Finite Group Databasebeta

← #19137993

Subgroups — #19137993

Order 5504.

814 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 157 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
2C2×#13063295C211yesmaximal, characteristic
3C22×(C2×D43)⋊C4C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D43C211yesmaximal, characteristic, complemented
5#14582828C212yesmaximal, complemented
6#14582817C212yesmaximal, complemented
7#14582777C212yesmaximal, complemented
8#14582776C211yesmaximal, characteristic, complemented
9#14582771C211yesmaximal, characteristic, complemented
10#14582758C212yesmaximal, complemented
11C43×#381C211yesmaximal, characteristic, complemented
12C24×D43C2211yescharacteristic
13C22×D172C2211yescharacteristic
14C22×D172C2211yescharacteristic
15C22×C22⋊D43C2211yescharacteristic
16C23×Dic43C2211yescharacteristic
17C22×C4×D43C2212yes—
18#13063308—24no—
19#13063295—24no—
20C2×Dic43⋊C4C2212yescomplemented
21C2×Dic43⋊C4C2212yescomplemented
22C2×C22⋊Dic43C2211yescharacteristic
23C2×(C2×D43)⋊C4—24no—
24C2×(C2×D43)⋊C4C2212yes—
25C2×(C2×D43)⋊C4C2212yes—
26C2×(C2×D43)⋊C4C2211yescharacteristic
27C2×(C2×D43)⋊C4C2212yes—
28C2×(C2×D43)⋊C4C2211yescharacteristic
29C2×(C2×D43)⋊C4C2212yescomplemented
30C2×(C2×D43)⋊C4C2212yescomplemented
31C22⋊C4×D43—24no—
32C43⋊#95C2212yescomplemented
33C43⋊#95C2212yescomplemented
34C23×C4×C43C2211yescharacteristic
35C2×C22⋊C4×C43C2211yescharacteristic
36C2×C22⋊C4×C43C2211yescharacteristic
37#95×C43C2212yescomplemented
38#95×C43C2212yescomplemented
39C24×C43C2311yescharacteristic
40C23×D43—22no—
41C23×D43—22no—
42C23×D43—24no—
43C23×D43—42no—
44C23×D43D414yes—
45C23×D43C2311yescharacteristic
46C23×D43C2311yescharacteristic
47C23×D43C2311yescharacteristic
48C22×C4×C43—22no—
49C22×C4×C43—22no—
50C22×C4×C43—22no—
51C22×C4×C43C2311yescharacteristic
52C22×C4×C43C2312yes—
53C22×C4×C43C2311yescharacteristic
54C22×C4×C43C2312yes—
55C2×C22⋊D43—22no—
56C2×C22⋊D43—22no—
57C2×C22⋊D43—22no—
58C22×Dic43—22no—
59C22×Dic43—42no—
60C22×Dic43C2312yes—
61C22×Dic43C2311yescharacteristic
62C22×Dic43C2312yes—
63C2×D172—22no—
64C2×D172—22no—
65C2×D172—24no—
66C2×D172—42no—
67C2×C4×D43—24no—
68C2×C4×D43—24no—
69C2×C4×D43—42no—
70C22⋊C4×C43—22no—
71C22⋊C4×C43—22no—
72C22⋊Dic43D414yescomplemented
73(C2×D43)⋊C4—24no—
74(C2×D43)⋊C4—24no—
75(C2×D43)⋊C4—24no—
76(C2×D43)⋊C4—24no—
77(C2×D43)⋊C4—22no—
78(C2×D43)⋊C4—22no—
79(C2×D43)⋊C4—42no—
80Dic43⋊C4—24no—
81Dic43⋊C4—42no—
82C23×C43—22no—
83C23×C43—22no—
84C23×C43C2×D412yes—
85C23×C43C2×D412yes—
86C23×C43#4112yes—
87C23×C43C2411yescharacteristic
88C22×D43—21no—
89C22×D43—21no—
90C22×D43—22no—
91C22×D43—28no—
92C22×D43—21no—
93C22×D43—22no—
94C22×D43—21no—
95C22×D43—22no—
96C22×D43—28no—
97C22×D43—21no—
98C22×D43—21no—
99C22×D43—22no—
100C22×D43—21no—
101C22×D43—22no—
102C22×D43—44no—
103C22×D43—48no—
104C22×D43—48no—
105C22×D43C2×D412yes—
106C22×D43C2×D412yes—
107C22×D43#4114yes—
108C2×C4×C43—22no—
109C2×C4×C43—22no—
110C2×C4×C43—22no—
111C2×C4×C43—21no—
112C2×C4×C43—21no—
113C2×C4×C43—22no—
114C2×C4×C43—22no—
115C2×C4×C43—22no—
116C2×C4×C43—22no—
117C2×C4×C43—22no—
118C2×C4×C43—44no—
119C22⋊D43—44no—
120C2×Dic43—22no—
121C2×Dic43—22no—
122C2×Dic43—21no—
123C2×Dic43—22no—
124C2×Dic43—21no—
125C2×Dic43—21no—
126C2×Dic43—22no—
127C2×Dic43—22no—
128C2×Dic43—44no—
129C2×Dic43C2×D414yes—
130D172—44no—
131D172—44no—
132C4×D43—48no—
133C22×C43—22no—
134C22×C43—22no—
135C22×C43—24no—
136C22×C43—24no—
137C22×C43#12114yes—
138C22×C43C22×D411yescharacteristic
139C22×C43C22×D411yescharacteristic
140C22×C43C2×#4112yes—
141C22×C43C2×#4111yescharacteristic
142C22×C43H(5,𝔽2)12yes—
143C2×D43—22no—
144C2×D43—22no—
145C2×D43—24no—
146C2×D43—24no—
147C2×D43—22no—
148C2×D43—22no—
149C2×D43—22no—
150C2×D43—416no—
151C2×D43—42no—
152C2×D43—42no—
153C2×D43—41no—
154C2×D43—41no—
155C2×D43—41no—
156C2×D43—42no—
157C2×D43—42no—
158C2×D43#12314yescomplemented
159C4×C43—42no—
160C4×C43—41no—
161C4×C43—41no—
162C4×C43—42no—
163Dic43—24no—
164Dic43—41no—
165Dic43—42no—
166#2412—431nomaximal, Sylow
167C2×C43—24no—
168C2×C43C2×#12111yescharacteristic
169C2×C43C2×#12311yescharacteristic
170C2×C43#53411yescharacteristic
171C2×C43#54612yes—
172C2×C43#55112yes—
173D43—24no—
174D43—42no—
175D43—81no—
176C2×#123—431no—
177C2×#120—431no—
178C22×C22⋊C4—431no—
179C22×C22⋊C4—431no—
180#397—432no—
181#394—432no—
182#386—431no—
183#386—432no—
184#385—431no—
185#381D4311yescharacteristic, complemented
186#380—432no—
187C43#241211yescharacteristic, complemented, Sylow
188C25—431no—
189C22×D4—431no—
190C22×D4—431no—
191C22×D4—431no—
192C23×C4—431no—
193C23×C4—432no—
194C23×C4C2×D4311yescharacteristic
195#123—864no—
196#120—864no—
197C2×C4⋊C4—432no—
198C2×C4⋊C4—432no—
199C2×C22⋊C4—431no—
200C2×C22⋊C4—432no—
201C2×C22⋊C4—432no—
202C2×C22⋊C4—432no—
203C2×C22⋊C4—431no—
204C2×C22⋊C4—431no—
205C2×C22⋊C4—432no—
206C2×C22⋊C4—432no—
207C2×C22⋊C4—864no—
208C2×C22⋊C4—864no—
209C2×C22⋊C4C2×D4311yescharacteristic
210C2×C22⋊C4C2×D4311yescharacteristic
211#95—432no—
212#95—432no—
213#95C2×D4312yescomplemented
214#95C2×D4312yescomplemented
215C24—431no—
216C24—431no—
217C24—434no—
218C24—431no—
219C24—864no—
220C24—862no—
221C24—862no—
222C24—1722no—
223C24C22×D4311yescharacteristic
224C2×D4—862no—
225C2×D4—862no—
226C2×D4—864no—
227C2×D4—862no—
228C2×D4—862no—
229C2×D4—862no—
230C2×D4—1722no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—432no—
235C22×C4—432no—
236C22×C4—431no—
237C22×C4—864no—
238C22×C4—864no—
239C22×C4—862no—
240C22×C4—1722no—
241C22×C4—1722no—
242C22×C4C22×D4311yescharacteristic
243C22×C4C22×D4312yes—
244C22×C4C22×D4312yes—
245C22×C4C22×D4311yescharacteristic
246C4⋊C4—864no—
247C4⋊C4—1722no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—434no—
251C22⋊C4—864no—
252C22⋊C4—864no—
253C22⋊C4—864no—
254C22⋊C4—864no—
255C22⋊C4—862no—
256C22⋊C4—862no—
257C22⋊C4—1722no—
258C23—22no—
259C23—22no—
260C23—432no—
261C23—434no—
262C23—432no—
263C23—868no—
264C23—861no—
265C23—861no—
266C23—862no—
267C23—868no—
268C23—861no—
269C23—862no—
270C23—861no—
271C23—862no—
272C23—861no—
273C23—861no—
274C23—862no—
275C23—861no—
276C23—862no—
277C23—1724no—
278C23—1728no—
279C23—1728no—
280C23#1062189812yes—
281C23D4×D4312yes—
282C23D4×D4312yes—
283C23C23×D4311yescharacteristic
284D4—1724no—
285D4—1724no—
286D4—1724no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—44no—
298C2×C4—434no—
299C2×C4—862no—
300C2×C4—862no—
301C2×C4—862no—
302C2×C4—861no—
303C2×C4—861no—
304C2×C4—861no—
305C2×C4—862no—
306C2×C4—862no—
307C2×C4—1728no—
308C2×C4—1724no—
309C22—24no—
310C22—22no—
311C22—22no—
312C22—24no—
313C22—434no—
314C22—864no—
315C22—862no—
316C22—862no—
317C22—864no—
318C22—862no—
319C22—862no—
320C22—862no—
321C22—17216no—
322C22—1722no—
323C22—1721no—
324C22—1721no—
325C22—1721no—
326C22—1722no—
327C22—1722no—
328C22—1722no—
329C22#1306329814yes—
330C22C2×D4×D4311yescharacteristic
331C22C2×D4×D4311yescharacteristic
332C22#41×D4312yes—
333C22C2×#1062189811yescharacteristic
334C22#1306337112yes—
335C4—42no—
336C4—41no—
337C4—41no—
338C4—42no—
339C4—864no—
340C4—1721no—
341C4—1722no—
342C2—24no—
343C2—864no—
344C2—3441no—
345C2—1722no—
346C2C2×#1306329811yescharacteristic
347C2#123×D4311yescharacteristic
348C2#1458346011yescharacteristic
349C2#1458354112yes—
350C2#1458357312yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC43×#381C2
hypercenter#381D43
hypocenterC43#2412
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C2×#13063308C2semidirect
3. C22×(C2×D43)⋊C4C2semidirect
4. C2×C22⋊C4×D43C2semidirect
5. #14582828C2semidirect
6. #14582817C2semidirect
7. #14582777C2semidirect
8. #14582776C2semidirect
9. #14582771C2semidirect
10. #14582758C2semidirect
11. C43×#381C2semidirect
20. C2×Dic43⋊C4C22semidirect
21. C2×Dic43⋊C4C22semidirect
29. C2×(C2×D43)⋊C4C22semidirect
30. C2×(C2×D43)⋊C4C22semidirect
32. C43⋊#95C22semidirect
33. C43⋊#95C22semidirect
37. #95×C43C22semidirect
38. #95×C43C22semidirect
72. C22⋊Dic43D4semidirect
158. C2×D43#123semidirect
185. #381D43semidirect
187. C43#2412semidirect
213. #95C2×D43semidirect
214. #95C2×D43semidirect