Finite Group Databasebeta

← #16828761

Subgroups — #16828761

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×#11745359C211yesmaximal, characteristic, complemented
2C2×#11745270C211yesmaximal, characteristic, complemented
3C22×#90782C211yesmaximal, characteristic, complemented
4#14271732C211yesmaximal, characteristic, complemented
5#14271309C214yesmaximal, complemented
6#14271298C214yesmaximal, complemented
7#14271157C211yesmaximal, characteristic, complemented
8#14271131C211yesmaximal, characteristic, complemented
9C29×#418C211yesmaximal, characteristic, complemented
10C22×D4×C29C2211yescharacteristic
11C22×C22⋊D29—22no—
12#11745359—24no—
13#11745359—24no—
14C2×#90807—21no—
15C2×#90807C2212yescomplemented
16C2×D4×D29—21no—
17C2×D4×D29C2212yescomplemented
18C2×#90805C2211yescharacteristic
19C2×#90805C2211yescharacteristic
20C22×D116C2211yescharacteristic, complemented
21C2×M16×C29C2211yescharacteristic, complemented
22C29×#117C2211yescharacteristic, complemented
23C2×#90782—24no—
24C2×#90782C2214yescomplemented
25C22×#6597C2211yescharacteristic, complemented
26#11745270—24no—
27C2×D232C2212yescomplemented
28C2×#90773C2212yescomplemented
29#11745257C2214yescomplemented
30#11745244—22no—
31#11745237—21no—
32#11745236—21no—
33C29×#102C2214yescomplemented
34#11745168C2214yescomplemented
35#11745158C2212yescomplemented
36#11745157C2212yescomplemented
37C24×C29—21no—
38C23×D29—22no—
39C23×D29—21no—
40C2×D4×C29—21no—
41C2×D4×C29—22no—
42C2×D4×C29D414yescomplemented
43C2×D4×C29C2312yes—
44C22×C4×C29C2311yescharacteristic
45C2×C22⋊D29—216no—
46C2×C22⋊D29—21no—
47C2×C22⋊D29—22no—
48C2×C22⋊D29—22no—
49C2×C22⋊D29D412yes—
50C22×Dic29—22no—
51#90807—28no—
52#90807—24no—
53#90807—24no—
54D4×D29—28no—
55D4×D29—24no—
56D4×D29—24no—
57#90805—22no—
58#90805—22no—
59#90805—22no—
60#90805D414yescomplemented
61C2×D116—22no—
62C2×D116—22no—
63C2×D116—21no—
64C2×D116—24no—
65C2×D116D411yescharacteristic
66C2×D116C2312yes—
67C2×D116C2311yescharacteristic
68C2×C4×D29—21no—
69C2×C4×D29C2312yes—
70C2×Dic58D411yescharacteristic
71C2×Dic58C2311yescharacteristic
72M16×C29—22no—
73C2×C8×C29C2312yes—
74C4⋊C4×C29C2312yes—
75C22⋊C4×C29—21no—
76C42×C29—21no—
77#90782—28no—
78#90781—24no—
79C2×#6597—22no—
80C2×#6597C2312yes—
81D232—24no—
82#90773—24no—
83C23×C29—22no—
84C23×C29—24no—
85C23×C29—82no—
86C23×C29C2×D411yescharacteristic
87C22×D29—21no—
88C22×D29—28no—
89C22×D29—22no—
90C22×D29—42no—
91C22×D29—84no—
92C22×D29—82no—
93C22×D29C2×D412yes—
94D4×C29—24no—
95D4×C29—22no—
96D4×C29C2×D414yes—
97C2×C4×C29—22no—
98C2×C4×C29C2×D412yes—
99C2×C4×C29C2×D411yescharacteristic
100C2×C4×C29C2×D412yes—
101C2×C4×C29C2411yescharacteristic
102C22⋊D29—28no—
103C22⋊D29—28no—
104C22⋊D29—24no—
105C22⋊D29—24no—
106C2×Dic29—28no—
107C2×Dic29—21no—
108C2×Dic29C2×D412yes—
109D116—22no—
110D116—21no—
111D116—22no—
112D116—22no—
113D116—42no—
114D116—84no—
115D116C2×D412yes—
116C4×D29—24no—
117C4×D29—22no—
118C4×D29—22no—
119Dic58—22no—
120Dic58—21no—
121Dic58C2×D412yes—
122C8×C29—22no—
123#6597—22no—
124#3008—291nomaximal, Sylow
125C22×C29—22no—
126C22×C29—24no—
127C22×C29—44no—
128C22×C29—84no—
129C22×C29#12012yes—
130C22×C29C22×D411yescharacteristic
131C2×D29—22no—
132C2×D29—22no—
133C2×D29—24no—
134C2×D29—42no—
135C2×D29—42no—
136C2×D29—44no—
137C2×D29—88no—
138C2×D29—84no—
139C4×C29—42no—
140C4×C29#12012yes—
141C4×C29C22×D411yescharacteristic
142C4×C29C22×D411yescharacteristic
143Dic29—22no—
144Dic29—22no—
145C2×H(5,𝔽2)—291no—
146C2×AGL(1,ℤ/8)—291no—
147C22×D8—291no—
148#535—291no—
149#449—294no—
150#447—294no—
151#418D2911yescharacteristic, complemented
152#416—291no—
153#408—291no—
154C2×C29—22no—
155C2×C29—44no—
156C2×C29C2×#12011yescharacteristic
157C2×C29#57612yes—
158D29—42no—
159D29—42no—
160D29—82no—
161H(5,𝔽2)—584no—
162H(5,𝔽2)—584no—
163C2×#41—291no—
164C2×#41—292no—
165C2×#41—291no—
166C2×#41—581no—
167C22×D4—292no—
168C22×D4—291no—
169C22×D4—581no—
170C22×D4—582no—
171C22×D4C2×D2911yescharacteristic
172AGL(1,ℤ/8)—584no—
173C2×SD16—292no—
174C2×D8—294no—
175C2×D8—292no—
176C2×D8—584no—
177C2×M16C2×D2911yescharacteristic, complemented
178C22×C8—291no—
179#127—581no—
180#124—581no—
181#121—294no—
182#120—582no—
183#117C2×D2911yescharacteristic, complemented
184#103—292no—
185#102—292no—
186#102C2×D2914yescomplemented
187C22⋊C8—294no—
188C29#300811yescharacteristic, complemented, Sylow
189C24—21no—
190C24—582no—
191C24—581no—
192#41—294no—
193#41—584no—
194#41—588no—
195#41—584no—
196#41—582no—
197#41—582no—
198#41—582no—
199C2×Q8—291no—
200C2×Q8—291no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—292no—
204C2×D4—291no—
205C2×D4—291no—
206C2×D4—292no—
207C2×D4—584no—
208C2×D4—582no—
209C2×D4—584no—
210C2×D4—588no—
211C2×D4—582no—
212C2×D4—5816no—
213C2×D4—581no—
214C2×D4—584no—
215C2×D4—582no—
216C2×D4—582no—
217C2×D4—581no—
218C2×D4D11614yescomplemented
219C2×D4C22×D2912yes—
220C22×C4—292no—
221C22×C4—582no—
222C22×C4—581no—
223C22×C4C22×D2911yescharacteristic
224SD16—584no—
225D8—584no—
226D8—588no—
227M16—22no—
228C2×C8—292no—
229C2×C8—582no—
230C2×C8C22×D2912yes—
231C4⋊C4C22×D2912yes—
232C22⋊C4—21no—
233C22⋊C4—584no—
234C42—21no—
235C23—22no—
236C23—24no—
237C23—82no—
238C23—292no—
239C23—588no—
240C23—581no—
241C23—582no—
242C23—1162no—
243C23—2322no—
244C23—2324no—
245C23D4×D2911yescharacteristic
246Q8—292no—
247Q8—581no—
248Q8—582no—
249D4—22no—
250D4—24no—
251D4—292no—
252D4—588no—
253D4—588no—
254D4—584no—
255D4—584no—
256D4—581no—
257D4—582no—
258D4—582no—
259D4—582no—
260D4—1162no—
261D4—2324no—
262D4C2×D11614yes—
263C2×C4—22no—
264C2×C4—292no—
265C2×C4—588no—
266C2×C4—581no—
267C2×C4—582no—
268C2×C4—582no—
269C2×C4—584no—
270C2×C4C2×D11612yes—
271C2×C4D4×D2912yes—
272C2×C4D4×D2911yescharacteristic
273C2×C4C23×D2911yescharacteristic
274C8—22no—
275C8—582no—
276C22—22no—
277C22—24no—
278C22—44no—
279C22—84no—
280C22—582no—
281C22—582no—
282C22—584no—
283C22—1162no—
284C22—1162no—
285C22—1164no—
286C22—2324no—
287C22—2328no—
288C22#1174524412yes—
289C22C2×D4×D2911yescharacteristic
290C4—42no—
291C4—582no—
292C4—582no—
293C4#1174524412yes—
294C4C22×D11611yescharacteristic
295C4C2×D4×D2911yescharacteristic
296C2—22no—
297C2—44no—
298C2—1162no—
299C2—1162no—
300C2—2322no—
301C2C2×#1174524411yescharacteristic
302C2#1427196712yes—
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#418C2
socleC22×C29C22×D4
hypercenter#418D29
hypocenterC29#3008
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#11745359C2semidirect
2. C2×#11745270C2semidirect
3. C22×#90782C2semidirect
4. #14271732C2semidirect
5. #14271309C2semidirect
6. #14271298C2semidirect
7. #14271157C2semidirect
8. #14271131C2semidirect
9. C29×#418C2semidirect
15. C2×#90807C22semidirect
17. C2×D4×D29C22semidirect
20. C22×D116C22semidirect
21. C2×M16×C29C22semidirect
22. C29×#117C22semidirect
24. C2×#90782C22semidirect
25. C22×#6597C22semidirect
27. C2×D232C22semidirect
28. C2×#90773C22semidirect
29. #11745257C22semidirect
33. C29×#102C22semidirect
34. #11745168C22semidirect
35. #11745158C22semidirect
36. #11745157C22semidirect
42. C2×D4×C29D4semidirect
60. #90805D4semidirect
151. #418D29semidirect
177. C2×M16C2×D29semidirect
183. #117C2×D29semidirect
186. #102C2×D29semidirect
188. C29#3008semidirect
218. C2×D4D116semidirect

Maximal subgroups

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

subgroupindex
C2×#117453592
C2×#117452702
C22×#907822
#142717322
#142713092
#142712982
#142711572
#142711312
C29×#4182
#300829