Finite Group Databasebeta

← #16828969

Subgroups — #16828969

Order 3712.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#11745276C212yesmaximal, complemented
2#127×D29C211yesmaximal, characteristic, complemented
3C4×#90809C211yesmaximal, characteristic, complemented
4#14271363C211yesmaximal, characteristic, complemented
5#14271361C212yesmaximal, complemented
6#14271359C211yesmaximal, characteristic, complemented
7#14271357C212yesmaximal, complemented
8#14271355C211yesmaximal, characteristic, complemented
9#14271185C211yesmaximal, characteristic, complemented
10#14271166C212yesmaximal, complemented
11C29×#460C211yesmaximal, characteristic, complemented
12C2×#90809C2211yescharacteristic
13C2×D4×D29—21no—
14C2×D4×D29—21no—
15C2×D4×D29C2212yes—
16C2×SD16×C29C2212yescomplemented
17C29×#127C2211yescharacteristic
18C4×Q8×C29C2211yescharacteristic, complemented
19Q8×Dic29C2211yescharacteristic, complemented
20C2×#90784C2212yescomplemented
21#11745302—21no—
22#11745302—21no—
23#11745302C2212yes—
24C2×#90782C2212yescomplemented
25#11745276—28no—
26C2×D232C2212yescomplemented
27C2×#90772C2212yes—
28#11745255—21no—
29#11745255C2211yescharacteristic, complemented
30#11745254—21no—
31#11745254C2211yescharacteristic
32#11745236C2211yescharacteristic
33C4×D116—21no—
34C4×D116C2211yescharacteristic, complemented
35C42×D29—21no—
36C42×D29C2211yescharacteristic
37C29×C4⋊C8C2211yescharacteristic, complemented
38C29×#102C2212yescomplemented
39#11745180C2212yescomplemented
40#11745169C2212yescomplemented
41#11745162C2212yescomplemented
42#11745157C2212yescomplemented
43C4⋊#6597C2211yescharacteristic, complemented
44C23×D29—22no—
45C2×Q8×C29C2311yescharacteristic
46C2×D4×C29—21no—
47C2×D4×C29—21no—
48C2×D4×C29C2312yes—
49C2×C22⋊D29—22no—
50C2×C22⋊D29—22no—
51C2×C22⋊D29—22no—
52#90809—22no—
53#90809D414yescomplemented
54D4×D29—24no—
55D4×D29—24no—
56D4×D29—44no—
57D4×D29—44no—
58C2×D116—21no—
59C2×D116—21no—
60C2×D116—21no—
61C2×D116C2312yes—
62C2×D116C2311yescharacteristic
63C2×C4×D29—21no—
64C2×C4×D29—21no—
65C2×C4×D29—21no—
66C2×C4×D29D411yescharacteristic
67C2×C4×D29D411yescharacteristic
68C2×C4×D29C2311yescharacteristic
69SD16×C29—24no—
70C2×C8×C29C2312yes—
71C4⋊C4×C29—21no—
72C4⋊C4×C29C2311yescharacteristic
73C42×C29—21no—
74C42×C29C2311yescharacteristic
75#90784—24no—
76#90782—24no—
77#90781—21no—
78#90781—21no—
79#90781—21no—
80C4⋊Dic29—21no—
81C4⋊Dic29C2311yescharacteristic
82C4×Dic29—21no—
83C4×Dic29—21no—
84C4×Dic29D411yescharacteristic
85C4×Dic29D411yescharacteristic
86C4×Dic29C2311yescharacteristic
87C2×#6597C2312yes—
88D232—24no—
89#90772—24no—
90C23×C29—22no—
91C22×D29—21no—
92C22×D29—22no—
93C22×D29—44no—
94C22×D29—44no—
95C22×D29—44no—
96C22×D29C2×D411yescharacteristic
97Q8×C29—21no—
98Q8×C29C2×D412yescomplemented
99D4×C29—22no—
100D4×C29—22no—
101D4×C29—42no—
102D4×C29—42no—
103C2×C4×C29—21no—
104C2×C4×C29—21no—
105C2×C4×C29C2×D411yescharacteristic
106C2×C4×C29C2×D411yescharacteristic
107C2×C4×C29C2411yescharacteristic
108C22⋊D29—44no—
109C22⋊D29—44no—
110C22⋊D29—44no—
111C2×Dic29—21no—
112C2×Dic29C2×D411yescharacteristic
113C2×Dic29C2×D411yescharacteristic
114C2×Dic29C2×D411yescharacteristic
115D116—24no—
116D116—22no—
117D116—21no—
118D116—22no—
119D116—42no—
120D116—42no—
121D116C2×D412yescomplemented
122C4×D29—22no—
123C4×D29—24no—
124C4×D29—22no—
125C4×D29—42no—
126C4×D29—42no—
127C4×D29C2×D412yes—
128C4×D29#4112yes—
129C8×C29—22no—
130#6597—22no—
131#3038—291nomaximal, Sylow
132C22×C29—42no—
133C22×C29—42no—
134C22×C29—42no—
135C22×C29C22×D411yescharacteristic
136C2×D29—21no—
137C2×D29—21no—
138C2×D29—22no—
139C2×D29—41no—
140C2×D29—41no—
141C2×D29—42no—
142C2×D29—42no—
143C2×D29—42no—
144C2×D29—88no—
145C2×D29#12112yes—
146C4×C29—22no—
147C4×C29—21no—
148C4×C29—41no—
149C4×C29AGL(1,ℤ/8)12yescomplemented
150C4×C29C22×D411yescharacteristic
151C4×C29C2×#4111yescharacteristic
152Dic29—21no—
153Dic29—41no—
154Dic29#12112yes—
155Dic29AGL(1,ℤ/8)12yescomplemented
156C2×AGL(1,ℤ/8)—292no—
157C2×#127—291no—
158C4×#41—291no—
159#460—292no—
160#460—291no—
161#460D2911yescharacteristic, complemented
162#459—291no—
163#459—291no—
164#459—292no—
165#423—291no—
166#418—292no—
167C2×C29—82no—
168C2×C29C2×#12111yescharacteristic
169C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
170C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
171D29—22no—
172D29—42no—
173D29—82no—
174C2×#41—291no—
175C22×D4—292no—
176C22×D4—581no—
177C22×D4—581no—
178AGL(1,ℤ/8)—588no—
179C2×SD16—292no—
180C2×SD16C2×D2912yescomplemented
181C2×D8—292no—
182C2×D8—292no—
183C2×M16—292no—
184#127—292no—
185#127—291no—
186#127—581no—
187#127—581no—
188#127C2×D2911yescharacteristic
189C4×Q8—291no—
190C4×Q8C2×D2911yescharacteristic, complemented
191C4×D4—291no—
192C4×D4—291no—
193C4×D4—581no—
194C4×D4—581no—
195#117—291no—
196#117—581no—
197C2×C42—291no—
198C2×C42—581no—
199C4⋊C8—292no—
200C4⋊C8—291no—
201C4⋊C8C2×D2911yescharacteristic, complemented
202#102—292no—
203#102—292no—
204#102—292no—
205#102C2×D2912yescomplemented
206C29#303811yescharacteristic, complemented, Sylow
207C24—582no—
208#41—294no—
209#41—582no—
210C2×Q8C22×D2911yescharacteristic
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—291no—
214C2×D4—292no—
215C2×D4—581no—
216C2×D4—584no—
217C2×D4—584no—
218C2×D4—582no—
219C2×D4—582no—
220C2×D4—582no—
221C2×D4—581no—
222C2×D4—581no—
223C2×D4—1164no—
224C2×D4—1164no—
225C2×D4C22×D2912yes—
226C22×C4—291no—
227C22×C4—291no—
228C22×C4—291no—
229C22×C4—581no—
230C22×C4—581no—
231C22×C4—581no—
232SD16—24no—
233SD16—584no—
234D8—584no—
235D8—584no—
236M16—584no—
237C2×C8—292no—
238C2×C8C22×D2912yes—
239C4⋊C4—21no—
240C4⋊C4—291no—
241C4⋊C4—581no—
242C4⋊C4C22×D2911yescharacteristic
243C22⋊C4—581no—
244C22⋊C4—581no—
245C22⋊C4—581no—
246C42—21no—
247C42—291no—
248C42—291no—
249C42—291no—
250C42—581no—
251C42—581no—
252C42C22×D2911yescharacteristic
253C23—22no—
254C23—291no—
255C23—581no—
256C23—582no—
257C23—1164no—
258C23—1164no—
259C23—1164no—
260Q8—21no—
261Q8D4×D2912yescomplemented
262D4—22no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—292no—
267D4—584no—
268D4—582no—
269D4—581no—
270D4—582no—
271D4—1162no—
272D4—1164no—
273D4—1164no—
274D4—1164no—
275D4—1162no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—291no—
279C2×C4—291no—
280C2×C4—292no—
281C2×C4—292no—
282C2×C4—291no—
283C2×C4—582no—
284C2×C4—584no—
285C2×C4—582no—
286C2×C4—581no—
287C2×C4—1162no—
288C2×C4—1162no—
289C2×C4D4×D2911yescharacteristic
290C2×C4D4×D2911yescharacteristic
291C2×C4C23×D2911yescharacteristic
292C8—22no—
293C8—582no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—292no—
298C22—581no—
299C22—581no—
300C22—582no—
301C22—1161no—
302C22—1161no—
303C22—1162no—
304C22—1162no—
305C22—1162no—
306C22—2328no—
307C22C2×D4×D2911yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—292no—
312C4—292no—
313C4—581no—
314C4—1161no—
315C4#1174527612yescomplemented
316C4C2×D4×D2911yescharacteristic
317C4#41×D2911yescharacteristic
318C2—82no—
319C2—582no—
320C2—1162no—
321C2—2322no—
322C2#121×D2911yescharacteristic
323C2AGL(1,ℤ/8)×D2911yescharacteristic
324C2C2×#1174527611yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#460C2
socleC22×C29C22×D4
hypercenter#460D29
hypocenterC29#3038
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#11745276C2semidirect
2. #127×D29C2semidirect
3. C4×#90809C2semidirect
4. #14271363C2semidirect
5. #14271361C2semidirect
6. #14271359C2semidirect
7. #14271357C2semidirect
8. #14271355C2semidirect
9. #14271185C2semidirect
10. #14271166C2semidirect
11. C29×#460C2semidirect
16. C2×SD16×C29C22semidirect
18. C4×Q8×C29C22semidirect
19. Q8×Dic29C22semidirect
20. C2×#90784C22semidirect
24. C2×#90782C22semidirect
26. C2×D232C22semidirect
29. #11745255C22semidirect
34. C4×D116C22semidirect
37. C29×C4⋊C8C22semidirect
38. C29×#102C22semidirect
39. #11745180C22semidirect
40. #11745169C22semidirect
41. #11745162C22semidirect
42. #11745157C22semidirect
43. C4⋊#6597C22semidirect
53. #90809D4semidirect
98. Q8×C29C2×D4semidirect
121. D116C2×D4semidirect
149. C4×C29AGL(1,ℤ/8)semidirect
155. Dic29AGL(1,ℤ/8)semidirect
161. #460D29semidirect
180. C2×SD16C2×D29semidirect
190. C4×Q8C2×D29semidirect
201. C4⋊C8C2×D29semidirect
205. #102C2×D29semidirect
206. C29#3038semidirect
261. Q8D4×D29semidirect
315. C4#11745276semidirect

Maximal subgroups

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

subgroupindex
C2×#117452762
#127×D292
C4×#908092
#142713632
#142713612
#142713592
#142713572
#142713552
#142711852
#142711662
C29×#4602
#303829