Finite Group Databasebeta

← #19137923

Subgroups — #19137923

Order 5504.

1116 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C23×C22⋊D43C211yesmaximal, characteristic
2C2×#13063309C212yesmaximal, complemented
3C22×C22⋊Dic43C211yesmaximal, characteristic, complemented
4#14582815C214yesmaximal, complemented
5#14582776C212yesmaximal, complemented
6#14582774C212yesmaximal, complemented
7#14582754C212yesmaximal, complemented
8C43×#379C211yesmaximal, characteristic, complemented
9C24×D43C2211yescharacteristic
10C22×C22⋊D43—24no—
11C22×C22⋊D43—28no—
12C22×C22⋊D43C2212yes—
13C22×C22⋊D43C2212yes—
14C23×Dic43C2211yescharacteristic
15C23×Dic43C2211yescharacteristic, complemented
16C23×Dic43C2211yescharacteristic, complemented
17#13063309—28no—
18C2×C4⋊Dic43C2214yescomplemented
19C2×C22⋊Dic43—24no—
20C2×C22⋊Dic43C2212yes—
21C2×C22⋊Dic43C2212yescomplemented
22C2×(C2×D43)⋊C4C2214yescomplemented
23C2×(C2×D43)⋊C4C2214yescomplemented
24C43⋊#95C2214yescomplemented
25C25×C43C2211yescharacteristic
26C2×C22⋊C4×C43C2212yes—
27C2×C22⋊C4×C43C2212yescomplemented
28C2×C22⋊C4×C43C2212yescomplemented
29C24×C43—24no—
30C24×C43—24no—
31C24×C43—42no—
32C24×C43D414yes—
33C24×C43C2311yescharacteristic
34C24×C43C2311yescharacteristic
35C24×C43C2311yescharacteristic
36C23×D43—22no—
37C23×D43—24no—
38C23×D43—24no—
39C23×D43—42no—
40C23×D43C2312yes—
41C22×C4×C43C2314yes—
42C2×C22⋊D43—232no—
43C2×C22⋊D43—24no—
44C2×C22⋊D43—24no—
45C2×C22⋊D43—24no—
46C2×C22⋊D43—24no—
47C2×C22⋊D43—24no—
48C2×C22⋊D43D418yes—
49C22×Dic43—24no—
50C22×Dic43—22no—
51C22×Dic43—22no—
52C22×Dic43—22no—
53C22×Dic43—24no—
54C22×Dic43—42no—
55C22×Dic43C2312yes—
56C22×Dic43C2312yes—
57C22×Dic43C2312yes—
58C22⋊C4×C43—24no—
59C22⋊C4×C43—24no—
60C22⋊C4×C43—42no—
61C22⋊Dic43—24no—
62C22⋊Dic43—24no—
63(C2×D43)⋊C4—28no—
64(C2×D43)⋊C4—44no—
65C4⋊Dic43—28no—
66C23×C43—22no—
67C23×C43—22no—
68C23×C43—22no—
69C23×C43—22no—
70C23×C43—28no—
71C23×C43—28no—
72C23×C43—22no—
73C23×C43—22no—
74C23×C43—22no—
75C23×C43—44no—
76C23×C43—416no—
77C23×C43C2×D412yes—
78C23×C43C2×D412yes—
79C23×C43C2×D412yes—
80C23×C43C2×D414yes—
81C23×C43#4114yes—
82C23×C43C2411yescharacteristic
83C22×D43—28no—
84C22×D43—22no—
85C22×D43—22no—
86C22×D43—24no—
87C22×D43—24no—
88C22×D43—416no—
89C22×D43—44no—
90C22×D43—44no—
91C22×D43C2×D414yes—
92C2×C4×C43—24no—
93C2×C4×C43—24no—
94C2×C4×C43—24no—
95C22⋊D43—216no—
96C22⋊D43—216no—
97C2×Dic43—28no—
98C2×Dic43—22no—
99C2×Dic43—22no—
100C2×Dic43—24no—
101C2×Dic43—22no—
102C2×Dic43—22no—
103C2×Dic43—24no—
104C2×Dic43—44no—
105C2×Dic43—44no—
106C2×Dic43C2×D414yes—
107C22×C43—22no—
108C22×C43—22no—
109C22×C43—22no—
110C22×C43—22no—
111C22×C43—28no—
112C22×C43—28no—
113C22×C43—22no—
114C22×C43—22no—
115C22×C43—22no—
116C22×C43—416no—
117C22×C43—44no—
118C22×C43#12014yes—
119C22×C43#12314yescomplemented
120C22×C43C22×D412yes—
121C22×C43C22×D411yescharacteristic
122C22×C43C2×#4112yes—
123C22×C43H(5,𝔽2)12yes—
124C2×D43—24no—
125C2×D43—28no—
126C2×D43—416no—
127C2×D43—42no—
128C2×D43—42no—
129C2×D43—44no—
130C4×C43—44no—
131Dic43—24no—
132Dic43—42no—
133Dic43—42no—
134#2405—431nomaximal, Sylow
135C2×C43—24no—
136C2×C43—24no—
137C2×C43—42no—
138C2×C43C2×#12011yescharacteristic
139C2×C43C2×#12311yescharacteristic
140C2×C43#53411yescharacteristic
141C2×C43#54614yes—
142D43—44no—
143C23×D4—431no—
144C2×#123—432no—
145C22×C22⋊C4—431no—
146#394—434no—
147#386—432no—
148#385—432no—
149#379—432no—
150#379D4311yescharacteristic, complemented
151C43#240511yescharacteristic, complemented, Sylow
152C25—431no—
153C25C2×D4311yescharacteristic
154C22×D4—432no—
155C22×D4—432no—
156C22×D4—868no—
157C22×D4—864no—
158C23×C4—431no—
159C23×C4—431no—
160C23×C4—431no—
161#123—868no—
162C2×C4⋊C4—434no—
163C2×C22⋊C4—432no—
164C2×C22⋊C4—432no—
165C2×C22⋊C4—434no—
166C2×C22⋊C4—434no—
167C2×C22⋊C4—864no—
168C2×C22⋊C4C2×D4312yes—
169C2×C22⋊C4C2×D4312yescomplemented
170C2×C22⋊C4C2×D4312yescomplemented
171#95—434no—
172C24—24no—
173C24—24no—
174C24—42no—
175C24—432no—
176C24—864no—
177C24—862no—
178C24—864no—
179C24—1722no—
180C24D17214yes—
181C24C22×D4311yescharacteristic
182C24C22×D4311yescharacteristic
183C24C22×D4311yescharacteristic
184C2×D4—438no—
185C2×D4—864no—
186C2×D4—864no—
187C2×D4—8632no—
188C2×D4—864no—
189C2×D4—864no—
190C2×D4—864no—
191C22×C4—432no—
192C22×C4—432no—
193C22×C4—432no—
194C22×C4—864no—
195C22×C4—862no—
196C22×C4—862no—
197C22×C4—862no—
198C22×C4—864no—
199C22×C4—1722no—
200C22×C4C22×D4314yes—
201C4⋊C4—868no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—42no—
205C22⋊C4—868no—
206C22⋊C4—864no—
207C22⋊C4—864no—
208C22⋊C4—1724no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—28no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—44no—
219C23—416no—
220C23—434no—
221C23—868no—
222C23—862no—
223C23—862no—
224C23—864no—
225C23—864no—
226C23—17216no—
227C23—1724no—
228C23—1724no—
229C23C2×D17212yes—
230C23C2×D17212yes—
231C23C2×D17212yes—
232C23D4×D4314yes—
233C23#1062190014yes—
234C23C23×D4311yescharacteristic
235D4—8616no—
236D4—8616no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—434no—
241C2×C4—868no—
242C2×C4—862no—
243C2×C4—862no—
244C2×C4—864no—
245C2×C4—862no—
246C2×C4—862no—
247C2×C4—864no—
248C2×C4—1724no—
249C2×C4—1724no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—28no—
255C22—28no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—416no—
260C22—44no—
261C22—864no—
262C22—868no—
263C22—17216no—
264C22—1722no—
265C22—1722no—
266C22—1724no—
267C22#1306329514yes—
268C22#1306330914yescomplemented
269C22C2×D4×D4312yes—
270C22C22×D17211yescharacteristic
271C22C2×#1062190012yes—
272C22#1306337012yes—
273C4—44no—
274C4—864no—
275C4—1722no—
276C4—1722no—
277C2—24no—
278C2—24no—
279C2—42no—
280C2—1724no—
281C2C2×#1306329511yescharacteristic
282C2C2×#1306330911yescharacteristic
283C2#1458346111yescharacteristic
284C2#1458353614yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC43×#379C2
hypercenter#379D43
hypocenterC43#2405
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#13063309C2semidirect
3. C22×C22⋊Dic43C2semidirect
4. #14582815C2semidirect
5. #14582776C2semidirect
6. #14582774C2semidirect
7. #14582754C2semidirect
8. C43×#379C2semidirect
15. C23×Dic43C22semidirect
16. C23×Dic43C22semidirect
18. C2×C4⋊Dic43C22semidirect
21. C2×C22⋊Dic43C22semidirect
22. C2×(C2×D43)⋊C4C22semidirect
23. C2×(C2×D43)⋊C4C22semidirect
24. C43⋊#95C22semidirect
27. C2×C22⋊C4×C43C22semidirect
28. C2×C22⋊C4×C43C22semidirect
119. C22×C43#123semidirect
150. #379D43semidirect
151. C43#2405semidirect
169. C2×C22⋊C4C2×D43semidirect
170. C2×C22⋊C4C2×D43semidirect
268. C22#13063309semidirect