Finite Group Databasebeta

← #36356544

Subgroups — #36356544

Order 22144.

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

normal subgroups only

This group has 928 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×#19149647C211yesmaximal, characteristic
2C2×#19149639C212yesmaximal, complemented
3C2×C22⋊C4×D173C212yesmaximal, complemented
4#26011379C212yesmaximal, complemented
5#26011371C211yesmaximal, characteristic, complemented
6#26011367C212yesmaximal, complemented
7#26011362C212yesmaximal, complemented
8C4×#14583803C212yesmaximal, complemented
9#390×C173C211yesmaximal, characteristic, complemented
10C24×D173C2211yescharacteristic
11C22×C22⋊D173C2212yes—
12C23×Dic173C2211yescharacteristic, complemented
13C22×C4×D173C2212yes—
14C22×C4×D173C2212yes—
15#19149647C418yescomplemented
16#19149639—28no—
17C2×#14583805C2212yescomplemented
18C2×C22⋊Dic173C2212yes—
19C2×C22⋊Dic173C2211yescharacteristic
20C2×C22⋊Dic173C2211yescharacteristic, complemented
21C2×#14583803C2212yes—
22C2×#14583803C2212yes—
23C2×#14583803C2212yes—
24C22⋊C4×D173—28no—
25C2×C4×Dic173C2212yescomplemented
26#19149586C2212yescomplemented
27#19149586C2212yescomplemented
28#19149586C2212yescomplemented
29C22×D4×C173C2211yescharacteristic
30C2×C4⋊C4×C173C2211yescharacteristic, complemented
31C2×C22⋊C4×C173C2212yes—
32C2×C22⋊C4×C173C2212yescomplemented
33C2×C42×C173C2211yescharacteristic, complemented
34C23×D173—24no—
35C23×D173—24no—
36C23×D173—41no—
37C23×D173—41no—
38C23×D173C2×C414yescomplemented
39C23×D173C2312yes—
40C2×C22⋊D173—24no—
41C2×C22⋊D173—24no—
42C2×C22⋊D173C2×C418yescomplemented
43C22×Dic173—24no—
44C22×Dic173—41no—
45C22×Dic173C2312yes—
46C22×Dic173C2312yes—
47C22×Dic173C2312yes—
48C2×C4×D173—24no—
49C2×C4×D173—24no—
50C2×C4×D173—24no—
51C2×C4×D173—24no—
52C2×C4×D173—42no—
53C2×C4×D173—42no—
54#14583805—24no—
55C22⋊Dic173—24no—
56C22⋊Dic173—41no—
57C22⋊Dic173C2×C414yescomplemented
58#14583803—24no—
59#14583803C2×C418yescomplemented
60#14583803D418yescomplemented
61C4×Dic173—24no—
62C24×C173C2311yescharacteristic
63C24×C173C2311yescharacteristic
64C2×D4×C173—41no—
65C2×D4×C173—41no—
66C2×D4×C173C2×C414yescomplemented
67C22×C4×C173C2312yes—
68C22×C4×C173C2311yescharacteristic
69C22×C4×C173C2312yes—
70C4⋊C4×C173—41no—
71C22⋊C4×C173—24no—
72C22⋊C4×C173—24no—
73C42×C173—41no—
74C22×D173—22no—
75C22×D173—22no—
76C22×D173—22no—
77C22×D173—22no—
78C22×D173—216no—
79C22×D173—44no—
80C22×D173—44no—
81C22×D173—48no—
82C22×D173—44no—
83C22×D173C22×C414yes—
84C22×D173C2×D414yes—
85C22×D173#4114yes—
86C22⋊D173—216no—
87C2×Dic173—22no—
88C2×Dic173—22no—
89C2×Dic173—22no—
90C2×Dic173—22no—
91C2×Dic173—22no—
92C2×Dic173—22no—
93C2×Dic173—22no—
94C2×Dic173—44no—
95C2×Dic173C22×C414yes—
96C2×Dic173C2×D414yes—
97C4×D173—48no—
98C4×D173—48no—
99C23×C173—22no—
100C23×C173—22no—
101C23×C173—22no—
102C23×C173—22no—
103C23×C173—21no—
104C23×C173—21no—
105C23×C173—21no—
106C23×C173—21no—
107C23×C173C22×C412yes—
108C23×C173C22×C412yes—
109C23×C173C2411yescharacteristic
110D4×C173—44no—
111C2×C4×C173—22no—
112C2×C4×C173—22no—
113C2×C4×C173—21no—
114C2×C4×C173—21no—
115C2×C4×C173—22no—
116C2×C4×C173—22no—
117C2×C4×C173—22no—
118C2×C4×C173C22×C412yes—
119C2×C4×C173C2×D414yes—
120C22×C173—24no—
121C22×C173—24no—
122C22×C173—22no—
123C22×C173—22no—
124C22×C173—41no—
125C22×C173—41no—
126C22×C173—42no—
127C22×C173—42no—
128C22×C173—41no—
129C22×C173—41no—
130C22×C173C23×C411yescharacteristic
131C22×C173C22×D412yes—
132C22×C173C2×#4112yes—
133C22×C173H(5,𝔽2)11yescharacteristic
134C22×C173H(5,𝔽2)11yescharacteristic
135C2×D173—24no—
136C2×D173—24no—
137C2×D173—24no—
138C2×D173—24no—
139C2×D173—24no—
140C2×D173—24no—
141C2×D173—416no—
142C2×D173C4×D418yescomplemented
143C4×C173—22no—
144C4×C173—24no—
145C4×C173—42no—
146Dic173—24no—
147Dic173—24no—
148Dic173—42no—
149C2×C173—42no—
150C2×C173—42no—
151C2×C173C2×C4×D412yes—
152C2×C173#51811yescharacteristic
153C2×C173D4211yescharacteristic
154C2×C173#54612yes—
155C2×C173#55111yescharacteristic
156D173—28no—
157C173#235811yescharacteristic, complemented, Sylow
158#2358—1731nomaximal, Sylow
159C2×#120—1731no—
160C2×C4×D4—1732no—
161C22×C22⋊C4—1732no—
162#390D17311yescharacteristic, complemented
163#388—1732no—
164#386—1731no—
165#386—1732no—
166#385—1732no—
167C4×C22⋊C4—1732no—
168C25—1731no—
169C22×D4—1732no—
170C22×D4C2×D17311yescharacteristic
171C23×C4—1731no—
172C23×C4—1732no—
173C23×C4—1732no—
174#120—1738no—
175C4×D4—3468no—
176C2×C4⋊C4—1732no—
177C2×C4⋊C4C2×D17311yescharacteristic, complemented
178C2×C22⋊C4—3468no—
179C2×C22⋊C4—1732no—
180C2×C22⋊C4—1732no—
181C2×C22⋊C4—1731no—
182C2×C22⋊C4—1732no—
183C2×C22⋊C4—1732no—
184C2×C22⋊C4—1731no—
185C2×C22⋊C4C2×D17312yes—
186C2×C22⋊C4C2×D17312yescomplemented
187C2×C42—1732no—
188C2×C42C2×D17311yescharacteristic, complemented
189#95—1732no—
190#95—1732no—
191#95—1732no—
192C24—3464no—
193C24—3464no—
194C24—1732no—
195C24—1734no—
196C24—6921no—
197C24—6921no—
198C24C22×D17311yescharacteristic
199C24C22×D17311yescharacteristic
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—3464no—
203C2×D4—3464no—
204C2×D4—1738no—
205C2×D4C4×D17314yescomplemented
206C22×C4—3464no—
207C22×C4—3464no—
208C22×C4—3464no—
209C22×C4—3464no—
210C22×C4—3464no—
211C22×C4—1732no—
212C22×C4—1732no—
213C22×C4—1732no—
214C22×C4—6922no—
215C22×C4—6922no—
216C22×C4—6921no—
217C22×C4C22×D17312yes—
218C22×C4C22×D17311yescharacteristic
219C22×C4C22×D17312yes—
220C4⋊C4—41no—
221C4⋊C4—3464no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—3464no—
225C22⋊C4—3464no—
226C22⋊C4—1734no—
227C22⋊C4—1738no—
228C22⋊C4—1738no—
229C22⋊C4—6921no—
230C42—41no—
231C42—3464no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—34616no—
241C23—3462no—
242C23—3462no—
243C23—3462no—
244C23—3462no—
245C23—1734no—
246C23—1734no—
247C23—1734no—
248C23—6924no—
249C23—6924no—
250C23—6924no—
251C23—6928no—
252C23C2×C4×D17312yes—
253C23C2×C4×D17312yes—
254C23C23×D17311yescharacteristic
255D4—44no—
256D4—34616no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—3462no—
265C2×C4—3462no—
266C2×C4—3462no—
267C2×C4—3462no—
268C2×C4—3462no—
269C2×C4—3462no—
270C2×C4—3462no—
271C2×C4—1734no—
272C2×C4—1734no—
273C2×C4—6928no—
274C2×C4—6928no—
275C2×C4—6924no—
276C2×C4C2×C4×D17312yes—
277C2×C4D4×D17314yes—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—22no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—42no—
287C22—42no—
288C22—3464no—
289C22—3464no—
290C22—3464no—
291C22—3464no—
292C22—3464no—
293C22—3464no—
294C22—1738no—
295C22—69216no—
296C22C22×C4×D17311yescharacteristic
297C22C2×D4×D17312yes—
298C22#41×D17312yes—
299C22#1914972411yescharacteristic
300C22#1914972411yescharacteristic
301C4—22no—
302C4—24no—
303C4—42no—
304C4—3464no—
305C4—3464no—
306C4—6922no—
307C2—42no—
308C2—42no—
309C2—3468no—
310C2C4×D4×D17312yes—
311C2#2601196411yescharacteristic
312C2#2601212611yescharacteristic
313C2#2601213012yes—
314C2#2601217111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#390×C173C2
socle, cosocleC23×C173C24
prime radicalC2×#19149647C2
hypercenter#390D173
hypocenterC173#2358
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#19149639C2semidirect
3. C2×C22⋊C4×D173C2semidirect
4. #26011379C2semidirect
5. #26011371C2semidirect
6. #26011367C2semidirect
7. #26011362C2semidirect
8. C4×#14583803C2semidirect
9. #390×C173C2semidirect
12. C23×Dic173C22semidirect
15. #19149647C4semidirect
17. C2×#14583805C22semidirect
20. C2×C22⋊Dic173C22semidirect
25. C2×C4×Dic173C22semidirect
26. #19149586C22semidirect
27. #19149586C22semidirect
28. #19149586C22semidirect
30. C2×C4⋊C4×C173C22semidirect
32. C2×C22⋊C4×C173C22semidirect
33. C2×C42×C173C22semidirect
38. C23×D173C2×C4semidirect
42. C2×C22⋊D173C2×C4semidirect
57. C22⋊Dic173C2×C4semidirect
59. #14583803C2×C4semidirect
60. #14583803D4semidirect
66. C2×D4×C173C2×C4semidirect
142. C2×D173C4×D4semidirect
157. C173#2358semidirect
162. #390D173semidirect
177. C2×C4⋊C4C2×D173semidirect
186. C2×C22⋊C4C2×D173semidirect
188. C2×C42C2×D173semidirect
205. C2×D4C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496472
C2×#191496392
C2×C22⋊C4×D1732
#260113792
#260113712
#260113672
#260113622
C4×#145838032
#390×C1732
#2358173