Finite Group Databasebeta

← #16831115

Subgroups — #16831115

Order 3712.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 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#14271833C211yesmaximal, characteristic, complemented
3D4×D116C211yesmaximal, characteristic, complemented
4#14271539C211yesmaximal, characteristic, complemented
5#14271415C212yesmaximal, complemented
6#14271364C211yesmaximal, characteristic, complemented
7#14271309C212yesmaximal, complemented
8#14271295C212yesmaximal, complemented
9C4×#90783C211yesmaximal, characteristic, complemented
10#14271240C211yesmaximal, characteristic, complemented
11C29×#459C211yesmaximal, characteristic, complemented
12C2×#90807C2212yescomplemented
13C2×D4×D29C2212yes—
14C22×D116—21no—
15C2×D8×C29C2212yes—
16C29×#127C2211yescharacteristic, complemented
17C4×D4×C29C2211yescharacteristic
18#11745302—21no—
19#11745301—22no—
20#11745300C2212yescomplemented
21D4×Dic29—21no—
22C2×#90783C2211yescharacteristic
23C2×#90782C2212yescomplemented
24C2×#90782C2212yescomplemented
25#11745292—21no—
26#11745273—28no—
27C2×#90773C2212yescomplemented
28C2×#90772C2212yescomplemented
29#11745257C2212yes—
30#11745244—22no—
31#11745236C2211yescharacteristic
32C4×D116C2211yescharacteristic
33C4×Dic58C2211yescharacteristic, complemented
34C29×C4⋊C8C2211yescharacteristic, complemented
35C29×#102C2212yescomplemented
36#11745180C2211yescharacteristic
37#11745169C2212yescomplemented
38#11745168C2212yescomplemented
39#11745158C2211yescharacteristic, complemented
40#11745156C2211yescharacteristic, complemented
41C4×#6597C2211yescharacteristic, complemented
42C23×D29—22no—
43C2×D4×C29—21no—
44C2×D4×C29—21no—
45C2×D4×C29C2312yes—
46C2×D4×C29C2311yescharacteristic
47C22×C4×C29—21no—
48C2×C22⋊D29—22no—
49C2×C22⋊D29—22no—
50C2×C22⋊D29—22no—
51C22×Dic29—22no—
52#90807—24no—
53#90807—42no—
54D4×D29—24no—
55D4×D29—42no—
56C2×D116—22no—
57C2×D116—21no—
58C2×D116—21no—
59C2×D116—44no—
60C2×D116D411yescharacteristic
61C2×D116C2312yes—
62C2×C4×D29C2312yes—
63C2×Dic58C2311yescharacteristic
64D8×C29—24no—
65C2×C8×C29C2312yes—
66C4⋊C4×C29C2311yescharacteristic
67C22⋊C4×C29—21no—
68C42×C29C2311yescharacteristic
69C22⋊Dic29—22no—
70#90783D414yescomplemented
71#90782—24no—
72#90782—42no—
73#90781—22no—
74#90781D412yes—
75C4⋊Dic29D411yescharacteristic
76C4⋊Dic29C2311yescharacteristic
77#90779—21no—
78C4×Dic29—21no—
79C2×#6597C2311yescharacteristic
80C2×#6597C2311yescharacteristic
81#90773—24no—
82#90772—24no—
83C23×C29—21no—
84C23×C29—22no—
85C22×D29—22no—
86C22×D29—44no—
87C22×D29—44no—
88C22×D29—82no—
89C22×D29C2×D412yes—
90D4×C29—22no—
91D4×C29—21no—
92D4×C29—22no—
93D4×C29—42no—
94D4×C29—81no—
95D4×C29C2×D412yes—
96C2×C4×C29—21no—
97C2×C4×C29—42no—
98C2×C4×C29C2×D411yescharacteristic
99C2×C4×C29C2×D411yescharacteristic
100C2×C4×C29C2411yescharacteristic
101C22⋊D29—44no—
102C22⋊D29—44no—
103C22⋊D29—82no—
104C2×Dic29—21no—
105C2×Dic29—21no—
106C2×Dic29—44no—
107C2×Dic29C2×D412yes—
108D116—22no—
109D116—22no—
110D116—22no—
111D116—44no—
112D116—42no—
113D116—81no—
114C4×D29—22no—
115C4×D29—22no—
116Dic58—21no—
117Dic58C2×D412yescomplemented
118C8×C29—22no—
119#6597—21no—
120#6597C2×D412yescomplemented
121#3274—291nomaximal, Sylow
122C22×C29—22no—
123C22×C29—41no—
124C22×C29—41no—
125C22×C29—42no—
126C22×C29—42no—
127C22×C29—42no—
128C22×C29C22×D411yescharacteristic
129C2×D29—22no—
130C2×D29—22no—
131C2×D29—22no—
132C2×D29—42no—
133C2×D29—42no—
134C2×D29—42no—
135C2×D29—88no—
136C4×C29—21no—
137C4×C29—41no—
138C4×C29AGL(1,ℤ/8)12yescomplemented
139C4×C29C22×D411yescharacteristic
140C4×C29H(5,𝔽2)11yescharacteristic
141Dic29—22no—
142Dic29—22no—
143Dic29—41no—
144C2×AGL(1,ℤ/8)—292no—
145#550—291no—
146D42—291no—
147#493—291no—
148#469—292no—
149#460—291no—
150#459D2911yescharacteristic, complemented
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×#41—292no—
163C22×D4—292no—
164C22×D4—581no—
165AGL(1,ℤ/8)—588no—
166C2×SD16—291no—
167C2×SD16—292no—
168C2×D8—292no—
169C2×D8—292no—
170C2×D8C2×D2912yes—
171C2×M16—292no—
172#127—291no—
173#127—581no—
174#127C2×D2911yescharacteristic, complemented
175#121—292no—
176#121—292no—
177#121—582no—
178#121—581no—
179#120—582no—
180C4×Q8—291no—
181C4×D4—291no—
182C4×D4—581no—
183C4×D4C2×D2911yescharacteristic
184#106—291no—
185C4⋊C8C2×D2911yescharacteristic, complemented
186#103—291no—
187#102—291no—
188#102—292no—
189#102C2×D2912yescomplemented
190C22⋊C8—292no—
191C4×C8—291no—
192C29#327411yescharacteristic, complemented, Sylow
193C24—582no—
194#41—584no—
195#41—1162no—
196C2×Q8—291no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—291no—
200C2×D4—292no—
201C2×D4—582no—
202C2×D4—582no—
203C2×D4—584no—
204C2×D4—582no—
205C2×D4—582no—
206C2×D4—581no—
207C2×D4—581no—
208C2×D4—1164no—
209C2×D4—1162no—
210C2×D4C22×D2912yes—
211C2×D4C22×D2911yescharacteristic
212C22×C4—21no—
213C22×C4—292no—
214C22×C4—582no—
215SD16—294no—
216SD16—584no—
217D8—24no—
218D8—584no—
219D8—1162no—
220M16—584no—
221C2×C8—291no—
222C2×C8—291no—
223C2×C8C22×D2912yes—
224C4⋊C4—291no—
225C4⋊C4—291no—
226C4⋊C4—581no—
227C4⋊C4C22×D2911yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—292no—
230C22⋊C4—582no—
231C22⋊C4—582no—
232C42—581no—
233C42C22×D2911yescharacteristic
234C23—21no—
235C23—22no—
236C23—292no—
237C23—582no—
238C23—1164no—
239C23—1164no—
240C23—2322no—
241Q8—292no—
242Q8—581no—
243D4—22no—
244D4—21no—
245D4—22no—
246D4—42no—
247D4—81no—
248D4—582no—
249D4—582no—
250D4—582no—
251D4—1164no—
252D4—1164no—
253D4—1164no—
254D4—1162no—
255D4—2321no—
256D4—2322no—
257D4D4×D2912yes—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—292no—
261C2×C4—581no—
262C2×C4—582no—
263C2×C4—582no—
264C2×C4—581no—
265C2×C4—1164no—
266C2×C4D4×D2911yescharacteristic
267C2×C4D4×D2911yescharacteristic
268C2×C4C23×D2911yescharacteristic
269C8—22no—
270C8—292no—
271C8—581no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—582no—
279C22—582no—
280C22—582no—
281C22—1162no—
282C22—1162no—
283C22—1162no—
284C22—2328no—
285C22C2×D4×D2911yescharacteristic
286C4—21no—
287C4—41no—
288C4—582no—
289C4—582no—
290C4—1161no—
291C4#1174527312yescomplemented
292C4C2×D4×D2911yescharacteristic
293C4#1174536411yescharacteristic
294C2—42no—
295C2—82no—
296C2—1162no—
297C2—2322no—
298C2#1427179711yescharacteristic
299C2C2×#1174527311yescharacteristic
300C2#1427196711yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#459C2
socleC22×C29C22×D4
hypercenter#459D29
hypocenterC29#3274
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#11745273C2semidirect
2. #14271833C2semidirect
3. D4×D116C2semidirect
4. #14271539C2semidirect
5. #14271415C2semidirect
6. #14271364C2semidirect
7. #14271309C2semidirect
8. #14271295C2semidirect
9. C4×#90783C2semidirect
10. #14271240C2semidirect
11. C29×#459C2semidirect
12. C2×#90807C22semidirect
16. C29×#127C22semidirect
20. #11745300C22semidirect
23. C2×#90782C22semidirect
24. C2×#90782C22semidirect
27. C2×#90773C22semidirect
28. C2×#90772C22semidirect
33. C4×Dic58C22semidirect
34. C29×C4⋊C8C22semidirect
35. C29×#102C22semidirect
37. #11745169C22semidirect
38. #11745168C22semidirect
39. #11745158C22semidirect
40. #11745156C22semidirect
41. C4×#6597C22semidirect
70. #90783D4semidirect
117. Dic58C2×D4semidirect
120. #6597C2×D4semidirect
138. C4×C29AGL(1,ℤ/8)semidirect
150. #459D29semidirect
174. #127C2×D29semidirect
185. C4⋊C8C2×D29semidirect
189. #102C2×D29semidirect
192. C29#3274semidirect
291. C4#11745273semidirect

Maximal subgroups

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

subgroupindex
C2×#117452732
#142718332
D4×D1162
#142715392
#142714152
#142713642
#142713092
#142712952
C4×#907832
#142712402
C29×#4592
#327429