Finite Group Databasebeta

← #36363499

Subgroups — #36363499

Order 22144.

462 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 137 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×#19149638C211yesmaximal, characteristic, complemented
2#26011812C214yesmaximal, complemented
3#26011807C212yesmaximal, complemented
4#26011802C212yesmaximal, complemented
5#26011531C212yesmaximal, complemented
6#26011496C212yesmaximal, complemented
7#26011455C211yesmaximal, characteristic, complemented
8C2×#121×C173C211yesmaximal, characteristic
9C2×#14583810C2212yes—
10#19149638—21no—
11#19149638—21no—
12#19149638C2212yes—
13#19149638C2212yescomplemented
14C2×C4⋊Dic173C2211yescharacteristic, complemented
15C2×C4⋊Dic173C2211yescharacteristic, complemented
16C2×C22⋊Dic173—21no—
17C2×C4×Dic173—21no—
18#19149612C2214yescomplemented
19#19149610C2214yescomplemented
20#19149600C2214yescomplemented
21#19149600C2214yescomplemented
22C22⋊#13063436C2212yescomplemented
23C22⋊#13063436C2212yescomplemented
24C22×D4×C173—21no—
25C22×D4×C173C2211yescharacteristic
26C23×C4×C173C2211yescharacteristic
27#121×C173—21no—
28#121×C173—21no—
29#121×C173C2212yes—
30#121×C173C2212yes—
31C2×C4⋊C4×C173C2211yescharacteristic
32C2×C22⋊C4×C173—21no—
33C22×Dic173—21no—
34C22×Dic173—21no—
35#14583810—24no—
36C2×#13063436C2314yes—
37C4⋊Dic173—21no—
38C4⋊Dic173—21no—
39C4⋊Dic173C2312yes—
40C4⋊Dic173C2312yes—
41C22⋊Dic173—22no—
42C22⋊Dic173—22no—
43C4×Dic173—22no—
44C4×Dic173—22no—
45C24×C173—21no—
46C24×C173D411yescharacteristic
47C2×D4×C173—22no—
48C2×D4×C173—22no—
49C2×D4×C173—21no—
50C2×D4×C173—24no—
51C2×D4×C173—44no—
52C2×D4×C173C2312yes—
53C22×C4×C173—24no—
54C22×C4×C173—21no—
55C22×C4×C173D411yescharacteristic
56C22×C4×C173D411yescharacteristic
57C22×C4×C173D411yescharacteristic
58C22×C4×C173C2311yescharacteristic
59C22×C4×C173C2311yescharacteristic
60C22×C4×C173C2311yescharacteristic
61C4⋊C4×C173—21no—
62C4⋊C4×C173C2312yes—
63C22⋊C4×C173—22no—
64C22⋊C4×C173—22no—
65C2×Dic173—22no—
66C2×Dic173—22no—
67C2×Dic173—42no—
68C2×Dic173—42no—
69#13063436—24no—
70C23×C173—22no—
71C23×C173—22no—
72C23×C173—22no—
73C23×C173—41no—
74C23×C173—42no—
75C23×C173—42no—
76C23×C173—42no—
77C23×C173C2×D411yescharacteristic
78C23×C173C2×D411yescharacteristic
79C23×C173C2×D411yescharacteristic
80D4×C173—24no—
81D4×C173—44no—
82D4×C173—44no—
83D4×C173—42no—
84C2×C4×C173—22no—
85C2×C4×C173—22no—
86C2×C4×C173—22no—
87C2×C4×C173—22no—
88C2×C4×C173—22no—
89C2×C4×C173—22no—
90C2×C4×C173—42no—
91C2×C4×C173C2×D411yescharacteristic
92C2×C4×C173C2×D411yescharacteristic
93C2×C4×C173C2×D411yescharacteristic
94C2×C4×C173#4114yes—
95C2×C4×C173C2411yescharacteristic
96C22×C173—21no—
97C22×C173—21no—
98C22×C173—21no—
99C22×C173—21no—
100C22×C173—21no—
101C22×C173—21no—
102C22×C173—21no—
103C22×C173—44no—
104C22×C173—42no—
105C22×C173—42no—
106C22×C173—42no—
107C22×C173—84no—
108C22×C173#12312yes—
109C22×C173AGL(1,ℤ/8)12yes—
110C22×C173C22×D411yescharacteristic
111C4×C173—21no—
112C4×C173—21no—
113C4×C173—42no—
114C4×C173#12312yes—
115C4×C173C2×#4112yes—
116Dic173—42no—
117Dic173—42no—
118C2×C173—22no—
119C2×C173—22no—
120C2×C173—41no—
121C2×C173—82no—
122C2×C173C2×#12311yescharacteristic
123C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
124C2×C173#57511yescharacteristic
125C173#309411yescharacteristic, complemented, Sylow
126#3094—1731nomaximal, Sylow
127C2×#121D17311yescharacteristic
128C2×#117—1731no—
129#482—1734no—
130#481—1732no—
131#480—1732no—
132#428—1732no—
133#418—1732no—
134#407—1731no—
135C22×D4—21no—
136C22×D4C2×D17311yescharacteristic
137C23×C4C2×D17311yescharacteristic
138C2×M16—1732no—
139#121—21no—
140#121—21no—
141#121C2×D17312yes—
142#121C2×D17312yes—
143#117—3461no—
144#117—3461no—
145#117—1732no—
146#117—1732no—
147C2×C4⋊C4—1731no—
148C2×C4⋊C4—1731no—
149C2×C4⋊C4C2×D17311yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—3461no—
152C2×C42—3461no—
153#107—1734no—
154#106—1734no—
155#102—1734no—
156#102—1734no—
157C22⋊C8—1732no—
158C22⋊C8—1732no—
159C24—21no—
160C24C22⋊D17311yescharacteristic
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—24no—
165C2×D4—44no—
166C2×D4C22×D17312yes—
167C22×C4—24no—
168C22×C4—21no—
169C22×C4—3461no—
170C22×C4—3461no—
171C22×C4C22⋊D17311yescharacteristic
172C22×C4C22⋊D17311yescharacteristic
173C22×C4C22⋊D17311yescharacteristic
174C22×C4C22×D17311yescharacteristic
175C22×C4C22×D17311yescharacteristic
176C22×C4C22×D17311yescharacteristic
177M16—3464no—
178C2×C8—1734no—
179C4⋊C4—21no—
180C4⋊C4—3461no—
181C4⋊C4—3461no—
182C4⋊C4—1732no—
183C4⋊C4—1732no—
184C4⋊C4C22×D17312yes—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—3462no—
188C22⋊C4—3462no—
189C42—3462no—
190C42—3462no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—41no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23C2×C22⋊D17311yescharacteristic
199C23C2×C22⋊D17311yescharacteristic
200C23C2×C22⋊D17311yescharacteristic
201D4—24no—
202D4—44no—
203D4—44no—
204D4—42no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—3462no—
213C2×C4—3462no—
214C2×C4—6922no—
215C2×C4—6922no—
216C2×C4C2×C22⋊D17311yescharacteristic
217C2×C4C2×C22⋊D17311yescharacteristic
218C2×C4C2×C22⋊D17311yescharacteristic
219C2×C4#1458382514yes—
220C2×C4C23×D17311yescharacteristic
221C8—3464no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—84no—
234C22#1914966012yes—
235C22#1914970512yes—
236C22C22×C22⋊D17311yescharacteristic
237C4—21no—
238C4—21no—
239C4—42no—
240C4—6922no—
241C4—6922no—
242C4#1914966012yes—
243C4C2×#1458382512yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—82no—
248C2C2×#1914966011yescharacteristic
249C2C2×#1914970511yescharacteristic
250C2#2601228911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting, prime radicalC2×#121×C173C2
socleC22×C173C22×D4
hypercenterC2×#121D173
hypocenterC173#3094
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#19149638C2semidirect
2. #26011812C2semidirect
3. #26011807C2semidirect
4. #26011802C2semidirect
5. #26011531C2semidirect
6. #26011496C2semidirect
7. #26011455C2semidirect
13. #19149638C22semidirect
14. C2×C4⋊Dic173C22semidirect
15. C2×C4⋊Dic173C22semidirect
18. #19149612C22semidirect
19. #19149610C22semidirect
20. #19149600C22semidirect
21. #19149600C22semidirect
22. C22⋊#13063436C22semidirect
23. C22⋊#13063436C22semidirect
125. C173#3094semidirect

Maximal subgroups

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

subgroupindex
C2×#191496382
#260118122
#260118072
#260118022
#260115312
#260114962
#260114552
C2×#121×C1732
#3094173