Finite Group Databasebeta

← C29×#12706 #20307606

Subgroups — C29×#12706 #20307606

Order 7424.

1000 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C29×#3485C211yesmaximal, characteristic
2C29×#2121C214yesmaximal, complemented
3C29×#1785C211yesmaximal, characteristic, complemented
4C29×#1782C211yesmaximal, characteristic, complemented
5C2×C29×H(5,𝔽2)C411yescharacteristic, complemented
6C22×#41×C29C2211yescharacteristic
7C29×#547—24no—
8C29×#546—24no—
9C29×#536—21no—
10C29×#534—21no—
11C29×#534C2212yes—
12C29×#525—21no—
13C2×C29×#123—21no—
14C2×C29×#123C412yescomplemented
15C29×#518C411yescharacteristic, complemented
16C2×C29×#101—21no—
17C2×C29×#100C2212yescomplemented
18C2×C29×#99—21no—
19C2×C29×#99C2212yescomplemented
20C29×#407—21no—
21C29×#381—21no—
22C29×#352—24no—
23C29×AΣL(1,𝔽16)—24no—
24C29×H(5,𝔽2)—42no—
25C29×H(5,𝔽2)—42no—
26C2×#41×C29—24no—
27C2×#41×C29—21no—
28C2×#41×C29—22no—
29C2×#41×C29—42no—
30C2×#41×C29—42no—
31C22×Q8×C29D411yescharacteristic
32C22×D4×C29—21no—
33C22×D4×C29—22no—
34C22×D4×C29C2×C411yescharacteristic
35C22×D4×C29C2×C412yes—
36C22×D4×C29D411yescharacteristic
37C22×D4×C29C2311yescharacteristic
38C23×C4×C29C2×C411yescharacteristic
39C23×C4×C29D411yescharacteristic
40C23×C4×C29D411yescharacteristic
41C2×M16×C29—22no—
42C29×C4⋊Q8—22no—
43C29×#124—22no—
44C29×#123—24no—
45C29×#123—22no—
46C29×#123—22no—
47C29×#123—22no—
48C29×#123—22no—
49C29×#123—24no—
50C29×#123D414yes—
51C29×C22⋊Q8—22no—
52C29×C22⋊Q8—24no—
53C29×#121—22no—
54C29×#121—24no—
55C29×#121—24no—
56C29×#120—22no—
57C29×#120—22no—
58C29×#120D414yes—
59C4×D4×C29—24no—
60C4×D4×C29—24no—
61C29×#117—21no—
62C2×C4⋊C4×C29—22no—
63C2×C4⋊C4×C29—22no—
64C2×C22⋊C4×C29—22no—
65C2×C22⋊C4×C29—21no—
66C2×C22⋊C4×C29—22no—
67C2×C22⋊C4×C29—22no—
68C2×C22⋊C4×C29C2×C412yes—
69C29×#101—42no—
70C29×#100—24no—
71C29×#99—24no—
72C29×#99—42no—
73C29×C22⋊C8—42no—
74C29×#95—42no—
75C24×C29—21no—
76C24×C29C22⋊C411yescharacteristic
77C24×C29C22⋊C412yes—
78C24×C29C22×C411yescharacteristic
79C24×C29C2×D411yescharacteristic
80#41×C29—42no—
81#41×C29—416no—
82#41×C29—48no—
83#41×C29—42no—
84C2×Q8×C29—21no—
85C2×Q8×C29—22no—
86C2×Q8×C29—22no—
87C2×Q8×C29—41no—
88C2×D4×C29—22no—
89C2×D4×C29—22no—
90C2×D4×C29—22no—
91C2×D4×C29—21no—
92C2×D4×C29—22no—
93C2×D4×C29—22no—
94C2×D4×C29—22no—
95C2×D4×C29—22no—
96C2×D4×C29—24no—
97C2×D4×C29—22no—
98C2×D4×C29—41no—
99C2×D4×C29—41no—
100C2×D4×C29—41no—
101C2×D4×C29—48no—
102C2×D4×C29—42no—
103C2×D4×C29—42no—
104C2×D4×C29—42no—
105C2×D4×C29—42no—
106C2×D4×C29C2×D414yes—
107C22×C4×C29—21no—
108C22×C4×C29—22no—
109C22×C4×C29—22no—
110C22×C4×C29—22no—
111C22×C4×C29—22no—
112C22×C4×C29—22no—
113C22×C4×C29—22no—
114C22×C4×C29—22no—
115C22×C4×C29—22no—
116C22×C4×C29—41no—
117C22×C4×C29—41no—
118C22×C4×C29—41no—
119C22×C4×C29—41no—
120C22×C4×C29—42no—
121C22×C4×C29C22⋊C411yescharacteristic
122C22×C4×C29C22⋊C411yescharacteristic
123C22×C4×C29C22⋊C411yescharacteristic
124C22×C4×C29C22⋊C412yes—
125C22×C4×C29C2×D411yescharacteristic
126M16×C29—42no—
127M16×C29—42no—
128C2×C8×C29—42no—
129C4⋊C4×C29—24no—
130C4⋊C4×C29—24no—
131C4⋊C4×C29—22no—
132C4⋊C4×C29—24no—
133C22⋊C4×C29—24no—
134C22⋊C4×C29—22no—
135C22⋊C4×C29—24no—
136C22⋊C4×C29—22no—
137C22⋊C4×C29—24no—
138C22⋊C4×C29—24no—
139C22⋊C4×C29—42no—
140C22⋊C4×C29—42no—
141C22⋊C4×C29C22⋊C414yes—
142C42×C29—22no—
143#12706C2911yesmaximal, characteristic, direct product factor, Sylow
144C23×C29—21no—
145C23×C29—21no—
146C23×C29—21no—
147C23×C29—21no—
148C23×C29—21no—
149C23×C29—21no—
150C23×C29—22no—
151C23×C29—42no—
152C23×C29—42no—
153C23×C29—81no—
154C23×C29—81no—
155C23×C29—81no—
156C23×C29—82no—
157C23×C29—82no—
158C23×C29#9912yes—
159C23×C29C2×C22⋊C412yes—
160C23×C29C2×C22⋊C411yescharacteristic
161C23×C29#12012yes—
162Q8×C29—44no—
163Q8×C29—42no—
164D4×C29—28no—
165D4×C29—44no—
166D4×C29—42no—
167D4×C29—44no—
168D4×C29—44no—
169D4×C29—44no—
170D4×C29—42no—
171D4×C29—42no—
172D4×C29—84no—
173C2×C4×C29—21no—
174C2×C4×C29—21no—
175C2×C4×C29—21no—
176C2×C4×C29—21no—
177C2×C4×C29—24no—
178C2×C4×C29—21no—
179C2×C4×C29—21no—
180C2×C4×C29—22no—
181C2×C4×C29—21no—
182C2×C4×C29—21no—
183C2×C4×C29—21no—
184C2×C4×C29—21no—
185C2×C4×C29—44no—
186C2×C4×C29—44no—
187C2×C4×C29—44no—
188C2×C4×C29—44no—
189C2×C4×C29—42no—
190C2×C4×C29—42no—
191C2×C4×C29—82no—
192C2×C4×C29—82no—
193C2×C4×C29—82no—
194C2×C4×C29—84no—
195C2×C4×C29#9912yes—
196C2×C4×C29#12012yes—
197C8×C29—82no—
198#3485C2×C2911yescharacteristic
199#2121C2×C2914yescomplemented
200#1785C2×C2911yescharacteristic, complemented
201#1782C2×C2911yescharacteristic, complemented
202C22×C29—22no—
203C22×C29—22no—
204C22×C29—22no—
205C22×C29—41no—
206C22×C29—42no—
207C22×C29—42no—
208C22×C29—42no—
209C22×C29—42no—
210C22×C29—42no—
211C22×C29—44no—
212C22×C29—42no—
213C22×C29—81no—
214C22×C29—81no—
215C22×C29—81no—
216C22×C29—81no—
217C22×C29—162no—
218C22×C29—162no—
219C22×C29#37911yescharacteristic
220C22×C29C2×#9911yescharacteristic
221C22×C29C2×#9911yescharacteristic
222C4×C29—22no—
223C4×C29—22no—
224C4×C29—44no—
225C4×C29—41no—
226C4×C29—41no—
227C4×C29—42no—
228C4×C29—162no—
229C2×H(5,𝔽2)C4×C2911yescharacteristic, complemented
230C22×#41C22×C2911yescharacteristic
231#547—24no—
232#546—24no—
233#536—21no—
234#534—21no—
235#534C22×C2912yes—
236#525—21no—
237C2×#123—21no—
238C2×#123C4×C2912yescomplemented
239#518C4×C2911yescharacteristic, complemented
240C2×#101—21no—
241C2×#100C22×C2912yescomplemented
242C2×#99—21no—
243C2×#99C22×C2912yescomplemented
244#407—21no—
245#381—21no—
246#352—24no—
247AΣL(1,𝔽16)—24no—
248C2×C29—21no—
249C2×C29—21no—
250C2×C29—21no—
251C2×C29—42no—
252C2×C29—42no—
253C2×C29—42no—
254C2×C29—82no—
255C2×C29#178111yescharacteristic
256H(5,𝔽2)—42no—
257H(5,𝔽2)—42no—
258C2×#41—24no—
259C2×#41—21no—
260C2×#41—22no—
261C2×#41—42no—
262C2×#41—42no—
263C22×Q8D4×C2911yescharacteristic
264C22×D4—21no—
265C22×D4—22no—
266C22×D4C2×C4×C2911yescharacteristic
267C22×D4C2×C4×C2912yes—
268C22×D4D4×C2911yescharacteristic
269C22×D4C23×C2911yescharacteristic
270C23×C4C2×C4×C2911yescharacteristic
271C23×C4D4×C2911yescharacteristic
272C23×C4D4×C2911yescharacteristic
273C2×M16—22no—
274C4⋊Q8—22no—
275#124—22no—
276#123—24no—
277#123—22no—
278#123—22no—
279#123—22no—
280#123—22no—
281#123—24no—
282#123D4×C2914yes—
283C22⋊Q8—22no—
284C22⋊Q8—24no—
285#121—22no—
286#121—24no—
287#121—24no—
288#120—22no—
289#120—22no—
290#120D4×C2914yes—
291C4×D4—24no—
292C4×D4—24no—
293#117—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4C2×C4×C2912yes—
301#101—42no—
302#100—24no—
303#99—24no—
304#99—42no—
305C22⋊C8—42no—
306#95—42no—
307C29#1270611yescharacteristic, direct product factor, Sylow
308C24—21no—
309C24C22⋊C4×C2911yescharacteristic
310C24C22⋊C4×C2912yes—
311C24C22×C4×C2911yescharacteristic
312C24C2×D4×C2911yescharacteristic
313#41—42no—
314#41—416no—
315#41—48no—
316#41—42no—
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—41no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—21no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—48no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4C2×D4×C2914yes—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4C22⋊C4×C2911yescharacteristic
355C22×C4C22⋊C4×C2911yescharacteristic
356C22×C4C22⋊C4×C2911yescharacteristic
357C22×C4C22⋊C4×C2912yes—
358C22×C4C2×D4×C2911yescharacteristic
359M16—42no—
360M16—42no—
361C2×C8—42no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—22no—
368C22⋊C4—24no—
369C22⋊C4—22no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4C22⋊C4×C2914yes—
375C42—22no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—22no—
383C23—42no—
384C23—42no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—82no—
389C23—82no—
390C23C29×#9912yes—
391C23C2×C22⋊C4×C2912yes—
392C23C2×C22⋊C4×C2911yescharacteristic
393C23C29×#12012yes—
394Q8—44no—
395Q8—42no—
396D4—28no—
397D4—44no—
398D4—42no—
399D4—44no—
400D4—44no—
401D4—44no—
402D4—42no—
403D4—42no—
404D4—84no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—84no—
427C2×C4C29×#9912yes—
428C2×C4C29×#12012yes—
429C8—82no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—41no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—44no—
440C22—42no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—162no—
446C22—162no—
447C22C29×#37911yescharacteristic
448C22C2×C29×#9911yescharacteristic
449C22C2×C29×#9911yescharacteristic
450C4—22no—
451C4—22no—
452C4—44no—
453C4—41no—
454C4—41no—
455C4—42no—
456C4—162no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—42no—
461C2—42no—
462C2—42no—
463C2—82no—
464C2C29×#178111yescharacteristic
subgroupG/H
center, socleC2×C29#1781
commutatorC24C22×C4×C29
Frattini, cosocleC22×D4C23×C29
prime radicalC29×#3485C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C29 ◁ C23×C29 ◁ C22×#41×C29 ◁ G

Complements

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

NH ≅ G/Ntype
2. C29×#2121C2semidirect
3. C29×#1785C2semidirect
4. C29×#1782C2semidirect
5. C2×C29×H(5,𝔽2)C4semidirect
14. C2×C29×#123C4semidirect
15. C29×#518C4semidirect
17. C2×C29×#100C22semidirect
19. C2×C29×#99C22semidirect
143. #12706C29direct (N × H)
199. #2121C2×C29semidirect
200. #1785C2×C29semidirect
201. #1782C2×C29semidirect
229. C2×H(5,𝔽2)C4×C29semidirect
238. C2×#123C4×C29semidirect
239. #518C4×C29semidirect
241. C2×#100C22×C29semidirect
243. C2×#99C22×C29semidirect
307. C29#12706direct (N × H)

Maximal subgroups

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

subgroupindex
C29×#34852
C29×#21212
C29×#17852
C29×#17822
#1270629