Finite Group Databasebeta

← #36365329

Subgroups — #36365329

Order 22144.

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

normal subgroups only

This group has 464 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#26012297C212yesmaximal, complemented
2#26012292C212yesmaximal, complemented
3#26011758C212yesmaximal, complemented
4#26011751C212yesmaximal, complemented
5#26011668C211yesmaximal, characteristic, complemented
6#26011663C211yesmaximal, characteristic, complemented
7#26011658C212yesmaximal, complemented
8#26011614C212yesmaximal, complemented
9#496×C173C211yesmaximal, characteristic, complemented
10#19149727C2211yescharacteristic, complemented
11#19149724C2211yescharacteristic
12#19149724C2212yescomplemented
13#19149710C2212yescomplemented
14#19149708—22no—
15#19149706—22no—
16#19149705C2212yes—
17#19149705C2212yescomplemented
18#19149705C2212yescomplemented
19#19149703—22no—
20#19149703—22no—
21#19149701C2212yes—
22#19149700—22no—
23#19149698—22no—
24SD16×D173—22no—
25D8×D173—22no—
26#19149685C2212yescomplemented
27#19149685C2212yescomplemented
28#19149615C2212yescomplemented
29#19149607C2212yescomplemented
30#19149607C2212yescomplemented
31#19149598C2211yescharacteristic, complemented
32#19149596C2212yescomplemented
33#19149596C2211yescharacteristic, complemented
34C2×SD16×C173C2212yescomplemented
35C2×D8×C173C2212yescomplemented
36#127×C173C2211yescharacteristic, complemented
37#124×C173C2211yescharacteristic, complemented
38#97×C173C2211yescharacteristic, complemented
39#14583827—22no—
40#14583826—41no—
41#14583826C2312yes—
42#14583826C2312yes—
43#14583825—22no—
44#14583825—22no—
45#14583825—22no—
46Q8×D173—22no—
47C2×C22⋊D173—41no—
48C2×C22⋊D173—42no—
49D4×D173—22no—
50D4×D173—22no—
51D4×D173—22no—
52Dic692D412yescomplemented
53#14583816—22no—
54#14583815D412yescomplemented
55#14583815D412yescomplemented
56#14583814—22no—
57#14583814—22no—
58#14583814—22no—
59#14583813—22no—
60D1384D412yescomplemented
61#14583811—22no—
62#14583811—22no—
63#14583811—22no—
64#14583810C2312yes—
65#14583810C2312yes—
66#14583809—22no—
67#14583809—22no—
68C8×D173—22no—
69C8×D173—22no—
70C2×Q8×C173C2311yescharacteristic
71C2×D4×C173—41no—
72C2×D4×C173C2311yescharacteristic
73C2×D4×C173C2312yes—
74SD16×C173—22no—
75SD16×C173—22no—
76D8×C173—22no—
77D8×C173—22no—
78C2×C8×C173C2312yes—
79C22⋊C4×C173—41no—
80C42×C173C2311yescharacteristic
81C22×D173—42no—
82C22×D173—41no—
83Dic346—41no—
84Dic346C2×D412yes—
85Dic346C2×D412yes—
86D692—41no—
87D692C2×D412yes—
88D692C2×D412yes—
89C22⋊D173—22no—
90C22⋊D173—22no—
91C22⋊D173—42no—
92C22⋊D173—42no—
93C22⋊D173—42no—
94C2×Dic173—42no—
95C2×Dic173—41no—
96C4×D173—22no—
97C4×D173—22no—
98C4×D173—42no—
99#13063436—22no—
100#13063436—22no—
101C23×C173—21no—
102C23×C173—22no—
103Q8×C173—22no—
104D4×C173—22no—
105D4×C173—22no—
106D4×C173—22no—
107D4×C173—82no—
108C2×C4×C173—21no—
109C2×C4×C173—21no—
110C2×C4×C173C2411yescharacteristic
111C8×C173C2×D412yes—
112C8×C173C2×D412yes—
113C22×C173—41no—
114C22×C173—42no—
115C22×C173—82no—
116C22×C173—81no—
117C22×C173H(5,𝔽2)11yescharacteristic
118C2×D173—22no—
119C2×D173—22no—
120C2×D173—162no—
121C2×D173—161no—
122C4×C173—41no—
123C4×C173—41no—
124C4×C173C22×D412yes—
125Dic173—22no—
126Dic173—22no—
127C2×C173—21no—
128C2×C173—82no—
129C2×C173—81no—
130C2×C173D4211yescharacteristic
131D173—42no—
132D173—42no—
133C173#328811yescharacteristic, complemented, Sylow
134#3288—1731nomaximal, Sylow
135#577—1732no—
136#576—1732no—
137#496D17311yescharacteristic, complemented
138#472—1732no—
139#471—1732no—
140#455—1731no—
141#454—1731no—
142#453—1732no—
143#444—1732no—
1442−1+4—1731no—
145H(5,𝔽2)—1731no—
146H(5,𝔽2)—1732no—
147#137—3462no—
148#137—1732no—
149AGL(1,ℤ/8)—3462no—
150AGL(1,ℤ/8)—3462no—
151AGL(1,ℤ/8)—3462no—
152AGL(1,ℤ/8)—1732no—
153AGL(1,ℤ/8)—1732no—
154AGL(1,ℤ/8)—1732no—
155#135—3462no—
156#135—3462no—
157#135—1732no—
158C2×SD16—3462no—
159C2×SD16C2×D17312yescomplemented
160C2×D8—3462no—
161C2×D8C2×D17312yescomplemented
162#131—1732no—
163#131—1732no—
164#127C2×D17311yescharacteristic, complemented
165#124C2×D17311yescharacteristic, complemented
166#108—1732no—
167#104—1732no—
168#104—1732no—
169#101—1731no—
170#100—1732no—
171#100—1731no—
172#97C2×D17311yescharacteristic, complemented
173#41—3462no—
174#41—3462no—
175#41—3462no—
176#41—3462no—
177#41—1732no—
178#41—1732no—
179#41—6921no—
180C2×Q8—3462no—
181C2×Q8C22×D17311yescharacteristic
182C2×D4—41no—
183C2×D4—3462no—
184C2×D4—3462no—
185C2×D4—3462no—
186C2×D4—6921no—
187C2×D4—6922no—
188C2×D4C22×D17311yescharacteristic
189C2×D4C22×D17312yes—
190Q16—3462no—
191Q16—1732no—
192SD16—22no—
193SD16—22no—
194SD16—3462no—
195SD16—3462no—
196SD16—3462no—
197SD16—3462no—
198SD16—1732no—
199SD16—1732no—
200D8—22no—
201D8—22no—
202D8—3462no—
203D8—3462no—
204D8—3462no—
205D8—1732no—
206M16—3462no—
207M16—3462no—
208M16—1732no—
209M16—1732no—
210C2×C8—3462no—
211C2×C8—3462no—
212C2×C8C22×D17312yes—
213C22⋊C4—41no—
214C42C22×D17311yescharacteristic
215C23—21no—
216C23—22no—
217C23—6922no—
218C23—6921no—
219Q8—22no—
220Q8—1732no—
221Q8—1732no—
222Q8—6921no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—82no—
227D4—3462no—
228D4—3462no—
229D4—1732no—
230D4—1732no—
231D4—6921no—
232D4—6922no—
233D4—6922no—
234D4—6922no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—3462no—
238C2×C4—3462no—
239C2×C4—6922no—
240C2×C4—6921no—
241C2×C4—6922no—
242C2×C4C23×D17311yescharacteristic
243C8—3462no—
244C8—3462no—
245C8D4×D17312yes—
246C8D4×D17312yes—
247C22—41no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—3462no—
252C22—3462no—
253C22—27682no—
254C22—27681no—
255C22#1914972411yescharacteristic
256C4—41no—
257C4—41no—
258C4—3462no—
259C4—3462no—
260C4C2×D4×D17312yes—
261C2—21no—
262C2—82no—
263C2—81no—
264C2—6922no—
265C2—6922no—
266C2#2601212611yescharacteristic
subgroupG/H
centerC2#26012126
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#496×C173C2
socleC2×C173D42
hypercenter#496D173
hypocenterC173#3288
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #26012297C2semidirect
2. #26012292C2semidirect
3. #26011758C2semidirect
4. #26011751C2semidirect
5. #26011668C2semidirect
6. #26011663C2semidirect
7. #26011658C2semidirect
8. #26011614C2semidirect
9. #496×C173C2semidirect
10. #19149727C22semidirect
12. #19149724C22semidirect
13. #19149710C22semidirect
17. #19149705C22semidirect
18. #19149705C22semidirect
26. #19149685C22semidirect
27. #19149685C22semidirect
28. #19149615C22semidirect
29. #19149607C22semidirect
30. #19149607C22semidirect
31. #19149598C22semidirect
32. #19149596C22semidirect
33. #19149596C22semidirect
34. C2×SD16×C173C22semidirect
35. C2×D8×C173C22semidirect
36. #127×C173C22semidirect
37. #124×C173C22semidirect
38. #97×C173C22semidirect
52. Dic692D4semidirect
54. #14583815D4semidirect
55. #14583815D4semidirect
60. D1384D4semidirect
133. C173#3288semidirect
137. #496D173semidirect
159. C2×SD16C2×D173semidirect
161. C2×D8C2×D173semidirect
164. #127C2×D173semidirect
165. #124C2×D173semidirect
172. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122972
#260122922
#260117582
#260117512
#260116682
#260116632
#260116582
#260116142
#496×C1732
#3288173