Finite Group Databasebeta

← #19137915

Subgroups — #19137915

Order 5504.

1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C22×#10621902C211yesmaximal, characteristic, complemented
2C2×#13063295C212yesmaximal, complemented
3C2×C22⋊C4×D43C211yesmaximal, characteristic, complemented
4#14582810C211yesmaximal, characteristic, complemented
5#14582809C211yesmaximal, characteristic, complemented
6#14582808C211yesmaximal, characteristic, complemented
7#14582777C212yesmaximal, complemented
8#14582777C212yesmaximal, complemented
9#14582775C212yesmaximal, complemented
10#14582761C211yesmaximal, characteristic, complemented
11C43×#393C211yesmaximal, characteristic, complemented
12C24×D43C2211yescharacteristic
13C2×#10621902—28no—
14C2×#10621902—24no—
15C22×D172C2212yes—
16C22×D172C2212yes—
17C22×D172C2211yescharacteristic, complemented
18C22×C22⋊D43C2212yes—
19C22×C4×D43C2211yescharacteristic
20C22×C4×D43C2211yescharacteristic
21C22×C4×D43C2212yescomplemented
22#13063295—28no—
23C2×C22⋊Dic43C2211yescharacteristic
24C2×(C2×D43)⋊C4C2211yescharacteristic
25C2×(C2×D43)⋊C4C2212yes—
26C2×(C2×D43)⋊C4C2211yescharacteristic
27C2×(C2×D43)⋊C4C2212yes—
28C2×(C2×D43)⋊C4C2212yescomplemented
29C2×(C2×D43)⋊C4C2212yescomplemented
30C22⋊C4×D43—24no—
31C43⋊#95C2211yescharacteristic, complemented
32C43⋊#95C2212yescomplemented
33C43⋊#95C2212yescomplemented
34C22×Q8×C43C2211yescharacteristic, complemented
35C2×C22⋊C4×C43C2211yescharacteristic
36C2×C22⋊C4×C43C2212yes—
37#95×C43C2211yescharacteristic, complemented
38#95×C43C2212yescomplemented
39C24×C43C2311yescharacteristic
40C23×D43—22no—
41C23×D43—22no—
42C23×D43—22no—
43C23×D43—24no—
44C23×D43—42no—
45C23×D43C2312yes—
46C23×D43C2311yescharacteristic
47C23×D43C2311yescharacteristic
48C2×Q8×C43—24no—
49C2×Q8×C43—22no—
50C22×C4×C43C2312yes—
51C22×C4×C43C2311yescharacteristic
52C22×C4×C43C2311yescharacteristic
53C22×C4×C43C2312yes—
54C2×C22⋊D43—24no—
55C2×C22⋊D43—24no—
56C2×C22⋊D43—42no—
57C22×Dic43C2312yes—
58C22×Dic43C2311yescharacteristic
59C22×Dic43C2311yescharacteristic
60#10621902—232no—
61C2×D172—22no—
62C2×D172—24no—
63C2×D172—24no—
64C2×D172—24no—
65C2×D172—24no—
66C2×D172—24no—
67C2×D172—42no—
68C2×D172D418yescomplemented
69C2×C4×D43—24no—
70C2×C4×D43—24no—
71C2×C4×D43—24no—
72C2×C4×D43—24no—
73C2×C4×D43—22no—
74C2×C4×D43—42no—
75C2×C4×D43D414yescomplemented
76C22⋊C4×C43—22no—
77C22⋊C4×C43—24no—
78C22⋊Dic43—22no—
79(C2×D43)⋊C4—22no—
80(C2×D43)⋊C4—24no—
81(C2×D43)⋊C4—22no—
82(C2×D43)⋊C4—24no—
83(C2×D43)⋊C4—42no—
84(C2×D43)⋊C4—42no—
85C23×C43—21no—
86C23×C43—22no—
87C23×C43—21no—
88C23×C43—22no—
89C23×C43#4112yescomplemented
90C23×C43C2411yescharacteristic
91C22×D43—21no—
92C22×D43—21no—
93C22×D43—22no—
94C22×D43—28no—
95C22×D43—21no—
96C22×D43—22no—
97C22×D43—22no—
98C22×D43—22no—
99C22×D43—22no—
100C22×D43—22no—
101C22×D43—21no—
102C22×D43—22no—
103C22×D43—22no—
104C22×D43—22no—
105C22×D43—48no—
106C22×D43—48no—
107C22×D43—44no—
108C22×D43—44no—
109C22×D43C2×D414yes—
110C22×D43C2×D414yes—
111C22×D43C2×D412yes—
112C22×D43#4112yescomplemented
113Q8×C43—28no—
114C2×C4×C43—22no—
115C2×C4×C43—22no—
116C2×C4×C43—22no—
117C2×C4×C43—21no—
118C2×C4×C43—22no—
119C2×C4×C43—22no—
120C2×C4×C43—22no—
121C2×C4×C43—22no—
122C2×C4×C43C2×D414yes—
123C2×C4×C43C2×D412yes—
124C22⋊D43—48no—
125C2×Dic43—22no—
126C2×Dic43—22no—
127C2×Dic43—21no—
128C2×Dic43—22no—
129C2×Dic43—22no—
130C2×Dic43—22no—
131C2×Dic43C2×D412yes—
132D172—28no—
133D172—216no—
134D172—48no—
135C4×D43—216no—
136C4×D43—28no—
137C4×D43—44no—
138C22×C43—22no—
139C22×C43—24no—
140C22×C43—41no—
141C22×C43—41no—
142C22×C43—42no—
143C22×C43C22×D411yescharacteristic
144C22×C43C22×D412yes—
145C22×C43C2×#4111yescharacteristic
146C22×C43C2×#4111yescharacteristic
147C22×C43H(5,𝔽2)12yes—
148C2×D43—22no—
149C2×D43—22no—
150C2×D43—24no—
151C2×D43—24no—
152C2×D43—24no—
153C2×D43—22no—
154C2×D43—24no—
155C2×D43—24no—
156C2×D43—24no—
157C2×D43—416no—
158C2×D43—41no—
159C2×D43—42no—
160C2×D43—42no—
161C2×D43—41no—
162C2×D43—42no—
163C2×D43—42no—
164C2×D43—42no—
165C2×D43#12014yes—
166C4×C43—24no—
167C4×C43—22no—
168C4×C43—41no—
169C4×C43—42no—
170Dic43—22no—
171Dic43—41no—
172Dic43—42no—
173#2404—431nomaximal, Sylow
174C2×C43—42no—
175C2×C43C2×#12011yescharacteristic
176C2×C43#52511yescharacteristic
177C2×C43#53511yescharacteristic
178C2×C43#54612yes—
179C2×C43#54612yes—
180D43—24no—
181D43—44no—
182D43—42no—
183C22×#41—431no—
184C2×#120—432no—
185C22×C22⋊C4—431no—
186#393—431no—
187#393D4311yescharacteristic, complemented
188#392—431no—
189#392—431no—
190#386—432no—
191#386—432no—
192#386—432no—
193#381—431no—
194C43#240411yescharacteristic, complemented, Sylow
195C25—431no—
196C2×#41—864no—
197C2×#41—868no—
198C22×Q8C2×D4311yescharacteristic, complemented
199C22×D4—432no—
200C22×D4—432no—
201C22×D4—431no—
202C22×D4—432no—
203C23×C4—431no—
204C23×C4—432no—
205C23×C4—431no—
206#120—868no—
207C2×C22⋊C4—431no—
208C2×C22⋊C4—431no—
209C2×C22⋊C4—431no—
210C2×C22⋊C4—432no—
211C2×C22⋊C4—432no—
212C2×C22⋊C4—432no—
213C2×C22⋊C4—432no—
214C2×C22⋊C4—864no—
215C2×C22⋊C4C2×D4311yescharacteristic
216C2×C22⋊C4C2×D4312yes—
217#95—432no—
218#95—431no—
219#95—432no—
220#95C2×D4311yescharacteristic, complemented
221#95C2×D4312yescomplemented
222C24—431no—
223C24—431no—
224C24—432no—
225C24—862no—
226C24—862no—
227C24—864no—
228C24—862no—
229C24—1722no—
230C24C22×D4311yescharacteristic
231#41—8632no—
232C2×Q8—22no—
233C2×Q8—24no—
234C2×D4—438no—
235C2×D4—864no—
236C2×D4—864no—
237C2×D4—864no—
238C2×D4—864no—
239C2×D4—864no—
240C2×D4—864no—
241C2×D4—864no—
242C2×D4—862no—
243C2×D4—1722no—
244C2×D4—1722no—
245C22×C4—431no—
246C22×C4—434no—
247C22×C4—432no—
248C22×C4—431no—
249C22×C4—864no—
250C22×C4—864no—
251C22×C4—864no—
252C22×C4—862no—
253C22×C4—864no—
254C22×C4—1722no—
255C22×C4C22×D4312yes—
256C22×C4C22×D4311yescharacteristic
257C22×C4C22×D4311yescharacteristic
258C22×C4C22×D4312yes—
259C22⋊C4—22no—
260C22⋊C4—24no—
261C22⋊C4—862no—
262C22⋊C4—864no—
263C22⋊C4—862no—
264C22⋊C4—862no—
265C22⋊C4—864no—
266C22⋊C4—1722no—
267C22⋊C4—1722no—
268C23—21no—
269C23—22no—
270C23—21no—
271C23—22no—
272C23—434no—
273C23—432no—
274C23—434no—
275C23—432no—
276C23—868no—
277C23—861no—
278C23—861no—
279C23—862no—
280C23—861no—
281C23—862no—
282C23—862no—
283C23—862no—
284C23—862no—
285C23—862no—
286C23—861no—
287C23—862no—
288C23—862no—
289C23—862no—
290C23—1728no—
291C23—1724no—
292C23—1728no—
293C23—1724no—
294C23#1062190212yescomplemented
295C23C23×D4311yescharacteristic
296Q8—28no—
297D4—8616no—
298D4—868no—
299D4—1728no—
300D4—1728no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—432no—
310C2×C4—868no—
311C2×C4—8616no—
312C2×C4—862no—
313C2×C4—861no—
314C2×C4—862no—
315C2×C4—862no—
316C2×C4—862no—
317C2×C4—862no—
318C2×C4—1724no—
319C2×C4D4×D4312yes—
320C2×C4D4×D4314yes—
321C22—22no—
322C22—24no—
323C22—41no—
324C22—41no—
325C22—42no—
326C22—434no—
327C22—862no—
328C22—862no—
329C22—864no—
330C22—864no—
331C22—864no—
332C22—862no—
333C22—864no—
334C22—864no—
335C22—864no—
336C22—17216no—
337C22—1721no—
338C22—1722no—
339C22—1721no—
340C22—1722no—
341C22—1722no—
342C22—1722no—
343C22—1722no—
344C22C2×D4×D4311yescharacteristic
345C22C2×D4×D4312yes—
346C22#41×D4311yescharacteristic
347C22C2×#1062190211yescharacteristic
348C22#1306337112yes—
349C4—22no—
350C4—24no—
351C4—41no—
352C4—42no—
353C4—862no—
354C4—1721no—
355C4—1722no—
356C2—42no—
357C2—864no—
358C2—1722no—
359C2—1724no—
360C2#120×D4311yescharacteristic
361C2#1458341211yescharacteristic
362C2#1458346711yescharacteristic
363C2#1458354012yes—
364C2#1458354212yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC43×#393C2
hypercenter#393D43
hypocenterC43#2404
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×#10621902C2semidirect
2. C2×#13063295C2semidirect
3. C2×C22⋊C4×D43C2semidirect
4. #14582810C2semidirect
5. #14582809C2semidirect
6. #14582808C2semidirect
7. #14582777C2semidirect
8. #14582777C2semidirect
9. #14582775C2semidirect
10. #14582761C2semidirect
11. C43×#393C2semidirect
17. C22×D172C22semidirect
21. C22×C4×D43C22semidirect
28. C2×(C2×D43)⋊C4C22semidirect
29. C2×(C2×D43)⋊C4C22semidirect
31. C43⋊#95C22semidirect
32. C43⋊#95C22semidirect
33. C43⋊#95C22semidirect
34. C22×Q8×C43C22semidirect
37. #95×C43C22semidirect
38. #95×C43C22semidirect
68. C2×D172D4semidirect
75. C2×C4×D43D4semidirect
89. C23×C43#41semidirect
112. C22×D43#41semidirect
187. #393D43semidirect
194. C43#2404semidirect
198. C22×Q8C2×D43semidirect
220. #95C2×D43semidirect
221. #95C2×D43semidirect
294. C23#10621902semidirect