Finite Group Databasebeta

← #36364343

Subgroups — #36364343

Order 22144.

420 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 115 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#26012001C211yesmaximal, characteristic
2C2×#19149659C211yesmaximal, characteristic, complemented
3#26011834C212yesmaximal, complemented
4#26011821C211yesmaximal, characteristic, complemented
5#26011733C212yesmaximal, complemented
6#26011719C212yesmaximal, complemented
7#26011654C212yesmaximal, complemented
8#26011649C211yesmaximal, characteristic, complemented
9#26011629C211yesmaximal, characteristic, complemented
10C22⋊C8×D173C211yesmaximal, characteristic, complemented
11#480×C173C211yesmaximal, characteristic, complemented
12C2×#14583825C2211yescharacteristic
13C22×Dic346C2211yescharacteristic
14C22×C4×D173C2211yescharacteristic
15C2×Dic692C2212yescomplemented
16C2×#14583814C2212yescomplemented
17C2×C8×D173C2212yes—
18#19149661—21no—
19#19149659—21no—
20#19149659—21no—
21#19149659C2212yes—
22#19149659C2212yes—
23#19149659C2211yescharacteristic
24#19149658C2211yescharacteristic
25#19149650C2211yescharacteristic
26#19149647—21no—
27D4×Dic173—21no—
28#19149639—21no—
29#19149637C2211yescharacteristic
30C2×C4⋊Dic173—21no—
31C2×#14583803—21no—
32#19149610C2212yescomplemented
33#19149604C2212yescomplemented
34#19149602C2212yescomplemented
35#19149602C2212yescomplemented
36C22⋊#13063436C2211yescharacteristic
37#19149591C2212yes—
38#121×C173C2211yescharacteristic
39C2×C4⋊C4×C173C2211yescharacteristic
40#107×C173C2212yescomplemented
41#102×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic
43C23×D173D411yescharacteristic
44#14583825—42no—
45C2×Dic346—21no—
46C2×Dic346—42no—
47C2×Dic346C2311yescharacteristic
48C2×Dic346C2312yes—
49C2×C22⋊D173—21no—
50C2×C22⋊D173—21no—
51C22×Dic173—21no—
52C22×Dic173—21no—
53C22×Dic173D411yescharacteristic
54C2×C4×D173—42no—
55C2×C4×D173D411yescharacteristic
56C2×C4×D173D411yescharacteristic
57C2×C4×D173C2311yescharacteristic
58C2×C4×D173C2311yescharacteristic
59Dic692—42no—
60#14583814—42no—
61C2×#13063436C2312yes—
62C8×D173—42no—
63C4⋊Dic173—22no—
64C4⋊Dic173—22no—
65C4⋊Dic173—21no—
66C4⋊Dic173C2311yescharacteristic
67#14583805—21no—
68C22⋊Dic173—21no—
69C22⋊Dic173—21no—
70#14583803—22no—
71#14583803—22no—
72#14583803—21no—
73#14583803—21no—
74C4×Dic173—21no—
75C2×D4×C173—21no—
76C2×D4×C173C2311yescharacteristic
77C22×C4×C173—21no—
78C22×C4×C173C2311yescharacteristic
79C2×C8×C173C2312yes—
80C4⋊C4×C173—21no—
81C4⋊C4×C173C2311yescharacteristic
82C4⋊C4×C173C2312yes—
83C22⋊C4×C173—21no—
84C22×D173—22no—
85C22×D173—22no—
86C22×D173—41no—
87C22×D173C2×D411yescharacteristic
88C22×D173C2×D411yescharacteristic
89Dic346—22no—
90Dic346—22no—
91Dic346—22no—
92Dic346—42no—
93C22⋊D173—42no—
94C22⋊D173—42no—
95C2×Dic173—21no—
96C2×Dic173—22no—
97C2×Dic173—21no—
98C2×Dic173—41no—
99C2×Dic173—42no—
100C2×Dic173—42no—
101C2×Dic173C2×D411yescharacteristic
102C2×Dic173C2×D411yescharacteristic
103C4×D173—22no—
104C4×D173—22no—
105C4×D173—41no—
106C4×D173—41no—
107#13063436—22no—
108C23×C173—21no—
109C23×C173C2×D411yescharacteristic
110D4×C173—22no—
111D4×C173—42no—
112C2×C4×C173—22no—
113C2×C4×C173—21no—
114C2×C4×C173—22no—
115C2×C4×C173—42no—
116C2×C4×C173C2×D411yescharacteristic
117C2×C4×C173C2411yescharacteristic
118C8×C173—22no—
119C22×C173—21no—
120C22×C173—21no—
121C22×C173—41no—
122C22×C173—41no—
123C22×C173—41no—
124C22×C173#13512yes—
125C22×C173C22×D411yescharacteristic
126C2×D173—21no—
127C2×D173—21no—
128C2×D173—21no—
129C2×D173—21no—
130C2×D173—21no—
131C2×D173—44no—
132C2×D173#13712yes—
133C4×C173—21no—
134C4×C173—42no—
135C4×C173—41no—
136C4×C173H(5,𝔽2)12yes—
137Dic173—21no—
138Dic173—21no—
139Dic173—41no—
140Dic173—42no—
141Dic173—41no—
142C2×C173—22no—
143C2×C173—81no—
144C2×C173#53411yescharacteristic
145C2×C173C2×#13511yescharacteristic
146C2×C173C2×#13711yescharacteristic
147D173—22no—
148D173—41no—
149C173#319011yescharacteristic, complemented, Sylow
150#3190—1731nomaximal, Sylow
151#525—1731no—
152C2×C22⋊Q8—1731no—
153#485—1732no—
154#483—1731no—
155#480D17311yescharacteristic, complemented
156#467—1732no—
157#465—1732no—
158#452—1732no—
159#451—1731no—
160#448—1731no—
161C2×C22⋊C8—1731no—
162C2×#41—1731no—
163C22×Q8—1731no—
164C23×C4—1731no—
165C2×Q16—1732no—
166C2×SD16—1732no—
167C22×C8—1732no—
168#123—3461no—
169C22⋊Q8—3461no—
170C22⋊Q8—3461no—
171C22⋊Q8—1731no—
172C22⋊Q8—1732no—
173C22⋊Q8—1732no—
174C22⋊Q8—1731no—
175#121—1731no—
176#121C2×D17311yescharacteristic
177#120—3461no—
178C4×D4—3461no—
179C4×D4—3461no—
180#117—1731no—
181C2×C4⋊C4—3461no—
182C2×C4⋊C4C2×D17311yescharacteristic
183C2×C22⋊C4—3461no—
184#107C2×D17312yescomplemented
185#106—1732no—
186#103—1732no—
187#103—1732no—
188#103—1732no—
189#102C2×D17312yescomplemented
190C22⋊C8—1731no—
191C22⋊C8—1732no—
192C22⋊C8C2×D17311yescharacteristic
193C24—1731no—
194#41—6922no—
195C2×Q8—3461no—
196C2×Q8—1732no—
197C2×Q8—1731no—
198C2×Q8—6922no—
199C2×D4—21no—
200C2×D4—3461no—
201C2×D4—3461no—
202C2×D4C22×D17311yescharacteristic
203C22×C4—21no—
204C22×C4—3461no—
205C22×C4—3461no—
206C22×C4—1731no—
207C22×C4—1731no—
208C22×C4—1731no—
209C22×C4—1731no—
210C22×C4—1731no—
211C22×C4—6922no—
212C22×C4C22×D17311yescharacteristic
213Q16—6922no—
214SD16—6922no—
215C2×C8—1732no—
216C2×C8—6922no—
217C2×C8C22×D17312yes—
218C4⋊C4—21no—
219C4⋊C4—3462no—
220C4⋊C4—3462no—
221C4⋊C4—3461no—
222C4⋊C4—3461no—
223C4⋊C4—1731no—
224C4⋊C4C22×D17312yes—
225C4⋊C4C22×D17311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—3462no—
228C22⋊C4—3462no—
229C22⋊C4—3461no—
230C22⋊C4—3461no—
231C22⋊C4—3461no—
232C22⋊C4—3461no—
233C42—3461no—
234C23—21no—
235C23—3462no—
236C23—3462no—
237C23—1731no—
238C23—1731no—
239C23—6921no—
240C23D4×D17311yescharacteristic
241Q8—3462no—
242Q8—3462no—
243Q8—3462no—
244Q8—6922no—
245D4—22no—
246D4—42no—
247D4—6922no—
248D4—6922no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—3462no—
254C2×C4—3462no—
255C2×C4—3461no—
256C2×C4—3461no—
257C2×C4—3462no—
258C2×C4—1731no—
259C2×C4—1731no—
260C2×C4—6921no—
261C2×C4—6921no—
262C2×C4—6921no—
263C2×C4—6922no—
264C2×C4—6922no—
265C2×C4D4×D17311yescharacteristic
266C2×C4C23×D17311yescharacteristic
267C8—22no—
268C8—3462no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—3461no—
275C22—3461no—
276C22—3461no—
277C22—3461no—
278C22—3461no—
279C22—1732no—
280C22—6924no—
281C22#1914969812yes—
282C22C2×D4×D17311yescharacteristic
283C4—21no—
284C4—42no—
285C4—41no—
286C4—3461no—
287C4—3461no—
288C4—6921no—
289C4—6921no—
290C4—6922no—
291C4#1914972512yes—
292C2—22no—
293C2—81no—
294C2—3462no—
295C2—6921no—
296C2#2601205211yescharacteristic
297C2C2×#1914969811yescharacteristic
298C2#137×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#480×C173C2
socleC22×C173C22×D4
prime radical#26012001C2
hypercenter#480D173
hypocenterC173#3190
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×#19149659C2semidirect
3. #26011834C2semidirect
4. #26011821C2semidirect
5. #26011733C2semidirect
6. #26011719C2semidirect
7. #26011654C2semidirect
8. #26011649C2semidirect
9. #26011629C2semidirect
10. C22⋊C8×D173C2semidirect
11. #480×C173C2semidirect
15. C2×Dic692C22semidirect
16. C2×#14583814C22semidirect
32. #19149610C22semidirect
33. #19149604C22semidirect
34. #19149602C22semidirect
35. #19149602C22semidirect
40. #107×C173C22semidirect
41. #102×C173C22semidirect
149. C173#3190semidirect
155. #480D173semidirect
184. #107C2×D173semidirect
189. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120012
C2×#191496592
#260118342
#260118212
#260117332
#260117192
#260116542
#260116492
#260116292
C22⋊C8×D1732
#480×C1732
#3190173