Finite Group Databasebeta

← #16831107

Subgroups — #16831107

Order 3712.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 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#14271835C211yesmaximal, characteristic, complemented
3#14271798C211yesmaximal, characteristic, complemented
4#14271539C211yesmaximal, characteristic, complemented
5#14271415C212yesmaximal, complemented
6#14271359C211yesmaximal, characteristic, complemented
7#14271307C212yesmaximal, complemented
8#14271298C212yesmaximal, complemented
9C4×#90784C211yesmaximal, characteristic, complemented
10#14271240C211yesmaximal, characteristic, complemented
11C29×#460C211yesmaximal, characteristic, complemented
12C2×#90809C2212yescomplemented
13C2×D4×D29—21no—
14C2×D4×D29C2212yes—
15C2×SD16×C29C2212yescomplemented
16C29×#127C2211yescharacteristic
17C4×Q8×C29C2211yescharacteristic, complemented
18C2×#90784C2211yescharacteristic
19#11745300—21no—
20#11745300C2212yes—
21#11745299—22no—
22C2×#90782C2212yescomplemented
23C2×#90782C2212yescomplemented
24#11745276—28no—
25C2×D232C2212yes—
26C2×#90772C2212yescomplemented
27#11745257—22no—
28#11745257C2212yescomplemented
29#11745236—21no—
30#11745236C2211yescharacteristic, complemented
31C4×D116—21no—
32C4×D116C2211yescharacteristic
33C4×D116C2211yescharacteristic
34C29×C4⋊C8C2211yescharacteristic, complemented
35C29×#102C2212yescomplemented
36#11745183C2211yescharacteristic, complemented
37#11745169C2212yescomplemented
38#11745168C2212yescomplemented
39#11745157C2211yescharacteristic
40#11745156C2211yescharacteristic, complemented
41C4×#6597C2211yescharacteristic, complemented
42C23×D29—22no—
43C2×Q8×C29C2311yescharacteristic
44C2×D4×C29—21no—
45C2×D4×C29—21no—
46C2×D4×C29C2312yes—
47C2×C22⋊D29—22no—
48C2×C22⋊D29—22no—
49#90809—24no—
50#90809—42no—
51D4×D29—24no—
52D4×D29—42no—
53D4×D29—44no—
54C2×D116—22no—
55C2×D116—21no—
56C2×D116—22no—
57C2×D116—21no—
58C2×D116D411yescharacteristic
59C2×D116C2312yes—
60C2×D116C2311yescharacteristic
61C2×C4×D29—21no—
62C2×C4×D29—22no—
63C2×C4×D29C2312yes—
64SD16×C29—24no—
65C2×C8×C29C2312yes—
66C4⋊C4×C29—21no—
67C4⋊C4×C29C2311yescharacteristic
68C42×C29—21no—
69C42×C29C2311yescharacteristic
70C22⋊Dic29—22no—
71#90784D414yescomplemented
72#90782—24no—
73#90782—42no—
74#90781—21no—
75#90781—22no—
76#90781D412yes—
77C4⋊Dic29D411yescharacteristic
78C4⋊Dic29C2311yescharacteristic
79C2×#6597C2311yescharacteristic
80C2×#6597C2311yescharacteristic
81D232—24no—
82#90772—24no—
83C23×C29—22no—
84C22×D29—21no—
85C22×D29—22no—
86C22×D29—44no—
87C22×D29—44no—
88C22×D29—82no—
89C22×D29C2×D412yes—
90Q8×C29—21no—
91Q8×C29C2×D412yescomplemented
92D4×C29—22no—
93D4×C29—22no—
94D4×C29—42no—
95D4×C29—81no—
96C2×C4×C29—21no—
97C2×C4×C29—21no—
98C2×C4×C29C2×D411yescharacteristic
99C2×C4×C29C2×D411yescharacteristic
100C2×C4×C29C2411yescharacteristic
101C22⋊D29—44no—
102C22⋊D29—44no—
103C2×Dic29—21no—
104C2×Dic29C2×D412yes—
105D116—22no—
106D116—22no—
107D116—21no—
108D116—22no—
109D116—44no—
110D116—42no—
111D116—81no—
112D116—82no—
113D116C2×D412yes—
114C4×D29—22no—
115C4×D29—22no—
116C4×D29—44no—
117C4×D29—42no—
118C8×C29—22no—
119#6597—21no—
120#6597C2×D412yescomplemented
121#3274—291nomaximal, Sylow
122C22×C29—42no—
123C22×C29—42no—
124C22×C29—42no—
125C22×C29C22×D411yescharacteristic
126C2×D29—22no—
127C2×D29—22no—
128C2×D29—22no—
129C2×D29—22no—
130C2×D29—41no—
131C2×D29—41no—
132C2×D29—42no—
133C2×D29—42no—
134C2×D29—42no—
135C2×D29—88no—
136C4×C29—22no—
137C4×C29—21no—
138C4×C29—41no—
139C4×C29AGL(1,ℤ/8)12yescomplemented
140C4×C29C22×D411yescharacteristic
141C4×C29H(5,𝔽2)11yescharacteristic
142Dic29—22no—
143Dic29—41no—
144C2×AGL(1,ℤ/8)—292no—
145#550—291no—
146D42—291no—
147#493—291no—
148#469—292no—
149#460D2911yescharacteristic, complemented
150#459—291no—
151#449—292no—
152#447—292no—
153C4×SD16—291no—
154#436—291no—
155C2×C29—82no—
156C2×C29D4211yescharacteristic
157C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
158C2×C29#57611yescharacteristic
159D29—42no—
160D29—42no—
161D29—82no—
162C2×#41—292no—
163C22×D4—292no—
164C22×D4—581no—
165AGL(1,ℤ/8)—588no—
166C2×SD16—291no—
167C2×SD16C2×D2912yescomplemented
168C2×D8—292no—
169C2×D8—292no—
170C2×D8—292no—
171C2×M16—292no—
172#127—291no—
173#127—581no—
174#127C2×D2911yescharacteristic
175#121—292no—
176#121—292no—
177#121—582no—
178#121—581no—
179#120—582no—
180C4×Q8C2×D2911yescharacteristic, complemented
181C4×D4—291no—
182C4×D4—291no—
183C4×D4—581no—
184#106—291no—
185C4⋊C8C2×D2911yescharacteristic, complemented
186#103—291no—
187#102—292no—
188#102—291no—
189#102C2×D2912yescomplemented
190C22⋊C8—292no—
191C4×C8—291no—
192C29#327411yescharacteristic, complemented, Sylow
193C24—582no—
194#41—584no—
195#41—1162no—
196C2×Q8C22×D2911yescharacteristic
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—291no—
200C2×D4—291no—
201C2×D4—292no—
202C2×D4—582no—
203C2×D4—582no—
204C2×D4—582no—
205C2×D4—584no—
206C2×D4—582no—
207C2×D4—581no—
208C2×D4—581no—
209C2×D4—1162no—
210C2×D4—1164no—
211C2×D4C22×D2912yes—
212C22×C4—292no—
213C22×C4—582no—
214C22×C4—581no—
215SD16—24no—
216SD16—294no—
217D8—584no—
218D8—584no—
219D8—1162no—
220M16—584no—
221C2×C8—291no—
222C2×C8—291no—
223C2×C8C22×D2912yes—
224C4⋊C4—21no—
225C4⋊C4—291no—
226C4⋊C4—291no—
227C4⋊C4C22×D2911yescharacteristic
228C22⋊C4—292no—
229C22⋊C4—582no—
230C22⋊C4—582no—
231C22⋊C4—581no—
232C42—21no—
233C42C22×D2911yescharacteristic
234C23—22no—
235C23—292no—
236C23—581no—
237C23—582no—
238C23—1164no—
239C23—1164no—
240C23—2322no—
241Q8—21no—
242Q8D4×D2912yescomplemented
243D4—22no—
244D4—22no—
245D4—42no—
246D4—81no—
247D4—292no—
248D4—582no—
249D4—581no—
250D4—582no—
251D4—582no—
252D4—1164no—
253D4—1164no—
254D4—1164no—
255D4—1162no—
256D4—2321no—
257D4—2322no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—292no—
261C2×C4—582no—
262C2×C4—582no—
263C2×C4—581no—
264C2×C4—1164no—
265C2×C4—1162no—
266C2×C4D4×D2911yescharacteristic
267C2×C4D4×D2911yescharacteristic
268C2×C4C23×D2911yescharacteristic
269C8—22no—
270C8—292no—
271C8—581no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—582no—
276C22—582no—
277C22—582no—
278C22—582no—
279C22—1161no—
280C22—1161no—
281C22—1162no—
282C22—1162no—
283C22—1162no—
284C22—2328no—
285C22C2×D4×D2911yescharacteristic
286C4—22no—
287C4—21no—
288C4—41no—
289C4—582no—
290C4—1161no—
291C4#1174527612yescomplemented
292C4C2×D4×D2911yescharacteristic
293C4#1174536411yescharacteristic
294C2—82no—
295C2—1162no—
296C2—1162no—
297C2—2322no—
298C2#1427179711yescharacteristic
299C2C2×#1174527611yescharacteristic
300C2#1427196711yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#460C2
socleC22×C29C22×D4
hypercenter#460D29
hypocenterC29#3274
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. #14271835C2semidirect
3. #14271798C2semidirect
4. #14271539C2semidirect
5. #14271415C2semidirect
6. #14271359C2semidirect
7. #14271307C2semidirect
8. #14271298C2semidirect
9. C4×#90784C2semidirect
10. #14271240C2semidirect
11. C29×#460C2semidirect
12. C2×#90809C22semidirect
15. C2×SD16×C29C22semidirect
17. C4×Q8×C29C22semidirect
22. C2×#90782C22semidirect
23. C2×#90782C22semidirect
26. C2×#90772C22semidirect
28. #11745257C22semidirect
30. #11745236C22semidirect
34. C29×C4⋊C8C22semidirect
35. C29×#102C22semidirect
36. #11745183C22semidirect
37. #11745169C22semidirect
38. #11745168C22semidirect
40. #11745156C22semidirect
41. C4×#6597C22semidirect
71. #90784D4semidirect
91. Q8×C29C2×D4semidirect
120. #6597C2×D4semidirect
139. C4×C29AGL(1,ℤ/8)semidirect
149. #460D29semidirect
167. C2×SD16C2×D29semidirect
180. C4×Q8C2×D29semidirect
185. C4⋊C8C2×D29semidirect
189. #102C2×D29semidirect
192. C29#3274semidirect
242. Q8D4×D29semidirect
291. C4#11745276semidirect

Maximal subgroups

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

subgroupindex
C2×#117452762
#142718352
#142717982
#142715392
#142714152
#142713592
#142713072
#142712982
C4×#907842
#142712402
C29×#4602
#327429