Finite Group Databasebeta

← #36365359

Subgroups — #36365359

Order 22144.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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#26012295C212yesmaximal, complemented
2#26012294C212yesmaximal, complemented
3#26011867C211yesmaximal, characteristic, complemented
4#26011760C212yesmaximal, complemented
5#26011759C212yesmaximal, complemented
6#26011660C212yesmaximal, complemented
7#26011658C211yesmaximal, characteristic, complemented
8#26011605C212yesmaximal, complemented
9#453×C173C211yesmaximal, characteristic, complemented
10#19149725C2212yescomplemented
11#19149724C2211yescharacteristic, complemented
12#19149707C2212yescomplemented
13#19149706—22no—
14#19149705—22no—
15#19149705C2212yescomplemented
16#19149704—22no—
17#19149704C2212yescomplemented
18#19149703—22no—
19#19149700—22no—
20#19149699—22no—
21#19149699C2212yes—
22C2×#14583811—22no—
23C2×#14583811C2212yescomplemented
24C2×#14583811C2212yescomplemented
25D8×D173—22no—
26#19149687C2212yescomplemented
27#19149686C2212yescomplemented
28#19149676C2211yescharacteristic, complemented
29#19149614C2212yescomplemented
30#19149607C2212yescomplemented
31#19149596C2211yescharacteristic, complemented
32#19149595C2212yescomplemented
33C4×#13063436C2211yescharacteristic, complemented
34H(5,𝔽2)×C173C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C173C2212yescomplemented
36#127×C173C2211yescharacteristic, complemented
37#104×C173C2212yescomplemented
38#100×C173C2211yescharacteristic, complemented
39#14583827—22no—
40#14583826—22no—
41#14583826C2312yes—
42#14583825—22no—
43C2×D692—22no—
44C2×D692—41no—
45C2×D692C2312yes—
46C2×C22⋊D173—41no—
47D4×D173—22no—
48D4×D173—22no—
49D4×D173—42no—
50#14583816D412yescomplemented
51#14583815—22no—
52#14583814—22no—
53#14583814D412yescomplemented
54#14583813—22no—
55#14583813D412yescomplemented
56D1384—22no—
57#14583811—22no—
58#14583811—22no—
59#14583811—22no—
60#14583811—22no—
61#14583811—22no—
62#14583811—22no—
63#14583811D412yescomplemented
64#14583810—22no—
65#14583810C2312yes—
66#14583809—22no—
67C2×#13063436—22no—
68C2×#13063436C2312yes—
69C8×D173—22no—
70#41×C173—22no—
71#41×C173C2312yes—
72C2×D4×C173—22no—
73C2×D4×C173—41no—
74C2×D4×C173—41no—
75C2×D4×C173C2311yescharacteristic
76C2×D4×C173C2311yescharacteristic
77SD16×C173—22no—
78D8×C173—22no—
79M16×C173C2312yes—
80C42×C173C2311yescharacteristic
81C22×D173—22no—
82C22×D173—41no—
83C22×D173—42no—
84Dic346C2×D412yes—
85D692—22no—
86D692—22no—
87D692—42no—
88D692—82no—
89D692C2×D412yes—
90C22⋊D173—22no—
91C22⋊D173—42no—
92C22⋊D173—42no—
93C2×Dic173—41no—
94C4×D173—22no—
95C4×D173—42no—
96#13063436—22no—
97#13063436C2×D412yes—
98#13063436C2×D412yes—
99C23×C173—21no—
100C23×C173—21no—
101C23×C173—41no—
102Q8×C173C2×D412yes—
103D4×C173—22no—
104D4×C173—22no—
105D4×C173—22no—
106D4×C173—42no—
107D4×C173—82no—
108D4×C173C2×D412yes—
109C2×C4×C173—21no—
110C2×C4×C173—22no—
111C2×C4×C173—41no—
112C2×C4×C173C2411yescharacteristic
113C8×C173—22no—
114C22×C173—22no—
115C22×C173—41no—
116C22×C173—41no—
117C22×C173—81no—
118C22×C173—81no—
119C22×C173—161no—
120C22×C173H(5,𝔽2)11yescharacteristic
121C2×D173—22no—
122C2×D173—42no—
123C2×D173—82no—
124C2×D173—161no—
125C2×D173—162no—
126C4×C173—22no—
127C4×C173—22no—
128C4×C173C22×D412yes—
129Dic173—22no—
130C2×C173—21no—
131C2×C173—42no—
132C2×C173—81no—
133C2×C173—81no—
134C2×C173D4211yescharacteristic
135D173—42no—
136D173—82no—
137C173#329211yescharacteristic, complemented, Sylow
138#3292—1731nomaximal, Sylow
139#576—1732no—
140#576—1732no—
141#493—1731no—
142#472—1732no—
143#472—1732no—
144#453—1731no—
145#453—1732no—
146#453D17311yescharacteristic, complemented
147#443—1732no—
148H(5,𝔽2)—1732no—
149H(5,𝔽2)—1731no—
150H(5,𝔽2)C2×D17311yescharacteristic, complemented
151AGL(1,ℤ/8)—3462no—
152AGL(1,ℤ/8)—3462no—
153AGL(1,ℤ/8)—3462no—
154AGL(1,ℤ/8)—3462no—
155AGL(1,ℤ/8)—1732no—
156AGL(1,ℤ/8)—1732no—
157AGL(1,ℤ/8)—1732no—
158AGL(1,ℤ/8)C2×D17312yescomplemented
159#135—3462no—
160#135—3462no—
161#135—1732no—
162C2×D8—3462no—
163C2×D8—3462no—
164C2×D8—1732no—
165C2×D8—1732no—
166#131—1732no—
167#131—1732no—
168#127—1731no—
169#127C2×D17311yescharacteristic, complemented
170#108—1732no—
171#104—1732no—
172#104C2×D17312yescomplemented
173#100—1731no—
174#100—1732no—
175#100C2×D17311yescharacteristic, complemented
176C4×C8—1731no—
177#41—22no—
178#41—3462no—
179#41—3462no—
180#41—3462no—
181#41—1732no—
182#41C22×D17312yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—3462no—
187C2×D4—3462no—
188C2×D4—3462no—
189C2×D4—1732no—
190C2×D4—6921no—
191C2×D4—6922no—
192C2×D4—6921no—
193C2×D4C22×D17311yescharacteristic
194C2×D4C22×D17311yescharacteristic
195Q16—1732no—
196SD16—22no—
197SD16—3462no—
198SD16—3462no—
199SD16—3462no—
200SD16—1732no—
201SD16—1732no—
202D8—22no—
203D8—3462no—
204D8—3462no—
205D8—3462no—
206D8—3462no—
207D8—3462no—
208D8—3462no—
209D8—3462no—
210D8—1732no—
211M16—3462no—
212M16—3462no—
213M16—1732no—
214M16C22×D17312yes—
215C2×C8—3462no—
216C2×C8—3462no—
217C2×C8—1732no—
218C42C22×D17311yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—3462no—
223C23—6922no—
224C23—6921no—
225Q8—1732no—
226Q8D4×D17312yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—3462no—
233D4—3462no—
234D4—3462no—
235D4—13842no—
236D4—1732no—
237D4—6922no—
238D4—6922no—
239D4—6922no—
240D4D4×D17312yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—3462no—
245C2×C4—6922no—
246C2×C4—6921no—
247C2×C4C23×D17311yescharacteristic
248C8—22no—
249C8—3462no—
250C8—1732no—
251C8—1732no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—3462no—
259C22—13842no—
260C22—6922no—
261C22—27682no—
262C22—27681no—
263C22#1914972411yescharacteristic
264C4—22no—
265C4—22no—
266C4—3462no—
267C4C2×D4×D17312yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—13842no—
273C2—6922no—
274C2#2601212311yescharacteristic
subgroupG/H
centerC2#26012123
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#453×C173C2
socleC2×C173D42
hypercenter#453D173
hypocenterC173#3292
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #26012295C2semidirect
2. #26012294C2semidirect
3. #26011867C2semidirect
4. #26011760C2semidirect
5. #26011759C2semidirect
6. #26011660C2semidirect
7. #26011658C2semidirect
8. #26011605C2semidirect
9. #453×C173C2semidirect
10. #19149725C22semidirect
11. #19149724C22semidirect
12. #19149707C22semidirect
15. #19149705C22semidirect
17. #19149704C22semidirect
23. C2×#14583811C22semidirect
24. C2×#14583811C22semidirect
26. #19149687C22semidirect
27. #19149686C22semidirect
28. #19149676C22semidirect
29. #19149614C22semidirect
30. #19149607C22semidirect
31. #19149596C22semidirect
32. #19149595C22semidirect
33. C4×#13063436C22semidirect
34. H(5,𝔽2)×C173C22semidirect
35. AGL(1,ℤ/8)×C173C22semidirect
36. #127×C173C22semidirect
37. #104×C173C22semidirect
38. #100×C173C22semidirect
50. #14583816D4semidirect
53. #14583814D4semidirect
55. #14583813D4semidirect
63. #14583811D4semidirect
137. C173#3292semidirect
146. #453D173semidirect
150. H(5,𝔽2)C2×D173semidirect
158. AGL(1,ℤ/8)C2×D173semidirect
169. #127C2×D173semidirect
172. #104C2×D173semidirect
175. #100C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122952
#260122942
#260118672
#260117602
#260117592
#260116602
#260116582
#260116052
#453×C1732
#3292173