Finite Group Databasebeta

← #36364359

Subgroups — #36364359

Order 22144.

514 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 113 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#26011811C212yesmaximal, complemented
3#26011808C212yesmaximal, complemented
4#26011744C211yesmaximal, characteristic, complemented
5#26011742C212yesmaximal, complemented
6#26011626C212yesmaximal, complemented
7#26011623C212yesmaximal, complemented
8#26011454C211yesmaximal, characteristic, complemented
9#469×C173C211yesmaximal, characteristic, complemented
10C2×D4×D173—22no—
11C2×D4×D173C2212yescomplemented
12C22×C4×D173C2211yescharacteristic
13C2×D1384C2211yescharacteristic, complemented
14C2×#14583811C2212yescomplemented
15C2×#14583810C2211yescharacteristic, complemented
16#19149654C2212yes—
17#19149653C2212yescomplemented
18#19149652—22no—
19#19149651—22no—
20#19149650C2212yes—
21C4⋊C4×D173C2212yescomplemented
22C22⋊C4×D173—22no—
23#19149611C2211yescharacteristic, complemented
24#19149610C2212yescomplemented
25#19149601C2212yescomplemented
26#19149600C2212yescomplemented
27#19149599C2212yescomplemented
28#19149591C2212yescomplemented
29#19149591C2212yescomplemented
30C2×D8×C173C2211yescharacteristic, complemented
31C2×M16×C173C2211yescharacteristic, complemented
32#121×C173C2212yescomplemented
33#106×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—
53#14583810—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—
64M16×C173—41no—
65C2×C8×C173C2311yescharacteristic
66C2×C8×C173C2311yescharacteristic
67C4⋊C4×C173C2312yes—
68C22⋊C4×C173—22no—
69C22×D173—24no—
70C22×D173—22no—
71C22×D173—41no—
72C22×D173—44no—
73C22×D173—44no—
74C22×D173—44no—
75C22×D173C2×D412yes—
76D692—22no—
77D692—22no—
78D692—44no—
79C22⋊D173—44no—
80C22⋊D173—44no—
81C22⋊D173—44no—
82C2×Dic173—22no—
83C2×Dic173—41no—
84C2×Dic173C2×D412yes—
85C4×D173—22no—
86C4×D173—22no—
87C4×D173—42no—
88C4×D173—44no—
89#13063436—22no—
90C23×C173—22no—
91C23×C173C2×D411yescharacteristic
92D4×C173—22no—
93D4×C173—22no—
94D4×C173—44no—
95C2×C4×C173—21no—
96C2×C4×C173—21no—
97C2×C4×C173—22no—
98C2×C4×C173C2×D411yescharacteristic
99C2×C4×C173C2411yescharacteristic
100C8×C173—21no—
101C8×C173—21no—
102C22×C173—21no—
103C22×C173—22no—
104C22×C173—42no—
105C22×C173—42no—
106C22×C173—42no—
107C22×C173C22×D411yescharacteristic
108C2×D173—22no—
109C2×D173—22no—
110C2×D173—44no—
111C2×D173—42no—
112C2×D173—42no—
113C2×D173—42no—
114C2×D173—88no—
115C2×D173AGL(1,ℤ/8)14yes—
116C4×C173—21no—
117C4×C173—42no—
118C4×C173H(5,𝔽2)11yescharacteristic
119C4×C173H(5,𝔽2)11yescharacteristic
120Dic173—22no—
121Dic173—42no—
122C2×C173—41no—
123C2×C173—82no—
124C2×C173#53411yescharacteristic
125C2×C173C2×AGL(1,ℤ/8)12yes—
126D173—24no—
127D173—82no—
128C173#319211yescharacteristic, complemented, Sylow
129#3192—1731nomaximal, Sylow
130C2×#121—1732no—
131#481—1732no—
132#481—1732no—
133#469—1731no—
134#469—1732no—
135#469D17311yescharacteristic, complemented
136#447—1732no—
137#447—1732no—
138#407—1731no—
139C22×D4—3462no—
140C22×D4—1732no—
141C23×C4—1731no—
142C2×D8—1732no—
143C2×D8—1731no—
144C2×D8C2×D17311yescharacteristic, complemented
145C2×M16—1731no—
146C2×M16C2×D17311yescharacteristic, complemented
147#121—3462no—
148#121—3462no—
149#121—1732no—
150#121—1732no—
151#121—1732no—
152#121C2×D17312yescomplemented
153C2×C4⋊C4—1732no—
154C2×C22⋊C4—3462no—
155#106—1731no—
156#106—1732no—
157#106C2×D17311yescharacteristic, complemented
158#102—1732no—
159#102—1732no—
160#102—1732no—
161#102C2×D17312yescomplemented
162C22⋊C8—1732no—
163C22⋊C8—1732no—
164C24—3462no—
165C24—1731no—
166C2×D4—22no—
167C2×D4—3462no—
168C2×D4—3462no—
169C2×D4—3462no—
170C2×D4—3462no—
171C2×D4—1732no—
172C2×D4—6922no—
173C2×D4—6922no—
174C2×D4—6928no—
175C2×D4C22×D17312yes—
176C22×C4—3462no—
177C22×C4—1732no—
178C22×C4—1731no—
179C22×C4—1732no—
180C22×C4—6921no—
181C22×C4—6921no—
182C22×C4C22×D17311yescharacteristic
183D8—41no—
184D8—6921no—
185D8—6922no—
186M16—41no—
187M16—6921no—
188C2×C8—1732no—
189C2×C8C22×D17311yescharacteristic
190C2×C8C22×D17311yescharacteristic
191C4⋊C4—3462no—
192C4⋊C4—1732no—
193C4⋊C4C22×D17312yes—
194C22⋊C4—22no—
195C22⋊C4—3462no—
196C22⋊C4—3462no—
197C22⋊C4—3462no—
198C23—22no—
199C23—3464no—
200C23—3462no—
201C23—1732no—
202C23—6921no—
203C23—6924no—
204C23—6924no—
205C23—6924no—
206C23D4×D17311yescharacteristic
207D4—22no—
208D4—22no—
209D4—44no—
210D4—3462no—
211D4—3462no—
212D4—6924no—
213D4—6924no—
214D4—6924no—
215D4—6924no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—3462no—
220C2×C4—3462no—
221C2×C4—3462no—
222C2×C4—1732no—
223C2×C4—6922no—
224C2×C4—6924no—
225C2×C4—6921no—
226C2×C4D4×D17311yescharacteristic
227C2×C4C23×D17311yescharacteristic
228C8—21no—
229C8—21no—
230C8—3462no—
231C22—22no—
232C22—21no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—3462no—
237C22—3462no—
238C22—13848no—
239C22—1734no—
240C22—6924no—
241C22—6922no—
242C22—6922no—
243C22—6922no—
244C22C2×D4×D17311yescharacteristic
245C4—21no—
246C4—42no—
247C4—3462no—
248C4—6922no—
249C4#1914972411yescharacteristic
250C4#1914972511yescharacteristic
251C2—41no—
252C2—82no—
253C2—3464no—
254C2—13842no—
255C2#2601204911yescharacteristic
256C2AGL(1,ℤ/8)×D17312yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#469×C173C2
socleC22×C173C22×D4
hypercenter#469D173
hypocenterC173#3192
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. #121×D173C2semidirect
2. #26011811C2semidirect
3. #26011808C2semidirect
4. #26011744C2semidirect
5. #26011742C2semidirect
6. #26011626C2semidirect
7. #26011623C2semidirect
8. #26011454C2semidirect
9. #469×C173C2semidirect
11. C2×D4×D173C22semidirect
13. C2×D1384C22semidirect
14. C2×#14583811C22semidirect
15. C2×#14583810C22semidirect
17. #19149653C22semidirect
21. C4⋊C4×D173C22semidirect
23. #19149611C22semidirect
24. #19149610C22semidirect
25. #19149601C22semidirect
26. #19149600C22semidirect
27. #19149599C22semidirect
28. #19149591C22semidirect
29. #19149591C22semidirect
30. C2×D8×C173C22semidirect
31. C2×M16×C173C22semidirect
32. #121×C173C22semidirect
33. #106×C173C22semidirect
34. #102×C173C22semidirect
128. C173#3192semidirect
135. #469D173semidirect
144. C2×D8C2×D173semidirect
146. C2×M16C2×D173semidirect
152. #121C2×D173semidirect
157. #106C2×D173semidirect
161. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#121×D1732
#260118112
#260118082
#260117442
#260117422
#260116262
#260116232
#260114542
#469×C1732
#3192173