Finite Group Databasebeta

← #36363486

Subgroups — #36363486

Order 22144.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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#26012002C211yesmaximal, characteristic
2#26011826C212yesmaximal, complemented
3#26011821C212yesmaximal, complemented
4#26011809C212yesmaximal, complemented
5#26011804C212yesmaximal, complemented
6#26011531C212yesmaximal, complemented
7#26011490C212yesmaximal, complemented
8#26011455C211yesmaximal, characteristic, complemented
9C22×C4⋊C4×C173C211yesmaximal, characteristic, complemented
10C2×#14583826C2211yescharacteristic
11C2×#14583810C2212yes—
12#19149662—21no—
13#19149658C2211yescharacteristic
14#19149658C2211yescharacteristic
15#19149653C2211yescharacteristic
16#19149653C2211yescharacteristic
17#19149648—21no—
18C4×C22⋊D173—21no—
19C4×C22⋊D173—21no—
20#19149638C2211yescharacteristic
21#19149612C2214yescomplemented
22#19149610C2214yescomplemented
23#19149602C2214yescomplemented
24#19149599C2214yescomplemented
25C22⋊#13063436C2212yes—
26C22⋊#13063436C2212yes—
27C23×C4×C173—21no—
28C23×C4×C173C2211yescharacteristic
29C2×C4⋊C4×C173—21no—
30C2×C4⋊C4×C173—21no—
31C2×C4⋊C4×C173—21no—
32C2×C4⋊C4×C173C2212yes—
33C2×C4⋊C4×C173C2212yes—
34C2×C4⋊C4×C173C2212yes—
35#14583826—24no—
36C2×Dic346C2311yescharacteristic
37C2×D692C2311yescharacteristic
38C2×C22⋊D173—21no—
39C2×C22⋊D173—21no—
40C2×C22⋊D173—21no—
41C2×C4×D173—21no—
42#14583810—24no—
43C2×#13063436C2314yes—
44C4⋊Dic173C2311yescharacteristic
45C4⋊Dic173C2311yescharacteristic
46#14583805—21no—
47#14583805—21no—
48C22⋊Dic173—21no—
49C22⋊Dic173—21no—
50C22⋊Dic173—21no—
51#14583803—21no—
52#14583803—21no—
53C4×Dic173—21no—
54C24×C173D411yescharacteristic
55C22×C4×C173—24no—
56C22×C4×C173—22no—
57C22×C4×C173—22no—
58C22×C4×C173—22no—
59C22×C4×C173—44no—
60C22×C4×C173D411yescharacteristic
61C22×C4×C173D411yescharacteristic
62C22×C4×C173D411yescharacteristic
63C22×C4×C173C2311yescharacteristic
64C22×C4×C173C2311yescharacteristic
65C22×C4×C173C2311yescharacteristic
66C4⋊C4×C173—22no—
67C4⋊C4×C173—22no—
68C4⋊C4×C173—22no—
69C4⋊C4×C173C2314yes—
70C22×D173—21no—
71Dic346—22no—
72D692—22no—
73C22⋊D173—42no—
74C22⋊D173—42no—
75C22⋊D173—42no—
76C2×Dic173—21no—
77C2×Dic173—21no—
78C2×Dic173—21no—
79C4×D173—42no—
80#13063436—24no—
81C23×C173—22no—
82C23×C173—22no—
83C23×C173—22no—
84C23×C173C2×D411yescharacteristic
85C23×C173C2×D411yescharacteristic
86C23×C173C2×D411yescharacteristic
87C2×C4×C173—22no—
88C2×C4×C173—22no—
89C2×C4×C173—22no—
90C2×C4×C173—22no—
91C2×C4×C173—22no—
92C2×C4×C173—24no—
93C2×C4×C173—44no—
94C2×C4×C173—44no—
95C2×C4×C173—44no—
96C2×C4×C173C2×D411yescharacteristic
97C2×C4×C173C2×D411yescharacteristic
98C2×C4×C173C2×D411yescharacteristic
99C2×C4×C173#4114yes—
100C2×C4×C173C2411yescharacteristic
101C22×C173—28no—
102C22×C173—21no—
103C22×C173—21no—
104C22×C173—21no—
105C22×C173—21no—
106C22×C173—21no—
107C22×C173—21no—
108C22×C173#12312yes—
109C22×C173AGL(1,ℤ/8)12yes—
110C22×C173#13712yes—
111C22×C173C22×D411yescharacteristic
112C2×D173—41no—
113C2×D173—41no—
114C2×D173—41no—
115C4×C173—21no—
116C4×C173—21no—
117C4×C173—44no—
118C4×C173#12312yes—
119C4×C173C2×#4112yes—
120Dic173—41no—
121Dic173—41no—
122Dic173—41no—
123C2×C173—22no—
124C2×C173—22no—
125C2×C173—22no—
126C2×C173C2×#12311yescharacteristic
127C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C173C2×#13711yescharacteristic
129D173—81no—
130C173#309211yescharacteristic, complemented, Sylow
131#3092—1731nomaximal, Sylow
132#525—1731no—
133C22×C4⋊C4D17311yescharacteristic, complemented
134#484—1732no—
135#483—1732no—
136#481—1732no—
137#480—1732no—
138#428—1732no—
139#417—1732no—
140#407—1731no—
141C2×#41—1731no—
142C23×C4—21no—
143C23×C4C2×D17311yescharacteristic
144C2×M16—1732no—
145#123—3461no—
146C22⋊Q8—1731no—
147C22⋊Q8—1731no—
148#121—1731no—
149#121—1731no—
150#120—3461no—
151C4×D4—3461no—
152C4×D4—3461no—
153#117—1731no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2×D17312yes—
158C2×C4⋊C4C2×D17312yes—
159C2×C4⋊C4C2×D17312yes—
160#107—1734no—
161#106—1734no—
162#103—1734no—
163#102—1734no—
164C22⋊C8—1732no—
165C22⋊C8—1732no—
166C24C22⋊D17311yescharacteristic
167#41—3464no—
168C2×Q8—1731no—
169C2×D4—3461no—
170C2×D4—3461no—
171C2×D4—3461no—
172C2×D4—1731no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—44no—
178C22×C4—3461no—
179C22×C4C22⋊D17311yescharacteristic
180C22×C4C22⋊D17311yescharacteristic
181C22×C4C22⋊D17311yescharacteristic
182C22×C4C22×D17311yescharacteristic
183C22×C4C22×D17311yescharacteristic
184C22×C4C22×D17311yescharacteristic
185M16—3464no—
186C2×C8—1734no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—3461no—
191C4⋊C4—3461no—
192C4⋊C4—1731no—
193C4⋊C4—1731no—
194C4⋊C4C22×D17314yes—
195C22⋊C4—3461no—
196C22⋊C4—3461no—
197C22⋊C4—3461no—
198C22⋊C4—3461no—
199C22⋊C4—3461no—
200C42—3461no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—3461no—
205C23C2×C22⋊D17311yescharacteristic
206C23C2×C22⋊D17311yescharacteristic
207C23C2×C22⋊D17311yescharacteristic
208Q8—3462no—
209D4—3462no—
210D4—6922no—
211D4—6922no—
212D4—6922no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—3461no—
223C2×C4—3461no—
224C2×C4—3461no—
225C2×C4—6922no—
226C2×C4C2×C22⋊D17311yescharacteristic
227C2×C4C2×C22⋊D17311yescharacteristic
228C2×C4C2×C22⋊D17311yescharacteristic
229C2×C4#1458382614yes—
230C2×C4C23×D17311yescharacteristic
231C8—3464no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—6921no—
240C22—6921no—
241C22—6921no—
242C22#1914966212yes—
243C22#1914970512yes—
244C22#1914971012yes—
245C22C22×C22⋊D17311yescharacteristic
246C4—21no—
247C4—21no—
248C4—44no—
249C4—6921no—
250C4—6921no—
251C4—6921no—
252C4#1914966212yes—
253C4C2×#1458382612yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—13841no—
258C2C2×#1914966211yescharacteristic
259C2C2×#1914970511yescharacteristic
260C2C2×#1914971011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC22×C4⋊C4×C173C2
socleC22×C173C22×D4
prime radical#26012002C2
hypercenterC22×C4⋊C4D173
hypocenterC173#3092
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #26011826C2semidirect
3. #26011821C2semidirect
4. #26011809C2semidirect
5. #26011804C2semidirect
6. #26011531C2semidirect
7. #26011490C2semidirect
8. #26011455C2semidirect
9. C22×C4⋊C4×C173C2semidirect
21. #19149612C22semidirect
22. #19149610C22semidirect
23. #19149602C22semidirect
24. #19149599C22semidirect
130. C173#3092semidirect
133. C22×C4⋊C4D173semidirect

Maximal subgroups

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

subgroupindex
#260120022
#260118262
#260118212
#260118092
#260118042
#260115312
#260114902
#260114552
C22×C4⋊C4×C1732
#3092173