Finite Group Databasebeta

← C29×#10256 #20305156

Subgroups — C29×#10256 #20305156

Order 7424.

580 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C29×#3246C211yesmaximal, characteristic, complemented
2C29×#3077C211yesmaximal, characteristic, complemented
3C29×#3036C211yesmaximal, characteristic
4C29×#1954C211yesmaximal, characteristic, complemented
5C29×#1493C211yesmaximal, characteristic, complemented
6C29×#1492C211yesmaximal, characteristic, complemented
7C29×#1292C211yesmaximal, characteristic, complemented
8C2×C29×AGL(1,ℤ/8)—21no—
9C29×#533C2211yescharacteristic
10C29×#532C2211yescharacteristic
11C2×C4×D4×C29C2211yescharacteristic
12C29×#478—22no—
13C29×#476—21no—
14C29×#476—21no—
15C29×#475—21no—
16C29×#474—21no—
17C29×#465—22no—
18C29×#463—22no—
19C29×#461—21no—
20C29×#461—21no—
21C29×#460—21no—
22C29×#459—21no—
23C29×#428—21no—
24C29×#423C2211yescharacteristic
25C29×#423C2211yescharacteristic
26C2×C29×C4⋊C8C2211yescharacteristic
27C29×#418—21no—
28C29×#417—21no—
29C2×C29×C22⋊C8—22no—
30C2×C4×C8×C29C2211yescharacteristic
31C29×Q8⋊C8—22no—
32C29×D4⋊C8—22no—
33C2×#41×C29—21no—
34C22×D4×C29D411yescharacteristic
35C23×C4×C29—21no—
36C29×AGL(1,ℤ/8)—42no—
37C29×AGL(1,ℤ/8)—42no—
38C2×SD16×C29—22no—
39C2×SD16×C29—22no—
40C2×D8×C29—22no—
41C2×M16×C29—21no—
42C2×M16×C29—21no—
43C22×C8×C29—21no—
44C22×C8×C29D412yes—
45C29×C4⋊Q8D411yescharacteristic
46C29×C4⋊Q8D411yescharacteristic
47C29×C4⋊Q8D411yescharacteristic
48C29×#127D411yescharacteristic
49C29×#125—21no—
50C29×#124—21no—
51C29×C22⋊Q8—21no—
52C29×C22⋊Q8—21no—
53C29×#121—21no—
54C29×#121—21no—
55C4×Q8×C29—21no—
56C4×Q8×C29—21no—
57C4×D4×C29—21no—
58C4×D4×C29—21no—
59C4×D4×C29—21no—
60C4×D4×C29—21no—
61C4×D4×C29—21no—
62C4×D4×C29—21no—
63C29×#117—21no—
64C2×C4⋊C4×C29D411yescharacteristic
65C2×C22⋊C4×C29—21no—
66C2×C42×C29C2311yescharacteristic
67C29×#107—22no—
68C29×#106—22no—
69C29×#106—22no—
70C29×C4⋊C8—21no—
71C29×C4⋊C8—21no—
72C29×C4⋊C8—21no—
73C29×C4⋊C8—21no—
74C29×C4⋊C8D412yes—
75C29×C4⋊C8D412yes—
76C29×#103—22no—
77C29×#103—22no—
78C29×#102—22no—
79C29×#102—22no—
80C29×#102—22no—
81C29×#102—22no—
82C29×C22⋊C8—44no—
83C4×C8×C29—21no—
84C4×C8×C29—21no—
85C24×C29—21no—
86#41×C29—41no—
87#41×C29—41no—
88#41×C29—41no—
89#41×C29—41no—
90C2×Q8×C29—21no—
91C2×Q8×C29—21no—
92C2×Q8×C29—21no—
93C2×D4×C29—21no—
94C2×D4×C29—21no—
95C2×D4×C29—21no—
96C2×D4×C29—21no—
97C2×D4×C29—21no—
98C2×D4×C29—21no—
99C2×D4×C29—21no—
100C2×D4×C29—21no—
101C2×D4×C29—41no—
102C2×D4×C29SD1612yescomplemented
103C22×C4×C29—21no—
104C22×C4×C29—42no—
105C22×C4×C29—42no—
106C22×C4×C29—41no—
107C22×C4×C29—41no—
108C22×C4×C29—41no—
109C22×C4×C29C2×D411yescharacteristic
110C22×C4×C29C2×D411yescharacteristic
111C22×C4×C29#4111yescharacteristic
112SD16×C29—44no—
113SD16×C29—82no—
114D8×C29—44no—
115M16×C29—41no—
116M16×C29—41no—
117M16×C29—41no—
118M16×C29—41no—
119C2×C8×C29—22no—
120C2×C8×C29—22no—
121C2×C8×C29—22no—
122C2×C8×C29—22no—
123C2×C8×C29—41no—
124C2×C8×C29—41no—
125C2×C8×C29—42no—
126C4⋊C4×C29—21no—
127C4⋊C4×C29—21no—
128C4⋊C4×C29—21no—
129C4⋊C4×C29—21no—
130C4⋊C4×C29—21no—
131C4⋊C4×C29—21no—
132C4⋊C4×C29—21no—
133C4⋊C4×C29—21no—
134C4⋊C4×C29—41no—
135C4⋊C4×C29—41no—
136C22⋊C4×C29—41no—
137C22⋊C4×C29—41no—
138C22⋊C4×C29—41no—
139C22⋊C4×C29—41no—
140C22⋊C4×C29—41no—
141C42×C29—41no—
142C42×C29C2×D411yescharacteristic
143C42×C29C2×D411yescharacteristic
144C42×C29C2×D411yescharacteristic
145C42×C29C2×D411yescharacteristic
146#10256C2911yesmaximal, characteristic, direct product factor, Sylow
147C23×C29—22no—
148C23×C29—42no—
149C23×C29—42no—
150C23×C29—41no—
151C23×C29—41no—
152C23×C29—41no—
153C23×C29#12111yescharacteristic
154Q8×C29—42no—
155Q8×C29—41no—
156Q8×C29—41no—
157Q8×C29—41no—
158Q8×C29—41no—
159D4×C29—22no—
160D4×C29—22no—
161D4×C29—42no—
162D4×C29—41no—
163D4×C29—41no—
164D4×C29—41no—
165D4×C29—41no—
166D4×C29—41no—
167D4×C29—41no—
168D4×C29—81no—
169D4×C29—81no—
170C2×C4×C29—21no—
171C2×C4×C29—21no—
172C2×C4×C29—41no—
173C2×C4×C29—42no—
174C2×C4×C29—41no—
175C2×C4×C29—41no—
176C2×C4×C29—41no—
177C2×C4×C29—41no—
178C2×C4×C29—82no—
179C2×C4×C29—81no—
180C2×C4×C29—81no—
181C2×C4×C29—81no—
182C2×C4×C29—81no—
183C2×C4×C29#12011yescharacteristic
184C2×C4×C29#12011yescharacteristic
185C2×C4×C29#12011yescharacteristic
186C2×C4×C29#12111yescharacteristic
187C2×C4×C29#12111yescharacteristic
188C2×C4×C29#12711yescharacteristic
189C2×C4×C29C2×SD1611yescharacteristic
190C2×C4×C29#13511yescharacteristic
191C2×C4×C29AGL(1,ℤ/8)11yescharacteristic
192C2×C4×C29#13711yescharacteristic
193C8×C29—42no—
194C8×C29—42no—
195C8×C29—42no—
196C8×C29—42no—
197#3246C2×C2911yescharacteristic, complemented
198#3077C2×C2911yescharacteristic, complemented
199#3036C2×C2911yescharacteristic
200#1954C2×C2911yescharacteristic, complemented
201#1493C2×C2911yescharacteristic, complemented
202#1492C2×C2911yescharacteristic, complemented
203#1292C2×C2911yescharacteristic, complemented
204C22×C29—21no—
205C22×C29—21no—
206C22×C29—44no—
207C22×C29—42no—
208C22×C29—82no—
209C22×C29—81no—
210C22×C29—81no—
211C22×C29—81no—
212C22×C29—81no—
213C22×C29—81no—
214C22×C29#39211yescharacteristic
215C22×C29#46511yescharacteristic
216C22×C29#46811yescharacteristic
217C4×C29—21no—
218C4×C29—21no—
219C4×C29—21no—
220C4×C29—81no—
221C4×C29—81no—
222C4×C29—81no—
223C4×C29—81no—
224C4×C29#44811yescharacteristic
225C4×C29#44911yescharacteristic
226C4×C29#45011yescharacteristic
227C4×C29#45211yescharacteristic
228C4×C29#45311yescharacteristic
229C4×C29#45411yescharacteristic
230C2×AGL(1,ℤ/8)—21no—
231#533C22×C2911yescharacteristic
232#532C22×C2911yescharacteristic
233C2×C4×D4C22×C2911yescharacteristic
234#478—22no—
235#476—21no—
236#476—21no—
237#475—21no—
238#474—21no—
239#465—22no—
240#463—22no—
241#461—21no—
242#461—21no—
243#460—21no—
244#459—21no—
245#428—21no—
246#423C22×C2911yescharacteristic
247#423C22×C2911yescharacteristic
248C2×C4⋊C8C22×C2911yescharacteristic
249#418—21no—
250#417—21no—
251C2×C22⋊C8—22no—
252C2×C4×C8C22×C2911yescharacteristic
253Q8⋊C8—22no—
254D4⋊C8—22no—
255C2×C29—21no—
256C2×C29—21no—
257C2×C29—82no—
258C2×C29—161no—
259C2×C29#200011yescharacteristic
260C2×C29#200611yescharacteristic
261C2×C29#201711yescharacteristic
262C2×#41—21no—
263C22×D4D4×C2911yescharacteristic
264C23×C4—21no—
265AGL(1,ℤ/8)—42no—
266AGL(1,ℤ/8)—42no—
267C2×SD16—22no—
268C2×SD16—22no—
269C2×D8—22no—
270C2×M16—21no—
271C2×M16—21no—
272C22×C8—21no—
273C22×C8D4×C2912yes—
274C4⋊Q8D4×C2911yescharacteristic
275C4⋊Q8D4×C2911yescharacteristic
276C4⋊Q8D4×C2911yescharacteristic
277#127D4×C2911yescharacteristic
278#125—21no—
279#124—21no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282#121—21no—
283#121—21no—
284C4×Q8—21no—
285C4×Q8—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290C4×D4—21no—
291C4×D4—21no—
292#117—21no—
293C2×C4⋊C4D4×C2911yescharacteristic
294C2×C22⋊C4—21no—
295C2×C42C23×C2911yescharacteristic
296#107—22no—
297#106—22no—
298#106—22no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8D4×C2912yes—
304C4⋊C8D4×C2912yes—
305#103—22no—
306#103—22no—
307#102—22no—
308#102—22no—
309#102—22no—
310#102—22no—
311C22⋊C8—44no—
312C4×C8—21no—
313C4×C8—21no—
314C29#1025611yescharacteristic, direct product factor, Sylow
315C24—21no—
316#41—41no—
317#41—41no—
318#41—41no—
319#41—41no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—41no—
332C2×D4SD16×C2912yescomplemented
333C22×C4—21no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4C2×D4×C2911yescharacteristic
340C22×C4C2×D4×C2911yescharacteristic
341C22×C4#41×C2911yescharacteristic
342SD16—44no—
343SD16—82no—
344D8—44no—
345M16—41no—
346M16—41no—
347M16—41no—
348M16—41no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—42no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C42—41no—
372C42C2×D4×C2911yescharacteristic
373C42C2×D4×C2911yescharacteristic
374C42C2×D4×C2911yescharacteristic
375C42C2×D4×C2911yescharacteristic
376C23—22no—
377C23—42no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23C29×#12111yescharacteristic
383Q8—42no—
384Q8—41no—
385Q8—41no—
386Q8—41no—
387Q8—41no—
388D4—22no—
389D4—22no—
390D4—42no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—41no—
397D4—81no—
398D4—81no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4C29×#12011yescharacteristic
413C2×C4C29×#12011yescharacteristic
414C2×C4C29×#12011yescharacteristic
415C2×C4C29×#12111yescharacteristic
416C2×C4C29×#12111yescharacteristic
417C2×C4C29×#12711yescharacteristic
418C2×C4C2×SD16×C2911yescharacteristic
419C2×C4C29×#13511yescharacteristic
420C2×C4C29×AGL(1,ℤ/8)11yescharacteristic
421C2×C4C29×#13711yescharacteristic
422C8—42no—
423C8—42no—
424C8—42no—
425C8—42no—
426C22—21no—
427C22—21no—
428C22—44no—
429C22—42no—
430C22—82no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22C29×#39211yescharacteristic
437C22C29×#46511yescharacteristic
438C22C29×#46811yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—81no—
446C4C29×#44811yescharacteristic
447C4C29×#44911yescharacteristic
448C4C29×#45011yescharacteristic
449C4C29×#45211yescharacteristic
450C4C29×#45311yescharacteristic
451C4C29×#45411yescharacteristic
452C2—21no—
453C2—21no—
454C2—82no—
455C2—161no—
456C2C29×#200011yescharacteristic
457C2C29×#200611yescharacteristic
458C2C29×#201711yescharacteristic
subgroupG/H
center, socleC22×C29#392
commutator, Frattini, cosocleC2×C42C23×C29
prime radicalC29×#3036C2
LCSG ▷ C2×C42 ▷ 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×#3246C2semidirect
2. C29×#3077C2semidirect
4. C29×#1954C2semidirect
5. C29×#1493C2semidirect
6. C29×#1492C2semidirect
7. C29×#1292C2semidirect
102. C2×D4×C29SD16semidirect
146. #10256C29direct (N × H)
197. #3246C2×C29semidirect
198. #3077C2×C29semidirect
200. #1954C2×C29semidirect
201. #1493C2×C29semidirect
202. #1492C2×C29semidirect
203. #1292C2×C29semidirect
314. C29#10256direct (N × H)
332. C2×D4SD16×C29semidirect

Maximal subgroups

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

subgroupindex
C29×#32462
C29×#30772
C29×#30362
C29×#19542
C29×#14932
C29×#14922
C29×#12922
#1025629