×C29112yes—
279#462C22×C2916yes—
280#462C22×C2916yes—
281#462C22×C2916yes—
282#461C22×C2916yes—
283#461C22×C2916yes—
284#460C22×C2916yes—
285#446C22×C2916yes—
286C8×Q8C22×C2912yes—
287#441C22×C2916yes—
288#440C22×C2916yes—
289C4×Q16C22×C2916yes—
290C4×SD16C22×C2916yes—
291#428C22×C2911yescharacteristic
292#423C22×C2913yes—
293#419C22×C2913yes—
294#417C22×C2913yes—
295C4×M16C22×C2913yes—
296C2×C29—22no—
297C2×C29—83no—
298C2×C29C2×#55011yescharacteristic
299C2×C29C22×#13711yescharacteristic
300C2×C29#358611yescharacteristic
301C2×#41—21no—
302C2×#41C23×C2913yes—
303C22×Q8D4×C2911yescharacteristic
304C22×Q8C23×C2913yes—
305#137—212no—
306#137—412no—
307C2×Q16C23×C2916yes—
308C2×Q16C23×C29112yes—
309C2×SD16C23×C2916yes—
310C2×SD16C23×C29112yes—
311C2×M16C23×C2913yes—
312C2×M16C23×C2911yescharacteristic
313C4⋊Q8—23no—
314C4⋊Q8—23no—
315C4⋊Q8C23×C2916yes—
316C4⋊Q8C23×C2916yes—
317C4⋊Q8C23×C2913yes—
318#127C23×C2913yes—
319#126—26no—
320#125—26no—
321#125—26no—
322#125—212no—
323#125—23no—
324#124—23no—
325#124C23×C2916yes—
326#123—26no—
327C22⋊Q8—23no—
328#121—23no—
329#121—23no—
330#121—23no—
331C4×Q8—26no—
332C4×Q8—22no—
333C4×Q8—21no—
334C4×Q8—23no—
335C4×Q8—23no—
336C4×Q8D4×C2911yescharacteristic
337C4×Q8D4×C2913yes—
338C4×Q8C23×C2916yes—
339C4×Q8C23×C2916yes—
340C4×Q8C23×C2913yes—
341C4×Q8C23×C2913yes—
342C4×Q8C23×C2911yescharacteristic
343C4×Q8C23×C2913yes—
344C4×D4—23no—
345C4×D4—23no—
346C4×D4—23no—
347C4×D4—23no—
348C4×D4—23no—
349C4×D4C23×C2913yes—
350C4×D4C23×C2913yes—
351#117—23no—
352#117—23no—
353#117C23×C2911yescharacteristic
354C2×C4⋊C4—23no—
355C2×C4⋊C4—23no—
356C2×C4⋊C4—23no—
357C2×C4⋊C4D4×C2913yes—
358C2×C4⋊C4C23×C2911yescharacteristic
359C2×C42—23no—
360C2×C42—23no—
361C2×C42C23×C2913yes—
362#107C23×C2912yes—
363#106C23×C2912yes—
364C4⋊C8C23×C2916yes—
365C4⋊C8C23×C2916yes—
366#103C23×C2916yes—
367#103C23×C2916yes—
368#103C23×C2916yes—
369#102C23×C2916yes—
370#97C23×C2916yes—
371C4×C8C23×C2916yes—
372C29#6140411yescharacteristic, direct product factor, Sylow
373#41—26no—
374#41—26no—
375#41—44no—
376C2×Q8—24no—
377C2×Q8—23no—
378C2×Q8—26no—
379C2×Q8—26no—
380C2×Q8C2×D4×C2913yes—
381C2×Q8C2×D4×C2911yescharacteristic
382C2×Q8C24×C2913yes—
383C2×Q8C24×C2916yes—
384C2×D4—23no—
385C2×D4—23no—
386C2×D4—23no—
387C2×D4C24×C2913yes—
388C22×C4—21no—
389C22×C4—23no—
390C22×C4—23no—
391C22×C4—23no—
392C22×C4—23no—
393C22×C4C2×D4×C2913yes—
394C22×C4C2×D4×C2913yes—
395C22×C4C24×C2911yescharacteristic
396Q16—212no—
397Q16—412no—
398SD16—212no—
399SD16—412no—
400M16—26no—
401M16#41×C2914yescomplemented
402C2×C8C24×C2916yes—
403C2×C8C24×C2912yes—
404C4⋊C4—26no—
405C4⋊C4—26no—
406C4⋊C4—26no—
407C4⋊C4—26no—
408C4⋊C4—26no—
409C4⋊C4—26no—
410C4⋊C4—21no—
411C4⋊C4—23no—
412C4⋊C4—23no—
413C4⋊C4—23no—
414C4⋊C4—23no—
415C4⋊C4C2×D4×C2913yes—
416C4⋊C4C2×D4×C2913yes—
417C4⋊C4C2×D4×C2916yes—
418C4⋊C4C24×C2912yes—
419C4⋊C4C24×C2911yescharacteristic
420C4⋊C4C24×C2911yescharacteristic
421C22⋊C4—21no—
422C22⋊C4—26no—
423C22⋊C4—23no—
424C22⋊C4—23no—
425C22⋊C4—26no—
426C42—21no—
427C42—23no—
428C42—23no—
429C42—26no—
430C42—26no—
431C42C2×D4×C2913yes—
432C42C2×D4×C2913yes—
433C42C24×C2913yes—
434C42C24×C2913yes—
435C23—23no—
436C23C22×D4×C2911yescharacteristic
437Q8—26no—
438Q8—26no—
439Q8—23no—
440Q8—26no—
441Q8—23no—
442Q8—46no—
443Q8C29×#13714yescomplemented
444D4—23no—
445D4—23no—
446D4—46no—
447D4—46no—
448D4—83no—
449C2×C4—26no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—23no—
454C2×C4—26no—
455C2×C4—43no—
456C2×C4—42no—
457C2×C4—46no—
458C2×C4—46no—
459C2×C4—46no—
460C2×C4—83no—
461C2×C4C22×D4×C2913yes—
462C2×C4C22×D4×C2913yes—
463C2×C4C22×D4×C2913yes—
464C2×C4C22×D4×C2911yescharacteristic
465C2×C4C22×D4×C2913yes—
466C2×C4C2×#41×C2912yes—
467C2×C4C29×H(5,𝔽2)12yes—
468C2×C4C25×C2911yescharacteristic
469C8—26no—
470C8C2×#41×C2914yes—
471C22—21no—
472C22—21no—
473C22—43no—
474C22—43no—
475C22—43no—
476C22C29×#55012yes—
477C22C23×D4×C2911yescharacteristic
478C4—23no—
479C4—23no—
480C4—23no—
481C4—43no—
482C4—46no—
483C4—41no—
484C4—42no—
485C4—41no—
486C4C29×#55012yes—
487C4C2×C29×#13716yes—
488C4C22×#41×C2911yescharacteristic
489C4C2×C29×H(5,𝔽2)11yescharacteristic
490C2—22no—
491C2—83no—
492C2C2×C29×#55011yescharacteristic
493C2C22×C29×#13711yescharacteristic
494C2C29×#358611yescharacteristic
subgroupG/H
center, socleC22×C29C23×D4
commutator, Frattini, cosocleC2×C4C25×C29
prime radicalC29×#3480C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C29 ◁ C2×C4×Q8×C29 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C29×#557C2semidirect
3. C29×#3400C2semidirect
4. C29×#3398C2semidirect
5. C29×#3395C2semidirect
6. C29×#3392C2semidirect
7. C29×#3152C2semidirect
8. C29×#3039C2semidirect
9. C29×#3037C2semidirect
10. Q8×M16×C29C2semidirect
11. C4×C29×#137C2semidirect
153. M16×C29#41semidirect
187. #61404C29direct (N × H)
196. Q8×C29#137semidirect
225. C2×#557C2×C29semidirect
226. #3400C2×C29semidirect
227. #3398C2×C29semidirect
228. #3395C2×C29semidirect
229. #3392C2×C29semidirect
230. #3152C2×C29semidirect
231. #3039C2×C29semidirect
232. #3037C2×C29semidirect
233. Q8×M16C2×C29semidirect
234. C4×#137C2×C29semidirect
372. C29#61404direct (N × H)
401. M16#41×C29semidirect
443. Q8C29×#137semidirect

Maximal subgroups

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

subgroupindex
C29×#34802
C2×C29×#5572
C29×#34002
C29×#33982
C29×#33952
C29×#33922
C29×#31522
C29×#30392
C29×#30372
Q8×M16×C292
C4×C29×#1372
#6140429