Finite Group Databasebeta

← #16831093

Subgroups — #16831093

Order 3712.

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

normal subgroups only

This group has 546 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×#11745273C212yesmaximal, complemented
2#14271817C211yesmaximal, characteristic, complemented
3D4×D116C211yesmaximal, characteristic, complemented
4#14271536C211yesmaximal, characteristic, complemented
5#14271410C212yesmaximal, complemented
6#14271368C211yesmaximal, characteristic, complemented
7#14271358C211yesmaximal, characteristic, complemented
8#14271315C212yesmaximal, complemented
9#14271309C212yesmaximal, complemented
10#14271242C211yesmaximal, characteristic, complemented
11C4×D8×C29C211yesmaximal, characteristic, complemented
12C2×#90807C2212yescomplemented
13C2×D4×D29C2212yes—
14C22×D116—21no—
15C2×D8×C29C2211yescharacteristic
16C4×D4×C29C2211yescharacteristic
17C4×D4×C29C2211yescharacteristic, complemented
18C2×#90783C2212yes—
19C2×#90782C2212yescomplemented
20#11745292—21no—
21#11745292—21no—
22C2×C4⋊Dic29—21no—
23#11745273—28no—
24C2×#90773C2212yescomplemented
25C2×#90773C2212yescomplemented
26C2×#90772C2212yescomplemented
27#11745259C2212yescomplemented
28#11745257C2212yes—
29#11745248—22no—
30#11745244—22no—
31#11745236C2211yescharacteristic
32C4×D116C2211yescharacteristic
33#11745231C2211yescharacteristic, complemented
34C29×#107C2211yescharacteristic, complemented
35C29×#102C2211yescharacteristic
36C29×#102C2211yescharacteristic, complemented
37C4×C8×C29C2211yescharacteristic, complemented
38#11745168C2212yescomplemented
39#11745158C2212yescomplemented
40#11745157C2212yescomplemented
41C4⋊#6597C2211yescharacteristic, complemented
42C23×D29—22no—
43C2×D4×C29C2311yescharacteristic
44C2×D4×C29C2311yescharacteristic
45C22×C4×C29—21no—
46C22×C4×C29—21no—
47C2×C22⋊D29—22no—
48C2×C22⋊D29—22no—
49C22×Dic29—22no—
50#90807—24no—
51#90807—42no—
52D4×D29—24no—
53D4×D29—42no—
54C2×D116—22no—
55C2×D116—21no—
56C2×D116—21no—
57C2×D116—44no—
58C2×D116D411yescharacteristic
59C2×D116C2312yes—
60C2×C4×D29C2312yes—
61C2×Dic58C2312yes—
62D8×C29D414yescomplemented
63C2×C8×C29C2311yescharacteristic
64C2×C8×C29C2311yescharacteristic
65C4⋊C4×C29C2311yescharacteristic
66C4⋊C4×C29C2311yescharacteristic
67C22⋊C4×C29—21no—
68C22⋊C4×C29—21no—
69C42×C29C2311yescharacteristic
70#90783—24no—
71#90782—24no—
72#90781—22no—
73#90781—22no—
74#90781D412yes—
75C4⋊Dic29—21no—
76C4⋊Dic29—21no—
77C4⋊Dic29—22no—
78C4⋊Dic29D411yescharacteristic
79C2×#6597C2312yes—
80#90773—24no—
81#90773—42no—
82#90772—24no—
83C23×C29—21no—
84C23×C29—21no—
85C22×D29—22no—
86C22×D29—44no—
87C22×D29—44no—
88C22×D29—82no—
89C22×D29C2×D412yes—
90D4×C29—21no—
91D4×C29—21no—
92D4×C29C2×D412yes—
93D4×C29C2×D412yescomplemented
94C2×C4×C29—21no—
95C2×C4×C29—21no—
96C2×C4×C29—42no—
97C2×C4×C29—42no—
98C2×C4×C29C2×D411yescharacteristic
99C2×C4×C29C2×D411yescharacteristic
100C2×C4×C29C2411yescharacteristic
101C22⋊D29—44no—
102C22⋊D29—44no—
103C2×Dic29—22no—
104C2×Dic29—44no—
105C2×Dic29C2×D412yes—
106D116—22no—
107D116—22no—
108D116—22no—
109D116—44no—
110D116—42no—
111D116—81no—
112C4×D29—22no—
113C4×D29—22no—
114Dic58—22no—
115Dic58—22no—
116C8×C29—21no—
117C8×C29C2×D412yescomplemented
118#6597—22no—
119#3272—291nomaximal, Sylow
120C22×C29—22no—
121C22×C29—22no—
122C22×C29—41no—
123C22×C29—41no—
124C22×C29—41no—
125C22×C29—41no—
126C22×C29C22×D411yescharacteristic
127C2×D29—22no—
128C2×D29—22no—
129C2×D29—22no—
130C2×D29—42no—
131C2×D29—42no—
132C2×D29—42no—
133C2×D29—88no—
134C4×C29—21no—
135C4×C29—41no—
136C4×C29—41no—
137C4×C29AGL(1,ℤ/8)12yescomplemented
138C4×C29C22×D411yescharacteristic
139C4×C29H(5,𝔽2)11yescharacteristic
140Dic29—22no—
141Dic29—42no—
142C2×AGL(1,ℤ/8)—292no—
143#547—291no—
144D42—291no—
145#492—291no—
146#468—292no—
147#461—291no—
148#459—291no—
149#450—292no—
150#449—292no—
151C4×D8D2911yescharacteristic, complemented
152#436—291no—
153C2×C29—42no—
154C2×C29—42no—
155C2×C29D4211yescharacteristic
156C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
157C2×C29#57711yescharacteristic
158D29—42no—
159D29—82no—
160C2×#41—292no—
161C22×D4—292no—
162C22×D4—581no—
163AGL(1,ℤ/8)—588no—
164C2×SD16—292no—
165C2×SD16—292no—
166C2×SD16—292no—
167C2×D8—292no—
168C2×D8C2×D2911yescharacteristic
169C2×M16—292no—
170C4⋊Q8—291no—
171#127—291no—
172#123—582no—
173C22⋊Q8—292no—
174#121—292no—
175#121—581no—
176#121—581no—
177#120—582no—
178C4×D4—291no—
179C4×D4C2×D2911yescharacteristic
180C4×D4C2×D2911yescharacteristic, complemented
181C2×C4⋊C4—581no—
182#107C2×D2911yescharacteristic, complemented
183C4⋊C8—291no—
184#103—292no—
185#102—292no—
186#102C2×D2911yescharacteristic
187#102C2×D2911yescharacteristic, complemented
188C22⋊C8—292no—
189C4×C8C2×D2911yescharacteristic, complemented
190C29#327211yescharacteristic, complemented, Sylow
191C24—582no—
192#41—584no—
193#41—1162no—
194C2×Q8—292no—
195C2×D4—291no—
196C2×D4—292no—
197C2×D4—582no—
198C2×D4—582no—
199C2×D4—584no—
200C2×D4—582no—
201C2×D4—581no—
202C2×D4—581no—
203C2×D4—1164no—
204C2×D4—1162no—
205C2×D4C22×D2911yescharacteristic
206C2×D4C22×D2911yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—292no—
210C22×C4—582no—
211SD16—584no—
212SD16—584no—
213SD16—1162no—
214D8—584no—
215D8D11614yescomplemented
216M16—584no—
217C2×C8—292no—
218C2×C8C22×D2911yescharacteristic
219C2×C8C22×D2911yescharacteristic
220C4⋊C4—291no—
221C4⋊C4—582no—
222C4⋊C4—581no—
223C4⋊C4—581no—
224C4⋊C4C22×D2911yescharacteristic
225C4⋊C4C22×D2911yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—292no—
229C22⋊C4—582no—
230C22⋊C4—582no—
231C42C22×D2911yescharacteristic
232C23—21no—
233C23—21no—
234C23—292no—
235C23—582no—
236C23—1164no—
237C23—1164no—
238C23—2322no—
239Q8—582no—
240Q8—582no—
241D4—21no—
242D4—21no—
243D4—582no—
244D4—582no—
245D4—582no—
246D4—1164no—
247D4—1164no—
248D4—1164no—
249D4—1162no—
250D4—2321no—
251D4C2×D11612yes—
252D4C2×D11612yescomplemented
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—292no—
258C2×C4—582no—
259C2×C4—582no—
260C2×C4—582no—
261C2×C4—1164no—
262C2×C4D4×D2911yescharacteristic
263C2×C4D4×D2911yescharacteristic
264C2×C4C23×D2911yescharacteristic
265C8—21no—
266C8—582no—
267C8C2×D11612yescomplemented
268C22—22no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—582no—
275C22—582no—
276C22—582no—
277C22—1162no—
278C22—1162no—
279C22—1162no—
280C22—2328no—
281C22C2×D4×D2911yescharacteristic
282C4—21no—
283C4—41no—
284C4—41no—
285C4—582no—
286C4—1162no—
287C4#1174527312yescomplemented
288C4C22×D11611yescharacteristic
289C4#1174536411yescharacteristic
290C2—42no—
291C2—42no—
292C2—1162no—
293C2—2322no—
294C2D4×D11611yescharacteristic
295C2C2×#1174527311yescharacteristic
296C2#1427197011yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC4×D8×C29C2
socleC22×C29C22×D4
hypercenterC4×D8D29
hypocenterC29#3272
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#11745273C2semidirect
2. #14271817C2semidirect
3. D4×D116C2semidirect
4. #14271536C2semidirect
5. #14271410C2semidirect
6. #14271368C2semidirect
7. #14271358C2semidirect
8. #14271315C2semidirect
9. #14271309C2semidirect
10. #14271242C2semidirect
11. C4×D8×C29C2semidirect
12. C2×#90807C22semidirect
17. C4×D4×C29C22semidirect
19. C2×#90782C22semidirect
24. C2×#90773C22semidirect
25. C2×#90773C22semidirect
26. C2×#90772C22semidirect
27. #11745259C22semidirect
33. #11745231C22semidirect
34. C29×#107C22semidirect
36. C29×#102C22semidirect
37. C4×C8×C29C22semidirect
38. #11745168C22semidirect
39. #11745158C22semidirect
40. #11745157C22semidirect
41. C4⋊#6597C22semidirect
62. D8×C29D4semidirect
93. D4×C29C2×D4semidirect
117. C8×C29C2×D4semidirect
137. C4×C29AGL(1,ℤ/8)semidirect
151. C4×D8D29semidirect
180. C4×D4C2×D29semidirect
182. #107C2×D29semidirect
187. #102C2×D29semidirect
189. C4×C8C2×D29semidirect
190. C29#3272semidirect
215. D8D116semidirect
252. D4C2×D116semidirect
267. C8C2×D116semidirect
287. C4#11745273semidirect

Maximal subgroups

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

subgroupindex
C2×#117452732
#142718172
D4×D1162
#142715362
#142714102
#142713682
#142713582
#142713152
#142713092
#142712422
C4×D8×C292
#327229