Finite Group Databasebeta

← #36364398

Subgroups — #36364398

Order 22144.

414 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 113 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#26012004C211yesmaximal, characteristic
2C2×#19149659C211yesmaximal, characteristic, complemented
3#26011834C212yesmaximal, complemented
4#26011809C211yesmaximal, characteristic, complemented
5#26011748C212yesmaximal, complemented
6#26011738C212yesmaximal, complemented
7#26011656C212yesmaximal, complemented
8#26011646C211yesmaximal, characteristic, complemented
9#26011632C211yesmaximal, characteristic, complemented
10#26011453C211yesmaximal, characteristic, complemented
11#484×C173C211yesmaximal, characteristic, complemented
12C2×#14583827C2211yescharacteristic
13C22×Dic346C2211yescharacteristic
14C22×C4×D173C2211yescharacteristic
15C2×#14583816C2212yescomplemented
16C2×#14583815C2212yescomplemented
17C2×#14583809C2212yes—
18#19149664—21no—
19#19149659—21no—
20#19149659—21no—
21#19149659C2212yes—
22#19149659C2212yes—
23#19149655C2211yescharacteristic
24#19149654C2211yescharacteristic
25#19149653C2211yescharacteristic
26#19149649—21no—
27#19149641—21no—
28#19149639—21no—
29#19149637C2211yescharacteristic
30C2×C4⋊Dic173—21no—
31C2×#14583803—21no—
32#19149610C2212yescomplemented
33#19149604C2212yescomplemented
34#19149602C2212yescomplemented
35#19149599C2212yescomplemented
36C22⋊#13063436C2211yescharacteristic
37#19149591C2212yes—
38C22⋊Q8×C173C2211yescharacteristic
39C2×C4⋊C4×C173C2211yescharacteristic
40#107×C173C2212yescomplemented
41#103×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic
43C23×D173D411yescharacteristic
44#14583827—42no—
45C2×Dic346—21no—
46C2×Dic346—42no—
47C2×Dic346C2312yes—
48C2×D692—21no—
49C2×D692—21no—
50C2×D692C2311yescharacteristic
51C2×C22⋊D173—21no—
52C22×Dic173—21no—
53C22×Dic173D411yescharacteristic
54C2×C4×D173—21no—
55C2×C4×D173—42no—
56C2×C4×D173D411yescharacteristic
57C2×C4×D173D411yescharacteristic
58C2×C4×D173C2311yescharacteristic
59C2×C4×D173C2311yescharacteristic
60#14583816—42no—
61#14583815—42no—
62#14583809—42no—
63C2×#13063436C2312yes—
64C4⋊Dic173—22no—
65C4⋊Dic173—22no—
66C4⋊Dic173C2311yescharacteristic
67#14583805—21no—
68#14583803—22no—
69#14583803—22no—
70#14583803—21no—
71#14583803—21no—
72#14583803—21no—
73#14583803—21no—
74C4×Dic173—21no—
75C2×Q8×C173C2311yescharacteristic
76C22×C4×C173—21no—
77C22×C4×C173C2311yescharacteristic
78C2×C8×C173C2312yes—
79C4⋊C4×C173—21no—
80C4⋊C4×C173—21no—
81C4⋊C4×C173C2312yes—
82C4⋊C4×C173C2311yescharacteristic
83C22⋊C4×C173—21no—
84C22×D173—22no—
85C22×D173—21no—
86C22×D173—41no—
87C22×D173—41no—
88C22×D173C2×D411yescharacteristic
89C22×D173C2×D411yescharacteristic
90Dic346—22no—
91Dic346—22no—
92Dic346—42no—
93D692—22no—
94D692—42no—
95D692—42no—
96C22⋊D173—42no—
97C2×Dic173—22no—
98C2×Dic173—21no—
99C2×Dic173—41no—
100C2×Dic173—42no—
101C2×Dic173C2×D411yescharacteristic
102C2×Dic173C2×D411yescharacteristic
103C4×D173—22no—
104C4×D173—22no—
105C4×D173—41no—
106C4×D173—41no—
107C4×D173—42no—
108#13063436—22no—
109C23×C173C2×D411yescharacteristic
110Q8×C173—22no—
111C2×C4×C173—22no—
112C2×C4×C173—21no—
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#13712yes—
122C22×C173C22×D411yescharacteristic
123C2×D173—21no—
124C2×D173—21no—
125C2×D173—21no—
126C2×D173—21no—
127C2×D173—21no—
128C2×D173—21no—
129C2×D173—44no—
130C2×D173—41no—
131C2×D173—41no—
132C2×D173—41no—
133C4×C173—21no—
134C4×C173—41no—
135C4×C173—42no—
136C4×C173—41no—
137C4×C173H(5,𝔽2)12yes—
138Dic173—21no—
139Dic173—21no—
140Dic173—42no—
141Dic173—41no—
142C2×C173—22no—
143C2×C173#53411yescharacteristic
144C2×C173C2×#13711yescharacteristic
145C2×C173#57511yescharacteristic
146D173—41no—
147D173—41no—
148D173—81no—
149C173#319611yescharacteristic, complemented, Sylow
150#3196—1731nomaximal, Sylow
151#525—1731no—
152C2×C22⋊Q8—1731no—
153#485—1732no—
154#484D17311yescharacteristic, complemented
155#481—1731no—
156#470—1732no—
157#468—1732no—
158#452—1732no—
159#451—1731no—
160#448—1731no—
161#407—1731no—
162C2×#41—1731no—
163C22×Q8—1731no—
164C23×C4—1731no—
165C2×Q16—1732no—
166C2×SD16—1732no—
167C2×M16—1732no—
168#123—3461no—
169C22⋊Q8—3461no—
170C22⋊Q8—3461no—
171C22⋊Q8—1731no—
172C22⋊Q8—1732no—
173C22⋊Q8—1732no—
174C22⋊Q8C2×D17311yescharacteristic
175#121—1731no—
176#121—1731no—
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#103C2×D17312yescomplemented
189#102—1732no—
190C22⋊C8—1731no—
191C22⋊C8—1732no—
192C22⋊C8C2×D17311yescharacteristic
193C24—1731no—
194#41—6922no—
195C2×Q8—3461no—
196C2×Q8—1732no—
197C2×Q8—6922no—
198C2×Q8C22×D17311yescharacteristic
199C2×D4—3461no—
200C2×D4—3461no—
201C2×D4—3461no—
202C2×D4—1731no—
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—
215M16—6922no—
216C2×C8—1732no—
217C2×C8C22×D17312yes—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—3462no—
221C4⋊C4—3462no—
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—3462no—
235C23—3461no—
236C23—1731no—
237C23—1731no—
238C23—6921no—
239C23—6921no—
240C23D4×D17311yescharacteristic
241Q8—22no—
242Q8—3462no—
243Q8—3462no—
244Q8—6922no—
245D4—3462no—
246D4—6922no—
247D4—6922no—
248D4—6922no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—3462no—
255C2×C4—3462no—
256C2×C4—3461no—
257C2×C4—3462no—
258C2×C4—1731no—
259C2×C4—1731no—
260C2×C4—6921no—
261C2×C4—6921no—
262C2×C4—6922no—
263C2×C4—6921no—
264C2×C4—6922no—
265C2×C4D4×D17311yescharacteristic
266C2×C4C23×D17311yescharacteristic
267C8—22no—
268C8—3462no—
269C22—21no—
270C22—21no—
271C22—3461no—
272C22—3461no—
273C22—3461no—
274C22—3461no—
275C22—3461no—
276C22—3461no—
277C22—6924no—
278C22—6921no—
279C22—6921no—
280C22—6921no—
281C22#1914971112yes—
282C22C2×D4×D17311yescharacteristic
283C4—21no—
284C4—42no—
285C4—41no—
286C4—41no—
287C4—3461no—
288C4—3461no—
289C4—6921no—
290C4—6922no—
291C4#1914972512yes—
292C2—22no—
293C2—13841no—
294C2—6921no—
295C2—6921no—
296C2#2601205211yescharacteristic
297C2C2×#1914971111yescharacteristic
298C2#2601228711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#484×C173C2
socleC22×C173C22×D4
prime radical#26012004C2
hypercenter#484D173
hypocenterC173#3196
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×#19149659C2semidirect
3. #26011834C2semidirect
4. #26011809C2semidirect
5. #26011748C2semidirect
6. #26011738C2semidirect
7. #26011656C2semidirect
8. #26011646C2semidirect
9. #26011632C2semidirect
10. #26011453C2semidirect
11. #484×C173C2semidirect
15. C2×#14583816C22semidirect
16. C2×#14583815C22semidirect
32. #19149610C22semidirect
33. #19149604C22semidirect
34. #19149602C22semidirect
35. #19149599C22semidirect
40. #107×C173C22semidirect
41. #103×C173C22semidirect
149. C173#3196semidirect
154. #484D173semidirect
184. #107C2×D173semidirect
188. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120042
C2×#191496592
#260118342
#260118092
#260117482
#260117382
#260116562
#260116462
#260116322
#260114532
#484×C1732
#3196173