Finite Group Databasebeta

← #19140724

Subgroups — #19140724

Order 5504.

598 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 137 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×#13063308C211yesmaximal, characteristic, complemented
2#14582830C211yesmaximal, characteristic, complemented
3#14582825C211yesmaximal, characteristic, complemented
4#14582813C212yesmaximal, complemented
5#14582813C211yesmaximal, characteristic, complemented
6#14582807C212yesmaximal, complemented
7#14582807C211yesmaximal, characteristic, complemented
8#14582799C212yesmaximal, complemented
9#14582793C212yesmaximal, complemented
10C4×C22⋊Dic43C211yesmaximal, characteristic, complemented
11C2×#121×C43C211yesmaximal, characteristic, complemented
12C22×C22⋊D43C2212yes—
13C22×C22⋊D43C2211yescharacteristic
14#13063308—24no—
15C2×Dic43⋊C4C2211yescharacteristic
16C2×Dic43⋊C4C2211yescharacteristic
17C2×Dic43⋊C4C2212yes—
18C2×C22⋊Dic43C2212yes—
19C2×C22⋊Dic43C2212yes—
20C2×C22⋊Dic43C2211yescharacteristic
21C2×C22⋊Dic43C2212yes—
22C2×C22⋊Dic43C2212yes—
23C2×(C2×D43)⋊C4C2211yescharacteristic
24C2×(C2×D43)⋊C4C2211yescharacteristic
25C2×(C2×D43)⋊C4C2212yes—
26C2×C4×Dic43C2212yes—
27C43⋊#95C2212yes—
28C43⋊#95C2212yes—
29C43⋊#95C2211yescharacteristic
30C43⋊#95C2211yescharacteristic
31C22×D4×C43C2212yes—
32C22×D4×C43C2211yescharacteristic
33C23×C4×C43C2211yescharacteristic
34#121×C43—24no—
35C2×C4⋊C4×C43C2211yescharacteristic
36C2×C22⋊C4×C43C2212yes—
37C24×C43C2312yes—
38C24×C43C2311yescharacteristic
39C23×D43C2311yescharacteristic
40C2×D4×C43—22no—
41C2×D4×C43—24no—
42C2×D4×C43—42no—
43C2×D4×C43—42no—
44C2×D4×C43—42no—
45C22×C4×C43—42no—
46C22×C4×C43D414yescomplemented
47C22×C4×C43C2312yes—
48C22×C4×C43C2311yescharacteristic
49C22×C4×C43C2311yescharacteristic
50C2×C22⋊D43—24no—
51C2×C22⋊D43—22no—
52C2×C22⋊D43—42no—
53C2×C22⋊D43—42no—
54C2×C22⋊D43—42no—
55C22×Dic43C2312yes—
56C22×Dic43C2311yescharacteristic
57C22×Dic43C2311yescharacteristic
58C22×Dic43C2311yescharacteristic
59C22×Dic43C2312yes—
60C4⋊C4×C43—22no—
61C22⋊C4×C43—24no—
62C22⋊Dic43—24no—
63C22⋊Dic43—24no—
64C22⋊Dic43—22no—
65C22⋊Dic43—42no—
66C22⋊Dic43—42no—
67(C2×D43)⋊C4—22no—
68(C2×D43)⋊C4—22no—
69(C2×D43)⋊C4—42no—
70Dic43⋊C4—22no—
71Dic43⋊C4—22no—
72Dic43⋊C4—42no—
73C4×Dic43—42no—
74C23×C43—22no—
75C23×C43—22no—
76C23×C43—21no—
77C23×C43—21no—
78C23×C43—22no—
79C23×C43—22no—
80C23×C43—22no—
81C23×C43—22no—
82C23×C43—22no—
83C23×C43—22no—
84C23×C43—22no—
85C23×C43C2×D412yes—
86C23×C43C2411yescharacteristic
87C22×D43—21no—
88C22×D43—22no—
89C22×D43—21no—
90C22×D43—22no—
91C22×D43—21no—
92D4×C43—44no—
93D4×C43—48no—
94C2×C4×C43—22no—
95C2×C4×C43—22no—
96C2×C4×C43—22no—
97C2×C4×C43—22no—
98C2×C4×C43—22no—
99C2×C4×C43—44no—
100C2×C4×C43C2×D412yes—
101C2×C4×C43C2×D412yes—
102C22⋊D43—48no—
103C22⋊D43—44no—
104C2×Dic43—22no—
105C2×Dic43—22no—
106C2×Dic43—21no—
107C2×Dic43—21no—
108C2×Dic43—21no—
109C2×Dic43—22no—
110C2×Dic43—22no—
111C2×Dic43—22no—
112C2×Dic43—21no—
113C2×Dic43—22no—
114C2×Dic43—21no—
115C2×Dic43—22no—
116C2×Dic43#4114yes—
117C22×C43—22no—
118C22×C43—22no—
119C22×C43—22no—
120C22×C43—42no—
121C22×C43—42no—
122C22×C43—42no—
123C22×C43—42no—
124C22×C43—42no—
125C22×C43—42no—
126C22×C43—42no—
127C22×C43—42no—
128C22×C43—42no—
129C22×C43C22×D411yescharacteristic
130C22×C43C2×#4112yes—
131C22×C43H(5,𝔽2)12yes—
132C22×C43H(5,𝔽2)11yescharacteristic
133C22×C43H(5,𝔽2)11yescharacteristic
134C2×D43—41no—
135C2×D43—41no—
136C2×D43—41no—
137C2×D43—42no—
138C2×D43—42no—
139C4×C43—22no—
140C4×C43—22no—
141C4×C43—42no—
142Dic43—24no—
143Dic43—41no—
144Dic43—41no—
145Dic43—41no—
146Dic43—42no—
147#2661—431nomaximal, Sylow
148C2×C43—42no—
149C2×C43—82no—
150C2×C43#53211yescharacteristic
151C2×C43#53411yescharacteristic
152C2×C43#53511yescharacteristic
153C2×C43#53812yes—
154C2×C43#54012yes—
155D43—81no—
156C2×#123—431no—
157C2×#121D4311yescharacteristic, complemented
158#398—431no—
159#397—431no—
160#394—431no—
161#394—432no—
162#392—431no—
163#392—432no—
164#390—432no—
165#388—432no—
166C4×C22⋊C4—431no—
167C43#266111yescharacteristic, complemented, Sylow
168C22×D4—431no—
169C22×D4—432no—
170C22×D4C2×D4312yes—
171C22×D4C2×D4311yescharacteristic
172C23×C4C2×D4311yescharacteristic
173#123—864no—
174#121—24no—
175C2×C4⋊C4—431no—
176C2×C4⋊C4—431no—
177C2×C4⋊C4—432no—
178C2×C4⋊C4C2×D4311yescharacteristic
179C2×C22⋊C4—431no—
180C2×C22⋊C4—432no—
181C2×C22⋊C4—432no—
182C2×C22⋊C4—431no—
183C2×C22⋊C4—431no—
184C2×C22⋊C4—432no—
185C2×C22⋊C4—432no—
186C2×C22⋊C4—432no—
187C2×C22⋊C4C2×D4312yes—
188C2×C42—432no—
189#95—431no—
190#95—431no—
191#95—432no—
192#95—432no—
193C24—431no—
194C24C22×D4312yes—
195C24C22×D4311yescharacteristic
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—864no—
202C2×D4—862no—
203C2×D4—1722no—
204C2×D4—1722no—
205C2×D4—1722no—
206C22×C4—42no—
207C22×C4—432no—
208C22×C4—432no—
209C22×C4—431no—
210C22×C4—431no—
211C22×C4—431no—
212C22×C4C22⋊D4314yescomplemented
213C22×C4C22×D4312yes—
214C22×C4C22×D4311yescharacteristic
215C22×C4C22×D4311yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—862no—
218C4⋊C4—862no—
219C4⋊C4—1722no—
220C22⋊C4—24no—
221C22⋊C4—864no—
222C22⋊C4—864no—
223C22⋊C4—862no—
224C22⋊C4—862no—
225C22⋊C4—862no—
226C22⋊C4—1722no—
227C22⋊C4—1722no—
228C22⋊C4—1722no—
229C42—1722no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—861no—
242C23—862no—
243C23—861no—
244C23—862no—
245C23—861no—
246C23C2×C22⋊D4312yes—
247C23C23×D4311yescharacteristic
248D4—44no—
249D4—48no—
250D4—1728no—
251D4—1724no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—434no—
259C2×C4—862no—
260C2×C4—862no—
261C2×C4—861no—
262C2×C4—861no—
263C2×C4—861no—
264C2×C4—862no—
265C2×C4—862no—
266C2×C4—862no—
267C2×C4—861no—
268C2×C4—862no—
269C2×C4—862no—
270C2×C4—861no—
271C2×C4C2×C22⋊D4312yes—
272C2×C4C2×C22⋊D4312yes—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—1721no—
286C22—1721no—
287C22—1721no—
288C22—1722no—
289C22—1722no—
290C22C22×C22⋊D4311yescharacteristic
291C22#41×D4312yes—
292C22#1306337012yes—
293C22#1306337011yescharacteristic
294C22#1306337011yescharacteristic
295C4—22no—
296C4—22no—
297C4—42no—
298C4—864no—
299C4—1721no—
300C4—1721no—
301C4—1721no—
302C4—1722no—
303C2—42no—
304C2—82no—
305C2—3441no—
306C2#1458344511yescharacteristic
307C2#1458345811yescharacteristic
308C2#1458346611yescharacteristic
309C2#1458348112yes—
310C2#1458349912yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC2×#121×C43C2
hypercenterC2×#121D43
hypocenterC43#2661
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#13063308C2semidirect
2. #14582830C2semidirect
3. #14582825C2semidirect
4. #14582813C2semidirect
5. #14582813C2semidirect
6. #14582807C2semidirect
7. #14582807C2semidirect
8. #14582799C2semidirect
9. #14582793C2semidirect
10. C4×C22⋊Dic43C2semidirect
11. C2×#121×C43C2semidirect
46. C22×C4×C43D4semidirect
157. C2×#121D43semidirect
167. C43#2661semidirect
212. C22×C4C22⋊D43semidirect