Finite Group Databasebeta

← #36364425

Subgroups — #36364425

Order 22144.

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

normal subgroups only

This group has 442 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#26012057C212yesmaximal, complemented
2#26011817C212yesmaximal, complemented
3#26011814C212yesmaximal, complemented
4#26011744C211yesmaximal, characteristic, complemented
5#26011725C212yesmaximal, complemented
6#26011626C212yesmaximal, complemented
7#26011623C212yesmaximal, complemented
8#26011459C211yesmaximal, characteristic, complemented
9#469×C173C211yesmaximal, characteristic, complemented
10C2×#14583826C2211yescharacteristic
11C2×D4×D173C2212yescomplemented
12C2×D1384C2211yescharacteristic
13C2×#14583811C2212yes—
14C22×#13063436C2211yescharacteristic, complemented
15#19149675—22no—
16#19149665—22no—
17#19149654C2212yes—
18#19149653C2212yescomplemented
19#19149650C2212yes—
20#19149649—22no—
21#19149647—22no—
22#19149637C2212yescomplemented
23#19149612C2212yescomplemented
24#19149611C2211yescharacteristic, complemented
25#19149601C2212yescomplemented
26#19149600C2212yescomplemented
27#19149599C2212yescomplemented
28#19149591C2212yescomplemented
29#19149591C2212yescomplemented
30C2×D8×C173C2211yescharacteristic
31C2×M16×C173C2211yescharacteristic, complemented
32#121×C173C2212yescomplemented
33#106×C173C2211yescharacteristic, complemented
34#102×C173C2212yescomplemented
35C23×D173—22no—
36#14583826—41no—
37#14583826—41no—
38C2×Dic346D411yescharacteristic
39C2×D692—22no—
40C2×D692D411yescharacteristic
41C2×D692C2312yes—
42C2×C22⋊D173—22no—
43C2×C22⋊D173—22no—
44C2×C22⋊D173—22no—
45C2×C22⋊D173D412yes—
46D4×D173—42no—
47D4×D173—42no—
48C2×C4×D173C2312yes—
49D1384—41no—
50#14583811—42no—
51C2×#13063436—41no—
52C2×#13063436C2312yes—
53C4⋊Dic173C2312yes—
54C22⋊Dic173—22no—
55#14583803—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—22no—
69C22×D173—44no—
70C22×D173—82no—
71C22×D173—82no—
72C22×D173C2×D412yes—
73Dic346—41no—
74D692—22no—
75D692—22no—
76D692—41no—
77D692—82no—
78C22⋊D173—42no—
79C22⋊D173—44no—
80C22⋊D173—82no—
81C22⋊D173—82no—
82C2×Dic173—22no—
83C2×Dic173C2×D412yes—
84C4×D173—22no—
85C4×D173—22no—
86#13063436—22no—
87C23×C173—22no—
88C23×C173C2×D411yescharacteristic
89D4×C173—22no—
90D4×C173—22no—
91D4×C173—82no—
92C2×C4×C173—21no—
93C2×C4×C173—21no—
94C2×C4×C173—22no—
95C2×C4×C173C2×D411yescharacteristic
96C2×C4×C173C2411yescharacteristic
97C8×C173—21no—
98C8×C173—21no—
99C22×C173—21no—
100C22×C173—22no—
101C22×C173—42no—
102C22×C173—42no—
103C22×C173—42no—
104C22×C173C22×D411yescharacteristic
105C2×D173—22no—
106C2×D173—22no—
107C2×D173—22no—
108C2×D173—42no—
109C2×D173—42no—
110C2×D173—42no—
111C2×D173—88no—
112C4×C173—21no—
113C4×C173—42no—
114C4×C173H(5,𝔽2)11yescharacteristic
115C4×C173H(5,𝔽2)11yescharacteristic
116Dic173—22no—
117Dic173—42no—
118C2×C173—41no—
119C2×C173—82no—
120C2×C173#53411yescharacteristic
121C2×C173#57612yes—
122D173—42no—
123D173—82no—
124C173#319911yescharacteristic, complemented, Sylow
125#3199—1731nomaximal, Sylow
126#535—1732no—
127#482—1732no—
128#482—1732no—
129#469—1731no—
130#469D17311yescharacteristic, complemented
131#466—1732no—
132#447—1732no—
133#447—1732no—
134#408—1731no—
135C2×#41—1731no—
136C22×D4—1732no—
137C2×D8—1732no—
138C2×D8—1731no—
139C2×D8C2×D17311yescharacteristic
140C2×M16C2×D17311yescharacteristic, complemented
141C22×C8—1731no—
142#127—3462no—
143#124—3462no—
144#121—1732no—
145#121—1732no—
146#121—1732no—
147#121C2×D17312yescomplemented
148#120—3462no—
149#120—3462no—
150#117—1732no—
151#107—1732no—
152#106—1731no—
153#106C2×D17311yescharacteristic, complemented
154#102—1732no—
155#102—1732no—
156#102—1732no—
157#102C2×D17312yescomplemented
158C22⋊C8—1732no—
159C22⋊C8—1732no—
160C24—3462no—
161#41—6921no—
162#41—6921no—
163C2×Q8—1731no—
164C2×D4—22no—
165C2×D4—3462no—
166C2×D4—3462no—
167C2×D4—3462no—
168C2×D4—3462no—
169C2×D4—1732no—
170C2×D4—1731no—
171C2×D4—1732no—
172C2×D4—6922no—
173C2×D4—6922no—
174C2×D4C22×D17312yes—
175C22×C4—1732no—
176C22×C4C22×D17311yescharacteristic
177D8—41no—
178D8—6921no—
179D8—6922no—
180M16—41no—
181C2×C8—1732no—
182C2×C8—6921no—
183C2×C8C22×D17311yescharacteristic
184C2×C8C22×D17311yescharacteristic
185C4⋊C4—1732no—
186C4⋊C4C22×D17312yes—
187C22⋊C4—22no—
188C22⋊C4—3462no—
189C22⋊C4—3462no—
190C22⋊C4—3462no—
191C22⋊C4—3462no—
192C42—3462no—
193C23—22no—
194C23—3462no—
195C23—13842no—
196C23—13842no—
197C23—1732no—
198C23—6924no—
199C23D4×D17311yescharacteristic
200Q8—6921no—
201D4—22no—
202D4—22no—
203D4—82no—
204D4—3462no—
205D4—3462no—
206D4—13842no—
207D4—13842no—
208D4—13842no—
209D4—6921no—
210D4—6922no—
211D4—6924no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—3462no—
216C2×C4—3462no—
217C2×C4—3462no—
218C2×C4—1732no—
219C2×C4D4×D17311yescharacteristic
220C2×C4C23×D17311yescharacteristic
221C8—21no—
222C8—21no—
223C8—3462no—
224C22—22no—
225C22—21no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—3462no—
230C22—3462no—
231C22—3462no—
232C22—13848no—
233C22—6922no—
234C22—6922no—
235C22—6922no—
236C22C2×D4×D17311yescharacteristic
237C4—21no—
238C4—42no—
239C4—3462no—
240C4—6922no—
241C4#1914972411yescharacteristic
242C4#1914972511yescharacteristic
243C2—41no—
244C2—82no—
245C2—13842no—
246C2—6922no—
247C2#2601204911yescharacteristic
248C2#2601229512yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#469×C173C2
socleC22×C173C22×D4
hypercenter#469D173
hypocenterC173#3199
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. #26012057C2semidirect
2. #26011817C2semidirect
3. #26011814C2semidirect
4. #26011744C2semidirect
5. #26011725C2semidirect
6. #26011626C2semidirect
7. #26011623C2semidirect
8. #26011459C2semidirect
9. #469×C173C2semidirect
11. C2×D4×D173C22semidirect
14. C22×#13063436C22semidirect
18. #19149653C22semidirect
22. #19149637C22semidirect
23. #19149612C22semidirect
24. #19149611C22semidirect
25. #19149601C22semidirect
26. #19149600C22semidirect
27. #19149599C22semidirect
28. #19149591C22semidirect
29. #19149591C22semidirect
31. C2×M16×C173C22semidirect
32. #121×C173C22semidirect
33. #106×C173C22semidirect
34. #102×C173C22semidirect
124. C173#3199semidirect
130. #469D173semidirect
140. C2×M16C2×D173semidirect
147. #121C2×D173semidirect
153. #106C2×D173semidirect
157. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120572
#260118172
#260118142
#260117442
#260117252
#260116262
#260116232
#260114592
#469×C1732
#3199173