Finite Group Databasebeta

← #36362022

Subgroups — #36362022

Order 22144.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C2×#19149725C211yesmaximal, characteristic
2#26011965C212yesmaximal, complemented
3#26011465C211yesmaximal, characteristic, complemented
4#26011464C212yesmaximal, complemented
5#99×D173C218yesmaximal, complemented
6#410×C173C211yesmaximal, characteristic, complemented
7#19149725—24no—
8#19149725—24no—
9C2×#14583827C412yescomplemented
10C2×#14583826C412yescomplemented
11C2×#14583826C2211yescharacteristic
12C22×D692C412yescomplemented
13C22×D692C2211yescharacteristic
14C2×D4×D173C412yescomplemented
15C2×D4×D173C2214yes—
16C4×D692—24no—
17#19149641—24no—
18#19149638C2212yescomplemented
19C22⋊C4×D173C2218yescomplemented
20#19149594C2214yescomplemented
21#19149593C2218yescomplemented
22C2×#41×C173C2211yescharacteristic
23#117×C173C2212yescomplemented
24#99×C173C2214yescomplemented
25C23×D173C2×C414yescomplemented
26C23×D173C2312yes—
27#14583827—28no—
28#14583826—24no—
29#14583826—24no—
30#14583826—22no—
31#14583826—22no—
32C2×Dic346C2×C411yescharacteristic, complemented
33C2×D692—24no—
34C2×D692—24no—
35C2×D692C2×C412yes—
36C2×D692C2×C412yes—
37C2×D692C2×C414yescomplemented
38C2×D692C2×C411yescharacteristic, complemented
39C2×D692D414yes—
40C2×D692D414yes—
41C2×C22⋊D173C2×C414yescomplemented
42C2×C22⋊D173C2312yes—
43D4×D173—216no—
44D4×D173—28no—
45C2×C4×D173—28no—
46C2×C4×D173C2×C412yescomplemented
47C2×C4×D173C2×C414yescomplemented
48C4⋊Dic173—22no—
49C22⋊Dic173C2314yes—
50#14583803—28no—
51C4×Dic173—22no—
52#41×C173—22no—
53#41×C173—22no—
54C2×Q8×C173C2×C411yescharacteristic, complemented
55C2×D4×C173C2×C411yescharacteristic, complemented
56C2×D4×C173C2312yes—
57C22×C4×C173C2×C412yescomplemented
58C22×C4×C173C2311yescharacteristic
59C4⋊C4×C173—22no—
60C22⋊C4×C173C2314yes—
61C42×C173—22no—
62C22×D173—28no—
63C22×D173—82no—
64C22×D173—84no—
65C22×D173C22×C412yes—
66C22×D173C22×C414yes—
67C22×D173C2×D418yes—
68Dic346—21no—
69Dic346—21no—
70D692—28no—
71D692—21no—
72D692—21no—
73D692C22⋊C418yes—
74D692C22⋊C414yes—
75D692C22⋊C414yes—
76C22⋊D173—24no—
77C22⋊D173—28no—
78C2×Dic173—24no—
79C2×Dic173C22×C412yes—
80C4×D173—28no—
81C4×D173—24no—
82C4×D173—88no—
83C23×C173C22×C412yes—
84C23×C173C2411yescharacteristic
85Q8×C173—22no—
86D4×C173—24no—
87D4×C173—22no—
88C2×C4×C173—22no—
89C2×C4×C173—22no—
90C2×C4×C173—24no—
91C2×C4×C173C22×C412yes—
92C2×C4×C173C22×C411yescharacteristic
93C2×C4×C173C22×C411yescharacteristic
94C2×C4×C173C2×D412yes—
95C2×C4×C173C2×D412yes—
96C22×C173—22no—
97C22×C173—41no—
98C22×C173—42no—
99C22×C173C23×C411yescharacteristic
100C22×C173C22×D412yes—
101C2×D173—22no—
102C2×D173—48no—
103C2×D173—44no—
104C2×D173—42no—
105C2×D173—88no—
106C2×D173C2×C22⋊C418yes—
107C4×C173—22no—
108C4×C173—44no—
109C4×C173C2×C22⋊C411yescharacteristic
110C4×C173C2×C22⋊C411yescharacteristic
111C4×C173C2×C22⋊C412yes—
112Dic173—22no—
113Dic173—44no—
114C2×C173—22no—
115C2×C173—21no—
116C2×C173—42no—
117C2×C173C22×C22⋊C411yescharacteristic
118D173—28no—
119D173—42no—
120C173#288311yescharacteristic, complemented, Sylow
121#2883—1731nomaximal, Sylow
122C2×H(5,𝔽2)—1731no—
123#518—1732no—
124#410—1731no—
125#410—1732no—
126#410D17311yescharacteristic, complemented
127C2×#99—1738no—
128H(5,𝔽2)—3464no—
129H(5,𝔽2)—3464no—
130C2×#41—1732no—
131C2×#41—1732no—
132C2×#41—1731no—
133C2×#41C2×D17311yescharacteristic
134C22×D4—1732no—
135C22×D4—1734no—
136C22×D4—1732no—
137C22×D4—1731no—
138C4×D4—3464no—
139C4×D4—3464no—
140#117—1732no—
141#117C2×D17312yescomplemented
142C2×C22⋊C4—1738no—
143#99—1734no—
144#99—1738no—
145#99C2×D17314yescomplemented
146C24—1734no—
147C24—1732no—
148#41—22no—
149#41—22no—
150#41—3464no—
151#41—3464no—
152#41—3468no—
153#41—3462no—
154#41—3462no—
155C2×Q8—1731no—
156C2×Q8C4×D17311yescharacteristic, complemented
157C2×D4—3464no—
158C2×D4—3464no—
159C2×D4—34616no—
160C2×D4—3468no—
161C2×D4—1734no—
162C2×D4—1732no—
163C2×D4—1734no—
164C2×D4—1731no—
165C2×D4—1732no—
166C2×D4—1732no—
167C2×D4—1734no—
168C2×D4—1734no—
169C2×D4C4×D17311yescharacteristic, complemented
170C2×D4C22×D17312yes—
171C22×C4—3468no—
172C22×C4—1734no—
173C22×C4—1732no—
174C22×C4C4×D17312yescomplemented
175C22×C4C22×D17311yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—3462no—
178C22⋊C4—3468no—
179C22⋊C4—1734no—
180C22⋊C4C22×D17314yes—
181C42—22no—
182C42—3462no—
183C23—3468no—
184C23—13844no—
185C23—13842no—
186C23—1732no—
187C23—1734no—
188C23—1738no—
189C23C2×C4×D17312yes—
190C23C23×D17311yescharacteristic
191Q8—22no—
192Q8—3461no—
193Q8—3461no—
194D4—24no—
195D4—22no—
196D4—3468no—
197D4—3468no—
198D4—3464no—
199D4—3461no—
200D4—3461no—
201D4—1734no—
202D4—1734no—
203D4—1738no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—3468no—
208C2×C4—3464no—
209C2×C4—3464no—
210C2×C4—13848no—
211C2×C4—1732no—
212C2×C4C2×C4×D17312yes—
213C2×C4C2×C4×D17311yescharacteristic
214C2×C4C2×C4×D17311yescharacteristic
215C2×C4D4×D17312yes—
216C2×C4D4×D17312yes—
217C22—22no—
218C22—41no—
219C22—42no—
220C22—3462no—
221C22—13848no—
222C22—1738no—
223C22—6928no—
224C22—6924no—
225C22—6922no—
226C22C22×C4×D17311yescharacteristic
227C22C2×D4×D17312yes—
228C4—22no—
229C4—44no—
230C4—3462no—
231C4—6924no—
232C4C22⋊C4×D17311yescharacteristic
233C4C22⋊C4×D17311yescharacteristic
234C4C22⋊C4×D17312yes—
235C2—22no—
236C2—21no—
237C2—42no—
238C2—3468no—
239C2—6922no—
240C2C2×C22⋊C4×D17311yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D173
commutatorC22×C173C23×C4
FrattiniC23C23×D173
Fitting#410×C173C2
socleC2×C173C22×C22⋊C4
prime radicalC2×#19149725C2
cosocleC23×C173C24
hypercenter#410D173
hypocenterC173#2883
LCSG ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #26011965C2semidirect
3. #26011465C2semidirect
4. #26011464C2semidirect
5. #99×D173C2semidirect
6. #410×C173C2semidirect
9. C2×#14583827C4semidirect
10. C2×#14583826C4semidirect
12. C22×D692C4semidirect
14. C2×D4×D173C4semidirect
18. #19149638C22semidirect
19. C22⋊C4×D173C22semidirect
20. #19149594C22semidirect
21. #19149593C22semidirect
23. #117×C173C22semidirect
24. #99×C173C22semidirect
25. C23×D173C2×C4semidirect
32. C2×Dic346C2×C4semidirect
37. C2×D692C2×C4semidirect
38. C2×D692C2×C4semidirect
41. C2×C22⋊D173C2×C4semidirect
46. C2×C4×D173C2×C4semidirect
47. C2×C4×D173C2×C4semidirect
54. C2×Q8×C173C2×C4semidirect
55. C2×D4×C173C2×C4semidirect
57. C22×C4×C173C2×C4semidirect
120. C173#2883semidirect
126. #410D173semidirect
141. #117C2×D173semidirect
145. #99C2×D173semidirect
156. C2×Q8C4×D173semidirect
169. C2×D4C4×D173semidirect
174. C22×C4C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497252
#260119652
#260114652
#260114642
#99×D1732
#410×C1732
#2883173