Finite Group Databasebeta

← #36364310

Subgroups — #36364310

Order 22144.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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#121×D173C212yesmaximal, complemented
2#26011806C212yesmaximal, complemented
3#26011803C212yesmaximal, complemented
4#26011727C211yesmaximal, characteristic, complemented
5#26011725C212yesmaximal, complemented
6#26011626C212yesmaximal, complemented
7#26011623C212yesmaximal, complemented
8C2×#19149591C211yesmaximal, characteristic, complemented
9#466×C173C211yesmaximal, characteristic, complemented
10C2×D4×D173—22no—
11C2×D4×D173C2212yescomplemented
12C22×C4×D173C2211yescharacteristic
13C2×D1384C2211yescharacteristic, complemented
14C2×#14583811C2212yescomplemented
15C22×#13063436C2211yescharacteristic, complemented
16#19149654C2212yes—
17#19149653C2212yescomplemented
18#19149652—22no—
19#19149651—22no—
20#19149650C2212yes—
21C4⋊C4×D173C2212yescomplemented
22C22⋊C4×D173—22no—
23#19149613C2211yescharacteristic, complemented
24#19149612C2212yescomplemented
25#19149601C2212yescomplemented
26#19149600C2212yescomplemented
27#19149599C2212yescomplemented
28#19149591C2212yescomplemented
29#19149591C2212yescomplemented
30C2×D8×C173C2211yescharacteristic, complemented
31C22×C8×C173C2211yescharacteristic, complemented
32#121×C173C2212yescomplemented
33#107×C173C2211yescharacteristic, complemented
34#102×C173C2212yescomplemented
35C23×D173—22no—
36C23×D173D411yescharacteristic
37C2×D692—22no—
38C2×D692C2312yes—
39C2×C22⋊D173—22no—
40C2×C22⋊D173—22no—
41C2×C22⋊D173—22no—
42D4×D173—42no—
43D4×D173—42no—
44D4×D173—48no—
45C22×Dic173D411yescharacteristic
46C2×C4×D173—22no—
47C2×C4×D173—41no—
48C2×C4×D173—41no—
49C2×C4×D173D412yes—
50C2×C4×D173C2312yes—
51D1384—41no—
52#14583811—42no—
53C2×#13063436—41no—
54C2×#13063436C2312yes—
55C4⋊Dic173C2312yes—
56#14583805—22no—
57C22⋊Dic173—22no—
58#14583803—22no—
59#14583803—22no—
60C2×D4×C173—22no—
61C2×D4×C173C2312yes—
62C22×C4×C173C2311yescharacteristic
63D8×C173—41no—
64C2×C8×C173—41no—
65C2×C8×C173C2311yescharacteristic
66C2×C8×C173C2311yescharacteristic
67C4⋊C4×C173C2312yes—
68C22⋊C4×C173—22no—
69C22×D173—22no—
70C22×D173—41no—
71C22×D173—41no—
72C22×D173—44no—
73C22×D173—44no—
74C22×D173—44no—
75C22×D173D814yescomplemented
76C22×D173C2×D412yes—
77D692—22no—
78D692—22no—
79D692—44no—
80C22⋊D173—44no—
81C22⋊D173—44no—
82C22⋊D173—44no—
83C2×Dic173—22no—
84C2×Dic173—41no—
85C2×Dic173C2×D412yes—
86C4×D173—22no—
87C4×D173—22no—
88C4×D173—42no—
89C4×D173—44no—
90#13063436—22no—
91C23×C173—22no—
92C23×C173C2×D411yescharacteristic
93D4×C173—22no—
94D4×C173—22no—
95D4×C173—44no—
96C2×C4×C173—21no—
97C2×C4×C173—21no—
98C2×C4×C173—22no—
99C2×C4×C173C2×D411yescharacteristic
100C2×C4×C173C2411yescharacteristic
101C8×C173—21no—
102C8×C173—21no—
103C22×C173—21no—
104C22×C173—21no—
105C22×C173—42no—
106C22×C173—42no—
107C22×C173—42no—
108C22×C173C2×D812yes—
109C22×C173C22×D411yescharacteristic
110C2×D173—22no—
111C2×D173—22no—
112C2×D173—44no—
113C2×D173—42no—
114C2×D173—42no—
115C2×D173—42no—
116C2×D173—88no—
117C2×D173C2×D814yes—
118C4×C173—21no—
119C4×C173—42no—
120C4×C173H(5,𝔽2)11yescharacteristic
121C4×C173H(5,𝔽2)11yescharacteristic
122Dic173—22no—
123Dic173—42no—
124C2×C173—22no—
125C2×C173—82no—
126C2×C173#53411yescharacteristic
127C2×C173C22×D811yescharacteristic
128C2×C173#57511yescharacteristic
129D173—24no—
130D173—82no—
131C173#318611yescharacteristic, complemented, Sylow
132#3186—1731nomaximal, Sylow
133C2×#121—1732no—
134#480—1732no—
135#480—1732no—
136#466—1731no—
137#466—1732no—
138#466D17311yescharacteristic, complemented
139#447—1732no—
140#447—1732no—
141C2×C22⋊C8—1731no—
142C22×D4—3462no—
143C22×D4—1732no—
144C23×C4—1731no—
145C2×D8—1732no—
146C2×D8—1731no—
147C2×D8C2×D17311yescharacteristic, complemented
148C22×C8—1731no—
149C22×C8C2×D17311yescharacteristic, complemented
150#121—3462no—
151#121—3462no—
152#121—1732no—
153#121—1732no—
154#121—1732no—
155#121C2×D17312yescomplemented
156C2×C4⋊C4—1732no—
157C2×C22⋊C4—3462no—
158#107—1731no—
159#107—1732no—
160#107C2×D17311yescharacteristic, complemented
161#102—1732no—
162#102—1732no—
163#102—1732no—
164#102C2×D17312yescomplemented
165C22⋊C8—1732no—
166C22⋊C8—1732no—
167C24—3462no—
168C24—1731no—
169C2×D4—22no—
170C2×D4—3462no—
171C2×D4—3462no—
172C2×D4—3462no—
173C2×D4—3462no—
174C2×D4—1732no—
175C2×D4—6922no—
176C2×D4—6922no—
177C2×D4—6928no—
178C2×D4C22×D17312yes—
179C22×C4—3462no—
180C22×C4—1732no—
181C22×C4—1731no—
182C22×C4—1732no—
183C22×C4—6921no—
184C22×C4—6921no—
185C22×C4C22×D17311yescharacteristic
186D8—41no—
187D8—6921no—
188D8—6922no—
189C2×C8—41no—
190C2×C8—1732no—
191C2×C8—6921no—
192C2×C8C22×D17311yescharacteristic
193C2×C8C22×D17311yescharacteristic
194C4⋊C4—3462no—
195C4⋊C4—1732no—
196C4⋊C4C22×D17312yes—
197C22⋊C4—22no—
198C22⋊C4—3462no—
199C22⋊C4—3462no—
200C22⋊C4—3462no—
201C23—22no—
202C23—3462no—
203C23—1732no—
204C23—1734no—
205C23—6921no—
206C23—6921no—
207C23—6924no—
208C23—6924no—
209C23—6924no—
210C23D4×D17311yescharacteristic
211D4—22no—
212D4—22no—
213D4—44no—
214D4—3462no—
215D4—3462no—
216D4—6924no—
217D4—6924no—
218D4—6924no—
219D4—6924no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—3462no—
224C2×C4—3462no—
225C2×C4—3462no—
226C2×C4—1732no—
227C2×C4—6922no—
228C2×C4—6924no—
229C2×C4—6921no—
230C2×C4D4×D17311yescharacteristic
231C2×C4C23×D17311yescharacteristic
232C8—21no—
233C8—21no—
234C8—3462no—
235C22—21no—
236C22—21no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—3462no—
241C22—3462no—
242C22—13848no—
243C22—1734no—
244C22—6924no—
245C22—6922no—
246C22—6922no—
247C22—6922no—
248C22D8×D17312yes—
249C22C2×D4×D17311yescharacteristic
250C4—21no—
251C4—42no—
252C4—3462no—
253C4—6922no—
254C4#1914972411yescharacteristic
255C4#1914972511yescharacteristic
256C2—22no—
257C2—82no—
258C2—3464no—
259C2—13842no—
260C2#2601204911yescharacteristic
261C2C2×D8×D17311yescharacteristic
262C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#466×C173C2
socleC22×C173C22×D4
hypercenter#466D173
hypocenterC173#3186
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. #121×D173C2semidirect
2. #26011806C2semidirect
3. #26011803C2semidirect
4. #26011727C2semidirect
5. #26011725C2semidirect
6. #26011626C2semidirect
7. #26011623C2semidirect
8. C2×#19149591C2semidirect
9. #466×C173C2semidirect
11. C2×D4×D173C22semidirect
13. C2×D1384C22semidirect
14. C2×#14583811C22semidirect
15. C22×#13063436C22semidirect
17. #19149653C22semidirect
21. C4⋊C4×D173C22semidirect
23. #19149613C22semidirect
24. #19149612C22semidirect
25. #19149601C22semidirect
26. #19149600C22semidirect
27. #19149599C22semidirect
28. #19149591C22semidirect
29. #19149591C22semidirect
30. C2×D8×C173C22semidirect
31. C22×C8×C173C22semidirect
32. #121×C173C22semidirect
33. #107×C173C22semidirect
34. #102×C173C22semidirect
75. C22×D173D8semidirect
131. C173#3186semidirect
138. #466D173semidirect
147. C2×D8C2×D173semidirect
149. C22×C8C2×D173semidirect
155. #121C2×D173semidirect
160. #107C2×D173semidirect
164. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#121×D1732
#260118062
#260118032
#260117272
#260117252
#260116262
#260116232
C2×#191495912
#466×C1732
#3186173