Finite Group Databasebeta

← #19140406

Subgroups — #19140406

Order 5504.

668 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13063302C211yesmaximal, characteristic, complemented
2C22×(C2×D43)⋊C4C211yesmaximal, characteristic
3#14582834C212yesmaximal, complemented
4#14582823C212yesmaximal, complemented
5#14582821C212yesmaximal, complemented
6#14582812C212yesmaximal, complemented
7#14582772C211yesmaximal, characteristic, complemented
8#14582757C211yesmaximal, characteristic, complemented
9#14582754C211yesmaximal, characteristic
10#14582754C211yesmaximal, characteristic
11C43×#385C211yesmaximal, characteristic, complemented
12C24×D43C2211yescharacteristic
13C22×Dic86C2211yescharacteristic, complemented
14C23×Dic43C2211yescharacteristic
15#13063302—24no—
16C2×C4⋊Dic43C2211yescharacteristic, complemented
17C2×Dic43⋊C4C2211yescharacteristic, complemented
18C2×Dic43⋊C4C2211yescharacteristic, complemented
19C2×Dic43⋊C4C2212yescomplemented
20C2×Dic43⋊C4C2212yescomplemented
21C2×C22⋊Dic43C2211yescharacteristic
22C2×C22⋊Dic43C2211yescharacteristic
23C2×(C2×D43)⋊C4—24no—
24C2×(C2×D43)⋊C4C2212yes—
25C2×(C2×D43)⋊C4C2212yes—
26C2×(C2×D43)⋊C4C2212yes—
27C2×(C2×D43)⋊C4C2212yes—
28C2×(C2×D43)⋊C4C2212yes—
29C2×(C2×D43)⋊C4C2212yes—
30C43⋊#95C2212yescomplemented
31C43⋊#95C2212yescomplemented
32C23×C4×C43C2211yescharacteristic
33C2×C4⋊C4×C43C2212yescomplemented
34C2×C22⋊C4×C43C2211yescharacteristic
35C2×C22⋊C4×C43C2211yescharacteristic
36#95×C43C2212yescomplemented
37C24×C43C2311yescharacteristic
38C23×D43—22no—
39C23×D43—22no—
40C23×D43—42no—
41C23×D43—42no—
42C23×D43Q814yescomplemented
43C23×D43C2312yes—
44C22×C4×C43—22no—
45C22×C4×C43—42no—
46C22×C4×C43C2311yescharacteristic
47C22×C4×C43C2312yes—
48C22×C4×C43C2312yes—
49C22×C4×C43C2311yescharacteristic
50C22×Dic43—42no—
51C22×Dic43D414yes—
52C22×Dic43C2312yes—
53C22×Dic43C2311yescharacteristic
54C22×Dic43C2311yescharacteristic
55C22×Dic43C2311yescharacteristic
56C22×Dic43C2311yescharacteristic
57C2×Dic86—22no—
58C2×Dic86—42no—
59C4⋊C4×C43—42no—
60C22⋊C4×C43—22no—
61C22⋊C4×C43—22no—
62C22⋊Dic43—22no—
63C22⋊Dic43—22no—
64(C2×D43)⋊C4—24no—
65(C2×D43)⋊C4—24no—
66(C2×D43)⋊C4—42no—
67(C2×D43)⋊C4—42no—
68(C2×D43)⋊C4—42no—
69(C2×D43)⋊C4—42no—
70C4⋊Dic43—22no—
71Dic43⋊C4—22no—
72Dic43⋊C4—22no—
73Dic43⋊C4—42no—
74Dic43⋊C4—42no—
75C23×C43—22no—
76C23×C43—22no—
77C23×C43C2×D412yes—
78C23×C43C2×Q812yes—
79C23×C43#4112yes—
80C23×C43C2411yescharacteristic
81C22×D43—28no—
82C22×D43—22no—
83C22×D43—22no—
84C22×D43—22no—
85C22×D43—22no—
86C22×D43—22no—
87C22×D43—22no—
88C22×D43—48no—
89C22×D43—44no—
90C22×D43—44no—
91C22×D43—48no—
92C22×D43C2×Q814yes—
93C2×C4×C43—21no—
94C2×C4×C43—21no—
95C2×C4×C43—22no—
96C2×C4×C43—22no—
97C2×C4×C43—22no—
98C2×C4×C43—22no—
99C2×C4×C43—22no—
100C2×C4×C43—22no—
101C2×C4×C43—22no—
102C2×C4×C43—22no—
103C2×C4×C43—44no—
104C2×Dic43—22no—
105C2×Dic43—22no—
106C2×Dic43—21no—
107C2×Dic43—21no—
108C2×Dic43—22no—
109C2×Dic43—22no—
110C2×Dic43—22no—
111C2×Dic43—22no—
112C2×Dic43—22no—
113C2×Dic43—44no—
114C2×Dic43C2×D412yes—
115C2×Dic43C2×D412yes—
116Dic86—44no—
117C22×C43—22no—
118C22×C43—22no—
119C22×C43—24no—
120C22×C43—24no—
121C22×C43C22⋊Q814yes—
122C22×C43C22×D411yescharacteristic
123C22×C43C22×Q811yescharacteristic
124C22×C43C2×#4111yescharacteristic
125C22×C43H(5,𝔽2)12yes—
126C22×C43H(5,𝔽2)12yes—
127C2×D43—24no—
128C2×D43—24no—
129C2×D43—24no—
130C2×D43—416no—
131C2×D43—42no—
132C2×D43—42no—
133C2×D43—42no—
134C2×D43—42no—
135C4×C43—41no—
136C4×C43—41no—
137C4×C43—42no—
138C4×C43—42no—
139Dic43—22no—
140Dic43—22no—
141Dic43—41no—
142Dic43—41no—
143Dic43—42no—
144#2626—431nomaximal, Sylow
145C2×C43—24no—
146C2×C43C2×C22⋊Q811yescharacteristic
147C2×C43#53411yescharacteristic
148C2×C43#53511yescharacteristic
149C2×C43#53812yes—
150C2×C43#54312yes—
151D43—44no—
152C2×C22⋊Q8—431no—
153C22×C22⋊C4—431no—
154#399—432no—
155#396—432no—
156#396—432no—
157#393—432no—
158#385—431no—
159#385D4311yescharacteristic, complemented
160#380—431no—
161#379—431no—
162#379—431no—
163C43#262611yescharacteristic, complemented, Sylow
164C25—431no—
165C22×Q8—431no—
166C23×C4—431no—
167C23×C4C2×D4311yescharacteristic
168C22⋊Q8—864no—
169C2×C4⋊C4—431no—
170C2×C4⋊C4—431no—
171C2×C4⋊C4—431no—
172C2×C4⋊C4—432no—
173C2×C4⋊C4—432no—
174C2×C4⋊C4C2×D4312yescomplemented
175C2×C22⋊C4—431no—
176C2×C22⋊C4—431no—
177C2×C22⋊C4—432no—
178C2×C22⋊C4—432no—
179C2×C22⋊C4—432no—
180C2×C22⋊C4—432no—
181C2×C22⋊C4—432no—
182C2×C22⋊C4—432no—
183C2×C22⋊C4—864no—
184C2×C22⋊C4C2×D4311yescharacteristic
185C2×C22⋊C4C2×D4311yescharacteristic
186#95—432no—
187#95—432no—
188#95C2×D4312yescomplemented
189C24—432no—
190C24—434no—
191C24—862no—
192C24—862no—
193C24—1722no—
194C24—1722no—
195C24C22×D4311yescharacteristic
196C2×Q8—862no—
197C2×Q8—1722no—
198C22×C4—22no—
199C22×C4—42no—
200C22×C4—431no—
201C22×C4—431no—
202C22×C4—434no—
203C22×C4—432no—
204C22×C4—431no—
205C22×C4—431no—
206C22×C4—1722no—
207C22×C4C22×D4312yes—
208C22×C4C22×D4312yes—
209C22×C4C22×D4311yescharacteristic
210C22×C4C22×D4311yescharacteristic
211C4⋊C4—42no—
212C4⋊C4—862no—
213C4⋊C4—862no—
214C4⋊C4—862no—
215C4⋊C4—1722no—
216C4⋊C4—1722no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—864no—
220C22⋊C4—864no—
221C22⋊C4—862no—
222C22⋊C4—862no—
223C22⋊C4—1722no—
224C22⋊C4—1722no—
225C22⋊C4—1722no—
226C22⋊C4—1722no—
227C23—22no—
228C23—22no—
229C23—434no—
230C23—868no—
231C23—862no—
232C23—862no—
233C23—862no—
234C23—862no—
235C23—862no—
236C23—862no—
237C23—1728no—
238C23—1724no—
239C23—1724no—
240C23—1728no—
241C23#1062189812yes—
242C23D4×D4312yes—
243C23Q8×D4312yes—
244C23C23×D4311yescharacteristic
245Q8—1724no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—44no—
257C2×C4—432no—
258C2×C4—432no—
259C2×C4—862no—
260C2×C4—862no—
261C2×C4—861no—
262C2×C4—861no—
263C2×C4—862no—
264C2×C4—862no—
265C2×C4—862no—
266C2×C4—862no—
267C2×C4—862no—
268C2×C4—1724no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—24no—
273C22—864no—
274C22—864no—
275C22—864no—
276C22—17216no—
277C22—1722no—
278C22—1722no—
279C22—1722no—
280C22—1722no—
281C22#1306330314yes—
282C22C2×D4×D4311yescharacteristic
283C22C2×Q8×D4311yescharacteristic
284C22C2×#1062189811yescharacteristic
285C22#1306337012yes—
286C22#1306337112yes—
287C4—41no—
288C4—42no—
289C4—41no—
290C4—42no—
291C4—862no—
292C4—862no—
293C4—1721no—
294C4—1721no—
295C4—1722no—
296C2—24no—
297C2—1724no—
298C2C2×#1306330311yescharacteristic
299C2#1458346011yescharacteristic
300C2#1458346411yescharacteristic
301C2#1458348212yes—
302C2#1458352412yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC43×#385C2
hypercenter#385D43
hypocenterC43#2626
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#13063302C2semidirect
3. #14582834C2semidirect
4. #14582823C2semidirect
5. #14582821C2semidirect
6. #14582812C2semidirect
7. #14582772C2semidirect
8. #14582757C2semidirect
11. C43×#385C2semidirect
13. C22×Dic86C22semidirect
16. C2×C4⋊Dic43C22semidirect
17. C2×Dic43⋊C4C22semidirect
18. C2×Dic43⋊C4C22semidirect
19. C2×Dic43⋊C4C22semidirect
20. C2×Dic43⋊C4C22semidirect
30. C43⋊#95C22semidirect
31. C43⋊#95C22semidirect
33. C2×C4⋊C4×C43C22semidirect
36. #95×C43C22semidirect
42. C23×D43Q8semidirect
159. #385D43semidirect
163. C43#2626semidirect
174. C2×C4⋊C4C2×D43semidirect
188. #95C2×D43semidirect