Finite Group Databasebeta

← #36364399

Subgroups — #36364399

Order 22144.

418 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 109 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#26012001C212yesmaximal, complemented
2#26011835C212yesmaximal, complemented
3#26011814C212yesmaximal, complemented
4#26011749C211yesmaximal, characteristic, complemented
5#26011735C212yesmaximal, complemented
6#26011654C212yesmaximal, complemented
7#26011633C212yesmaximal, complemented
8#26011454C211yesmaximal, characteristic, complemented
9#469×C173C211yesmaximal, characteristic, complemented
10C2×#14583825C2212yes—
11C22×C4×D173C2211yescharacteristic
12C2×Dic692C2211yescharacteristic, complemented
13C2×#14583814C2212yescomplemented
14C2×#14583810C2211yescharacteristic
15#19149661—22no—
16#19149659C2212yes—
17#19149658C2212yes—
18#19149650C2212yes—
19#19149647—22no—
20D4×Dic173—22no—
21#19149639—22no—
22#19149637C2212yes—
23#19149612C2212yescomplemented
24#19149611C2211yescharacteristic, complemented
25#19149604C2212yescomplemented
26#19149602C2212yescomplemented
27#19149600C2212yescomplemented
28#19149591C2212yes—
29#19149591C2212yes—
30C2×D8×C173C2211yescharacteristic, complemented
31C2×M16×C173C2211yescharacteristic
32#121×C173C2212yes—
33#106×C173C2211yescharacteristic, complemented
34#102×C173C2212yescomplemented
35C23×D173D411yescharacteristic
36#14583825—42no—
37#14583825—42no—
38C2×Dic346C2312yes—
39C2×C22⋊D173—22no—
40C2×C22⋊D173—22no—
41C22×Dic173—22no—
42C22×Dic173D411yescharacteristic
43C2×C4×D173—41no—
44C2×C4×D173—41no—
45C2×C4×D173D412yes—
46C2×C4×D173C2312yes—
47Dic692—41no—
48#14583814—42no—
49#14583810—41no—
50C2×#13063436C2312yes—
51C4⋊Dic173—22no—
52C4⋊Dic173C2312yes—
53#14583805—22no—
54C22⋊Dic173—22no—
55C22⋊Dic173—22no—
56#14583803—22no—
57#14583803—22no—
58C4×Dic173—22no—
59C2×D4×C173—22no—
60C2×D4×C173C2312yes—
61C22×C4×C173C2311yescharacteristic
62D8×C173—41no—
63M16×C173—41no—
64C2×C8×C173C2311yescharacteristic
65C2×C8×C173C2311yescharacteristic
66C4⋊C4×C173C2312yes—
67C22⋊C4×C173—22no—
68C22×D173—41no—
69C22×D173—42no—
70C22×D173C2×D412yes—
71Dic346—22no—
72Dic346—22no—
73C22⋊D173—44no—
74C22⋊D173—44no—
75C2×Dic173—22no—
76C2×Dic173—22no—
77C2×Dic173—41no—
78C2×Dic173—44no—
79C2×Dic173C2×D412yes—
80C4×D173—22no—
81C4×D173—22no—
82C4×D173—42no—
83#13063436—22no—
84C23×C173—22no—
85C23×C173C2×D411yescharacteristic
86D4×C173—22no—
87D4×C173—22no—
88D4×C173—44no—
89C2×C4×C173—21no—
90C2×C4×C173—21no—
91C2×C4×C173—22no—
92C2×C4×C173C2×D411yescharacteristic
93C2×C4×C173C2411yescharacteristic
94C8×C173—21no—
95C8×C173—21no—
96C22×C173—21no—
97C22×C173—22no—
98C22×C173—42no—
99C22×C173—42no—
100C22×C173—42no—
101C22×C173C22×D411yescharacteristic
102C2×D173—22no—
103C2×D173—22no—
104C2×D173—22no—
105C2×D173—44no—
106C4×C173—21no—
107C4×C173—42no—
108C4×C173H(5,𝔽2)11yescharacteristic
109C4×C173H(5,𝔽2)11yescharacteristic
110Dic173—22no—
111Dic173—42no—
112Dic173—42no—
113C2×C173—41no—
114C2×C173—82no—
115C2×C173#53411yescharacteristic
116C2×C173#57512yes—
117D173—42no—
118C173#319711yescharacteristic, complemented, Sylow
119#3197—1731nomaximal, Sylow
120#525—1732no—
121#485—1732no—
122#482—1732no—
123#470—1731no—
124#469D17311yescharacteristic, complemented
125#468—1732no—
126#452—1732no—
127#449—1732no—
128#407—1731no—
129C2×#41—1732no—
130C23×C4—1731no—
131C2×Q16—1731no—
132C2×SD16—1732no—
133C2×D8C2×D17311yescharacteristic, complemented
134C2×M16—1731no—
135C2×M16C2×D17311yescharacteristic
136#123—3462no—
137C22⋊Q8—1732no—
138C22⋊Q8—1732no—
139#121—1732no—
140#121C2×D17312yes—
141#120—3462no—
142C4×D4—3462no—
143C4×D4—3462no—
144#117—1732no—
145#107—1732no—
146#106—1731no—
147#106C2×D17311yescharacteristic, complemented
148#103—1732no—
149#103—1732no—
150#102—1732no—
151#102C2×D17312yescomplemented
152C22⋊C8—1732no—
153C22⋊C8—1732no—
154C24—1731no—
155#41—6922no—
156#41—6922no—
157C2×Q8—1732no—
158C2×D4—22no—
159C2×D4—3462no—
160C2×D4—3462no—
161C2×D4C22×D17312yes—
162C22×C4—3462no—
163C22×C4—1732no—
164C22×C4—1731no—
165C22×C4—1732no—
166C22×C4—6921no—
167C22×C4—6921no—
168C22×C4C22×D17311yescharacteristic
169Q16—6921no—
170SD16—6922no—
171D8—41no—
172M16—41no—
173M16—6921no—
174C2×C8—1732no—
175C2×C8C22×D17311yescharacteristic
176C2×C8C22×D17311yescharacteristic
177C4⋊C4—3462no—
178C4⋊C4—3462no—
179C4⋊C4—1732no—
180C4⋊C4C22×D17312yes—
181C22⋊C4—22no—
182C22⋊C4—3462no—
183C22⋊C4—3462no—
184C22⋊C4—3462no—
185C22⋊C4—3462no—
186C42—3462no—
187C23—22no—
188C23—1732no—
189C23—6921no—
190C23—6922no—
191C23D4×D17311yescharacteristic
192Q8—3462no—
193Q8—3462no—
194D4—22no—
195D4—22no—
196D4—44no—
197D4—6924no—
198D4—6924no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—3462no—
203C2×C4—3462no—
204C2×C4—3462no—
205C2×C4—3462no—
206C2×C4—1732no—
207C2×C4—6922no—
208C2×C4—6921no—
209C2×C4—6924no—
210C2×C4D4×D17311yescharacteristic
211C2×C4C23×D17311yescharacteristic
212C8—21no—
213C8—21no—
214C8—3462no—
215C22—22no—
216C22—21no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—3462no—
221C22—3462no—
222C22—3462no—
223C22—6924no—
224C22C2×D4×D17311yescharacteristic
225C4—21no—
226C4—42no—
227C4—3462no—
228C4—6922no—
229C4—6922no—
230C4#1914972411yescharacteristic
231C4#1914972511yescharacteristic
232C2—41no—
233C2—82no—
234C2—6922no—
235C2#2601204911yescharacteristic
236C2#2601228712yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#469×C173C2
socleC22×C173C22×D4
hypercenter#469D173
hypocenterC173#3197
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #26012001C2semidirect
2. #26011835C2semidirect
3. #26011814C2semidirect
4. #26011749C2semidirect
5. #26011735C2semidirect
6. #26011654C2semidirect
7. #26011633C2semidirect
8. #26011454C2semidirect
9. #469×C173C2semidirect
12. C2×Dic692C22semidirect
13. C2×#14583814C22semidirect
23. #19149612C22semidirect
24. #19149611C22semidirect
25. #19149604C22semidirect
26. #19149602C22semidirect
27. #19149600C22semidirect
30. C2×D8×C173C22semidirect
33. #106×C173C22semidirect
34. #102×C173C22semidirect
118. C173#3197semidirect
124. #469D173semidirect
133. C2×D8C2×D173semidirect
147. #106C2×D173semidirect
151. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120012
#260118352
#260118142
#260117492
#260117352
#260116542
#260116332
#260114542
#469×C1732
#3197173