Finite Group Databasebeta

← #16828886

Subgroups — #16828886

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#14271675C212yesmaximal, complemented
3#14271670C211yesmaximal, characteristic, complemented
4#14271354C214yesmaximal, complemented
5#14271351C214yesmaximal, complemented
6#14271136C212yesmaximal, complemented
7C29×#410C211yesmaximal, characteristic, complemented
8C2×#41×C29C2211yescharacteristic
9#11745365—22no—
10#11745364—22no—
11#41×D29—24no—
12#11745362—21no—
13#11745359—21no—
14#11745359C2214yes—
15C2×#90805—22no—
16C2×#90805C2212yes—
17C22×C4×D29C2211yescharacteristic, complemented
18C29×#117C2212yescomplemented
19#11745308—21no—
20#11745300—21no—
21#11745299C2212yescomplemented
22#11745297C2212yescomplemented
23C4×C22⋊D29—22no—
24#11745288C2211yescharacteristic, complemented
25#11745259—22no—
26#11745257—22no—
27#11745248C2214yescomplemented
28#11745244C2214yescomplemented
29C4×D116—24no—
30C29×#99C2214yescomplemented
31#11745154C2218yescomplemented
32C23×D29C2311yescharacteristic
33#41×C29—22no—
34#41×C29—22no—
35C2×Q8×C29D411yescharacteristic, complemented
36C2×D4×C29D411yescharacteristic, complemented
37C2×D4×C29C2312yes—
38C22×C4×C29D412yescomplemented
39C22×C4×C29C2311yescharacteristic
40C2×C22⋊D29—24no—
41C2×C22⋊D29—22no—
42C2×C22⋊D29C2314yes—
43C22×Dic29C2311yescharacteristic
44#90809—24no—
45Q8×D29—24no—
46#90807—24no—
47#90807—28no—
48D4×D29—24no—
49D4×D29—28no—
50#90805—24no—
51#90805—24no—
52#90805—22no—
53#90805—24no—
54#90805—22no—
55#90805—22no—
56#90805—22no—
57C2×D116—22no—
58C2×D116—24no—
59C2×D116D412yescomplemented
60C2×C4×D29—24no—
61C2×C4×D29—21no—
62C2×C4×D29—21no—
63C2×C4×D29—22no—
64C2×C4×D29—22no—
65C2×C4×D29D414yescomplemented
66C2×Dic58—22no—
67C2×Dic58D412yescomplemented
68C4⋊C4×C29—22no—
69C22⋊C4×C29C2314yes—
70C42×C29—22no—
71C22⋊Dic29—22no—
72C22⋊Dic29C2312yes—
73#90781—24no—
74#90781—22no—
75#90781—24no—
76C4⋊Dic29—21no—
77C4⋊Dic29—24no—
78#90779—22no—
79C4×Dic29—21no—
80C23×C29C2×D412yes—
81C23×C29C2411yescharacteristic
82C22×D29—22no—
83C22×D29—21no—
84C22×D29—24no—
85C22×D29—42no—
86C22×D29C2×D414yes—
87Q8×C29—22no—
88D4×C29—24no—
89D4×C29—22no—
90C2×C4×C29—22no—
91C2×C4×C29—22no—
92C2×C4×C29—24no—
93C2×C4×C29C2×D412yes—
94C2×C4×C29C2×D411yescharacteristic
95C2×C4×C29C2×D411yescharacteristic
96C2×C4×C29C2×D412yes—
97C2×C4×C29C2×D412yes—
98C22⋊D29—28no—
99C22⋊D29—24no—
100C22⋊D29—24no—
101C22⋊D29—44no—
102C2×Dic29—24no—
103C2×Dic29—22no—
104C2×Dic29—21no—
105C2×Dic29—22no—
106C2×Dic29C2×D414yes—
107D116—24no—
108D116—22no—
109D116—22no—
110D116—48no—
111C4×D29—28no—
112C4×D29—24no—
113C4×D29—24no—
114C4×D29—41no—
115C4×D29—42no—
116C4×D29—41no—
117Dic58—24no—
118Dic58—22no—
119Dic58—22no—
120#3025—291nomaximal, Sylow
121C22×C29—22no—
122C22×C29—41no—
123C22×C29—42no—
124C22×C29C22×D411yescharacteristic
125C22×C29C22×D412yes—
126C2×D29—24no—
127C2×D29—22no—
128C2×D29—24no—
129C2×D29—41no—
130C2×D29—42no—
131C2×D29—41no—
132C2×D29—44no—
133C2×D29—84no—
134C4×C29—22no—
135C4×C29—44no—
136C4×C29#12011yescharacteristic
137C4×C29#12012yes—
138C4×C29#12011yescharacteristic
139Dic29—24no—
140Dic29—42no—
141Dic29—41no—
142#585—291no—
143#525—291no—
144#525—292no—
145#458—294no—
146B(4,𝔽2)—294no—
147#410—292no—
148#410D2911yescharacteristic, complemented
149C2×C29—21no—
150C2×C29—22no—
151C2×C29—42no—
152C2×C29C2×#12011yescharacteristic
153D29—42no—
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×C4—291no—
165#123—292no—
166#123—294no—
167C22⋊Q8—582no—
168C22⋊Q8—581no—
169#121—582no—
170#121—581no—
171#120—292no—
172#120—294no—
173C4×D4—584no—
174C4×D4—582no—
175#117—291no—
176#117C2×D2912yescomplemented
177#99—298no—
178#99C2×D2914yescomplemented
179C29#302511yescharacteristic, complemented, Sylow
180C24—291no—
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—584no—
191#41—582no—
192#41—582no—
193C2×Q8—292no—
194C2×Q8—582no—
195C2×Q8—584no—
196C2×Q8D11611yescharacteristic, complemented
197C2×D4—294no—
198C2×D4—292no—
199C2×D4—582no—
200C2×D4—584no—
201C2×D4—588no—
202C2×D4—584no—
203C2×D4—584no—
204C2×D4—582no—
205C2×D4D11611yescharacteristic, complemented
206C2×D4C22×D2912yes—
207C22×C4—294no—
208C22×C4—291no—
209C22×C4—584no—
210C22×C4—581no—
211C22×C4—581no—
212C22×C4—582no—
213C22×C4—582no—
214C22×C4D11612yescomplemented
215C22×C4C22×D2911yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—584no—
218C4⋊C4—581no—
219C4⋊C4—582no—
220C22⋊C4—292no—
221C22⋊C4—584no—
222C22⋊C4—584no—
223C22⋊C4—582no—
224C22⋊C4—582no—
225C22⋊C4C22×D2914yes—
226C42—22no—
227C42—581no—
228C23—294no—
229C23—584no—
230C23—581no—
231C23—582no—
232C23—1162no—
233C23C2×D11612yes—
234C23C23×D2911yescharacteristic
235Q8—22no—
236Q8—584no—
237Q8—582no—
238Q8—582no—
239D4—24no—
240D4—22no—
241D4—584no—
242D4—588no—
243D4—584no—
244D4—584no—
245D4—582no—
246D4—582no—
247D4—1168no—
248D4—1164no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—294no—
253C2×C4—588no—
254C2×C4—584no—
255C2×C4—584no—
256C2×C4—584no—
257C2×C4—582no—
258C2×C4—581no—
259C2×C4—582no—
260C2×C4—1161no—
261C2×C4—1162no—
262C2×C4—1161no—
263C2×C4C2×D11612yes—
264C2×C4C2×D11611yescharacteristic
265C2×C4C2×D11611yescharacteristic
266C2×C4D4×D2912yes—
267C2×C4D4×D2912yes—
268C22—22no—
269C22—41no—
270C22—42no—
271C22—584no—
272C22—582no—
273C22—584no—
274C22—1164no—
275C22—1162no—
276C22—1161no—
277C22—1161no—
278C22—2324no—
279C22C22×D11611yescharacteristic
280C22C2×D4×D2912yes—
281C4—22no—
282C4—44no—
283C4—584no—
284C4—1162no—
285C4—1161no—
286C4#1174524411yescharacteristic
287C4#1174524412yes—
288C4#1174524411yescharacteristic
289C2—22no—
290C2—21no—
291C2—42no—
292C2—1162no—
293C2—1164no—
294C2C2×#1174524411yescharacteristic
subgroupG/H
centerC4#11745244
commutator, cosocleC23×C29C24
FrattiniC23C23×D29
FittingC29×#410C2
socleC2×C29C2×#120
hypercenter#410D29
hypocenterC29#3025
LCSG ▷ C23×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #14271999C2semidirect
2. #14271675C2semidirect
3. #14271670C2semidirect
4. #14271354C2semidirect
5. #14271351C2semidirect
6. #14271136C2semidirect
7. C29×#410C2semidirect
17. C22×C4×D29C22semidirect
18. C29×#117C22semidirect
21. #11745299C22semidirect
22. #11745297C22semidirect
24. #11745288C22semidirect
27. #11745248C22semidirect
28. #11745244C22semidirect
30. C29×#99C22semidirect
31. #11745154C22semidirect
35. C2×Q8×C29D4semidirect
36. C2×D4×C29D4semidirect
38. C22×C4×C29D4semidirect
59. C2×D116D4semidirect
65. C2×C4×D29D4semidirect
67. C2×Dic58D4semidirect
148. #410D29semidirect
176. #117C2×D29semidirect
178. #99C2×D29semidirect
179. C29#3025semidirect
196. C2×Q8D116semidirect
205. C2×D4D116semidirect
214. C22×C4D116semidirect

Maximal subgroups

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

subgroupindex
#142719992
#142716752
#142716702
#142713542
#142713512
#142711362
C29×#4102
#302529