Finite Group Databasebeta

← #36362954

Subgroups — #36362954

Order 22144.

766 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 145 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#26012327C211yesmaximal, characteristic, complemented
2#26012002C212yesmaximal, complemented
3#26011680C214yesmaximal, complemented
4#26011677C214yesmaximal, complemented
5#26011465C211yesmaximal, characteristic, complemented
6#26011464C212yesmaximal, complemented
7#525×C173C211yesmaximal, characteristic, complemented
8#19149727—21no—
9#19149726—22no—
10#19149725—22no—
11#19149724—21no—
12#19149724C2214yes—
13C2×#14583826—22no—
14C2×#14583826C2212yes—
15#41×D173—24no—
16#19149662C2214yescomplemented
17#19149658—22no—
18#19149653—22no—
19#19149648C2214yescomplemented
20C4×C22⋊D173—24no—
21#19149638C2212yescomplemented
22#19149594C2214yescomplemented
23#19149593C2218yescomplemented
24C2×#41×C173C2211yescharacteristic
25C23×C4×C173C2211yescharacteristic, complemented
26#123×C173C2212yescomplemented
27C22⋊Q8×C173—21no—
28#121×C173—21no—
29#120×C173C2212yescomplemented
30C4×D4×C173—22no—
31#117×C173C2211yescharacteristic, complemented
32#14583827—24no—
33#14583826—24no—
34#14583826—24no—
35#14583826—22no—
36#14583826—22no—
37#14583826—24no—
38#14583826—22no—
39#14583826—22no—
40#14583825—24no—
41#14583825—28no—
42C2×Dic346—22no—
43C2×Dic346D412yescomplemented
44Q8×D173—24no—
45C2×D692—22no—
46C2×D692D412yescomplemented
47C2×C22⋊D173—24no—
48C2×C22⋊D173—24no—
49C2×C22⋊D173C2314yes—
50D4×D173—24no—
51D4×D173—28no—
52C2×C4×D173—24no—
53C2×C4×D173D414yescomplemented
54C4⋊Dic173—22no—
55#14583805—24no—
56C22⋊Dic173—24no—
57C22⋊Dic173C2314yes—
58#14583803—24no—
59C4×Dic173—22no—
60C24×C173C2311yescharacteristic
61#41×C173—22no—
62#41×C173—22no—
63C2×Q8×C173D411yescharacteristic, complemented
64C2×D4×C173—22no—
65C2×D4×C173D411yescharacteristic, complemented
66C2×D4×C173C2312yes—
67C22×C4×C173—22no—
68C22×C4×C173—22no—
69C22×C4×C173—21no—
70C22×C4×C173—21no—
71C22×C4×C173D412yescomplemented
72C22×C4×C173C2311yescharacteristic
73C22×C4×C173C2311yescharacteristic
74C4⋊C4×C173—21no—
75C4⋊C4×C173—22no—
76C22⋊C4×C173—22no—
77C22⋊C4×C173—22no—
78C22⋊C4×C173C2312yes—
79C42×C173—21no—
80C22×D173—24no—
81C22×D173C2×D414yes—
82Dic346—24no—
83Dic346—22no—
84Dic346—22no—
85D692—24no—
86D692—22no—
87D692—22no—
88C22⋊D173—28no—
89C22⋊D173—24no—
90C22⋊D173—24no—
91C22⋊D173—48no—
92C2×Dic173—24no—
93C2×Dic173—24no—
94C2×Dic173C2×D414yes—
95C4×D173—28no—
96C4×D173—24no—
97C4×D173—24no—
98C23×C173—22no—
99C23×C173—21no—
100C23×C173—42no—
101C23×C173C2×D412yes—
102C23×C173C2411yescharacteristic
103Q8×C173—22no—
104D4×C173—24no—
105D4×C173—22no—
106D4×C173—44no—
107C2×C4×C173—22no—
108C2×C4×C173—22no—
109C2×C4×C173—22no—
110C2×C4×C173—21no—
111C2×C4×C173—22no—
112C2×C4×C173—41no—
113C2×C4×C173—42no—
114C2×C4×C173—41no—
115C2×C4×C173C2×D412yes—
116C2×C4×C173C2×D412yes—
117C2×C4×C173C2×D412yes—
118C2×C4×C173C2×D411yescharacteristic
119C2×C4×C173C2×D411yescharacteristic
120C22×C173—22no—
121C22×C173—24no—
122C22×C173—22no—
123C22×C173—41no—
124C22×C173—42no—
125C22×C173—41no—
126C22×C173—42no—
127C22×C173—41no—
128C22×C173C22×D412yes—
129C22×C173C22×D411yescharacteristic
130C2×D173—24no—
131C2×D173—44no—
132C2×D173—84no—
133C4×C173—22no—
134C4×C173—42no—
135C4×C173—41no—
136C4×C173#12011yescharacteristic
137C4×C173#12012yes—
138C4×C173#12011yescharacteristic
139Dic173—24no—
140Dic173—44no—
141C2×C173—22no—
142C2×C173—21no—
143C2×C173—42no—
144C2×C173—42no—
145C2×C173C2×#12011yescharacteristic
146D173—44no—
147C173#302511yescharacteristic, complemented, Sylow
148#3025—1731nomaximal, Sylow
149#585—1731no—
150#525—1732no—
151#525D17311yescharacteristic, complemented
152#458—1734no—
153B(4,𝔽2)—1734no—
154#410—1731no—
155#410—1732no—
1562−1+4—3462no—
1572−1+4—3461no—
158H(5,𝔽2)—3462no—
159H(5,𝔽2)—3461no—
160H(5,𝔽2)—1734no—
161C2×#41—3464no—
162C2×#41—3462no—
163C2×#41—1732no—
164C2×#41C2×D17311yescharacteristic
165C23×C4C2×D17311yescharacteristic, complemented
166#123—1734no—
167#123C2×D17312yescomplemented
168C22⋊Q8—21no—
169C22⋊Q8—3462no—
170#121—21no—
171#121—3462no—
172#120—1734no—
173#120C2×D17312yescomplemented
174C4×D4—22no—
175C4×D4—3464no—
176#117—1732no—
177#117C2×D17311yescharacteristic, complemented
178#99—1734no—
179#99—1738no—
180C24C22×D17311yescharacteristic
181#41—22no—
182#41—22no—
183#41—3464no—
184#41—3464no—
185#41—3464no—
186#41—3464no—
187#41—3468no—
188#41—3462no—
189#41—3462no—
190#41—3462no—
191#41—3464no—
192#41—3462no—
193C2×Q8—3462no—
194C2×Q8—3464no—
195C2×Q8—1732no—
196C2×Q8C22⋊D17311yescharacteristic, complemented
197C2×D4—22no—
198C2×D4—3462no—
199C2×D4—3464no—
200C2×D4—3468no—
201C2×D4—3464no—
202C2×D4—3464no—
203C2×D4—1734no—
204C2×D4—1732no—
205C2×D4C22⋊D17311yescharacteristic, complemented
206C2×D4C22×D17312yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—3464no—
212C22×C4—1734no—
213C22×C4C22⋊D17312yescomplemented
214C22×C4C22×D17311yescharacteristic
215C22×C4C22×D17311yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—3464no—
219C4⋊C4—3462no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—3464no—
223C22⋊C4—3464no—
224C22⋊C4—1734no—
225C22⋊C4C22×D17312yes—
226C42—21no—
227C42—3462no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—3464no—
232C23—1734no—
233C23C2×C22⋊D17312yes—
234C23C23×D17311yescharacteristic
235Q8—22no—
236Q8—3464no—
237Q8—3462no—
238Q8—3462no—
239D4—24no—
240D4—22no—
241D4—44no—
242D4—3464no—
243D4—3468no—
244D4—3464no—
245D4—3464no—
246D4—3462no—
247D4—3462no—
248D4—6928no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—3468no—
258C2×C4—3464no—
259C2×C4—3464no—
260C2×C4—3464no—
261C2×C4—3464no—
262C2×C4—1734no—
263C2×C4D4×D17312yes—
264C2×C4D4×D17312yes—
265C2×C4C2×C22⋊D17312yes—
266C2×C4C2×C22⋊D17311yescharacteristic
267C2×C4C2×C22⋊D17311yescharacteristic
268C22—22no—
269C22—24no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—3464no—
277C22—13844no—
278C22—6924no—
279C22C2×D4×D17312yes—
280C22C22×C22⋊D17311yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—3464no—
285C4—6924no—
286C4#1914964711yescharacteristic
287C4#1914964712yes—
288C4#1914964711yescharacteristic
289C2—21no—
290C2—22no—
291C2—42no—
292C2—42no—
293C2—6924no—
294C2C2×#1914964711yescharacteristic
subgroupG/H
centerC4#19149647
commutator, cosocleC23×C173C24
FrattiniC23C23×D173
Fitting#525×C173C2
socleC2×C173C2×#120
hypercenter#525D173
hypocenterC173#3025
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #26012327C2semidirect
2. #26012002C2semidirect
3. #26011680C2semidirect
4. #26011677C2semidirect
5. #26011465C2semidirect
6. #26011464C2semidirect
7. #525×C173C2semidirect
16. #19149662C22semidirect
19. #19149648C22semidirect
21. #19149638C22semidirect
22. #19149594C22semidirect
23. #19149593C22semidirect
25. C23×C4×C173C22semidirect
26. #123×C173C22semidirect
29. #120×C173C22semidirect
31. #117×C173C22semidirect
43. C2×Dic346D4semidirect
46. C2×D692D4semidirect
53. C2×C4×D173D4semidirect
63. C2×Q8×C173D4semidirect
65. C2×D4×C173D4semidirect
71. C22×C4×C173D4semidirect
147. C173#3025semidirect
151. #525D173semidirect
165. C23×C4C2×D173semidirect
167. #123C2×D173semidirect
173. #120C2×D173semidirect
177. #117C2×D173semidirect
196. C2×Q8C22⋊D173semidirect
205. C2×D4C22⋊D173semidirect
213. C22×C4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260123272
#260120022
#260116802
#260116772
#260114652
#260114642
#525×C1732
#3025173