Finite Group Databasebeta

← #36364423

Subgroups — #36364423

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#26011818C212yesmaximal, complemented
3#26011813C212yesmaximal, complemented
4#26011742C212yesmaximal, complemented
5#26011727C211yesmaximal, characteristic, complemented
6#26011626C212yesmaximal, complemented
7#26011623C212yesmaximal, complemented
8#26011457C211yesmaximal, characteristic, complemented
9#466×C173C211yesmaximal, characteristic, complemented
10C2×#14583826C2211yescharacteristic
11C2×D4×D173C2212yescomplemented
12C2×D1384C2211yescharacteristic
13C2×#14583811C2212yes—
14C2×#14583810C2211yescharacteristic, complemented
15#19149675—22no—
16#19149665—22no—
17#19149654C2212yes—
18#19149653C2212yescomplemented
19#19149650C2212yes—
20#19149649—22no—
21#19149647—22no—
22#19149637C2212yescomplemented
23#19149613C2211yescharacteristic, complemented
24#19149610C2212yescomplemented
25#19149601C2212yescomplemented
26#19149600C2212yescomplemented
27#19149599C2212yescomplemented
28#19149591C2212yescomplemented
29#19149591C2212yescomplemented
30C2×D8×C173C2211yescharacteristic
31C22×C8×C173C2211yescharacteristic, complemented
32#121×C173C2212yescomplemented
33#107×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—
51#14583810—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—
63C2×C8×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—21no—
101C22×C173—21no—
102C22×C173—42no—
103C22×C173—42no—
104C22×C173—42no—
105C22×C173C22×D411yescharacteristic
106C2×D173—22no—
107C2×D173—22no—
108C2×D173—22no—
109C2×D173—42no—
110C2×D173—42no—
111C2×D173—42no—
112C2×D173—88no—
113C4×C173—21no—
114C4×C173—42no—
115C4×C173H(5,𝔽2)11yescharacteristic
116C4×C173H(5,𝔽2)11yescharacteristic
117Dic173—22no—
118Dic173—42no—
119C2×C173—41no—
120C2×C173—82no—
121C2×C173#53411yescharacteristic
122C2×C173#57611yescharacteristic
123C2×C173#57611yescharacteristic
124D173—42no—
125D173—82no—
126C173#319911yescharacteristic, complemented, Sylow
127#3199—1731nomaximal, Sylow
128#535—1732no—
129#482—1732no—
130#482—1732no—
131#469—1732no—
132#466—1731no—
133#466D17311yescharacteristic, complemented
134#447—1732no—
135#447—1732no—
136#408—1731no—
137C2×#41—1731no—
138C22×D4—1732no—
139C2×D8—1732no—
140C2×D8—1731no—
141C2×D8C2×D17311yescharacteristic
142C2×M16—1731no—
143C22×C8C2×D17311yescharacteristic, complemented
144#127—3462no—
145#124—3462no—
146#121—1732no—
147#121—1732no—
148#121—1732no—
149#121C2×D17312yescomplemented
150#120—3462no—
151#120—3462no—
152#117—1732no—
153#107—1731no—
154#107C2×D17311yescharacteristic, complemented
155#106—1732no—
156#102—1732no—
157#102—1732no—
158#102—1732no—
159#102C2×D17312yescomplemented
160C22⋊C8—1732no—
161C22⋊C8—1732no—
162C24—3462no—
163#41—6921no—
164#41—6921no—
165C2×Q8—1731no—
166C2×D4—22no—
167C2×D4—3462no—
168C2×D4—3462no—
169C2×D4—3462no—
170C2×D4—3462no—
171C2×D4—1732no—
172C2×D4—1731no—
173C2×D4—1732no—
174C2×D4—6922no—
175C2×D4—6922no—
176C2×D4C22×D17312yes—
177C22×C4—1732no—
178C22×C4C22×D17311yescharacteristic
179D8—41no—
180D8—6921no—
181D8—6922no—
182M16—6921no—
183C2×C8—41no—
184C2×C8—1732no—
185C2×C8C22×D17311yescharacteristic
186C2×C8C22×D17311yescharacteristic
187C4⋊C4—1732no—
188C4⋊C4C22×D17312yes—
189C22⋊C4—22no—
190C22⋊C4—3462no—
191C22⋊C4—3462no—
192C22⋊C4—3462no—
193C22⋊C4—3462no—
194C42—3462no—
195C23—22no—
196C23—3462no—
197C23—13842no—
198C23—13842no—
199C23—1732no—
200C23—6924no—
201C23D4×D17311yescharacteristic
202Q8—6921no—
203D4—22no—
204D4—22no—
205D4—82no—
206D4—3462no—
207D4—3462no—
208D4—13842no—
209D4—13842no—
210D4—13842no—
211D4—6921no—
212D4—6922no—
213D4—6924no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—3462no—
218C2×C4—3462no—
219C2×C4—3462no—
220C2×C4—1732no—
221C2×C4D4×D17311yescharacteristic
222C2×C4C23×D17311yescharacteristic
223C8—21no—
224C8—21no—
225C8—3462no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—3462no—
233C22—3462no—
234C22—3462no—
235C22—13848no—
236C22—6922no—
237C22—6922no—
238C22—6922no—
239C22C2×D4×D17311yescharacteristic
240C4—21no—
241C4—42no—
242C4—3462no—
243C4—6922no—
244C4#1914972411yescharacteristic
245C4#1914972511yescharacteristic
246C2—41no—
247C2—82no—
248C2—13842no—
249C2—6922no—
250C2#2601204911yescharacteristic
251C2#2601229211yescharacteristic
252C2#2601229311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#466×C173C2
socleC22×C173C22×D4
hypercenter#466D173
hypocenterC173#3199
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #26012057C2semidirect
2. #26011818C2semidirect
3. #26011813C2semidirect
4. #26011742C2semidirect
5. #26011727C2semidirect
6. #26011626C2semidirect
7. #26011623C2semidirect
8. #26011457C2semidirect
9. #466×C173C2semidirect
11. C2×D4×D173C22semidirect
14. C2×#14583810C22semidirect
18. #19149653C22semidirect
22. #19149637C22semidirect
23. #19149613C22semidirect
24. #19149610C22semidirect
25. #19149601C22semidirect
26. #19149600C22semidirect
27. #19149599C22semidirect
28. #19149591C22semidirect
29. #19149591C22semidirect
31. C22×C8×C173C22semidirect
32. #121×C173C22semidirect
33. #107×C173C22semidirect
34. #102×C173C22semidirect
126. C173#3199semidirect
133. #466D173semidirect
143. C22×C8C2×D173semidirect
149. #121C2×D173semidirect
154. #107C2×D173semidirect
159. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120572
#260118182
#260118132
#260117422
#260117272
#260116262
#260116232
#260114572
#466×C1732
#3199173