Finite Group Databasebeta

← #19138244

Subgroups — #19138244

Order 5504.

1088 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 169 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×#13063311C212yesmaximal, complemented
3C2×#13063302C211yesmaximal, characteristic, complemented
4#14582815C212yesmaximal, complemented
5#14582814C212yesmaximal, complemented
6#14582799C212yesmaximal, complemented
7#14582755C211yesmaximal, characteristic, complemented
8#14582754C211yesmaximal, characteristic, complemented
9#14582754C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic43C211yesmaximal, characteristic, complemented
11C43×#379C211yesmaximal, characteristic, complemented
12C24×D43C2211yescharacteristic
13C22×Dic86C2211yescharacteristic, complemented
14C22×C22⋊D43—28no—
15C22×C22⋊D43—24no—
16C22×C22⋊D43C2212yes—
17C22×C22⋊D43C2212yes—
18C23×Dic43C2211yescharacteristic
19#13063311—28no—
20#13063302—24no—
21C2×C4⋊Dic43C2211yescharacteristic, complemented
22C2×Dic43⋊C4C2211yescharacteristic, complemented
23C2×Dic43⋊C4C2211yescharacteristic, complemented
24C2×C22⋊Dic43C2211yescharacteristic
25C2×C22⋊Dic43C2211yescharacteristic
26C2×C22⋊Dic43C2212yescomplemented
27C2×C22⋊Dic43C2212yescomplemented
28C2×(C2×D43)⋊C4C2212yescomplemented
29C2×(C2×D43)⋊C4C2212yescomplemented
30C2×(C2×D43)⋊C4C2212yescomplemented
31C2×(C2×D43)⋊C4C2212yescomplemented
32C2×C4×Dic43C2212yescomplemented
33C43⋊#95C2212yescomplemented
34C25×C43C2211yescharacteristic
35C2×C22⋊C4×C43C2211yescharacteristic
36C2×C22⋊C4×C43C2211yescharacteristic
37C2×C22⋊C4×C43C2212yescomplemented
38C2×C22⋊C4×C43C2212yescomplemented
39C24×C43—24no—
40C24×C43—22no—
41C24×C43—22no—
42C24×C43—22no—
43C24×C43—42no—
44C24×C43C2312yes—
45C24×C43C2311yescharacteristic
46C23×D43—22no—
47C23×D43—24no—
48C23×D43—22no—
49C23×D43—22no—
50C23×D43—42no—
51C23×D43C2312yes—
52C22×C4×C43C2312yes—
53C22×C4×C43C2311yescharacteristic
54C22×C4×C43C2311yescharacteristic
55C2×C22⋊D43—232no—
56C2×C22⋊D43—24no—
57C2×C22⋊D43—24no—
58C2×C22⋊D43—24no—
59C2×C22⋊D43—24no—
60C2×C22⋊D43—24no—
61C2×C22⋊D43D418yes—
62C22×Dic43—24no—
63C22×Dic43D414yes—
64C22×Dic43C2312yes—
65C22×Dic43C2311yescharacteristic
66C22×Dic43C2311yescharacteristic
67C22×Dic43C2311yescharacteristic
68C22×Dic43C2311yescharacteristic
69C2×Dic86—22no—
70C2×Dic86—24no—
71C22⋊C4×C43—22no—
72C22⋊C4×C43—24no—
73C22⋊C4×C43—22no—
74C22⋊C4×C43—42no—
75C22⋊Dic43—22no—
76C22⋊Dic43—24no—
77C22⋊Dic43—22no—
78C22⋊Dic43—42no—
79(C2×D43)⋊C4—24no—
80(C2×D43)⋊C4—24no—
81(C2×D43)⋊C4—42no—
82(C2×D43)⋊C4—42no—
83C4⋊Dic43—22no—
84Dic43⋊C4—22no—
85Dic43⋊C4—22no—
86C4×Dic43—24no—
87C23×C43—22no—
88C23×C43—22no—
89C23×C43—28no—
90C23×C43—22no—
91C23×C43—22no—
92C23×C43—22no—
93C23×C43—22no—
94C23×C43—22no—
95C23×C43—22no—
96C23×C43—48no—
97C23×C43—48no—
98C23×C43—44no—
99C23×C43—44no—
100C23×C43C2×D414yes—
101C23×C43C2×D412yes—
102C23×C43#4112yes—
103C23×C43#4112yes—
104C23×C43C2411yescharacteristic
105C22×D43—28no—
106C22×D43—22no—
107C22×D43—22no—
108C22×D43—22no—
109C22×D43—22no—
110C22×D43—22no—
111C22×D43—22no—
112C22×D43—48no—
113C22×D43—48no—
114C22×D43—44no—
115C22×D43—44no—
116C22×D43C2×D414yes—
117C2×C4×C43—22no—
118C2×C4×C43—21no—
119C2×C4×C43—21no—
120C2×C4×C43—22no—
121C2×C4×C43—22no—
122C2×C4×C43—22no—
123C2×C4×C43—22no—
124C22⋊D43—216no—
125C22⋊D43—216no—
126C2×Dic43—22no—
127C2×Dic43—28no—
128C2×Dic43—22no—
129C2×Dic43—21no—
130C2×Dic43—21no—
131C2×Dic43—22no—
132C2×Dic43—22no—
133C2×Dic43—22no—
134C2×Dic43—22no—
135C2×Dic43C2×D412yes—
136C2×Dic43C2×D414yes—
137C2×Dic43C2×D412yes—
138Dic86—44no—
139C22×C43—22no—
140C22×C43—22no—
141C22×C43—24no—
142C22×C43—24no—
143C22×C43—24no—
144C22×C43—24no—
145C22×C43—24no—
146C22×C43—416no—
147C22×C43—42no—
148C22×C43—42no—
149C22×C43—42no—
150C22×C43—42no—
151C22×C43#12414yescomplemented
152C22×C43C22×D411yescharacteristic
153C22×C43C22×D412yes—
154C22×C43C2×#4111yescharacteristic
155C22×C43C2×#4111yescharacteristic
156C22×C43H(5,𝔽2)12yes—
157C2×D43—24no—
158C2×D43—24no—
159C2×D43—24no—
160C2×D43—416no—
161C2×D43—42no—
162C2×D43—42no—
163C2×D43—42no—
164C2×D43—42no—
165C4×C43—41no—
166C4×C43—41no—
167C4×C43—42no—
168Dic43—22no—
169Dic43—41no—
170Dic43—41no—
171Dic43—42no—
172Dic43#12014yes—
173#2435—431nomaximal, Sylow
174C2×C43—24no—
175C2×C43—44no—
176C2×C43C2×#12011yescharacteristic
177C2×C43C2×#12411yescharacteristic
178C2×C43#53511yescharacteristic
179C2×C43#54612yes—
180C2×C43#54612yes—
181D43—44no—
182C23×D4—431no—
183C2×#124—432no—
184C2×C22⋊Q8—431no—
185#394—432no—
186#394—432no—
187#390—432no—
188#379—431no—
189#379—431no—
190#379—431no—
191#379D4311yescharacteristic, complemented
192C4×C22⋊C4—431no—
193C43#243511yescharacteristic, complemented, Sylow
194C25—431no—
195C25C2×D4311yescharacteristic
196C22×Q8—431no—
197C22×D4—432no—
198C22×D4—432no—
199C22×D4—864no—
200C22×D4—868no—
201C23×C4—431no—
202#124—868no—
203C22⋊Q8—864no—
204C2×C4⋊C4—431no—
205C2×C4⋊C4—431no—
206C2×C4⋊C4—431no—
207C2×C22⋊C4—431no—
208C2×C22⋊C4—432no—
209C2×C22⋊C4—431no—
210C2×C22⋊C4—432no—
211C2×C22⋊C4—432no—
212C2×C22⋊C4—432no—
213C2×C22⋊C4—432no—
214C2×C22⋊C4—432no—
215C2×C22⋊C4C2×D4311yescharacteristic
216C2×C22⋊C4C2×D4311yescharacteristic
217C2×C22⋊C4C2×D4312yescomplemented
218C2×C22⋊C4C2×D4312yescomplemented
219C2×C42—432no—
220#95—432no—
221C24—24no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—42no—
226C24—432no—
227C24—864no—
228C24—862no—
229C24—862no—
230C24—862no—
231C24—1722no—
232C24C22×D4312yes—
233C24C22×D4311yescharacteristic
234C2×Q8—862no—
235C2×Q8—864no—
236C2×D4—438no—
237C2×D4—864no—
238C2×D4—864no—
239C2×D4—864no—
240C2×D4—864no—
241C2×D4—8632no—
242C2×D4—864no—
243C22×C4—431no—
244C22×C4—434no—
245C22×C4—431no—
246C22×C4—432no—
247C22×C4—431no—
248C22×C4—431no—
249C22×C4—864no—
250C22×C4C22×D4312yes—
251C22×C4C22×D4311yescharacteristic
252C22×C4C22×D4311yescharacteristic
253C4⋊C4—862no—
254C4⋊C4—862no—
255C4⋊C4—862no—
256C22⋊C4—22no—
257C22⋊C4—24no—
258C22⋊C4—22no—
259C22⋊C4—42no—
260C22⋊C4—864no—
261C22⋊C4—864no—
262C22⋊C4—862no—
263C22⋊C4—862no—
264C22⋊C4—864no—
265C22⋊C4—1722no—
266C22⋊C4—1722no—
267C22⋊C4—1722no—
268C42—864no—
269C23—22no—
270C23—22no—
271C23—28no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—48no—
279C23—48no—
280C23—44no—
281C23—44no—
282C23—434no—
283C23—868no—
284C23—862no—
285C23—862no—
286C23—862no—
287C23—862no—
288C23—862no—
289C23—862no—
290C23—1728no—
291C23—1728no—
292C23—1724no—
293C23—1724no—
294C23#1062189812yes—
295C23D4×D4314yes—
296C23D4×D4312yes—
297C23#1062190012yes—
298C23C23×D4311yescharacteristic
299Q8—1724no—
300D4—8616no—
301D4—8616no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—434no—
310C2×C4—432no—
311C2×C4—432no—
312C2×C4—868no—
313C2×C4—862no—
314C2×C4—862no—
315C2×C4—862no—
316C2×C4—861no—
317C2×C4—861no—
318C2×C4—862no—
319C2×C4—862no—
320C2×C4—862no—
321C22—22no—
322C22—22no—
323C22—24no—
324C22—24no—
325C22—24no—
326C22—24no—
327C22—24no—
328C22—416no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—864no—
334C22—864no—
335C22—864no—
336C22—17216no—
337C22—1722no—
338C22—1722no—
339C22—1722no—
340C22—1722no—
341C22#1306331114yescomplemented
342C22C2×D4×D4311yescharacteristic
343C22C2×D4×D4312yes—
344C22C2×#1062190011yescharacteristic
345C22C2×#1062189811yescharacteristic
346C22#1306337012yes—
347C4—41no—
348C4—41no—
349C4—42no—
350C4—434no—
351C4—862no—
352C4—1721no—
353C4—1721no—
354C4—1722no—
355C2—24no—
356C2—44no—
357C2—1724no—
358C2#120×D4311yescharacteristic
359C2C2×#1306331111yescharacteristic
360C2#1458346411yescharacteristic
361C2#1458353612yes—
362C2#1458353912yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC43×#379C2
hypercenter#379D43
hypocenterC43#2435
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#13063311C2semidirect
3. C2×#13063302C2semidirect
4. #14582815C2semidirect
5. #14582814C2semidirect
6. #14582799C2semidirect
7. #14582755C2semidirect
8. #14582754C2semidirect
9. #14582754C2semidirect
10. C22⋊C4×Dic43C2semidirect
11. C43×#379C2semidirect
13. C22×Dic86C22semidirect
21. C2×C4⋊Dic43C22semidirect
22. C2×Dic43⋊C4C22semidirect
23. C2×Dic43⋊C4C22semidirect
26. C2×C22⋊Dic43C22semidirect
27. C2×C22⋊Dic43C22semidirect
28. C2×(C2×D43)⋊C4C22semidirect
29. C2×(C2×D43)⋊C4C22semidirect
30. C2×(C2×D43)⋊C4C22semidirect
31. C2×(C2×D43)⋊C4C22semidirect
32. C2×C4×Dic43C22semidirect
33. C43⋊#95C22semidirect
37. C2×C22⋊C4×C43C22semidirect
38. C2×C22⋊C4×C43C22semidirect
151. C22×C43#124semidirect
191. #379D43semidirect
193. C43#2435semidirect
217. C2×C22⋊C4C2×D43semidirect
218. C2×C22⋊C4C2×D43semidirect
341. C22#13063311semidirect