Finite Group Databasebeta

← #16828732

Subgroups — #16828732

Order 3712.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#90805C211yesmaximal, characteristic, complemented
2C2×#11745313C211yesmaximal, characteristic, complemented
3C2×#11745311C211yesmaximal, characteristic, complemented
4#14271323C214yesmaximal, complemented
5#14271306C214yesmaximal, complemented
6#14271171C211yesmaximal, characteristic, complemented
7#14271167C211yesmaximal, characteristic, complemented
8#14271127C211yesmaximal, characteristic, complemented
9C29×#525C211yesmaximal, characteristic, complemented
10C2×#41×C29C2211yescharacteristic
11C23×C4×C29C2211yescharacteristic
12C22×C22⋊D29—21no—
13C2×#90805—22no—
14C2×#90805—24no—
15C2×#90805—24no—
16C2×#90805C2214yes—
17C22×D116C2211yescharacteristic
18C22×C4×D29—21no—
19C22×Dic58C2211yescharacteristic
20C29×#123—21no—
21C29×C22⋊Q8C2212yescomplemented
22C29×#121C2212yescomplemented
23C29×#120—21no—
24C4×D4×C29—22no—
25C29×#117C2211yescharacteristic
26#11745313—24no—
27#11745311—24no—
28C2×#90785C2212yescomplemented
29C2×#90784C2212yescomplemented
30C2×#90783C2212yescomplemented
31C2×#90782C2212yescomplemented
32C2×#90777C2211yescharacteristic
33C2×#90777C2211yescharacteristic
34C22⋊#6597C2214yescomplemented
35#11745158C2214yescomplemented
36#11745157C2214yescomplemented
37C24×C29D411yescharacteristic
38C23×D29—21no—
39#41×C29—22no—
40#41×C29—22no—
41C2×Q8×C29C2311yescharacteristic
42C2×D4×C29—22no—
43C2×D4×C29—21no—
44C2×D4×C29C2311yescharacteristic
45C22×C4×C29—22no—
46C22×C4×C29—22no—
47C22×C4×C29—21no—
48C22×C4×C29D412yes—
49C22×C4×C29D411yescharacteristic
50C22×C4×C29C2312yes—
51C22×C4×C29C2311yescharacteristic
52C2×C22⋊D29—28no—
53C2×C22⋊D29—24no—
54C22×Dic29—21no—
55#90805—216no—
56#90805—28no—
57#90805—28no—
58C2×D116—22no—
59C2×D116—21no—
60C2×D116D414yescomplemented
61C2×D116C2312yes—
62C2×C4×D29—28no—
63C2×C4×D29—24no—
64C2×Dic58—22no—
65C2×Dic58—21no—
66C2×Dic58D414yescomplemented
67C2×Dic58C2312yes—
68C4⋊C4×C29—22no—
69C4⋊C4×C29C2312yes—
70C22⋊C4×C29—21no—
71C22⋊C4×C29—22no—
72C22⋊C4×C29—22no—
73C42×C29—21no—
74#90785—24no—
75#90784—24no—
76#90783—24no—
77#90782—24no—
78#90777—22no—
79#90777—22no—
80C2×#6597C2312yes—
81C2×#6597C2312yes—
82C23×C29—22no—
83C23×C29—21no—
84C23×C29—42no—
85C23×C29C2×D412yes—
86C23×C29C2×D411yescharacteristic
87C22×D29—24no—
88C22×D29—22no—
89C22×D29—44no—
90Q8×C29—21no—
91Q8×C29—21no—
92D4×C29—21no—
93D4×C29—21no—
94D4×C29—44no—
95D4×C29—42no—
96C2×C4×C29—22no—
97C2×C4×C29—24no—
98C2×C4×C29—22no—
99C2×C4×C29—22no—
100C2×C4×C29—22no—
101C2×C4×C29—21no—
102C2×C4×C29—42no—
103C2×C4×C29C2×D412yes—
104C2×C4×C29C2×D412yes—
105C2×C4×C29C2×D412yes—
106C2×C4×C29C2×D411yescharacteristic
107C2×C4×C29C2411yescharacteristic
108C22⋊D29—216no—
109C2×Dic29—24no—
110C2×Dic29—22no—
111D116—24no—
112D116—22no—
113D116C2×D414yes—
114C4×D29—216no—
115Dic58—24no—
116Dic58—22no—
117Dic58C2×D414yes—
118#6597—22no—
119#6597—22no—
120#3005—291nomaximal, Sylow
121C22×C29—22no—
122C22×C29—22no—
123C22×C29—22no—
124C22×C29—22no—
125C22×C29—44no—
126C22×C29—41no—
127C22×C29—42no—
128C22×C29#12012yes—
129C22×C29C22×D411yescharacteristic
130C2×D29—24no—
131C2×D29—48no—
132C2×D29—44no—
133C4×C29—22no—
134C4×C29—41no—
135C4×C29—42no—
136C4×C29#12012yes—
137C4×C29C22×D411yescharacteristic
138C4×C29C22×D411yescharacteristic
139Dic29—24no—
140C22×#41—291no—
141C2×#137—291no—
142C2×AGL(1,ℤ/8)—291no—
143#525D2911yescharacteristic, complemented
144#452—294no—
145#449—294no—
146#419—291no—
147#418—291no—
148#407—291no—
149C2×C29—22no—
150C2×C29—42no—
151C2×C29—81no—
152C2×C29C2×#12011yescharacteristic
153C2×C29#57512yes—
154D29—44no—
155C2×#41—294no—
156C2×#41—584no—
157C2×#41—584no—
158C2×#41—582no—
159C2×#41C2×D2911yescharacteristic
160C22×Q8—291no—
161C22×D4—291no—
162C22×D4—581no—
163C23×C4—581no—
164C23×C4C2×D2911yescharacteristic
165#137—584no—
166AGL(1,ℤ/8)—584no—
167C2×Q16—292no—
168C2×SD16—292no—
169C2×SD16—292no—
170C2×D8—292no—
171C2×M16—291no—
172C2×M16—291no—
173#123—21no—
174C22⋊Q8C2×D2912yescomplemented
175#121C2×D2912yescomplemented
176#120—21no—
177C4×D4—22no—
178#117C2×D2911yescharacteristic
179#103—294no—
180#102—294no—
181C22⋊C8—294no—
182C29#300511yescharacteristic, complemented, Sylow
183C24—581no—
184C24C22⋊D2911yescharacteristic
185#41—22no—
186#41—22no—
187#41—588no—
188#41—588no—
189#41—5816no—
190C2×Q8—294no—
191C2×Q8—292no—
192C2×Q8—582no—
193C2×Q8—581no—
194C2×Q8C22×D2911yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—294no—
198C2×D4—292no—
199C2×D4—588no—
200C2×D4—584no—
201C2×D4—582no—
202C2×D4—581no—
203C2×D4C22×D2911yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—588no—
208C22×C4—581no—
209C22×C4—584no—
210C22×C4C22⋊D2912yes—
211C22×C4C22⋊D2911yescharacteristic
212C22×C4C22×D2912yes—
213C22×C4C22×D2911yescharacteristic
214Q16—584no—
215SD16—584no—
216SD16—584no—
217D8—584no—
218M16—582no—
219M16—582no—
220C2×C8—292no—
221C2×C8—292no—
222C4⋊C4—22no—
223C4⋊C4C22×D2912yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—584no—
232C23—582no—
233C23—1164no—
234C23C2×C22⋊D2912yes—
235C23C2×C22⋊D2911yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—294no—
239Q8—584no—
240Q8—582no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—294no—
246D4—5816no—
247D4—584no—
248D4—582no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—5816no—
257C2×C4—584no—
258C2×C4—582no—
259C2×C4D4×D2912yes—
260C2×C4D4×D2912yes—
261C2×C4C2×C22⋊D2912yes—
262C2×C4C2×C22⋊D2911yescharacteristic
263C2×C4C23×D2911yescharacteristic
264C8—582no—
265C8—582no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—41no—
272C22—42no—
273C22—584no—
274C22—1168no—
275C22—1164no—
276C22#1174529912yes—
277C22C22×C22⋊D2911yescharacteristic
278C4—22no—
279C4—41no—
280C4—42no—
281C4—584no—
282C4#1174529912yes—
283C4C2×D4×D2911yescharacteristic
284C4C2×D4×D2911yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—1164no—
289C2C2×#1174529911yescharacteristic
290C2#1427196112yes—
subgroupG/H
centerC22C22×C22⋊D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#525C2
socleC22×C29C22×D4
hypercenter#525D29
hypocenterC29#3005
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#90805C2semidirect
2. C2×#11745313C2semidirect
3. C2×#11745311C2semidirect
4. #14271323C2semidirect
5. #14271306C2semidirect
6. #14271171C2semidirect
7. #14271167C2semidirect
8. #14271127C2semidirect
9. C29×#525C2semidirect
21. C29×C22⋊Q8C22semidirect
22. C29×#121C22semidirect
28. C2×#90785C22semidirect
29. C2×#90784C22semidirect
30. C2×#90783C22semidirect
31. C2×#90782C22semidirect
34. C22⋊#6597C22semidirect
35. #11745158C22semidirect
36. #11745157C22semidirect
60. C2×D116D4semidirect
66. C2×Dic58D4semidirect
143. #525D29semidirect
174. C22⋊Q8C2×D29semidirect
175. #121C2×D29semidirect
182. C29#3005semidirect

Maximal subgroups

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

subgroupindex
C22×#908052
C2×#117453132
C2×#117453112
#142713232
#142713062
#142711712
#142711672
#142711272
C29×#5252
#300529