Finite Group Databasebeta

← #450×D173 #36362742

Subgroups — #450×D173 #36362742

Order 22144.

996 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 155 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×D4×D173C211yesmaximal, characteristic
2C2×SD16×D173C212yesmaximal, complemented
3C22⋊Q8×D173C211yesmaximal, characteristic, complemented
4#26011644C211yesmaximal, characteristic, complemented
5#26011643C212yesmaximal, complemented
6#26011642C211yesmaximal, characteristic, complemented
7#26011641C211yesmaximal, characteristic, complemented
8#26011640C212yesmaximal, complemented
9#102×D173C212yesmaximal, complemented
10C22⋊C8×D173C211yesmaximal, characteristic, complemented
11#450×C173C211yesmaximal, characteristic, complemented
12C24×D173—21no—
13C2×Q8×D173C2211yescharacteristic, complemented
14C22×D692C2211yescharacteristic
15C22×C22⋊D173—21no—
16C2×D4×D173—28no—
17C2×D4×D173—21no—
18C2×D4×D173—21no—
19C2×D4×D173C2212yes—
20C2×D4×D173C2212yes—
21C22×C4×D173C2211yescharacteristic
22C2×#14583815C2212yescomplemented
23C2×#14583814C2212yescomplemented
24C2×#14583813C2212yescomplemented
25SD16×D173—28no—
26C2×C8×D173C2212yescomplemented
27#19149659C2211yescharacteristic, complemented
28#19149658C2211yescharacteristic, complemented
29#19149657—21no—
30#19149656—21no—
31#19149655C2211yescharacteristic, complemented
32C4⋊C4×D173—21no—
33C4⋊C4×D173C2211yescharacteristic, complemented
34C22⋊C4×D173—21no—
35#19149601C2212yescomplemented
36#19149600C2212yescomplemented
37#19149599C2212yescomplemented
38C22⋊#13063436C2211yescharacteristic, complemented
39#19149591C2212yescomplemented
40C22×D4×C173C2211yescharacteristic
41C2×SD16×C173C2212yescomplemented
42C22⋊Q8×C173C2211yescharacteristic, complemented
43#102×C173C2212yescomplemented
44C22⋊C8×C173C2211yescharacteristic, complemented
45C23×D173—28no—
46C23×D173—22no—
47C23×D173—22no—
48C23×D173—21no—
49C23×D173—44no—
50C23×D173—44no—
51C23×D173D411yescharacteristic
52C2×Dic346—21no—
53C2×Dic346C2311yescharacteristic, complemented
54Q8×D173—24no—
55C2×D692—24no—
56C2×D692—21no—
57C2×D692C2312yes—
58C2×C22⋊D173—28no—
59C2×C22⋊D173—22no—
60C2×C22⋊D173—22no—
61D4×D173—28no—
62D4×D173—28no—
63D4×D173—24no—
64D4×D173—24no—
65D4×D173—24no—
66D4×D173D418yes—
67C22×Dic173D411yescharacteristic
68C2×C4×D173—24no—
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173D411yescharacteristic
72C2×C4×D173D411yescharacteristic
73C2×C4×D173C2311yescharacteristic
74C2×C4×D173C2311yescharacteristic
75#14583815—24no—
76#14583814—24no—
77#14583813—24no—
78C2×#13063436C2312yescomplemented
79C8×D173—24no—
80C4⋊Dic173—21no—
81C4⋊Dic173C2311yescharacteristic, complemented
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85C22⋊Dic173—21no—
86#14583803—21no—
87#14583803—21no—
88C24×C173—21no—
89C2×Q8×C173C2311yescharacteristic, complemented
90C2×D4×C173—24no—
91C2×D4×C173—21no—
92C2×D4×C173C2312yes—
93C22×C4×C173C2311yescharacteristic
94SD16×C173—24no—
95C2×C8×C173C2312yescomplemented
96C4⋊C4×C173—21no—
97C4⋊C4×C173C2311yescharacteristic, complemented
98C22⋊C4×C173—21no—
99C22×D173—22no—
100C22×D173—22no—
101C22×D173—28no—
102C22×D173—24no—
103C22×D173—22no—
104C22×D173—28no—
105C22×D173—44no—
106C22×D173—432no—
107C22×D173—42no—
108C22×D173—42no—
109C22×D173—44no—
110C22×D173—44no—
111C22×D173—44no—
112C22×D173SD1614yescomplemented
113C22×D173C2×D411yescharacteristic
114C22×D173C2×D411yescharacteristic
115Dic346—22no—
116Dic346—42no—
117D692—24no—
118D692—22no—
119D692C2×D414yes—
120C22⋊D173—28no—
121C22⋊D173—28no—
122C2×Dic173—22no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C4×D173—22no—
128C4×D173—22no—
129C4×D173—22no—
130C4×D173—42no—
131C4×D173—42no—
132C4×D173C2×D414yes—
133#13063436—22no—
134C23×C173—24no—
135C23×C173—22no—
136C23×C173—42no—
137C23×C173—42no—
138C23×C173C2×D411yescharacteristic
139Q8×C173—22no—
140D4×C173—24no—
141D4×C173—22no—
142D4×C173C2×D414yes—
143C2×C4×C173—22no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2411yescharacteristic
148C8×C173—22no—
149C22×C173—21no—
150C22×C173—21no—
151C22×C173—24no—
152C22×C173—48no—
153C22×C173—42no—
154C22×C173—42no—
155C22×C173C2×SD1612yescomplemented
156C22×C173C22×D411yescharacteristic
157C2×D173—21no—
158C2×D173—21no—
159C2×D173—28no—
160C2×D173—24no—
161C2×D173—416no—
162C2×D173—48no—
163C2×D173—42no—
164C2×D173—42no—
165C2×D173—416no—
166C2×D173#12012yes—
167C2×D173C2×SD1612yes—
168C2×D173C2×SD1612yescomplemented
169C2×D173AGL(1,ℤ/8)12yes—
170C4×C173—21no—
171C4×C173—41no—
172C4×C173—41no—
173C4×C173C22×D412yes—
174Dic173—21no—
175Dic173—41no—
176Dic173—41no—
177Dic173#12012yes—
178C2×C173—22no—
179C2×C173—44no—
180C2×C173C2×#12011yescharacteristic
181C2×C173C22×SD1611yescharacteristic
182C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
183D173—22no—
184D173—44no—
185D173#45014yesdirect product factor
186C173C2×#45011yescharacteristic, complemented, Sylow
187C2×#450—1731nomaximal, Sylow
188C23×D4—1731no—
189C22×SD16—1732no—
190C2×C22⋊Q8—1731no—
191#450—1731no—
192#450—1732no—
193#450—1731no—
194#450—1732no—
195#450—1731no—
196#450D17311yescharacteristic, direct product factor
197C2×#102—1732no—
198C2×C22⋊C8—1731no—
199C25—3461no—
200C22×Q8—1731no—
201C22×D4—3468no—
202C22×D4—3461no—
203C22×D4—3461no—
204C22×D4—3461no—
205C22×D4—1732no—
206C22×D4—1732no—
207C22×D4—1731no—
208C22×D4C2×D17311yescharacteristic
209C23×C4—1731no—
210C2×SD16—3468no—
211C2×SD16—1732no—
212C2×SD16—1732no—
213C2×SD16—1732no—
214C2×SD16C2×D17312yescomplemented
215C22×C8—1732no—
216C22⋊Q8—3461no—
217C22⋊Q8—3461no—
218C22⋊Q8—1731no—
219C22⋊Q8—1731no—
220C22⋊Q8—1731no—
221C22⋊Q8C2×D17311yescharacteristic, complemented
222C2×C4⋊C4—3461no—
223C2×C4⋊C4—1731no—
224C2×C22⋊C4—3461no—
225#102—1732no—
226#102—1732no—
227#102—1732no—
228#102C2×D17312yescomplemented
229C22⋊C8—1731no—
230C22⋊C8—1732no—
231C22⋊C8C2×D17311yescharacteristic, complemented
232C24—21no—
233C24—3468no—
234C24—3462no—
235C24—3462no—
236C24—3461no—
237C24—1731no—
238C24—6924no—
239C24—6924no—
240C2×Q8—3461no—
241C2×Q8—3464no—
242C2×Q8—1731no—
243C2×Q8C22×D17311yescharacteristic, complemented
244C2×D4—24no—
245C2×D4—21no—
246C2×D4—3464no—
247C2×D4—3468no—
248C2×D4—3468no—
249C2×D4—3464no—
250C2×D4—3468no—
251C2×D4—3464no—
252C2×D4—3464no—
253C2×D4—3462no—
254C2×D4—3462no—
255C2×D4—3461no—
256C2×D4—1738no—
257C2×D4—1732no—
258C2×D4C22×D17312yes—
259C22×C4—3464no—
260C22×C4—3461no—
261C22×C4—3461no—
262C22×C4—1731no—
263C22×C4—1731no—
264C22×C4—1731no—
265C22×C4—1731no—
266C22×C4—1731no—
267C22×C4C22×D17311yescharacteristic
268SD16—24no—
269SD16—3464no—
270SD16—3464no—
271SD16—3464no—
272C2×C8—3464no—
273C2×C8—1732no—
274C2×C8C22×D17312yescomplemented
275C4⋊C4—21no—
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—1731no—
281C4⋊C4C22×D17311yescharacteristic, complemented
282C22⋊C4—21no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C23—24no—
287C23—22no—
288C23—42no—
289C23—42no—
290C23—3462no—
291C23—3462no—
292C23—3464no—
293C23—3468no—
294C23—3468no—
295C23—3462no—
296C23—1734no—
297C23—1731no—
298C23—1731no—
299C23—6922no—
300C23—69232no—
301C23—6924no—
302C23—6924no—
303C23—6924no—
304C23—6924no—
305C23—6922no—
306C23D4×D17311yescharacteristic
307Q8—22no—
308Q8—3462no—
309Q8—6922no—
310D4—24no—
311D4—22no—
312D4—3464no—
313D4—3468no—
314D4—3468no—
315D4—3462no—
316D4—1734no—
317D4D4×D17314yes—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—3462no—
322C2×C4—3462no—
323C2×C4—3462no—
324C2×C4—3462no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—1731no—
328C2×C4—1731no—
329C2×C4—1734no—
330C2×C4—6922no—
331C2×C4—6922no—
332C2×C4D4×D17311yescharacteristic
333C2×C4C23×D17311yescharacteristic
334C8—22no—
335C8—3462no—
336C22—21no—
337C22—21no—
338C22—24no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22—3468no—
343C22—3461no—
344C22—3461no—
345C22—3464no—
346C22—1732no—
347C22—1732no—
348C22—1732no—
349C22—1732no—
350C22—6928no—
351C22—69216no—
352C22—69216no—
353C22—6922no—
354C22—6922no—
355C22SD16×D17312yescomplemented
356C22C2×D4×D17311yescharacteristic
357C4—21no—
358C4—41no—
359C4—41no—
360C4—3461no—
361C4—1732no—
362C4—6921no—
363C4—6921no—
364C4C2×D4×D17312yes—
365C2—22no—
366C2—44no—
367C2—3462no—
368C2—1734no—
369C2—6924no—
370C2#120×D17311yescharacteristic
371C2C2×SD16×D17311yescharacteristic
372C2AGL(1,ℤ/8)×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
prime radicalC22×D4×D173C2
hypercenter#450D173
hypocenterC173C2×#450
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×SD16×D173C2semidirect
3. C22⋊Q8×D173C2semidirect
4. #26011644C2semidirect
5. #26011643C2semidirect
6. #26011642C2semidirect
7. #26011641C2semidirect
8. #26011640C2semidirect
9. #102×D173C2semidirect
10. C22⋊C8×D173C2semidirect
11. #450×C173C2semidirect
13. C2×Q8×D173C22semidirect
22. C2×#14583815C22semidirect
23. C2×#14583814C22semidirect
24. C2×#14583813C22semidirect
26. C2×C8×D173C22semidirect
27. #19149659C22semidirect
28. #19149658C22semidirect
31. #19149655C22semidirect
33. C4⋊C4×D173C22semidirect
35. #19149601C22semidirect
36. #19149600C22semidirect
37. #19149599C22semidirect
38. C22⋊#13063436C22semidirect
39. #19149591C22semidirect
41. C2×SD16×C173C22semidirect
42. C22⋊Q8×C173C22semidirect
43. #102×C173C22semidirect
44. C22⋊C8×C173C22semidirect
53. C2×Dic346C23semidirect
78. C2×#13063436C23semidirect
81. C4⋊Dic173C23semidirect
89. C2×Q8×C173C23semidirect
95. C2×C8×C173C23semidirect
97. C4⋊C4×C173C23semidirect
112. C22×D173SD16semidirect
155. C22×C173C2×SD16semidirect
168. C2×D173C2×SD16semidirect
185. D173#450direct (N × H)
186. C173C2×#450semidirect
196. #450D173direct (N × H)
214. C2×SD16C2×D173semidirect
221. C22⋊Q8C2×D173semidirect
228. #102C2×D173semidirect
231. C22⋊C8C2×D173semidirect
243. C2×Q8C22×D173semidirect
274. C2×C8C22×D173semidirect
281. C4⋊C4C22×D173semidirect
355. C22SD16×D173semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×D4×D1732
C2×SD16×D1732
C22⋊Q8×D1732
#260116442
#260116432
#260116422
#260116412
#260116402
#102×D1732
C22⋊C8×D1732
#450×C1732
C2×#450173