Finite Group Databasebeta

← #16831092

Subgroups — #16831092

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×#11745311C212yesmaximal, complemented
2D4×D116C211yesmaximal, characteristic, complemented
3#14271537C211yesmaximal, characteristic, complemented
4#14271411C212yesmaximal, complemented
5#14271365C211yesmaximal, characteristic, complemented
6#14271358C211yesmaximal, characteristic, complemented
7#14271313C212yesmaximal, complemented
8#14271308C212yesmaximal, complemented
9C4×#90782C211yesmaximal, characteristic, complemented
10#14271241C211yesmaximal, characteristic, complemented
11C29×#547C211yesmaximal, characteristic, complemented
12C2×#41×C29C2212yescomplemented
13C2×D4×D29—21no—
14C22×D116C2212yes—
15C29×C4⋊Q8C2211yescharacteristic, complemented
16C29×#123—22no—
17C29×C22⋊Q8C2212yescomplemented
18C29×#121—21no—
19C4×D4×C29C2211yescharacteristic
20C4×D4×C29C2211yescharacteristic, complemented
21C2×C4⋊C4×C29—21no—
22#11745311—28no—
23C2×#90784C2212yes—
24C2×#90784C2212yescomplemented
25C2×#90784C2212yescomplemented
26C2×#90782C2211yescharacteristic
27C2×#90782C2212yescomplemented
28#11745292C2212yes—
29C2×#90777C2212yescomplemented
30#11745257—21no—
31#11745244—22no—
32#11745236C2211yescharacteristic
33C4×D116C2211yescharacteristic
34#11745183C2212yescomplemented
35#11745180C2211yescharacteristic, complemented
36C22⋊#6597C2212yescomplemented
37#11745157C2211yescharacteristic
38#11745157C2212yescomplemented
39#11745155C2211yescharacteristic, complemented
40C4⋊#6597C2211yescharacteristic, complemented
41C4×#6597C2211yescharacteristic, complemented
42C23×D29—22no—
43#41×C29—24no—
44#41×C29—42no—
45C2×Q8×C29C2312yes—
46C2×D4×C29—22no—
47C2×D4×C29D411yescharacteristic
48C2×D4×C29C2311yescharacteristic
49C22×C4×C29—21no—
50C22×C4×C29—22no—
51C22×C4×C29C2312yes—
52C2×C22⋊D29—22no—
53D4×D29—44no—
54C2×D116—24no—
55C2×D116—22no—
56C2×D116—21no—
57C2×D116—21no—
58C2×D116—42no—
59C2×D116C2312yes—
60C2×D116C2311yescharacteristic
61C2×C4×D29—21no—
62C4⋊C4×C29—21no—
63C4⋊C4×C29—22no—
64C4⋊C4×C29—21no—
65C4⋊C4×C29D411yescharacteristic
66C4⋊C4×C29C2311yescharacteristic
67C22⋊C4×C29—22no—
68C22⋊C4×C29—21no—
69C22⋊C4×C29D412yes—
70C42×C29C2311yescharacteristic
71#90784—24no—
72#90784—24no—
73#90784—42no—
74#90782—24no—
75#90782D414yescomplemented
76#90781—22no—
77#90781—21no—
78C4⋊Dic29C2311yescharacteristic
79#90777—24no—
80C2×#6597C2312yes—
81C2×#6597C2311yescharacteristic
82C2×#6597C2311yescharacteristic
83C23×C29—21no—
84C23×C29C2×D412yes—
85C22×D29—21no—
86C22×D29—22no—
87C22×D29—44no—
88C22×D29—44no—
89C22×D29—82no—
90Q8×C29—22no—
91Q8×C29—22no—
92D4×C29—22no—
93D4×C29—21no—
94D4×C29—44no—
95D4×C29C2×D412yescomplemented
96C2×C4×C29—22no—
97C2×C4×C29—22no—
98C2×C4×C29—21no—
99C2×C4×C29—22no—
100C2×C4×C29—44no—
101C2×C4×C29—42no—
102C2×C4×C29C2×D411yescharacteristic
103C2×C4×C29C2×D412yes—
104C2×C4×C29C2×D411yescharacteristic
105C2×C4×C29C2411yescharacteristic
106C22⋊D29—44no—
107C2×Dic29—21no—
108D116—22no—
109D116—21no—
110D116—22no—
111D116—44no—
112D116—42no—
113D116—81no—
114D116C2×D412yes—
115C4×D29—42no—
116#6597—22no—
117#6597—21no—
118#6597C2×D412yescomplemented
119#3272—291nomaximal, Sylow
120C22×C29—22no—
121C22×C29—22no—
122C22×C29—22no—
123C22×C29—22no—
124C22×C29—41no—
125C22×C29—41no—
126C22×C29C22×D411yescharacteristic
127C2×D29—22no—
128C2×D29—41no—
129C2×D29—41no—
130C2×D29—42no—
131C2×D29—42no—
132C2×D29—42no—
133C2×D29—88no—
134C4×C29—22no—
135C4×C29—21no—
136C4×C29—42no—
137C4×C29—41no—
138C4×C29AGL(1,ℤ/8)12yescomplemented
139C4×C29C22×D411yescharacteristic
140C4×C29H(5,𝔽2)11yescharacteristic
141Dic29—41no—
142C2×AGL(1,ℤ/8)—292no—
143#547D2911yescharacteristic, complemented
144D42—291no—
145#492—291no—
146#468—292no—
147#461—291no—
148#459—291no—
149#450—292no—
150#449—292no—
151C4×D8—291no—
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×#41C2×D2912yescomplemented
161C22×D4—292no—
162C22×D4—581no—
163AGL(1,ℤ/8)—588no—
164C2×SD16—292no—
165C2×SD16—292no—
166C2×SD16—292no—
167C2×D8—291no—
168C2×D8—292no—
169C2×M16—292no—
170C4⋊Q8C2×D2911yescharacteristic, complemented
171#127—291no—
172#123—22no—
173C22⋊Q8C2×D2912yescomplemented
174#121—21no—
175#121—292no—
176#121—581no—
177#120—582no—
178C4×D4—291no—
179C4×D4C2×D2911yescharacteristic
180C4×D4C2×D2911yescharacteristic, complemented
181C2×C4⋊C4—21no—
182#107—291no—
183C4⋊C8—291no—
184#103—292no—
185#102—291no—
186#102—291no—
187#102—292no—
188C22⋊C8—292no—
189C4×C8—291no—
190C29#327211yescharacteristic, complemented, Sylow
191C24—582no—
192#41—24no—
193#41—42no—
194C2×Q8C22×D2912yes—
195C2×D4—22no—
196C2×D4—291no—
197C2×D4—292no—
198C2×D4—584no—
199C2×D4—582no—
200C2×D4—582no—
201C2×D4—581no—
202C2×D4—581no—
203C2×D4—1162no—
204C2×D4—1164no—
205C2×D4C22⋊D2911yescharacteristic
206C2×D4C22×D2911yescharacteristic
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—581no—
210C22×C4C22×D2912yes—
211SD16—584no—
212SD16—584no—
213SD16—1162no—
214D8—294no—
215D8—584no—
216M16—584no—
217C2×C8—291no—
218C2×C8—291no—
219C2×C8—292no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—291no—
224C4⋊C4C22⋊D2911yescharacteristic
225C4⋊C4C22×D2911yescharacteristic
226C22⋊C4—22no—
227C22⋊C4—21no—
228C22⋊C4—581no—
229C22⋊C4—582no—
230C22⋊C4C22⋊D2912yes—
231C42C22×D2911yescharacteristic
232C23—21no—
233C23—581no—
234C23—582no—
235C23—1164no—
236C23—1164no—
237C23—2322no—
238C23C2×C22⋊D2912yes—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—21no—
243D4—44no—
244D4—292no—
245D4—582no—
246D4—581no—
247D4—582no—
248D4—1164no—
249D4—1164no—
250D4—1162no—
251D4—2321no—
252D4D4×D2912yescomplemented
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—581no—
260C2×C4—1162no—
261C2×C4C2×C22⋊D2911yescharacteristic
262C2×C4C2×C22⋊D2912yes—
263C2×C4C2×C22⋊D2911yescharacteristic
264C2×C4C23×D2911yescharacteristic
265C8—292no—
266C8—582no—
267C8—581no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—582no—
275C22—1161no—
276C22—1161no—
277C22—1162no—
278C22—1162no—
279C22—1162no—
280C22—2328no—
281C22C22×C22⋊D2911yescharacteristic
282C4—22no—
283C4—21no—
284C4—42no—
285C4—41no—
286C4—1161no—
287C4#1174531112yescomplemented
288C4C2×D4×D2911yescharacteristic
289C4#1174535911yescharacteristic
290C2—42no—
291C2—42no—
292C2—1162no—
293C2—2322no—
294C2D4×C22⋊D2911yescharacteristic
295C2C2×#1174531111yescharacteristic
296C2#1427197111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#547C2
socleC22×C29C22×D4
hypercenter#547D29
hypocenterC29#3272
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#11745311C2semidirect
2. D4×D116C2semidirect
3. #14271537C2semidirect
4. #14271411C2semidirect
5. #14271365C2semidirect
6. #14271358C2semidirect
7. #14271313C2semidirect
8. #14271308C2semidirect
9. C4×#90782C2semidirect
10. #14271241C2semidirect
11. C29×#547C2semidirect
12. C2×#41×C29C22semidirect
15. C29×C4⋊Q8C22semidirect
17. C29×C22⋊Q8C22semidirect
20. C4×D4×C29C22semidirect
24. C2×#90784C22semidirect
25. C2×#90784C22semidirect
27. C2×#90782C22semidirect
29. C2×#90777C22semidirect
34. #11745183C22semidirect
35. #11745180C22semidirect
36. C22⋊#6597C22semidirect
38. #11745157C22semidirect
39. #11745155C22semidirect
40. C4⋊#6597C22semidirect
41. C4×#6597C22semidirect
75. #90782D4semidirect
95. D4×C29C2×D4semidirect
118. #6597C2×D4semidirect
138. C4×C29AGL(1,ℤ/8)semidirect
143. #547D29semidirect
160. C2×#41C2×D29semidirect
170. C4⋊Q8C2×D29semidirect
173. C22⋊Q8C2×D29semidirect
180. C4×D4C2×D29semidirect
190. C29#3272semidirect
252. D4D4×D29semidirect
287. C4#11745311semidirect

Maximal subgroups

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

subgroupindex
C2×#117453112
D4×D1162
#142715372
#142714112
#142713652
#142713582
#142713132
#142713082
C4×#907822
#142712412
C29×#5472
#327229