Finite Group Databasebeta

← C29×#9368 #20304268

Subgroups — C29×#9368 #20304268

Order 7424.

774 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C29×#3006C211yesmaximal, characteristic
2C29×#2341C211yesmaximal, characteristic, complemented
3C29×#1964C211yesmaximal, characteristic, complemented
4C29×#1789C211yesmaximal, characteristic, complemented
5C29×#1507C211yesmaximal, characteristic, complemented
6C29×#1305C211yesmaximal, characteristic, complemented
7C29×#1297C211yesmaximal, characteristic, complemented
8C22×#41×C29C2211yescharacteristic
9C2×C29×AGL(1,ℤ/8)C412yescomplemented
10C2×C29×#121C2211yescharacteristic
11C2×C4×Q8×C29—21no—
12C29×#449—22no—
13C29×#448—21no—
14C29×#447—21no—
15C2×C29×#104—22no—
16C29×#417C412yescomplemented
17C2×C29×#99—22no—
18C29×#407C2211yescharacteristic
19C29×#407C2211yescharacteristic
20C29×#390—21no—
21C29×#390—22no—
22C29×#380—21no—
23C29×#380C2211yescharacteristic
24C4×C22⋊C4×C29—21no—
25C4×C22⋊C4×C29C2211yescharacteristic
26C4×C22⋊C4×C29C2211yescharacteristic
27C29×#327—21no—
28C29×#327—21no—
29C2×#41×C29—21no—
30C2×#41×C29—21no—
31C2×#41×C29—44no—
32C2×#41×C29C2×C412yes—
33C2×#41×C29D412yes—
34C22×Q8×C29D411yescharacteristic
35C22×D4×C29—21no—
36C22×D4×C29—21no—
37C22×D4×C29C2×C411yescharacteristic
38C22×D4×C29D411yescharacteristic
39C23×C4×C29—21no—
40C23×C4×C29C2311yescharacteristic
41C29×AGL(1,ℤ/8)—44no—
42C2×SD16×C29—22no—
43C2×D8×C29—22no—
44C2×M16×C29—21no—
45C2×M16×C29C2×C412yes—
46C29×#121—21no—
47C29×#121—21no—
48C29×#121—21no—
49C29×#121—21no—
50C4×Q8×C29—44no—
51C2×C4⋊C4×C29—22no—
52C2×C4⋊C4×C29—21no—
53C2×C4⋊C4×C29C2×C411yescharacteristic
54C2×C22⋊C4×C29—21no—
55C2×C22⋊C4×C29—22no—
56C2×C22⋊C4×C29—21no—
57C2×C22⋊C4×C29—21no—
58C2×C22⋊C4×C29—22no—
59C2×C22⋊C4×C29D412yes—
60C2×C42×C29—22no—
61C2×C42×C29—21no—
62C2×C42×C29—21no—
63C29×#104—44no—
64C29×#103—22no—
65C29×#102—22no—
66C29×#99—44no—
67C29×C22⋊C8—21no—
68C29×C22⋊C8—21no—
69C29×C22⋊C8—21no—
70C29×C22⋊C8—21no—
71C29×#95—21no—
72C29×#95—21no—
73C29×#95D411yescharacteristic
74C29×#95D411yescharacteristic
75C24×C29—21no—
76C24×C29—21no—
77C24×C29C2×D411yescharacteristic
78#41×C29—24no—
79#41×C29—24no—
80#41×C29—42no—
81#41×C29—42no—
82#41×C29—44no—
83#41×C29—44no—
84C2×Q8×C29—24no—
85C2×Q8×C29—21no—
86C2×Q8×C29C22⋊C412yes—
87C2×D4×C29—22no—
88C2×D4×C29—22no—
89C2×D4×C29—21no—
90C2×D4×C29—21no—
91C2×D4×C29—21no—
92C2×D4×C29—42no—
93C2×D4×C29—44no—
94C2×D4×C29—42no—
95C2×D4×C29—42no—
96C2×D4×C29—42no—
97C2×D4×C29—41no—
98C2×D4×C29—41no—
99C2×D4×C29C22⋊C412yes—
100C22×C4×C29—21no—
101C22×C4×C29—21no—
102C22×C4×C29—22no—
103C22×C4×C29—22no—
104C22×C4×C29—21no—
105C22×C4×C29—21no—
106C22×C4×C29—22no—
107C22×C4×C29—42no—
108C22×C4×C29—44no—
109C22×C4×C29—41no—
110C22×C4×C29C22×C411yescharacteristic
111C22×C4×C29C2×D411yescharacteristic
112C22×C4×C29C2×D411yescharacteristic
113C22×C4×C29C2×D411yescharacteristic
114C22×C4×C29#4111yescharacteristic
115C22×C4×C29#4111yescharacteristic
116SD16×C29—44no—
117D8×C29—44no—
118M16×C29—42no—
119M16×C29—42no—
120C2×C8×C29—22no—
121C2×C8×C29—41no—
122C4⋊C4×C29—21no—
123C4⋊C4×C29—21no—
124C4⋊C4×C29—44no—
125C4⋊C4×C29—42no—
126C22⋊C4×C29—22no—
127C22⋊C4×C29—42no—
128C22⋊C4×C29—44no—
129C22⋊C4×C29—41no—
130C22⋊C4×C29—41no—
131C22⋊C4×C29—41no—
132C22⋊C4×C29—42no—
133C22⋊C4×C29—82no—
134C42×C29—44no—
135C42×C29—42no—
136C42×C29—42no—
137#9368C2911yesmaximal, characteristic, direct product factor, Sylow
138C23×C29—21no—
139C23×C29—21no—
140C23×C29—22no—
141C23×C29—41no—
142C23×C29—41no—
143C23×C29—42no—
144C23×C29—42no—
145C23×C29—42no—
146C23×C29—41no—
147C23×C29—41no—
148C23×C29—41no—
149C23×C29—81no—
150C23×C29—81no—
151C23×C29C4×D411yescharacteristic
152C23×C29#12011yescharacteristic
153C23×C29#12111yescharacteristic
154Q8×C29—24no—
155Q8×C29—22no—
156Q8×C29#9914yescomplemented
157D4×C29—22no—
158D4×C29—22no—
159D4×C29—44no—
160D4×C29—44no—
161D4×C29—42no—
162D4×C29—42no—
163D4×C29—81no—
164D4×C29—82no—
165D4×C29—81no—
166D4×C29—81no—
167C2×C4×C29—24no—
168C2×C4×C29—22no—
169C2×C4×C29—22no—
170C2×C4×C29—22no—
171C2×C4×C29—22no—
172C2×C4×C29—22no—
173C2×C4×C29—41no—
174C2×C4×C29—41no—
175C2×C4×C29—41no—
176C2×C4×C29—44no—
177C2×C4×C29—44no—
178C2×C4×C29—41no—
179C2×C4×C29—41no—
180C2×C4×C29—41no—
181C2×C4×C29—42no—
182C2×C4×C29—42no—
183C2×C4×C29—41no—
184C2×C4×C29—42no—
185C2×C4×C29—42no—
186C2×C4×C29—41no—
187C2×C4×C29—42no—
188C2×C4×C29—41no—
189C2×C4×C29—81no—
190C2×C4×C29—81no—
191C2×C4×C29C2×C22⋊C411yescharacteristic
192C2×C4×C29C4×D411yescharacteristic
193C2×C4×C29#12111yescharacteristic
194C2×C4×C29#12311yescharacteristic
195C8×C29—42no—
196C8×C29—81no—
197#3006C2×C2911yescharacteristic
198#2341C2×C2911yescharacteristic, complemented
199#1964C2×C2911yescharacteristic, complemented
200#1789C2×C2911yescharacteristic, complemented
201#1507C2×C2911yescharacteristic, complemented
202#1305C2×C2911yescharacteristic, complemented
203#1297C2×C2911yescharacteristic, complemented
204C22×C29—21no—
205C22×C29—21no—
206C22×C29—21no—
207C22×C29—21no—
208C22×C29—21no—
209C22×C29—21no—
210C22×C29—21no—
211C22×C29—21no—
212C22×C29—42no—
213C22×C29—42no—
214C22×C29—42no—
215C22×C29—81no—
216C22×C29—81no—
217C22×C29—82no—
218C22×C29—82no—
219C22×C29—81no—
220C22×C29—81no—
221C22×C29—81no—
222C22×C29—162no—
223C22×C29#38611yescharacteristic
224C22×C29#45311yescharacteristic
225C22×C29#45411yescharacteristic
226C4×C29—21no—
227C4×C29—24no—
228C4×C29—21no—
229C4×C29—21no—
230C4×C29—42no—
231C4×C29—82no—
232C4×C29—81no—
233C4×C29—82no—
234C4×C29—81no—
235C4×C29C2×#9912yes—
236C22×#41C22×C2911yescharacteristic
237C2×AGL(1,ℤ/8)C4×C2912yescomplemented
238C2×#121C22×C2911yescharacteristic
239C2×C4×Q8—21no—
240#449—22no—
241#448—21no—
242#447—21no—
243C2×#104—22no—
244#417C4×C2912yescomplemented
245C2×#99—22no—
246#407C22×C2911yescharacteristic
247#407C22×C2911yescharacteristic
248#390—21no—
249#390—22no—
250#380—21no—
251#380C22×C2911yescharacteristic
252C4×C22⋊C4—21no—
253C4×C22⋊C4C22×C2911yescharacteristic
254C4×C22⋊C4C22×C2911yescharacteristic
255#327—21no—
256#327—21no—
257C2×C29—21no—
258C2×C29—21no—
259C2×C29—41no—
260C2×C29—41no—
261C2×C29—82no—
262C2×C29—161no—
263C2×C29#188311yescharacteristic
264C2×C29#189811yescharacteristic
265C2×C29#190611yescharacteristic
266C2×#41—21no—
267C2×#41—21no—
268C2×#41—44no—
269C2×#41C2×C4×C2912yes—
270C2×#41D4×C2912yes—
271C22×Q8D4×C2911yescharacteristic
272C22×D4—21no—
273C22×D4—21no—
274C22×D4C2×C4×C2911yescharacteristic
275C22×D4D4×C2911yescharacteristic
276C23×C4—21no—
277C23×C4C23×C2911yescharacteristic
278AGL(1,ℤ/8)—44no—
279C2×SD16—22no—
280C2×D8—22no—
281C2×M16—21no—
282C2×M16C2×C4×C2912yes—
283#121—21no—
284#121—21no—
285#121—21no—
286#121—21no—
287C4×Q8—44no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4C2×C4×C2911yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4D4×C2912yes—
297C2×C42—22no—
298C2×C42—21no—
299C2×C42—21no—
300#104—44no—
301#103—22no—
302#102—22no—
303#99—44no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—21no—
308#95—21no—
309#95—21no—
310#95D4×C2911yescharacteristic
311#95D4×C2911yescharacteristic
312C29#936811yescharacteristic, direct product factor, Sylow
313C24—21no—
314C24—21no—
315C24C2×D4×C2911yescharacteristic
316#41—24no—
317#41—24no—
318#41—42no—
319#41—42no—
320#41—44no—
321#41—44no—
322C2×Q8—24no—
323C2×Q8—21no—
324C2×Q8C22⋊C4×C2912yes—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—42no—
331C2×D4—44no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4C22⋊C4×C2912yes—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—41no—
348C22×C4C22×C4×C2911yescharacteristic
349C22×C4C2×D4×C2911yescharacteristic
350C22×C4C2×D4×C2911yescharacteristic
351C22×C4C2×D4×C2911yescharacteristic
352C22×C4#41×C2911yescharacteristic
353C22×C4#41×C2911yescharacteristic
354SD16—44no—
355D8—44no—
356M16—42no—
357M16—42no—
358C2×C8—22no—
359C2×C8—41no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—44no—
363C4⋊C4—42no—
364C22⋊C4—22no—
365C22⋊C4—42no—
366C22⋊C4—44no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—42no—
371C22⋊C4—82no—
372C42—44no—
373C42—42no—
374C42—42no—
375C23—21no—
376C23—21no—
377C23—22no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—81no—
387C23—81no—
388C23C4×D4×C2911yescharacteristic
389C23C29×#12011yescharacteristic
390C23C29×#12111yescharacteristic
391Q8—24no—
392Q8—22no—
393Q8C29×#9914yescomplemented
394D4—22no—
395D4—22no—
396D4—44no—
397D4—44no—
398D4—42no—
399D4—42no—
400D4—81no—
401D4—82no—
402D4—81no—
403D4—81no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4C2×C22⋊C4×C2911yescharacteristic
429C2×C4C4×D4×C2911yescharacteristic
430C2×C4C29×#12111yescharacteristic
431C2×C4C29×#12311yescharacteristic
432C8—42no—
433C8—81no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—81no—
446C22—81no—
447C22—82no—
448C22—82no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—162no—
453C22C29×#38611yescharacteristic
454C22C29×#45311yescharacteristic
455C22C29×#45411yescharacteristic
456C4—21no—
457C4—24no—
458C4—21no—
459C4—21no—
460C4—42no—
461C4—82no—
462C4—81no—
463C4—82no—
464C4—81no—
465C4C2×C29×#9912yes—
466C2—21no—
467C2—21no—
468C2—41no—
469C2—41no—
470C2—82no—
471C2—161no—
472C2C29×#188311yescharacteristic
473C2C29×#189811yescharacteristic
474C2C29×#190611yescharacteristic
subgroupG/H
center, socleC22×C29#386
commutatorC22×C4C22×C4×C29
Frattini, cosocleC23×C4C23×C29
prime radicalC29×#3006C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C29 ◁ C23×C4×C29 ◁ G

Complements

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

NH ≅ G/Ntype
2. C29×#2341C2semidirect
3. C29×#1964C2semidirect
4. C29×#1789C2semidirect
5. C29×#1507C2semidirect
6. C29×#1305C2semidirect
7. C29×#1297C2semidirect
9. C2×C29×AGL(1,ℤ/8)C4semidirect
16. C29×#417C4semidirect
137. #9368C29direct (N × H)
156. Q8×C29#99semidirect
198. #2341C2×C29semidirect
199. #1964C2×C29semidirect
200. #1789C2×C29semidirect
201. #1507C2×C29semidirect
202. #1305C2×C29semidirect
203. #1297C2×C29semidirect
237. C2×AGL(1,ℤ/8)C4×C29semidirect
244. #417C4×C29semidirect
312. C29#9368direct (N × H)
393. Q8C29×#99semidirect

Maximal subgroups

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

subgroupindex
C29×#30062
C29×#23412
C29×#19642
C29×#17892
C29×#15072
C29×#13052
C29×#12972
#936829