Finite Group Databasebeta

← #16828758

Subgroups — #16828758

Order 3712.

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

normal subgroups only

This group has 784 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
1C2×#11745364C211yesmaximal, characteristic, complemented
2C2×#11745294C211yesmaximal, characteristic, complemented
3C22×#90782C211yesmaximal, characteristic, complemented
4#14271306C214yesmaximal, complemented
5#14271296C214yesmaximal, complemented
6#14271167C211yesmaximal, characteristic, complemented
7#14271157C211yesmaximal, characteristic, complemented
8#14271130C211yesmaximal, characteristic, complemented
9C29×#535C211yesmaximal, characteristic, complemented
10C2×#41×C29C2211yescharacteristic
11C22×D4×C29C2211yescharacteristic, complemented
12#11745364—24no—
13#11745364—24no—
14C2×#90809—21no—
15C2×D4×D29—21no—
16C2×D4×D29—22no—
17C2×#90805C2211yescharacteristic
18C2×#90805C2212yescomplemented
19C22×D116C2211yescharacteristic
20C22×D116C2212yescomplemented
21C29×#127—21no—
22C29×#124—21no—
23C29×#121C2214yescomplemented
24C29×#120—22no—
25C29×#117C2211yescharacteristic, complemented
26C2×#90783C2212yescomplemented
27#11745294—24no—
28C2×#90782—24no—
29C2×#90782C2212yescomplemented
30C2×#90782C2214yescomplemented
31C2×#90777C2211yescharacteristic, complemented
32C22×#6597C2211yescharacteristic, complemented
33C22⋊#6597C2214yescomplemented
34#11745158C2212yescomplemented
35#11745157C2212yescomplemented
36#11745157C2214yescomplemented
37C24×C29—21no—
38C23×D29—21no—
39C23×D29—22no—
40#41×C29—22no—
41#41×C29—22no—
42C2×Q8×C29D411yescharacteristic
43C2×D4×C29—24no—
44C2×D4×C29—22no—
45C2×D4×C29—22no—
46C2×D4×C29—21no—
47C2×D4×C29D412yes—
48C2×D4×C29D411yescharacteristic
49C2×D4×C29C2312yes—
50C22×C4×C29C2312yes—
51C22×C4×C29C2311yescharacteristic
52C2×C22⋊D29—21no—
53C2×C22⋊D29—22no—
54#90809—28no—
55D4×D29—216no—
56D4×D29—28no—
57#90805—22no—
58#90805—24no—
59#90805—24no—
60#90805D414yescomplemented
61C2×D116—21no—
62C2×D116—22no—
63C2×D116—22no—
64C2×D116—24no—
65C2×D116—24no—
66C2×D116D414yescomplemented
67C2×D116C2312yes—
68C2×D116C2311yescharacteristic
69C2×C4×D29—21no—
70C2×C4×D29—22no—
71C2×Dic58C2311yescharacteristic
72C4⋊C4×C29C2312yes—
73C22⋊C4×C29—21no—
74C22⋊C4×C29—24no—
75C42×C29—21no—
76#90783—24no—
77#90782—28no—
78#90782—24no—
79#90777—22no—
80C2×#6597—22no—
81C2×#6597C2312yes—
82C2×#6597C2312yes—
83C23×C29—22no—
84C23×C29—42no—
85C23×C29—82no—
86C23×C29C2×D412yes—
87C23×C29C2×D411yescharacteristic
88C22×D29—21no—
89C22×D29—22no—
90C22×D29—28no—
91C22×D29—24no—
92C22×D29—82no—
93C22×D29—84no—
94Q8×C29—22no—
95D4×C29—24no—
96D4×C29—22no—
97D4×C29—22no—
98D4×C29—22no—
99D4×C29—42no—
100D4×C29—84no—
101C2×C4×C29—22no—
102C2×C4×C29—22no—
103C2×C4×C29—22no—
104C2×C4×C29C2×D412yes—
105C2×C4×C29C2×D412yes—
106C2×C4×C29C2×D412yes—
107C2×C4×C29C2×D411yescharacteristic
108C2×C4×C29C2411yescharacteristic
109C22⋊D29—28no—
110C22⋊D29—24no—
111C2×Dic29—21no—
112D116—28no—
113D116—24no—
114D116—21no—
115D116—22no—
116D116C2×D412yes—
117D116C2×D414yes—
118C4×D29—28no—
119C4×D29—24no—
120Dic58—21no—
121Dic58C2×D412yes—
122#6597—22no—
123#6597—22no—
124#3008—291nomaximal, Sylow
125C22×C29—22no—
126C22×C29—22no—
127C22×C29—24no—
128C22×C29—42no—
129C22×C29—44no—
130C22×C29—84no—
131C22×C29#12012yes—
132C22×C29C22×D411yescharacteristic
133C2×D29—24no—
134C2×D29—22no—
135C2×D29—44no—
136C2×D29—42no—
137C2×D29—84no—
138C2×D29—88no—
139C4×C29—22no—
140C4×C29—42no—
141C4×C29#12012yes—
142C4×C29C22×D411yescharacteristic
143C4×C29C22×D411yescharacteristic
144Dic29—22no—
145C2×H(5,𝔽2)—291no—
146C2×AGL(1,ℤ/8)—291no—
147C22×D8—291no—
148#535D2911yescharacteristic, complemented
149#449—294no—
150#447—294no—
151#418—291no—
152#416—291no—
153#408—291no—
154C2×C29—22no—
155C2×C29—42no—
156C2×C29—82no—
157C2×C29C2×#12011yescharacteristic
158C2×C29#57612yes—
159D29—44no—
160D29—42no—
161H(5,𝔽2)—584no—
162H(5,𝔽2)—584no—
163C2×#41—291no—
164C2×#41—292no—
165C2×#41—581no—
166C2×#41C2×D2911yescharacteristic
167C22×D4—291no—
168C22×D4—292no—
169C22×D4—582no—
170C22×D4—581no—
171C22×D4C2×D2911yescharacteristic, complemented
172AGL(1,ℤ/8)—584no—
173C2×SD16—292no—
174C2×D8—294no—
175C2×D8—292no—
176C2×D8—584no—
177C2×M16—291no—
178C22×C8—291no—
179#127—21no—
180#124—21no—
181#121C2×D2914yescomplemented
182#120—22no—
183#117C2×D2911yescharacteristic, complemented
184#103—292no—
185#102—292no—
186#102—294no—
187C22⋊C8—294no—
188C29#300811yescharacteristic, complemented, Sylow
189C24—21no—
190C24—582no—
191C24—581no—
192#41—22no—
193#41—22no—
194#41—294no—
195#41—584no—
196#41—584no—
197#41—588no—
198#41—582no—
199C2×Q8—291no—
200C2×Q8C22⋊D2911yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—294no—
206C2×D4—292no—
207C2×D4—291no—
208C2×D4—584no—
209C2×D4—584no—
210C2×D4—5816no—
211C2×D4—588no—
212C2×D4—582no—
213C2×D4—581no—
214C2×D4—582no—
215C2×D4—582no—
216C2×D4—581no—
217C2×D4C22⋊D2912yes—
218C2×D4C22⋊D2911yescharacteristic
219C2×D4C22×D2912yes—
220C22×C4—582no—
221C22×C4—581no—
222C22×C4C22×D2912yes—
223C22×C4C22×D2911yescharacteristic
224SD16—584no—
225D8—588no—
226D8—584no—
227M16—582no—
228C2×C8—292no—
229C2×C8—292no—
230C2×C8—582no—
231C4⋊C4C22×D2912yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—82no—
238C23—588no—
239C23—582no—
240C23—581no—
241C23—584no—
242C23—2322no—
243C23—2324no—
244C23C2×C22⋊D2912yes—
245C23C2×C22⋊D2911yescharacteristic
246Q8—22no—
247Q8—292no—
248Q8—581no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—84no—
255D4—294no—
256D4—292no—
257D4—588no—
258D4—588no—
259D4—584no—
260D4—584no—
261D4—581no—
262D4—582no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—588no—
267C2×C4—581no—
268C2×C4—584no—
269C2×C4D4×D2912yes—
270C2×C4D4×D2912yes—
271C2×C4C2×C22⋊D2912yes—
272C2×C4C2×C22⋊D2911yescharacteristic
273C2×C4C23×D2911yescharacteristic
274C8—582no—
275C8—582no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—584no—
283C22—582no—
284C22—1164no—
285C22—1162no—
286C22—2324no—
287C22—2328no—
288C22#1174529912yes—
289C22C22×C22⋊D2911yescharacteristic
290C4—22no—
291C4—42no—
292C4—582no—
293C4#1174529912yes—
294C4C2×D4×D2911yescharacteristic
295C4C2×D4×D2911yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—1164no—
300C2—1162no—
301C2C2×#1174529911yescharacteristic
302C2#1427196612yes—
subgroupG/H
centerC22C22×C22⋊D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#535C2
socleC22×C29C22×D4
hypercenter#535D29
hypocenterC29#3008
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#11745364C2semidirect
2. C2×#11745294C2semidirect
3. C22×#90782C2semidirect
4. #14271306C2semidirect
5. #14271296C2semidirect
6. #14271167C2semidirect
7. #14271157C2semidirect
8. #14271130C2semidirect
9. C29×#535C2semidirect
11. C22×D4×C29C22semidirect
18. C2×#90805C22semidirect
20. C22×D116C22semidirect
23. C29×#121C22semidirect
25. C29×#117C22semidirect
26. C2×#90783C22semidirect
29. C2×#90782C22semidirect
30. C2×#90782C22semidirect
31. C2×#90777C22semidirect
32. C22×#6597C22semidirect
33. C22⋊#6597C22semidirect
34. #11745158C22semidirect
35. #11745157C22semidirect
36. #11745157C22semidirect
60. #90805D4semidirect
66. C2×D116D4semidirect
148. #535D29semidirect
171. C22×D4C2×D29semidirect
181. #121C2×D29semidirect
183. #117C2×D29semidirect
188. C29#3008semidirect

Maximal subgroups

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

subgroupindex
C2×#117453642
C2×#117452942
C22×#907822
#142713062
#142712962
#142711672
#142711572
#142711302
C29×#5352
#300829