Finite Group Databasebeta

← #16831112

Subgroups — #16831112

Order 3712.

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

normal subgroups only

This group has 556 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#14271539C211yesmaximal, characteristic, complemented
4#14271414C212yesmaximal, complemented
5#14271363C211yesmaximal, characteristic, complemented
6#14271355C211yesmaximal, characteristic, complemented
7#14271308C212yesmaximal, complemented
8#14271296C212yesmaximal, complemented
9C4×#90784C211yesmaximal, characteristic, complemented
10#14271241C211yesmaximal, characteristic, complemented
11C29×#550C211yesmaximal, characteristic, complemented
12C2×#41×C29C2212yescomplemented
13C2×D4×D29—21no—
14C22×D116C2212yes—
15C29×#127—21no—
16C29×#127C2211yescharacteristic, complemented
17C29×#121—22no—
18C29×#121C2212yescomplemented
19C4×Q8×C29C2211yescharacteristic, complemented
20C4×D4×C29—21no—
21C4×D4×C29C2211yescharacteristic
22#11745311—28no—
23C2×#90784C2211yescharacteristic
24C2×#90784C2212yescomplemented
25C2×#90782C2212yes—
26C2×#90782C2212yescomplemented
27C2×#90782C2212yescomplemented
28#11745292C2212yes—
29C2×#90777C2212yescomplemented
30#11745257—21no—
31#11745244—22no—
32#11745236C2211yescharacteristic
33C4×D116C2211yescharacteristic
34#11745183C2211yescharacteristic, complemented
35#11745180C2212yescomplemented
36C22⋊#6597C2212yescomplemented
37#11745157C2211yescharacteristic
38#11745157C2212yescomplemented
39#11745156C2211yescharacteristic, complemented
40C4⋊#6597C2211yescharacteristic, complemented
41C4×#6597C2211yescharacteristic, complemented
42C23×D29—22no—
43#41×C29—24no—
44#41×C29—42no—
45C2×Q8×C29C2311yescharacteristic
46C2×D4×C29—22no—
47C2×D4×C29—22no—
48C2×D4×C29—21no—
49C2×D4×C29—21no—
50C2×D4×C29D411yescharacteristic
51C2×D4×C29C2312yes—
52C22×C4×C29—22no—
53C22×C4×C29C2312yes—
54C2×C22⋊D29—22no—
55D4×D29—44no—
56C2×D116—24no—
57C2×D116—22no—
58C2×D116—21no—
59C2×D116—21no—
60C2×D116—42no—
61C2×D116C2312yes—
62C2×D116C2311yescharacteristic
63C2×C4×D29—21no—
64C4⋊C4×C29—21no—
65C4⋊C4×C29D411yescharacteristic
66C4⋊C4×C29C2311yescharacteristic
67C22⋊C4×C29—22no—
68C22⋊C4×C29D412yes—
69C42×C29—21no—
70C42×C29C2311yescharacteristic
71#90784—24no—
72#90784D414yescomplemented
73#90782—24no—
74#90782—24no—
75#90782—42no—
76#90781—22no—
77#90781—21no—
78C4⋊Dic29C2311yescharacteristic
79#90777—24no—
80C2×#6597C2312yes—
81C2×#6597C2311yescharacteristic
82C2×#6597C2311yescharacteristic
83C23×C29—22no—
84C23×C29C2×D412yes—
85C22×D29—21no—
86C22×D29—22no—
87C22×D29—44no—
88C22×D29—44no—
89C22×D29—82no—
90Q8×C29—21no—
91Q8×C29C2×D412yescomplemented
92D4×C29—22no—
93D4×C29—22no—
94D4×C29—22no—
95D4×C29—44no—
96D4×C29—42no—
97D4×C29—81no—
98D4×C29—82no—
99C2×C4×C29—21no—
100C2×C4×C29—22no—
101C2×C4×C29—22no—
102C2×C4×C29—21no—
103C2×C4×C29—44no—
104C2×C4×C29C2×D411yescharacteristic
105C2×C4×C29C2×D412yes—
106C2×C4×C29C2×D411yescharacteristic
107C2×C4×C29C2411yescharacteristic
108C22⋊D29—44no—
109C2×Dic29—21no—
110D116—22no—
111D116—21no—
112D116—22no—
113D116—44no—
114D116—42no—
115D116—81no—
116D116C2×D412yes—
117C4×D29—42no—
118#6597—22no—
119#6597—21no—
120#6597C2×D412yescomplemented
121#3274—291nomaximal, Sylow
122C22×C29—22no—
123C22×C29—22no—
124C22×C29—22no—
125C22×C29—42no—
126C22×C29—42no—
127C22×C29—42no—
128C22×C29C22×D411yescharacteristic
129C2×D29—22no—
130C2×D29—41no—
131C2×D29—41no—
132C2×D29—42no—
133C2×D29—42no—
134C2×D29—42no—
135C2×D29—88no—
136C4×C29—22no—
137C4×C29—22no—
138C4×C29—21no—
139C4×C29—41no—
140C4×C29AGL(1,ℤ/8)12yescomplemented
141C4×C29C22×D411yescharacteristic
142C4×C29H(5,𝔽2)11yescharacteristic
143Dic29—41no—
144C2×AGL(1,ℤ/8)—292no—
145#550D2911yescharacteristic, complemented
146D42—291no—
147#493—291no—
148#469—292no—
149#460—291no—
150#459—291no—
151#449—292no—
152#447—292no—
153C4×SD16—291no—
154#436—291no—
155C2×C29—42no—
156C2×C29—82no—
157C2×C29D4211yescharacteristic
158C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C29#57611yescharacteristic
160D29—42no—
161D29—82no—
162C2×#41C2×D2912yescomplemented
163C22×D4—292no—
164C22×D4—581no—
165AGL(1,ℤ/8)—588no—
166C2×SD16—291no—
167C2×SD16—292no—
168C2×D8—292no—
169C2×D8—292no—
170C2×D8—292no—
171C2×M16—292no—
172#127—21no—
173#127—291no—
174#127C2×D2911yescharacteristic, complemented
175#121—22no—
176#121—292no—
177#121—581no—
178#121C2×D2912yescomplemented
179#120—582no—
180C4×Q8C2×D2911yescharacteristic, complemented
181C4×D4—21no—
182C4×D4—291no—
183C4×D4C2×D2911yescharacteristic
184#106—291no—
185C4⋊C8—291no—
186#103—291no—
187#102—292no—
188#102—291no—
189#102—292no—
190C22⋊C8—292no—
191C4×C8—291no—
192C29#327411yescharacteristic, complemented, Sylow
193C24—582no—
194#41—24no—
195#41—42no—
196C2×Q8C22×D2911yescharacteristic
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—291no—
202C2×D4—292no—
203C2×D4—584no—
204C2×D4—582no—
205C2×D4—582no—
206C2×D4—581no—
207C2×D4—581no—
208C2×D4—1162no—
209C2×D4—1164no—
210C2×D4C22⋊D2911yescharacteristic
211C2×D4C22×D2912yes—
212C22×C4—22no—
213C22×C4—581no—
214C22×C4C22×D2912yes—
215SD16—294no—
216SD16—584no—
217D8—584no—
218D8—584no—
219D8—1162no—
220M16—584no—
221C2×C8—291no—
222C2×C8—291no—
223C2×C8—292no—
224C4⋊C4—21no—
225C4⋊C4—291no—
226C4⋊C4C22⋊D2911yescharacteristic
227C4⋊C4C22×D2911yescharacteristic
228C22⋊C4—22no—
229C22⋊C4—581no—
230C22⋊C4—582no—
231C22⋊C4C22⋊D2912yes—
232C42—21no—
233C42C22×D2911yescharacteristic
234C23—22no—
235C23—581no—
236C23—582no—
237C23—1164no—
238C23—1164no—
239C23—2322no—
240C23C2×C22⋊D2912yes—
241Q8—21no—
242Q8D4×D2912yescomplemented
243D4—22no—
244D4—22no—
245D4—22no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—82no—
250D4—292no—
251D4—582no—
252D4—581no—
253D4—582no—
254D4—1164no—
255D4—1164no—
256D4—1162no—
257D4—2321no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—581no—
264C2×C4—1162no—
265C2×C4C2×C22⋊D2911yescharacteristic
266C2×C4C2×C22⋊D2912yes—
267C2×C4C2×C22⋊D2911yescharacteristic
268C2×C4C23×D2911yescharacteristic
269C8—292no—
270C8—582no—
271C8—581no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—582no—
279C22—1161no—
280C22—1161no—
281C22—1162no—
282C22—1162no—
283C22—1162no—
284C22—2328no—
285C22C22×C22⋊D2911yescharacteristic
286C4—22no—
287C4—22no—
288C4—21no—
289C4—41no—
290C4—1161no—
291C4#1174531112yescomplemented
292C4C2×D4×D2911yescharacteristic
293C4#1174535911yescharacteristic
294C2—42no—
295C2—82no—
296C2—1162no—
297C2—2322no—
298C2D4×C22⋊D2911yescharacteristic
299C2C2×#1174531111yescharacteristic
300C2#1427196611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#550C2
socleC22×C29C22×D4
hypercenter#550D29
hypocenterC29#3274
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#11745311C2semidirect
2. D4×D116C2semidirect
3. #14271539C2semidirect
4. #14271414C2semidirect
5. #14271363C2semidirect
6. #14271355C2semidirect
7. #14271308C2semidirect
8. #14271296C2semidirect
9. C4×#90784C2semidirect
10. #14271241C2semidirect
11. C29×#550C2semidirect
12. C2×#41×C29C22semidirect
16. C29×#127C22semidirect
18. C29×#121C22semidirect
19. C4×Q8×C29C22semidirect
24. C2×#90784C22semidirect
26. C2×#90782C22semidirect
27. C2×#90782C22semidirect
29. C2×#90777C22semidirect
34. #11745183C22semidirect
35. #11745180C22semidirect
36. C22⋊#6597C22semidirect
38. #11745157C22semidirect
39. #11745156C22semidirect
40. C4⋊#6597C22semidirect
41. C4×#6597C22semidirect
72. #90784D4semidirect
91. Q8×C29C2×D4semidirect
120. #6597C2×D4semidirect
140. C4×C29AGL(1,ℤ/8)semidirect
145. #550D29semidirect
162. C2×#41C2×D29semidirect
174. #127C2×D29semidirect
178. #121C2×D29semidirect
180. C4×Q8C2×D29semidirect
192. C29#3274semidirect
242. Q8D4×D29semidirect
291. 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
#142715392
#142714142
#142713632
#142713552
#142713082
#142712962
C4×#907842
#142712412
C29×#5502
#327429