Finite Group Databasebeta

← #36363198

Subgroups — #36363198

Order 22144.

492 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#19149685C211yesmaximal, characteristic, complemented
2#26012065C211yesmaximal, characteristic, complemented
3#26012058C211yesmaximal, characteristic
4#26011734C214yesmaximal, complemented
5#26011717C214yesmaximal, complemented
6#26011529C211yesmaximal, characteristic, complemented
7#26011500C211yesmaximal, characteristic, complemented
8#26011496C211yesmaximal, characteristic, complemented
9C22×SD16×C173C211yesmaximal, characteristic, complemented
10C2×#14583826C2211yescharacteristic
11#19149685—24no—
12C2×#14583810C2211yescharacteristic
13C2×#14583809C2212yescomplemented
14C2×C8×D173C2212yescomplemented
15#19149677—21no—
16#19149675—21no—
17#19149668—21no—
18#19149666—21no—
19#19149662—22no—
20#19149655C2214yescomplemented
21#19149650C2214yes—
22#19149648—22no—
23#19149638C2211yescharacteristic
24#19149638C2211yescharacteristic
25#19149613C2212yescomplemented
26#19149611C2212yescomplemented
27#19149603C2214yescomplemented
28#19149600C2214yes—
29C22×Q8×C173C2211yescharacteristic
30C22×D4×C173C2211yescharacteristic
31C2×SD16×C173—24no—
32C2×SD16×C173C2212yes—
33C2×SD16×C173C2212yes—
34C22×C8×C173C2211yescharacteristic
35#14583826—22no—
36#14583826—22no—
37C2×Dic346D411yescharacteristic
38C2×D692D411yescharacteristic
39C2×C22⋊D173—24no—
40C2×C22⋊D173D412yes—
41C2×C4×D173C2312yes—
42#14583810—22no—
43#14583809—24no—
44C2×#13063436C2312yes—
45C8×D173—24no—
46C4⋊Dic173C2312yes—
47C4⋊Dic173C2312yes—
48#14583805—24no—
49C22⋊Dic173—24no—
50C22⋊Dic173—21no—
51C22⋊Dic173—21no—
52#14583803—24no—
53C4×Dic173—21no—
54C4×Dic173—21no—
55C24×C173—21no—
56C2×Q8×C173—21no—
57C2×Q8×C173—22no—
58C2×Q8×C173—22no—
59C2×Q8×C173C2312yes—
60C2×D4×C173—22no—
61C2×D4×C173—21no—
62C2×D4×C173—22no—
63C2×D4×C173C2312yes—
64C22×C4×C173—21no—
65C22×C4×C173C2311yescharacteristic
66SD16×C173—24no—
67SD16×C173—24no—
68C2×C8×C173D414yescomplemented
69C2×C8×C173C2311yescharacteristic
70C2×C8×C173C2311yescharacteristic
71C22×D173C2×D412yes—
72Dic346—22no—
73D692—22no—
74C22⋊D173—24no—
75C22⋊D173—84no—
76C2×Dic173—22no—
77C2×Dic173—22no—
78C2×Dic173C2×D412yes—
79C4×D173—22no—
80C4×D173—22no—
81#13063436—22no—
82C23×C173—22no—
83C23×C173—42no—
84C23×C173—82no—
85C23×C173C2×D411yescharacteristic
86Q8×C173—22no—
87Q8×C173—22no—
88Q8×C173—42no—
89D4×C173—22no—
90D4×C173—22no—
91D4×C173—42no—
92C2×C4×C173—22no—
93C2×C4×C173—42no—
94C2×C4×C173C2×D411yescharacteristic
95C2×C4×C173C2×D412yes—
96C2×C4×C173#4112yes—
97C2×C4×C173C2411yescharacteristic
98C8×C173C2×D412yes—
99C8×C173C2×D412yes—
100C22×C173—22no—
101C22×C173—42no—
102C22×C173—44no—
103C22×C173—84no—
104C22×C173#12112yes—
105C22×C173C22×D411yescharacteristic
106C2×D173—22no—
107C2×D173—24no—
108C4×C173—42no—
109C4×C173#12112yes—
110C4×C173C22×D411yescharacteristic
111C4×C173C2×#4111yescharacteristic
112Dic173—22no—
113Dic173—42no—
114Dic173—42no—
115C2×C173—22no—
116C2×C173—82no—
117C2×C173C2×#12111yescharacteristic
118C2×C173#57712yes—
119D173—42no—
120C173#305711yescharacteristic, complemented, Sylow
121#3057—1731nomaximal, Sylow
122C22×SD16D17311yescharacteristic, complemented
123C2×#131—1731no—
124#536—1731no—
125#535—1731no—
126#468—1734no—
127#465—1734no—
128#427—1731no—
129#419—1731no—
130#418—1731no—
131C2×#41—1731no—
132C22×Q8C2×D17311yescharacteristic
133C22×D4C2×D17311yescharacteristic
134C2×SD16—24no—
135C2×SD16C2×D17312yes—
136C2×SD16C2×D17312yes—
137#131—3464no—
138C2×M16—1731no—
139C2×M16—1732no—
140C22×C8—1732no—
141C22×C8C2×D17311yescharacteristic
142C4⋊Q8—3461no—
143#127—3461no—
144#124—3461no—
145#124—3461no—
146#123—3462no—
147C22⋊Q8—1734no—
148#121—1734no—
149#120—3462no—
150#117—1731no—
151#117—1731no—
152#107—1732no—
153#106—1732no—
154#103—1734no—
155#102—1734no—
156C24—21no—
157#41—3462no—
158#41—3462no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—1731no—
163C2×Q8C22×D17312yes—
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—3464no—
168C2×D4—1732no—
169C2×D4—1731no—
170C2×D4C22×D17312yes—
171C22×C4—21no—
172C22×C4—1732no—
173C22×C4C22×D17311yescharacteristic
174SD16—24no—
175SD16—24no—
176M16—3462no—
177M16—3464no—
178C2×C8—3464no—
179C2×C8—1732no—
180C2×C8C22⋊D17314yescomplemented
181C2×C8C22×D17311yescharacteristic
182C2×C8C22×D17311yescharacteristic
183C4⋊C4—3464no—
184C4⋊C4—1732no—
185C4⋊C4—1732no—
186C22⋊C4—3464no—
187C22⋊C4—3461no—
188C22⋊C4—3461no—
189C22⋊C4—3464no—
190C42—3461no—
191C42—3461no—
192C23—22no—
193C23—42no—
194C23—82no—
195C23—1732no—
196C23D4×D17311yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—42no—
200Q8—3462no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—3464no—
205D4—3462no—
206D4—13844no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—3462no—
210C2×C4—3462no—
211C2×C4—3462no—
212C2×C4—3462no—
213C2×C4—1732no—
214C2×C4D4×D17311yescharacteristic
215C2×C4C2×C22⋊D17312yes—
216C2×C4#1458382512yes—
217C2×C4C23×D17311yescharacteristic
218C8—3462no—
219C8C2×C22⋊D17312yes—
220C8C2×C22⋊D17312yes—
221C22—22no—
222C22—42no—
223C22—44no—
224C22—84no—
225C22—3462no—
226C22—3464no—
227C22#1914965012yes—
228C22C2×D4×D17311yescharacteristic
229C4—42no—
230C4—3462no—
231C4—6922no—
232C4—6922no—
233C4#1914965012yes—
234C4C22×C22⋊D17311yescharacteristic
235C4C2×#1458382511yescharacteristic
236C2—22no—
237C2—82no—
238C2—6922no—
239C2C2×#1914965011yescharacteristic
240C2#2601229712yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC22×SD16×C173C2
socleC22×C173C22×D4
prime radical#26012058C2
hypercenterC22×SD16D173
hypocenterC173#3057
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#19149685C2semidirect
2. #26012065C2semidirect
4. #26011734C2semidirect
5. #26011717C2semidirect
6. #26011529C2semidirect
7. #26011500C2semidirect
8. #26011496C2semidirect
9. C22×SD16×C173C2semidirect
13. C2×#14583809C22semidirect
14. C2×C8×D173C22semidirect
20. #19149655C22semidirect
25. #19149613C22semidirect
26. #19149611C22semidirect
27. #19149603C22semidirect
68. C2×C8×C173D4semidirect
120. C173#3057semidirect
122. C22×SD16D173semidirect
180. C2×C8C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496852
#260120652
#260120582
#260117342
#260117172
#260115292
#260115002
#260114962
C22×SD16×C1732
#3057173