Finite Group Databasebeta

← #19137922

Subgroups — #19137922

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×#13063308C212yesmaximal, complemented
3#14582814C214yesmaximal, complemented
4#14582778C212yesmaximal, complemented
5#14582771C212yesmaximal, complemented
6#14582755C211yesmaximal, characteristic, complemented
7#14582754C212yesmaximal, complemented
8C22×C22⋊C4×C43C211yesmaximal, characteristic, complemented
9C24×D43C2211yescharacteristic
10C22×C22⋊D43—24no—
11C22×C22⋊D43—28no—
12C22×C22⋊D43C2212yes—
13C22×C22⋊D43C2212yes—
14C23×Dic43C2211yescharacteristic
15#13063308—28no—
16C2×Dic43⋊C4C2214yescomplemented
17C2×C22⋊Dic43C2212yes—
18C2×C22⋊Dic43C2212yescomplemented
19C2×C22⋊Dic43C2212yescomplemented
20C2×(C2×D43)⋊C4C2214yescomplemented
21C2×(C2×D43)⋊C4C2214yescomplemented
22C43⋊#95C2214yescomplemented
23C25×C43C2211yescharacteristic
24C23×C4×C43C2211yescharacteristic, complemented
25C23×C4×C43C2211yescharacteristic, complemented
26C2×C22⋊C4×C43—24no—
27C2×C22⋊C4×C43C2212yes—
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×C43—22no—
42C22×C4×C43—22no—
43C22×C4×C43—24no—
44C22×C4×C43—42no—
45C22×C4×C43C2312yes—
46C22×C4×C43C2312yes—
47C2×C22⋊D43—232no—
48C2×C22⋊D43—24no—
49C2×C22⋊D43—24no—
50C2×C22⋊D43—24no—
51C2×C22⋊D43—24no—
52C2×C22⋊D43—24no—
53C2×C22⋊D43D418yes—
54C22×Dic43—24no—
55C22×Dic43—22no—
56C22×Dic43C2314yes—
57C22×Dic43C2312yes—
58C22⋊C4×C43—24no—
59C22⋊C4×C43—24no—
60C22⋊Dic43—24no—
61C22⋊Dic43—24no—
62C22⋊Dic43—42no—
63(C2×D43)⋊C4—28no—
64(C2×D43)⋊C4—44no—
65Dic43⋊C4—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×D414yes—
78C23×C43C2×D412yes—
79C23×C43C2×D412yes—
80C23×C43C2×D412yes—
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—22no—
94C2×C4×C43—22no—
95C2×C4×C43—24no—
96C2×C4×C43—44no—
97C2×C4×C43—44no—
98C22⋊D43—216no—
99C22⋊D43—216no—
100C2×Dic43—28no—
101C2×Dic43—22no—
102C2×Dic43—22no—
103C2×Dic43—24no—
104C2×Dic43—24no—
105C2×Dic43—24no—
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—42no—
131C4×C43—42no—
132Dic43—24no—
133Dic43—44no—
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⋊C4D4311yescharacteristic, complemented
146#394—434no—
147#386—432no—
148#385—432no—
149#379—431no—
150#379—432no—
151C43#240511yescharacteristic, complemented, Sylow
152C25—431no—
153C25C2×D4311yescharacteristic
154C22×D4—432no—
155C22×D4—432no—
156C22×D4—868no—
157C22×D4—864no—
158C23×C4—431no—
159C23×C4C2×D4311yescharacteristic, complemented
160C23×C4C2×D4311yescharacteristic, complemented
161#123—868no—
162C2×C4⋊C4—434no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—432no—
165C2×C22⋊C4—432no—
166C2×C22⋊C4—434no—
167C2×C22⋊C4—432no—
168C2×C22⋊C4—434no—
169C2×C22⋊C4C2×D4312yes—
170C2×C22⋊C4C2×D4312yescomplemented
171#95—434no—
172C24—24no—
173C24—24no—
174C24—42no—
175C24—432no—
176C24—864no—
177C24—862no—
178C24—864no—
179C24—1722no—
180C24C22⋊D4314yes—
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—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—432no—
196C22×C4—434no—
197C22×C4—864no—
198C22×C4—862no—
199C22×C4C22×D4312yes—
200C22×C4C22×D4312yes—
201C4⋊C4—868no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—868no—
205C22⋊C4—864no—
206C22⋊C4—864no—
207C22⋊C4—1724no—
208C22⋊C4—1722no—
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—
229C23#1062189814yes—
230C23D4×D4314yes—
231C23C2×C22⋊D4312yes—
232C23C2×C22⋊D4312yes—
233C23C2×C22⋊D4312yes—
234C23C23×D4311yescharacteristic
235D4—8616no—
236D4—8616no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—434no—
244C2×C4—868no—
245C2×C4—862no—
246C2×C4—862no—
247C2×C4—864no—
248C2×C4—864no—
249C2×C4—864no—
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#1306329314yes—
268C22#1306330814yescomplemented
269C22C2×D4×D4312yes—
270C22C22×C22⋊D4311yescharacteristic
271C22C2×#1062189812yes—
272C22#1306337012yes—
273C4—42no—
274C4—42no—
275C4—864no—
276C4—1724no—
277C2—24no—
278C2—24no—
279C2—42no—
280C2—1724no—
281C2C2×#1306329311yescharacteristic
282C2C2×#1306330811yescharacteristic
283C2#1458345811yescharacteristic
284C2#1458353914yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC22×C22⋊C4×C43C2
hypercenterC22×C22⋊C4D43
hypocenterC43#2405
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#13063308C2semidirect
3. #14582814C2semidirect
4. #14582778C2semidirect
5. #14582771C2semidirect
6. #14582755C2semidirect
7. #14582754C2semidirect
8. C22×C22⋊C4×C43C2semidirect
16. C2×Dic43⋊C4C22semidirect
18. C2×C22⋊Dic43C22semidirect
19. C2×C22⋊Dic43C22semidirect
20. C2×(C2×D43)⋊C4C22semidirect
21. C2×(C2×D43)⋊C4C22semidirect
22. C43⋊#95C22semidirect
24. C23×C4×C43C22semidirect
25. C23×C4×C43C22semidirect
28. C2×C22⋊C4×C43C22semidirect
119. C22×C43#123semidirect
145. C22×C22⋊C4D43semidirect
151. C43#2405semidirect
159. C23×C4C2×D43semidirect
160. C23×C4C2×D43semidirect
170. C2×C22⋊C4C2×D43semidirect
268. C22#13063308semidirect