Finite Group Databasebeta

← #16828883

Subgroups — #16828883

Order 3712.

766 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#14271999C211yesmaximal, characteristic, complemented
2#14271674C212yesmaximal, complemented
3#14271352C214yesmaximal, complemented
4#14271349C214yesmaximal, complemented
5#14271137C211yesmaximal, characteristic, complemented
6#14271136C212yesmaximal, complemented
7C29×#525C211yesmaximal, characteristic, complemented
8C2×#41×C29C2211yescharacteristic
9C23×C4×C29C2211yescharacteristic, complemented
10#11745365—22no—
11#11745364—22no—
12#41×D29—24no—
13#11745362—21no—
14#11745359—21no—
15#11745359C2214yes—
16C2×#90805—22no—
17C2×#90805C2212yes—
18C29×#123C2212yescomplemented
19C29×C22⋊Q8—21no—
20C29×#121—21no—
21C29×#120C2212yescomplemented
22C4×D4×C29—22no—
23C29×#117C2211yescharacteristic, complemented
24#11745315C2214yescomplemented
25#11745292—22no—
26#11745291C2214yescomplemented
27C4×C22⋊D29—24no—
28#11745288C2212yescomplemented
29#11745286—22no—
30#11745182C2214yescomplemented
31#11745154C2218yescomplemented
32C24×C29C2311yescharacteristic
33#41×C29—22no—
34#41×C29—22no—
35C2×Q8×C29D411yescharacteristic, complemented
36C2×D4×C29—22no—
37C2×D4×C29D411yescharacteristic, complemented
38C2×D4×C29C2312yes—
39C22×C4×C29—22no—
40C22×C4×C29—22no—
41C22×C4×C29—21no—
42C22×C4×C29—21no—
43C22×C4×C29D412yescomplemented
44C22×C4×C29C2311yescharacteristic
45C22×C4×C29C2311yescharacteristic
46C2×C22⋊D29—24no—
47C2×C22⋊D29—24no—
48C2×C22⋊D29C2314yes—
49#90809—24no—
50Q8×D29—24no—
51#90807—24no—
52#90807—28no—
53D4×D29—24no—
54D4×D29—28no—
55#90805—24no—
56#90805—24no—
57#90805—22no—
58#90805—24no—
59#90805—22no—
60#90805—22no—
61#90805—22no—
62C2×D116—22no—
63C2×D116D412yescomplemented
64C2×C4×D29—24no—
65C2×C4×D29D414yescomplemented
66C2×Dic58—22no—
67C2×Dic58D412yescomplemented
68C4⋊C4×C29—21no—
69C4⋊C4×C29—22no—
70C22⋊C4×C29—22no—
71C22⋊C4×C29—22no—
72C22⋊C4×C29C2312yes—
73C42×C29—21no—
74C22⋊Dic29—24no—
75C22⋊Dic29C2314yes—
76#90781—24no—
77C4⋊Dic29—22no—
78#90779—24no—
79C4×Dic29—22no—
80C23×C29—22no—
81C23×C29—21no—
82C23×C29—42no—
83C23×C29C2×D412yes—
84C23×C29C2411yescharacteristic
85C22×D29—24no—
86C22×D29C2×D414yes—
87Q8×C29—22no—
88D4×C29—24no—
89D4×C29—22no—
90D4×C29—44no—
91C2×C4×C29—22no—
92C2×C4×C29—22no—
93C2×C4×C29—22no—
94C2×C4×C29—22no—
95C2×C4×C29—21no—
96C2×C4×C29—41no—
97C2×C4×C29—42no—
98C2×C4×C29—41no—
99C2×C4×C29C2×D412yes—
100C2×C4×C29C2×D412yes—
101C2×C4×C29C2×D412yes—
102C2×C4×C29C2×D411yescharacteristic
103C2×C4×C29C2×D411yescharacteristic
104C22⋊D29—28no—
105C22⋊D29—24no—
106C22⋊D29—24no—
107C22⋊D29—48no—
108C2×Dic29—24no—
109C2×Dic29—24no—
110C2×Dic29C2×D414yes—
111D116—24no—
112D116—22no—
113D116—22no—
114C4×D29—28no—
115C4×D29—24no—
116C4×D29—24no—
117Dic58—24no—
118Dic58—22no—
119Dic58—22no—
120#3025—291nomaximal, Sylow
121C22×C29—24no—
122C22×C29—22no—
123C22×C29—22no—
124C22×C29—41no—
125C22×C29—42no—
126C22×C29—42no—
127C22×C29—41no—
128C22×C29—41no—
129C22×C29C22×D412yes—
130C22×C29C22×D411yescharacteristic
131C2×D29—24no—
132C2×D29—44no—
133C2×D29—84no—
134C4×C29—22no—
135C4×C29—42no—
136C4×C29—41no—
137C4×C29#12011yescharacteristic
138C4×C29#12012yes—
139C4×C29#12011yescharacteristic
140Dic29—24no—
141Dic29—44no—
142#585—291no—
143#525—292no—
144#525D2911yescharacteristic, complemented
145#458—294no—
146B(4,𝔽2)—294no—
147#410—291no—
148#410—292no—
149C2×C29—22no—
150C2×C29—21no—
151C2×C29—42no—
152C2×C29—42no—
153C2×C29C2×#12011yescharacteristic
154D29—44no—
1552−1+4—582no—
1562−1+4—581no—
157H(5,𝔽2)—294no—
158H(5,𝔽2)—582no—
159H(5,𝔽2)—581no—
160C2×#41—292no—
161C2×#41—584no—
162C2×#41—582no—
163C2×#41C2×D2911yescharacteristic
164C23×C4C2×D2911yescharacteristic, complemented
165#123—294no—
166#123C2×D2912yescomplemented
167C22⋊Q8—21no—
168C22⋊Q8—582no—
169#121—21no—
170#121—582no—
171#120—294no—
172#120C2×D2912yescomplemented
173C4×D4—22no—
174C4×D4—584no—
175#117—292no—
176#117C2×D2911yescharacteristic, complemented
177#99—294no—
178#99—298no—
179C29#302511yescharacteristic, complemented, Sylow
180C24C22×D2911yescharacteristic
181#41—22no—
182#41—22no—
183#41—584no—
184#41—584no—
185#41—588no—
186#41—584no—
187#41—584no—
188#41—582no—
189#41—582no—
190#41—582no—
191#41—582no—
192#41—584no—
193C2×Q8—292no—
194C2×Q8—582no—
195C2×Q8—584no—
196C2×Q8C22⋊D2911yescharacteristic, complemented
197C2×D4—22no—
198C2×D4—294no—
199C2×D4—292no—
200C2×D4—582no—
201C2×D4—588no—
202C2×D4—584no—
203C2×D4—584no—
204C2×D4—584no—
205C2×D4C22⋊D2911yescharacteristic, complemented
206C2×D4C22×D2912yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—294no—
212C22×C4—584no—
213C22×C4C22⋊D2912yescomplemented
214C22×C4C22×D2911yescharacteristic
215C22×C4C22×D2911yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—584no—
219C4⋊C4—582no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—294no—
223C22⋊C4—584no—
224C22⋊C4—584no—
225C22⋊C4C22×D2912yes—
226C42—21no—
227C42—582no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—294no—
232C23—584no—
233C23C2×C22⋊D2912yes—
234C23C23×D2911yescharacteristic
235Q8—22no—
236Q8—584no—
237Q8—582no—
238Q8—582no—
239D4—24no—
240D4—22no—
241D4—44no—
242D4—584no—
243D4—588no—
244D4—584no—
245D4—584no—
246D4—582no—
247D4—582no—
248D4—1168no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—294no—
258C2×C4—588no—
259C2×C4—584no—
260C2×C4—584no—
261C2×C4—584no—
262C2×C4—584no—
263C2×C4D4×D2912yes—
264C2×C4D4×D2912yes—
265C2×C4C2×C22⋊D2912yes—
266C2×C4C2×C22⋊D2911yescharacteristic
267C2×C4C2×C22⋊D2911yescharacteristic
268C22—22no—
269C22—24no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—584no—
277C22—1164no—
278C22—2324no—
279C22C2×D4×D2912yes—
280C22C22×C22⋊D2911yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—584no—
285C4—1164no—
286C4#1174529911yescharacteristic
287C4#1174529912yes—
288C4#1174529911yescharacteristic
289C2—21no—
290C2—22no—
291C2—42no—
292C2—42no—
293C2—1164no—
294C2C2×#1174529911yescharacteristic
subgroupG/H
centerC4#11745299
commutator, cosocleC23×C29C24
FrattiniC23C23×D29
FittingC29×#525C2
socleC2×C29C2×#120
hypercenter#525D29
hypocenterC29#3025
LCSG ▷ C23×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #14271999C2semidirect
2. #14271674C2semidirect
3. #14271352C2semidirect
4. #14271349C2semidirect
5. #14271137C2semidirect
6. #14271136C2semidirect
7. C29×#525C2semidirect
9. C23×C4×C29C22semidirect
18. C29×#123C22semidirect
21. C29×#120C22semidirect
23. C29×#117C22semidirect
24. #11745315C22semidirect
26. #11745291C22semidirect
28. #11745288C22semidirect
30. #11745182C22semidirect
31. #11745154C22semidirect
35. C2×Q8×C29D4semidirect
37. C2×D4×C29D4semidirect
43. C22×C4×C29D4semidirect
63. C2×D116D4semidirect
65. C2×C4×D29D4semidirect
67. C2×Dic58D4semidirect
144. #525D29semidirect
164. C23×C4C2×D29semidirect
166. #123C2×D29semidirect
172. #120C2×D29semidirect
176. #117C2×D29semidirect
179. C29#3025semidirect
196. C2×Q8C22⋊D29semidirect
205. C2×D4C22⋊D29semidirect
213. C22×C4C22⋊D29semidirect

Maximal subgroups

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

subgroupindex
#142719992
#142716742
#142713522
#142713492
#142711372
#142711362
C29×#5252
#302529