Finite Group Databasebeta

← #36361842

Subgroups — #36361842

Order 22144.

1224 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 201 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
1C23×C4×D173C211yesmaximal, characteristic, complemented
2C2×#19149650C212yesmaximal, complemented
3C2×#19149647C212yesmaximal, complemented
4#26011371C212yesmaximal, complemented
5#26011367C214yesmaximal, complemented
6#26011351C211yesmaximal, characteristic, complemented
7#26011348C212yesmaximal, complemented
8C2×#121×C173C211yesmaximal, characteristic, complemented
9C24×D173C2211yescharacteristic
10C22×C22⋊D173C2214yes—
11C23×Dic173C2211yescharacteristic
12C22×C4×D173—24no—
13C22×C4×D173—28no—
14C22×C4×D173C2212yes—
15C22×C4×D173C2212yes—
16#19149650—28no—
17#19149647—28no—
18C2×C4⋊Dic173C2212yescomplemented
19C2×C22⋊Dic173C2212yes—
20C2×C22⋊Dic173C2214yescomplemented
21C2×#14583803C2214yes—
22#19149586C2214yescomplemented
23#19149586C2212yescomplemented
24C22×D4×C173C2212yes—
25C22×D4×C173C2211yescharacteristic, complemented
26C23×C4×C173C2211yescharacteristic
27#121×C173—24no—
28C2×C4⋊C4×C173C2211yescharacteristic, complemented
29C2×C22⋊C4×C173C2212yescomplemented
30C23×D173—24no—
31C23×D173—24no—
32C23×D173—22no—
33C23×D173—22no—
34C23×D173D414yescomplemented
35C23×D173C2312yes—
36C2×C22⋊D173—28no—
37C2×C22⋊D173—28no—
38C2×C22⋊D173—28no—
39C22×Dic173—24no—
40C22×Dic173—22no—
41C22×Dic173C2314yes—
42C22×Dic173C2312yes—
43C2×C4×D173—232no—
44C2×C4×D173—24no—
45C2×C4×D173—24no—
46C2×C4×D173—24no—
47C2×C4×D173—24no—
48C2×C4×D173—24no—
49C2×C4×D173D418yescomplemented
50C4⋊Dic173—24no—
51C22⋊Dic173—28no—
52C22⋊Dic173—24no—
53#14583803—28no—
54C24×C173C2312yes—
55C24×C173C2311yescharacteristic
56C2×D4×C173—24no—
57C2×D4×C173—24no—
58C2×D4×C173—24no—
59C2×D4×C173—24no—
60C2×D4×C173—22no—
61C22×C4×C173—24no—
62C22×C4×C173D414yescomplemented
63C22×C4×C173C2312yes—
64C22×C4×C173C2311yescharacteristic
65C22×C4×C173C2311yescharacteristic
66C4⋊C4×C173—22no—
67C22⋊C4×C173—24no—
68C22×D173—28no—
69C22×D173—216no—
70C22×D173—28no—
71C22×D173—28no—
72C22×D173—22no—
73C22×D173—22no—
74C22×D173—22no—
75C22×D173—22no—
76C22×D173—216no—
77C22×D173C2×D414yes—
78C22×D173C2×D414yes—
79C22×D173#4114yes—
80C22⋊D173—416no—
81C2×Dic173—28no—
82C2×Dic173—22no—
83C2×Dic173—22no—
84C2×Dic173—24no—
85C2×Dic173—24no—
86C2×Dic173—24no—
87C2×Dic173C2×D414yes—
88C4×D173—216no—
89C4×D173—216no—
90C23×C173—22no—
91C23×C173—22no—
92C23×C173—22no—
93C23×C173—24no—
94C23×C173—22no—
95C23×C173—24no—
96C23×C173—22no—
97C23×C173C2×D412yes—
98C23×C173C2×D412yes—
99C23×C173#4112yes—
100C23×C173C2411yescharacteristic
101D4×C173—48no—
102D4×C173—44no—
103C2×C4×C173—22no—
104C2×C4×C173—28no—
105C2×C4×C173—22no—
106C2×C4×C173—22no—
107C2×C4×C173—22no—
108C2×C4×C173C2×D414yes—
109C2×C4×C173C2×D412yes—
110C2×C4×C173C2×D412yes—
111C22×C173—22no—
112C22×C173—24no—
113C22×C173—22no—
114C22×C173—24no—
115C22×C173—42no—
116C22×C173—42no—
117C22×C173—44no—
118C22×C173—42no—
119C22×C173—44no—
120C22×C173#12114yescomplemented
121C22×C173C22×D412yes—
122C22×C173C22×D411yescharacteristic
123C22×C173C22×D411yescharacteristic
124C22×C173C2×#4112yes—
125C22×C173C2×#4111yescharacteristic
126C2×D173—232no—
127C2×D173—24no—
128C2×D173—24no—
129C2×D173—24no—
130C2×D173—24no—
131C2×D173—24no—
132C2×D173—24no—
133C2×D173#12118yescomplemented
134C4×C173—22no—
135C4×C173—42no—
136C4×C173#12014yescomplemented
137Dic173—24no—
138Dic173—44no—
139C2×C173—24no—
140C2×C173—82no—
141C2×C173C2×#12011yescharacteristic
142C2×C173C2×#12112yes—
143C2×C173C2×#12111yescharacteristic
144C2×C173#52511yescharacteristic
145C2×C173#52512yes—
146D173—28no—
147C173#284711yescharacteristic, complemented, Sylow
148#2847—1731nomaximal, Sylow
149C24×C4—1731no—
150C2×#121—1732no—
151C2×#121D17311yescharacteristic, complemented
152C2×#120—1732no—
153#386—1732no—
154#386—1734no—
155#380—1731no—
156#380—1732no—
157C25—1731no—
158C22×D4—1734no—
159C22×D4C2×D17312yes—
160C22×D4C2×D17311yescharacteristic, complemented
161C23×C4—3468no—
162C23×C4—3464no—
163C23×C4—1731no—
164C23×C4—1732no—
165C23×C4—1732no—
166C23×C4C2×D17311yescharacteristic
167#121—24no—
168#121—3468no—
169#120—3468no—
170C2×C4⋊C4—1732no—
171C2×C4⋊C4C2×D17311yescharacteristic, complemented
172C2×C22⋊C4—1732no—
173C2×C22⋊C4—1734no—
174C2×C22⋊C4—1734no—
175C2×C22⋊C4C2×D17312yescomplemented
176#95—1732no—
177#95—1734no—
178C24—3464no—
179C24—3464no—
180C24—3462no—
181C24—3462no—
182C24—1734no—
183C24—1732no—
184C24C22×D17312yes—
185C24C22×D17311yescharacteristic
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—3468no—
192C2×D4—3468no—
193C2×D4—3468no—
194C22×C4—24no—
195C22×C4—34632no—
196C22×C4—3464no—
197C22×C4—3464no—
198C22×C4—3462no—
199C22×C4—3464no—
200C22×C4—3464no—
201C22×C4—3464no—
202C22×C4—3464no—
203C22×C4—1734no—
204C22×C4—1732no—
205C22×C4—1738no—
206C22×C4C22⋊D17314yescomplemented
207C22×C4C22×D17312yes—
208C22×C4C22×D17311yescharacteristic
209C22×C4C22×D17311yescharacteristic
210C4⋊C4—22no—
211C4⋊C4—3464no—
212C22⋊C4—24no—
213C22⋊C4—3468no—
214C22⋊C4—3464no—
215C22⋊C4—3468no—
216C23—22no—
217C23—22no—
218C23—24no—
219C23—22no—
220C23—22no—
221C23—24no—
222C23—22no—
223C23—3468no—
224C23—3468no—
225C23—34616no—
226C23—34616no—
227C23—3468no—
228C23—3462no—
229C23—3462no—
230C23—3462no—
231C23—3462no—
232C23—1734no—
233C23—1734no—
234C23—1734no—
235C23D4×D17312yes—
236C23C2×C22⋊D17312yes—
237C23#1458382512yes—
238C23C23×D17311yescharacteristic
239D4—48no—
240D4—44no—
241D4—69216no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—34616no—
248C2×C4—3468no—
249C2×C4—3462no—
250C2×C4—3462no—
251C2×C4—34616no—
252C2×C4—3464no—
253C2×C4—3464no—
254C2×C4—3464no—
255C2×C4—1734no—
256C2×C4D4×D17314yes—
257C2×C4C2×C22⋊D17312yes—
258C2×C4C2×C22⋊D17312yes—
259C22—22no—
260C22—24no—
261C22—22no—
262C22—24no—
263C22—42no—
264C22—44no—
265C22—42no—
266C22—42no—
267C22—44no—
268C22—34632no—
269C22—3464no—
270C22—3464no—
271C22—3464no—
272C22—3464no—
273C22—3464no—
274C22—3464no—
275C22—1738no—
276C22#1914965014yescomplemented
277C22C2×D4×D17312yes—
278C22C2×D4×D17311yescharacteristic
279C22C22×C22⋊D17311yescharacteristic
280C22#41×D17312yes—
281C22C2×#1458382511yescharacteristic
282C4—22no—
283C4—42no—
284C4—3464no—
285C4—6924no—
286C4#1914964714yescomplemented
287C2—24no—
288C2—82no—
289C2—3468no—
290C2C2×#1914964711yescharacteristic
291C2#121×D17312yes—
292C2C2×#1914965011yescharacteristic
293C2#2601199811yescharacteristic
294C2#2601200112yes—
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
FittingC2×#121×C173C2
hypercenterC2×#121D173
hypocenterC173#2847
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C23×C4×D173C2semidirect
2. C2×#19149650C2semidirect
3. C2×#19149647C2semidirect
4. #26011371C2semidirect
5. #26011367C2semidirect
6. #26011351C2semidirect
7. #26011348C2semidirect
8. C2×#121×C173C2semidirect
18. C2×C4⋊Dic173C22semidirect
20. C2×C22⋊Dic173C22semidirect
22. #19149586C22semidirect
23. #19149586C22semidirect
25. C22×D4×C173C22semidirect
28. C2×C4⋊C4×C173C22semidirect
29. C2×C22⋊C4×C173C22semidirect
34. C23×D173D4semidirect
49. C2×C4×D173D4semidirect
62. C22×C4×C173D4semidirect
120. C22×C173#121semidirect
133. C2×D173#121semidirect
136. C4×C173#120semidirect
147. C173#2847semidirect
151. C2×#121D173semidirect
160. C22×D4C2×D173semidirect
171. C2×C4⋊C4C2×D173semidirect
175. C2×C22⋊C4C2×D173semidirect
206. C22×C4C22⋊D173semidirect
276. C22#19149650semidirect
286. C4#19149647semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1732
C2×#191496502
C2×#191496472
#260113712
#260113672
#260113512
#260113482
C2×#121×C1732
#2847173