Finite Group Databasebeta

← #36365360

Subgroups — #36365360

Order 22144.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#26012294C212yesmaximal, complemented
2#26012293C211yesmaximal, characteristic, complemented
3#26011866C211yesmaximal, characteristic, complemented
4#26011761C211yesmaximal, characteristic, complemented
5#26011759C212yesmaximal, complemented
6#26011758C211yesmaximal, characteristic, complemented
7#26011659C212yesmaximal, complemented
8#26011657C212yesmaximal, complemented
9#26011608C211yesmaximal, characteristic, complemented
10#26011606C211yesmaximal, characteristic, complemented
11#576×C173C211yesmaximal, characteristic, complemented
12#19149725C2212yescomplemented
13#19149706—22no—
14#19149705—22no—
15#19149705C2212yescomplemented
16#19149704—22no—
17#19149704C2212yescomplemented
18#19149704C2212yescomplemented
19#19149702—21no—
20#19149699—22no—
21#19149699C2211yescharacteristic
22C2×D1384C2211yescharacteristic, complemented
23C2×#14583811—22no—
24C2×#14583811C2212yescomplemented
25D8×D173—21no—
26#19149686C2212yescomplemented
27#19149685C2211yescharacteristic, complemented
28#19149675C2212yescomplemented
29#19149615C2211yescharacteristic, complemented
30#19149614C2211yescharacteristic, complemented
31#19149606C2212yescomplemented
32#19149606C2211yescharacteristic, complemented
33#19149605C2211yescharacteristic, complemented
34#19149596C2212yescomplemented
35#19149595C2212yescomplemented
36C8×Dic173C2211yescharacteristic, complemented
37H(5,𝔽2)×C173C2212yescomplemented
38AGL(1,ℤ/8)×C173—22no—
39AGL(1,ℤ/8)×C173C2212yescomplemented
40#135×C173—21no—
41#135×C173C2211yescharacteristic
42C2×D8×C173—21no—
43C2×D8×C173C2211yescharacteristic, complemented
44#131×C173C2211yescharacteristic, complemented
45#14583827—22no—
46#14583826—22no—
47#14583826C2311yescharacteristic
48C2×D692—22no—
49C2×D692C2312yes—
50C2×C22⋊D173—42no—
51D4×D173—22no—
52D4×D173—42no—
53#14583816D411yescharacteristic, complemented
54#14583814—22no—
55#14583814D411yescharacteristic, complemented
56#14583813—22no—
57#14583813—22no—
58#14583813D411yescharacteristic, complemented
59D1384—21no—
60D1384—21no—
61#14583811—22no—
62#14583811—22no—
63#14583811—22no—
64#14583811—22no—
65#14583811—22no—
66#14583811D411yescharacteristic, complemented
67#14583810—22no—
68#14583810C2312yes—
69#14583810C2311yescharacteristic
70#14583809—21no—
71C2×#13063436—22no—
72C2×#13063436C2311yescharacteristic
73C8×D173—21no—
74C4×Dic173C2311yescharacteristic
75#41×C173—22no—
76#41×C173—22no—
77#41×C173C2311yescharacteristic
78#41×C173C2312yes—
79C2×D4×C173—22no—
80C2×D4×C173—22no—
81C2×D4×C173—42no—
82C2×D4×C173C2312yes—
83Q16×C173D411yescharacteristic, complemented
84SD16×C173—22no—
85SD16×C173D412yescomplemented
86D8×C173—22no—
87D8×C173—21no—
88D8×C173—21no—
89D8×C173D411yescharacteristic, complemented
90M16×C173—21no—
91M16×C173C2311yescharacteristic
92C2×C8×C173—21no—
93C2×C8×C173C2311yescharacteristic
94C22×D173—22no—
95C22×D173—42no—
96Dic346C2×D411yescharacteristic
97D692—22no—
98D692—22no—
99D692—42no—
100D692C2×D411yescharacteristic
101C22⋊D173—21no—
102C22⋊D173—42no—
103C22⋊D173—84no—
104C2×Dic173—21no—
105C4×D173—21no—
106C4×D173—42no—
107#13063436—22no—
108#13063436—21no—
109#13063436C2×D411yescharacteristic
110#13063436C2×D411yescharacteristic
111C23×C173—22no—
112C23×C173—42no—
113Q8×C173C2×D411yescharacteristic
114Q8×C173C2×D412yes—
115D4×C173—22no—
116D4×C173—21no—
117D4×C173—22no—
118D4×C173—22no—
119D4×C173—42no—
120D4×C173—42no—
121D4×C173C2×D411yescharacteristic
122D4×C173C2×D412yes—
123C2×C4×C173—22no—
124C2×C4×C173—21no—
125C2×C4×C173—42no—
126C2×C4×C173C2411yescharacteristic
127C8×C173—21no—
128C8×C173C2×D411yescharacteristic
129C8×C173C2×D411yescharacteristic
130C22×C173—21no—
131C22×C173—22no—
132C22×C173—42no—
133C22×C173—82no—
134C22×C173—162no—
135C22×C173H(5,𝔽2)11yescharacteristic
136C2×D173—21no—
137C2×D173—42no—
138C2×D173—82no—
139C2×D173—162no—
140C4×C173—21no—
141C4×C173—22no—
142C4×C173C22×D411yescharacteristic
143C4×C173C22×D411yescharacteristic
144Dic173—22no—
145Dic173—21no—
146C2×C173—21no—
147C2×C173—42no—
148C2×C173—41no—
149C2×C173—82no—
150C2×C173D4211yescharacteristic
151D173—41no—
152D173—82no—
153C173#329211yescharacteristic, complemented, Sylow
154#3292—1731nomaximal, Sylow
155#576—1732no—
156#576—1731no—
157#576D17311yescharacteristic, complemented
158#493—1731no—
159#472—1732no—
160#472—1731no—
161#472—1731no—
162#453—1732no—
163#453—1732no—
164#443—1731no—
165#443—1731no—
166H(5,𝔽2)—1732no—
167H(5,𝔽2)C2×D17312yescomplemented
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—3462no—
170AGL(1,ℤ/8)—3462no—
171AGL(1,ℤ/8)—3462no—
172AGL(1,ℤ/8)—1732no—
173AGL(1,ℤ/8)—1732no—
174AGL(1,ℤ/8)—1732no—
175AGL(1,ℤ/8)C2×D17312yescomplemented
176#135—21no—
177#135—3462no—
178#135—3461no—
179#135—1731no—
180#135C2×D17311yescharacteristic
181C2×D8—21no—
182C2×D8—3462no—
183C2×D8—3461no—
184C2×D8—1732no—
185C2×D8—1731no—
186C2×D8C2×D17311yescharacteristic, complemented
187#131—1732no—
188#131—1731no—
189#131C2×D17311yescharacteristic, complemented
190#127—1732no—
191#108—1731no—
192#108—1731no—
193#104—1731no—
194#104—1732no—
195#104—1731no—
196#100—1732no—
197#100—1732no—
198C4×C8—1731no—
199#41—22no—
200#41—22no—
201#41—3462no—
202#41—3462no—
203#41—1731no—
204#41C22×D17311yescharacteristic
205#41C22×D17312yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—3462no—
210C2×D4—3462no—
211C2×D4—1732no—
212C2×D4—6922no—
213C2×D4—6922no—
214C2×D4C22×D17312yes—
215Q16—1731no—
216Q16C22⋊D17311yescharacteristic, complemented
217SD16—22no—
218SD16—3462no—
219SD16—3462no—
220SD16—3462no—
221SD16—1731no—
222SD16—1731no—
223SD16C22⋊D17312yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—3462no—
228D8—3462no—
229D8—3462no—
230D8—3462no—
231D8—3462no—
232D8—3461no—
233D8—3461no—
234D8—1731no—
235D8C22⋊D17311yescharacteristic, complemented
236M16—21no—
237M16—3462no—
238M16—3461no—
239M16—1732no—
240M16—1731no—
241M16C22×D17311yescharacteristic
242C2×C8—21no—
243C2×C8—3462no—
244C2×C8—3461no—
245C2×C8—1731no—
246C2×C8C22×D17311yescharacteristic
247C42—1731no—
248C23—22no—
249C23—42no—
250C23—3462no—
251C23—6922no—
252Q8—1731no—
253Q8D4×D17311yescharacteristic
254Q8C2×C22⋊D17312yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—3461no—
262D4—3462no—
263D4—3462no—
264D4—13844no—
265D4—1731no—
266D4—6922no—
267D4—6922no—
268D4D4×D17311yescharacteristic
269D4C2×C22⋊D17312yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—3461no—
274C2×C4—3461no—
275C2×C4—6922no—
276C2×C4C23×D17311yescharacteristic
277C8—21no—
278C8—3462no—
279C8—3461no—
280C8—1731no—
281C8—1731no—
282C8C2×C22⋊D17311yescharacteristic
283C8C2×C22⋊D17311yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—3461no—
290C22—13842no—
291C22—6922no—
292C22—27682no—
293C22#1914972411yescharacteristic
294C4—21no—
295C4—22no—
296C4—3462no—
297C4—3461no—
298C4C2×D4×D17311yescharacteristic
299C4C22×C22⋊D17311yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—13842no—
305C2—6921no—
306C2D4×C22⋊D17311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#576×C173C2
socleC2×C173D42
hypercenter#576D173
hypocenterC173#3292
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #26012294C2semidirect
2. #26012293C2semidirect
3. #26011866C2semidirect
4. #26011761C2semidirect
5. #26011759C2semidirect
6. #26011758C2semidirect
7. #26011659C2semidirect
8. #26011657C2semidirect
9. #26011608C2semidirect
10. #26011606C2semidirect
11. #576×C173C2semidirect
12. #19149725C22semidirect
15. #19149705C22semidirect
17. #19149704C22semidirect
18. #19149704C22semidirect
22. C2×D1384C22semidirect
24. C2×#14583811C22semidirect
26. #19149686C22semidirect
27. #19149685C22semidirect
28. #19149675C22semidirect
29. #19149615C22semidirect
30. #19149614C22semidirect
31. #19149606C22semidirect
32. #19149606C22semidirect
33. #19149605C22semidirect
34. #19149596C22semidirect
35. #19149595C22semidirect
36. C8×Dic173C22semidirect
37. H(5,𝔽2)×C173C22semidirect
39. AGL(1,ℤ/8)×C173C22semidirect
43. C2×D8×C173C22semidirect
44. #131×C173C22semidirect
53. #14583816D4semidirect
55. #14583814D4semidirect
58. #14583813D4semidirect
66. #14583811D4semidirect
83. Q16×C173D4semidirect
85. SD16×C173D4semidirect
89. D8×C173D4semidirect
153. C173#3292semidirect
157. #576D173semidirect
167. H(5,𝔽2)C2×D173semidirect
175. AGL(1,ℤ/8)C2×D173semidirect
186. C2×D8C2×D173semidirect
189. #131C2×D173semidirect
216. Q16C22⋊D173semidirect
223. SD16C22⋊D173semidirect
235. D8C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260122942
#260122932
#260118662
#260117612
#260117592
#260117582
#260116592
#260116572
#260116082
#260116062
#576×C1732
#3292173