Finite Group Databasebeta

← #36362942

Subgroups — #36362942

Order 22144.

704 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 161 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×#19149709C212yesmaximal, complemented
2#26012062C211yesmaximal, characteristic, complemented
3#26011674C214yesmaximal, complemented
4#26011669C214yesmaximal, complemented
5#26011508C212yesmaximal, complemented
6#26011475C211yesmaximal, characteristic, complemented
7#585×C173C211yesmaximal, characteristic
8C22×Dic346C2211yescharacteristic, complemented
9#19149709—24no—
10#19149709C2218yescomplemented
11C2×#14583816—22no—
12C2×#14583814—22no—
13C2×#14583810C2212yescomplemented
14#19149677C2212yescomplemented
15#19149666C2212yescomplemented
16#19149660—22no—
17#19149658—22no—
18#19149638C2211yescharacteristic, complemented
19#19149606C2218yescomplemented
20#19149598C2212yescomplemented
21#19149596C2212yescomplemented
222−1+4×C173—22no—
232−1+4×C173C2212yes—
24H(5,𝔽2)×C173—22no—
25H(5,𝔽2)×C173—22no—
26H(5,𝔽2)×C173C2212yes—
27C2×#41×C173—24no—
28C2×#41×C173—22no—
29C2×#41×C173C2212yes—
30C2×#41×C173C2211yescharacteristic
31C2×Dic346—21no—
32C2×Dic346—22no—
33C2×Dic346—22no—
34C2×Dic346C2312yes—
35C22×Dic173—21no—
36#14583816—28no—
37#14583814—28no—
38#14583810—22no—
39#14583810C2314yes—
40C2×#13063436—22no—
41C4⋊Dic173—21no—
42#14583805—24no—
43C22⋊Dic173—21no—
44C22⋊Dic173—24no—
45C4×Dic173C2312yes—
46#41×C173—24no—
47#41×C173—28no—
48#41×C173—24no—
49#41×C173—24no—
50#41×C173—24no—
51#41×C173—22no—
52#41×C173—22no—
53#41×C173—24no—
54#41×C173—22no—
55#41×C173—22no—
56#41×C173D414yes—
57#41×C173C2314yes—
58C2×Q8×C173—22no—
59C2×Q8×C173—24no—
60C2×Q8×C173D412yes—
61C2×Q8×C173C2311yescharacteristic
62C2×D4×C173—28no—
63C2×D4×C173—22no—
64C2×D4×C173—24no—
65C2×D4×C173—22no—
66C2×D4×C173—24no—
67C2×D4×C173D412yes—
68C2×D4×C173D412yes—
69C2×D4×C173C2311yescharacteristic
70C22×C4×C173—24no—
71C22×C4×C173—22no—
72C22×C4×C173D412yes—
73C22×C4×C173C2311yescharacteristic
74Dic346—24no—
75Dic346—41no—
76Dic346—41no—
77C2×Dic173—22no—
78C2×Dic173—22no—
79C2×Dic173—42no—
80#13063436—24no—
81C23×C173—24no—
82C23×C173—22no—
83C23×C173C2×D412yes—
84C23×C173C2×D411yescharacteristic
85Q8×C173—24no—
86Q8×C173—22no—
87Q8×C173—22no—
88Q8×C173C2×D414yes—
89D4×C173—28no—
90D4×C173—24no—
91D4×C173—24no—
92D4×C173—24no—
93D4×C173—24no—
94D4×C173—22no—
95D4×C173—22no—
96D4×C173C2×D414yes—
97C2×C4×C173—24no—
98C2×C4×C173—28no—
99C2×C4×C173—22no—
100C2×C4×C173—24no—
101C2×C4×C173—22no—
102C2×C4×C173—22no—
103C2×C4×C173—24no—
104C2×C4×C173C2×D412yes—
105C2×C4×C173C2×D411yescharacteristic
106C2×C4×C173C2×D412yes—
107C2×C4×C173C2×D412yes—
108C2×C4×C173C2411yescharacteristic
109C22×C173—22no—
110C22×C173—24no—
111C22×C173—41no—
112C22×C173—42no—
113C22×C173—82no—
114C22×C173—84no—
115C22×C173#12012yes—
116C22×C173C22×D411yescharacteristic
117C4×C173—24no—
118C4×C173—22no—
119C4×C173#12011yescharacteristic
120C4×C173#12011yescharacteristic
121C4×C173C22×D412yes—
122Dic173—42no—
123Dic173—42no—
124C2×C173—21no—
125C2×C173—22no—
126C2×C173—42no—
127C2×C173—44no—
128C2×C173C2×#12011yescharacteristic
129C173#302211yescharacteristic, complemented, Sylow
130#3022—1731nomaximal, Sylow
131#585D17311yescharacteristic
132C2×#137—1732no—
133#536—1731no—
134#456—1734no—
135#455—1734no—
136#421—1732no—
137#413—1731no—
1382−1+4—22no—
1392−1+4C2×D17312yes—
140H(5,𝔽2)—22no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D17312yes—
143C2×#41—24no—
144C2×#41—22no—
145C2×#41C2×D17312yes—
146C2×#41C2×D17311yescharacteristic
147C22×Q8—1731no—
148#137—3464no—
149#137—1738no—
150C2×Q16—3462no—
151C2×SD16—3462no—
152C2×M16—1732no—
153C4⋊Q8—1732no—
154#124—1732no—
155#123—3462no—
156C22⋊Q8—3462no—
157#117—1731no—
158#104—1738no—
159#101—1732no—
160#100—1732no—
161#41—24no—
162#41—28no—
163#41—24no—
164#41—24no—
165#41—24no—
166#41—22no—
167#41—22no—
168#41—24no—
169#41—22no—
170#41—22no—
171#41C22⋊D17314yes—
172#41C22×D17314yes—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—3461no—
176C2×Q8—3462no—
177C2×Q8—3462no—
178C2×Q8—1732no—
179C2×Q8C22⋊D17312yes—
180C2×Q8C22×D17311yescharacteristic
181C2×D4—28no—
182C2×D4—22no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4C22⋊D17312yes—
187C2×D4C22⋊D17312yes—
188C2×D4C22×D17311yescharacteristic
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—3461no—
192C22×C4C22⋊D17312yes—
193C22×C4C22×D17311yescharacteristic
194Q16—3468no—
195SD16—3468no—
196M16—3462no—
197M16—1734no—
198C2×C8—3462no—
199C4⋊C4—3461no—
200C4⋊C4—3464no—
201C22⋊C4—3461no—
202C22⋊C4—3464no—
203C42—1732no—
204C23—24no—
205C23—22no—
206C23C2×C22⋊D17312yes—
207C23C2×C22⋊D17311yescharacteristic
208Q8—24no—
209Q8—22no—
210Q8—22no—
211Q8—3464no—
212Q8—6921no—
213Q8—6921no—
214Q8C2×C22⋊D17314yes—
215D4—28no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4C2×C22⋊D17314yes—
223C2×C4—24no—
224C2×C4—28no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—3462no—
231C2×C4—3462no—
232C2×C4—6922no—
233C2×C4C2×C22⋊D17312yes—
234C2×C4C2×C22⋊D17311yescharacteristic
235C2×C4C2×C22⋊D17312yes—
236C2×C4C2×C22⋊D17312yes—
237C2×C4C23×D17311yescharacteristic
238C8—3464no—
239C22—22no—
240C22—24no—
241C22—41no—
242C22—42no—
243C22—82no—
244C22—84no—
245C22#1914964812yes—
246C22C22×C22⋊D17311yescharacteristic
247C4—24no—
248C4—22no—
249C4—6922no—
250C4—6922no—
251C4#1914964811yescharacteristic
252C4#1914964811yescharacteristic
253C4C22×C22⋊D17312yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—44no—
258C2C2×#1914964811yescharacteristic
subgroupG/H
centerC2C2×#19149648
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting, prime radical#585×C173C2
socleC2×C173C2×#120
hypercenter#585D173
hypocenterC173#3022
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#19149709C2semidirect
2. #26012062C2semidirect
3. #26011674C2semidirect
4. #26011669C2semidirect
5. #26011508C2semidirect
6. #26011475C2semidirect
8. C22×Dic346C22semidirect
10. #19149709C22semidirect
13. C2×#14583810C22semidirect
14. #19149677C22semidirect
15. #19149666C22semidirect
18. #19149638C22semidirect
19. #19149606C22semidirect
20. #19149598C22semidirect
21. #19149596C22semidirect
129. C173#3022semidirect

Maximal subgroups

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

subgroupindex
C2×#191497092
#260120622
#260116742
#260116692
#260115082
#260114752
#585×C1732
#3022173