Finite Group Databasebeta

← #19137918

Subgroups — #19137918

Order 5504.

1116 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 197 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×#13063306C212yesmaximal, complemented
2C22×C22⋊Dic43C211yesmaximal, characteristic, complemented
3#14582813C214yesmaximal, complemented
4#14582779C212yesmaximal, complemented
5#14582772C212yesmaximal, complemented
6#14582755C212yesmaximal, complemented
7#14582755C211yesmaximal, characteristic, complemented
8C23×D4×C43C211yesmaximal, characteristic
9C23×Dic43C2211yescharacteristic, complemented
10C23×Dic43C2211yescharacteristic, complemented
11#13063306—28no—
12C2×Dic43⋊C4C2214yescomplemented
13C2×C22⋊Dic43—24no—
14C2×C22⋊Dic43C2212yes—
15C2×C22⋊Dic43C2212yes—
16C2×C22⋊Dic43C2212yescomplemented
17C2×C22⋊Dic43C2214yescomplemented
18C2×C22⋊Dic43C2212yescomplemented
19C2×C22⋊Dic43C2212yescomplemented
20C2×C22⋊Dic43C2214yescomplemented
21C43⋊#95C2214yescomplemented
22C25×C43C2211yescharacteristic
23C25×C43C2211yescharacteristic
24C22×D4×C43—24no—
25C22×D4×C43—28no—
26C22×D4×C43C2212yes—
27C22×D4×C43C2212yes—
28C23×C4×C43C2211yescharacteristic
29C24×C43—24no—
30C24×C43—24no—
31C24×C43—22no—
32C24×C43—24no—
33C24×C43—24no—
34C24×C43—42no—
35C24×C43—42no—
36C24×C43D414yes—
37C24×C43C2311yescharacteristic
38C24×C43C2312yes—
39C24×C43C2311yescharacteristic
40C24×C43C2311yescharacteristic
41C2×D4×C43—232no—
42C2×D4×C43—24no—
43C2×D4×C43—24no—
44C2×D4×C43—24no—
45C2×D4×C43—24no—
46C2×D4×C43—24no—
47C2×D4×C43D418yes—
48C22×C4×C43—24no—
49C22×C4×C43—22no—
50C22×C4×C43C2312yes—
51C22×Dic43—22no—
52C22×Dic43—22no—
53C22×Dic43—24no—
54C22×Dic43—42no—
55C22×Dic43C2312yes—
56C22×Dic43C2314yes—
57C22×Dic43C2312yes—
58C22⋊Dic43—24no—
59C22⋊Dic43—24no—
60C22⋊Dic43—28no—
61C22⋊Dic43—24no—
62C22⋊Dic43—24no—
63C22⋊Dic43—42no—
64C22⋊Dic43—44no—
65Dic43⋊C4—28no—
66C23×C43—22no—
67C23×C43—22no—
68C23×C43—22no—
69C23×C43—22no—
70C23×C43—28no—
71C23×C43—28no—
72C23×C43—28no—
73C23×C43—22no—
74C23×C43—22no—
75C23×C43—22no—
76C23×C43—22no—
77C23×C43—24no—
78C23×C43—22no—
79C23×C43—24no—
80C23×C43—44no—
81C23×C43—416no—
82C23×C43—416no—
83C23×C43—44no—
84C23×C43—44no—
85C23×C43C2×D412yes—
86C23×C43C2×D414yes—
87C23×C43C2×D414yes—
88C23×C43C2×D412yes—
89C23×C43C2×D412yes—
90C23×C43#4114yes—
91C23×C43C2411yescharacteristic
92D4×C43—216no—
93D4×C43—216no—
94C2×C4×C43—28no—
95C2×C4×C43—22no—
96C2×C4×C43—22no—
97C2×C4×C43C2×D414yes—
98C2×Dic43—24no—
99C2×Dic43—24no—
100C2×Dic43—22no—
101C2×Dic43—22no—
102C2×Dic43—24no—
103C2×Dic43—24no—
104C2×Dic43—24no—
105C2×Dic43—44no—
106C2×Dic43—44no—
107C22×C43—22no—
108C22×C43—22no—
109C22×C43—22no—
110C22×C43—22no—
111C22×C43—28no—
112C22×C43—28no—
113C22×C43—24no—
114C22×C43—22no—
115C22×C43—22no—
116C22×C43—22no—
117C22×C43—28no—
118C22×C43—416no—
119C22×C43—416no—
120C22×C43—42no—
121C22×C43—42no—
122C22×C43—44no—
123C22×C43—44no—
124C22×C43#12014yes—
125C22×C43#12314yescomplemented
126C22×C43C22×D411yescharacteristic
127C22×C43C22×D412yes—
128C22×C43C2×#4112yes—
129C22×C43H(5,𝔽2)12yes—
130C4×C43—24no—
131Dic43—42no—
132Dic43—42no—
133Dic43—44no—
134#2405—431nomaximal, Sylow
135C2×C43—24no—
136C2×C43—24no—
137C2×C43—42no—
138C2×C43—44no—
139C2×C43C2×#12011yescharacteristic
140C2×C43C2×#12311yescharacteristic
141C2×C43#53411yescharacteristic
142C2×C43#54614yes—
143C23×D4D4311yescharacteristic
144C2×#123—432no—
145C22×C22⋊C4—431no—
146#394—434no—
147#386—432no—
148#385—432no—
149#379—432no—
150#379—431no—
151C43#240511yescharacteristic, complemented, Sylow
152C25C2×D4311yescharacteristic
153C25C2×D4311yescharacteristic
154C22×D4—24no—
155C22×D4—28no—
156C22×D4C2×D4312yes—
157C22×D4C2×D4312yes—
158C23×C4—431no—
159C23×C4—431no—
160C23×C4C2×D4311yescharacteristic
161#123—868no—
162C2×C4⋊C4—434no—
163C2×C22⋊C4—432no—
164C2×C22⋊C4—432no—
165C2×C22⋊C4—432no—
166C2×C22⋊C4—434no—
167C2×C22⋊C4—432no—
168C2×C22⋊C4—434no—
169C2×C22⋊C4—432no—
170C2×C22⋊C4—864no—
171#95—434no—
172C24—24no—
173C24—24no—
174C24—24no—
175C24—22no—
176C24—24no—
177C24—42no—
178C24—42no—
179C24C22⋊D4314yes—
180C24C22×D4311yescharacteristic
181C24C22×D4312yes—
182C24C22×D4311yescharacteristic
183C24C22×D4311yescharacteristic
184C2×D4—232no—
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4C22⋊D4318yes—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—432no—
194C22×C4—434no—
195C22×C4—432no—
196C22×C4—862no—
197C22×C4—862no—
198C22×C4—864no—
199C22×C4—1722no—
200C22×C4C22×D4312yes—
201C4⋊C4—868no—
202C22⋊C4—864no—
203C22⋊C4—864no—
204C22⋊C4—864no—
205C22⋊C4—864no—
206C22⋊C4—868no—
207C22⋊C4—1722no—
208C22⋊C4—1724no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—28no—
215C23—28no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—24no—
221C23—22no—
222C23—24no—
223C23—44no—
224C23—416no—
225C23—416no—
226C23—44no—
227C23—44no—
228C23#1062190014yes—
229C23C2×C22⋊D4312yes—
230C23C2×C22⋊D4314yes—
231C23C2×C22⋊D4314yes—
232C23C2×C22⋊D4312yes—
233C23C2×C22⋊D4312yes—
234C23C23×D4311yescharacteristic
235D4—216no—
236D4—216no—
237C2×C4—28no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—864no—
241C2×C4—862no—
242C2×C4—862no—
243C2×C4—864no—
244C2×C4—864no—
245C2×C4—864no—
246C2×C4—864no—
247C2×C4—1724no—
248C2×C4—1724no—
249C2×C4C2×C22⋊D4314yes—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—28no—
255C22—28no—
256C22—22no—
257C22—24no—
258C22—22no—
259C22—22no—
260C22—28no—
261C22—416no—
262C22—416no—
263C22—42no—
264C22—42no—
265C22—44no—
266C22—44no—
267C22#1306329414yes—
268C22#1306330614yescomplemented
269C22C22×C22⋊D4311yescharacteristic
270C22C22×C22⋊D4312yes—
271C22C2×#1062190012yes—
272C22#1306337012yes—
273C4—24no—
274C4—1722no—
275C4—1722no—
276C4—1724no—
277C2—24no—
278C2—24no—
279C2—42no—
280C2—44no—
281C2C2×#1306329411yescharacteristic
282C2C2×#1306330611yescharacteristic
283C2#1458345811yescharacteristic
284C2#1458354314yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC23×D4×C43C2
hypercenterC23×D4D43
hypocenterC43#2405
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#13063306C2semidirect
2. C22×C22⋊Dic43C2semidirect
3. #14582813C2semidirect
4. #14582779C2semidirect
5. #14582772C2semidirect
6. #14582755C2semidirect
7. #14582755C2semidirect
9. C23×Dic43C22semidirect
10. C23×Dic43C22semidirect
12. C2×Dic43⋊C4C22semidirect
16. C2×C22⋊Dic43C22semidirect
17. C2×C22⋊Dic43C22semidirect
18. C2×C22⋊Dic43C22semidirect
19. C2×C22⋊Dic43C22semidirect
20. C2×C22⋊Dic43C22semidirect
21. C43⋊#95C22semidirect
125. C22×C43#123semidirect
151. C43#2405semidirect
268. C22#13063306semidirect