Finite Group Databasebeta

← #19147924

Subgroups — #19147924

Order 5504.

1556 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10621898C211yesmaximal, characteristic, complemented
2#14583477C214yesmaximal, complemented
3#14583473C212yesmaximal, complemented
4#14583470C212yesmaximal, complemented
5#14583466C212yesmaximal, complemented
6#14583463C212yesmaximal, complemented
7#14583458C214yesmaximal, complemented
8#14583445C214yesmaximal, complemented
9#14583410C214yesmaximal, complemented
10#14583406C214yesmaximal, complemented
11C2×#13063284C211yesmaximal, characteristic, complemented
12C22×#41×C43C211yesmaximal, characteristic, complemented
13C2×#10621898—28no—
14C2×#10621898C2212yes—
15C2×#10621898C2212yes—
16C2×#10621898C2212yescomplemented
17C2×#10621898C2212yescomplemented
18C22×Dic86C2211yescharacteristic, complemented
19C22×D172C2211yescharacteristic, complemented
20C22×C22⋊D43C2212yes—
21C22×C4×D43C2212yescomplemented
22#13063323C2214yescomplemented
23#13063321C2214yescomplemented
24#13063314C2214yescomplemented
25#13063312C2214yescomplemented
26#13063308C2218yescomplemented
27#13063306C2218yescomplemented
28#13063304C2218yescomplemented
29#13063301C2218yescomplemented
30#13063299C2218yescomplemented
31#13063297C22116yescomplemented
32#13063296C2218yescomplemented
33#13063294C2218yescomplemented
34#13063293C2218yescomplemented
35C4×C22⋊D43C22116yescomplemented
36#13063284C2212yescomplemented
37#13063284C2212yescomplemented
38#13063284C2212yescomplemented
39#13063284C2212yescomplemented
40C2×C4⋊Dic43C2212yescomplemented
41C2×C22⋊Dic43C2212yes—
42C2×C4×Dic43C2212yescomplemented
43C2×#41×C43—28no—
44C2×#41×C43C2212yes—
45C2×#41×C43C2212yes—
46C2×#41×C43C2212yescomplemented
47C2×#41×C43C2212yescomplemented
48C22×Q8×C43C2211yescharacteristic, complemented
49C22×D4×C43C2212yes—
50C22×D4×C43C2211yescharacteristic, complemented
51C23×C4×C43C2211yescharacteristic
52C23×C4×C43C2212yescomplemented
53C24×C43C2312yes—
54C24×C43C2311yescharacteristic
55C23×D43C2312yes—
56#41×C43—28no—
57#41×C43—28no—
58#41×C43—28no—
59#41×C43—28no—
60C2×Q8×C43—24no—
61C2×Q8×C43C2312yes—
62C2×Q8×C43C2312yes—
63C2×D4×C43—24no—
64C2×D4×C43—28no—
65C2×D4×C43C2314yes—
66C2×D4×C43C2314yes—
67C2×D4×C43C2312yes—
68C2×D4×C43C2312yes—
69C22×C4×C43—28no—
70C22×C4×C43D418yescomplemented
71C22×C4×C43C2312yes—
72C22×C4×C43C2314yes—
73C22×C4×C43C2314yes—
74C22×C4×C43C2311yescharacteristic
75C22×C4×C43C2311yescharacteristic
76C22×C4×C43C2311yescharacteristic
77C22×C4×C43C2311yescharacteristic
78C22×C4×C43C2311yescharacteristic
79C22×C4×C43C2311yescharacteristic
80C2×C22⋊D43—28no—
81C2×C22⋊D43C2314yes—
82C2×C22⋊D43C2314yes—
83C2×C22⋊D43C23116yes—
84C22×Dic43C2312yes—
85C22×Dic43C2314yes—
86#10621898—28no—
87#10621898—28no—
88#10621898—28no—
89#10621898—28no—
90C2×D172—24no—
91C2×D172C2312yes—
92C2×D172C2312yes—
93C2×C4×D43—28no—
94C2×C4×D43C2314yes—
95C2×C4×D43C2314yes—
96C2×Dic86—24no—
97C2×Dic86C2312yes—
98C2×Dic86C2312yes—
99C22⋊Dic43C2314yes—
100C22⋊Dic43C2314yes—
101C22⋊Dic43C23116yes—
102(C2×D43)⋊C4C23116yes—
103C4⋊Dic43C2314yes—
104C4⋊Dic43C2314yes—
105Dic43⋊C4C23116yes—
106C4×Dic43C2314yes—
107C4×Dic43C2314yes—
108C23×C43—24no—
109C23×C43—42no—
110C23×C43—44no—
111C23×C43C2×D414yes—
112C23×C43C2414yes—
113C23×C43C2411yescharacteristic
114C23×C43C2411yescharacteristic
115C23×C43C2411yescharacteristic
116C22×D43—24no—
117C22×D43—44no—
118C22×D43C2414yes—
119Q8×C43—24no—
120Q8×C43—24no—
121D4×C43—28no—
122D4×C43—28no—
123D4×C43—24no—
124D4×C43—24no—
125C2×C4×C43—24no—
126C2×C4×C43—28no—
127C2×C4×C43—28no—
128C2×C4×C43C2×D414yes—
129C2×C4×C43C2×D414yes—
130C2×C4×C43C2×D414yes—
131C2×C4×C43C2×D414yes—
132C2×C4×C43C2×D414yes—
133C2×C4×C43C2×D414yes—
134C2×C4×C43C2414yes—
135C2×C4×C43C2411yescharacteristic
136C2×C4×C43C2411yescharacteristic
137C2×C4×C43C2411yescharacteristic
138C2×C4×C43C2411yescharacteristic
139C22⋊D43—28no—
140C22⋊D43—28no—
141C22⋊D43—416no—
142C2×Dic43—24no—
143C2×Dic43—44no—
144C2×Dic43C2418yes—
145C2×Dic43C2414yes—
146D172—24no—
147D172—24no—
148C4×D43—28no—
149C4×D43—28no—
150Dic86—24no—
151Dic86—24no—
152C22×C43—22no—
153C22×C43—22no—
154C22×C43—22no—
155C22×C43—24no—
156C22×C43—28no—
157C22×C43—44no—
158C22×C43—48no—
159C22×C43C22×D412yes—
160C22×C43C22×D412yes—
161C22×C43C22×D412yes—
162C22×C43C2511yescharacteristic
163C2×D43—24no—
164C2×D43—28no—
165C2×D43—48no—
166C4×C43—24no—
167C4×C43C22×D412yes—
168C4×C43C22×D412yes—
169C4×C43C22×D412yes—
170C4×C43C22×D412yes—
171Dic43—24no—
172Dic43—28no—
173#3449—431nomaximal, Sylow
174C2×C43—22no—
175C2×C43—22no—
176C2×C43—22no—
177C2×C43—44no—
178C2×C43C23×D411yescharacteristic
179C2×C43#58512yes—
180D43—44no—
181C22×#41—431no—
182C22×#41D4311yescharacteristic, complemented
183#537—434no—
184#536—432no—
185#536—432no—
186#535—432no—
187#535—432no—
188#534—434no—
189#532—434no—
190#525—434no—
191#525—434no—
192C2×#117—431no—
193C43#344911yescharacteristic, complemented, Sylow
194C2×#41—28no—
195C2×#41—432no—
196C2×#41—432no—
197C2×#41—432no—
198C2×#41—432no—
199C2×#41—868no—
200C2×#41C2×D4312yes—
201C2×#41C2×D4312yes—
202C2×#41C2×D4312yescomplemented
203C2×#41C2×D4312yescomplemented
204C22×Q8—431no—
205C22×Q8C2×D4311yescharacteristic, complemented
206C22×D4—432no—
207C22×D4—431no—
208C22×D4C2×D4312yes—
209C22×D4C2×D4311yescharacteristic, complemented
210C23×C4—432no—
211C23×C4C2×D4311yescharacteristic
212C23×C4C2×D4312yescomplemented
213C4⋊Q8—434no—
214#127—434no—
215#124—434no—
216#124—434no—
217#123—438no—
218#123—438no—
219C22⋊Q8—438no—
220C22⋊Q8—438no—
221#121—438no—
222#121—4316no—
223#121—438no—
224#120—438no—
225#120—438no—
226C4×D4—4316no—
227#117—432no—
228#117—432no—
229#117—432no—
230#117—432no—
231C2×C4⋊C4—432no—
232C2×C22⋊C4—432no—
233C2×C42—432no—
234C24—432no—
235C24C22×D4312yes—
236C24C22×D4311yescharacteristic
237#41—28no—
238#41—28no—
239#41—28no—
240#41—28no—
241#41—868no—
242#41—868no—
243#41—868no—
244#41—868no—
245C2×Q8—24no—
246C2×Q8—432no—
247C2×Q8—432no—
248C2×Q8—864no—
249C2×Q8C22×D4312yes—
250C2×Q8C22×D4312yes—
251C2×D4—28no—
252C2×D4—24no—
253C2×D4—434no—
254C2×D4—434no—
255C2×D4—4316no—
256C2×D4—432no—
257C2×D4—432no—
258C2×D4—868no—
259C2×D4—864no—
260C2×D4C22×D4314yes—
261C2×D4C22×D4314yes—
262C2×D4C22×D4312yes—
263C2×D4C22×D4312yes—
264C22×C4—28no—
265C22×C4—432no—
266C22×C4—434no—
267C22×C4—434no—
268C22×C4—434no—
269C22×C4—868no—
270C22×C4C22⋊D4318yescomplemented
271C22×C4C22×D4314yes—
272C22×C4C22×D4312yes—
273C22×C4C22×D4314yes—
274C22×C4C22×D4311yescharacteristic
275C22×C4C22×D4311yescharacteristic
276C22×C4C22×D4311yescharacteristic
277C22×C4C22×D4311yescharacteristic
278C22×C4C22×D4311yescharacteristic
279C22×C4C22×D4311yescharacteristic
280C4⋊C4—434no—
281C4⋊C4—434no—
282C4⋊C4—4316no—
283C22⋊C4—434no—
284C22⋊C4—434no—
285C22⋊C4—4316no—
286C22⋊C4—4316no—
287C42—434no—
288C42—434no—
289C23—24no—
290C23—42no—
291C23—44no—
292C23—434no—
293C23—864no—
294C23—1724no—
295C23C2×C22⋊D4314yes—
296C23C23×D4314yes—
297C23C23×D4311yescharacteristic
298C23C23×D4311yescharacteristic
299C23C23×D4311yescharacteristic
300Q8—24no—
301Q8—24no—
302Q8—864no—
303Q8—864no—
304D4—28no—
305D4—28no—
306D4—24no—
307D4—24no—
308D4—868no—
309D4—868no—
310D4—864no—
311D4—864no—
312D4—17216no—
313C2×C4—24no—
314C2×C4—28no—
315C2×C4—28no—
316C2×C4—438no—
317C2×C4—434no—
318C2×C4—864no—
319C2×C4—868no—
320C2×C4—868no—
321C2×C4—1724no—
322C2×C4C2×C22⋊D4314yes—
323C2×C4C2×C22⋊D4314yes—
324C2×C4C2×C22⋊D4314yes—
325C2×C4C2×C22⋊D4314yes—
326C2×C4C2×C22⋊D4314yes—
327C2×C4C2×C22⋊D4314yes—
328C2×C4C23×D4314yes—
329C2×C4C23×D4311yescharacteristic
330C2×C4C23×D4311yescharacteristic
331C2×C4C23×D4311yescharacteristic
332C2×C4C23×D4311yescharacteristic
333C22—22no—
334C22—22no—
335C22—22no—
336C22—24no—
337C22—28no—
338C22—44no—
339C22—48no—
340C22—864no—
341C22—868no—
342C22—1728no—
343C22C22×C22⋊D4312yes—
344C22C22×C22⋊D4312yes—
345C22C22×C22⋊D4312yes—
346C22C24×D4311yescharacteristic
347C4—24no—
348C4—864no—
349C4—868no—
350C4C22×C22⋊D4312yes—
351C4C22×C22⋊D4312yes—
352C4C22×C22⋊D4312yes—
353C4C22×C22⋊D4312yes—
354C2—22no—
355C2—22no—
356C2—22no—
357C2—44no—
358C2—1724no—
359C2C23×C22⋊D4311yescharacteristic
360C2#1458373512yes—
subgroupG/H
centerC2×C4C23×D43
commutator, socleC22×C43C25
FrattiniC22C24×D43
FittingC22×#41×C43C2
hypercenterC22×#41D43
hypocenterC43#3449
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C2×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×#10621898C2semidirect
2. #14583477C2semidirect
3. #14583473C2semidirect
4. #14583470C2semidirect
5. #14583466C2semidirect
6. #14583463C2semidirect
7. #14583458C2semidirect
8. #14583445C2semidirect
9. #14583410C2semidirect
10. #14583406C2semidirect
11. C2×#13063284C2semidirect
12. C22×#41×C43C2semidirect
16. C2×#10621898C22semidirect
17. C2×#10621898C22semidirect
18. C22×Dic86C22semidirect
19. C22×D172C22semidirect
21. C22×C4×D43C22semidirect
22. #13063323C22semidirect
23. #13063321C22semidirect
24. #13063314C22semidirect
25. #13063312C22semidirect
26. #13063308C22semidirect
27. #13063306C22semidirect
28. #13063304C22semidirect
29. #13063301C22semidirect
30. #13063299C22semidirect
31. #13063297C22semidirect
32. #13063296C22semidirect
33. #13063294C22semidirect
34. #13063293C22semidirect
35. C4×C22⋊D43C22semidirect
36. #13063284C22semidirect
37. #13063284C22semidirect
38. #13063284C22semidirect
39. #13063284C22semidirect
40. C2×C4⋊Dic43C22semidirect
42. C2×C4×Dic43C22semidirect
46. C2×#41×C43C22semidirect
47. C2×#41×C43C22semidirect
48. C22×Q8×C43C22semidirect
50. C22×D4×C43C22semidirect
52. C23×C4×C43C22semidirect
70. C22×C4×C43D4semidirect
182. C22×#41D43semidirect
193. C43#3449semidirect
202. C2×#41C2×D43semidirect
203. C2×#41C2×D43semidirect
205. C22×Q8C2×D43semidirect
209. C22×D4C2×D43semidirect
212. C23×C4C2×D43semidirect
270. C22×C4C22⋊D43semidirect