Finite Group Databasebeta

← #483×D173 #36363437

Subgroups — #483×D173 #36363437

Order 22144.

460 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C22⋊Q8×D173C211yesmaximal, characteristic
2C2×C4⋊C4×D173C211yesmaximal, characteristic
3#26011823C212yesmaximal, complemented
4#26011822C211yesmaximal, characteristic, complemented
5#26011821C211yesmaximal, characteristic, complemented
6#26011820C212yesmaximal, complemented
7#26011819C211yesmaximal, characteristic, complemented
8#106×D173C212yesmaximal, complemented
9#103×D173C212yesmaximal, complemented
10C22⋊C8×D173C211yesmaximal, characteristic
11#483×C173C211yesmaximal, characteristic, complemented
12C2×Q8×D173C2211yescharacteristic
13C22×C4×D173—21no—
14C22×C4×D173C2211yescharacteristic
15C2×C8×D173C2212yes—
16#19149659C2211yescharacteristic
17#19149658C2211yescharacteristic
18#19149657—21no—
19#19149656—21no—
20#19149655C2211yescharacteristic
21C2×C4⋊Dic173C2211yescharacteristic
22C2×#14583805—21no—
23C4⋊C4×D173—21no—
24C4⋊C4×D173—21no—
25C4⋊C4×D173—21no—
26C4⋊C4×D173C2212yes—
27C4⋊C4×D173C2212yes—
28C4⋊C4×D173C2211yescharacteristic
29C22⋊C4×D173—21no—
30#19149611C2212yescomplemented
31#19149610C2212yescomplemented
32#19149610C2212yescomplemented
33#19149604C2212yescomplemented
34#19149603C2212yescomplemented
35#19149602C2212yescomplemented
36C22⋊#13063436C2211yescharacteristic
37#19149591C2212yes—
38C22⋊Q8×C173C2211yescharacteristic
39C2×C4⋊C4×C173C2211yescharacteristic
40#106×C173C2212yescomplemented
41#103×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic
43C23×D173D411yescharacteristic
44C2×Dic346—21no—
45C2×Dic346C2311yescharacteristic
46Q8×D173—24no—
47C22×Dic173—21no—
48C22×Dic173D411yescharacteristic
49C2×C4×D173—24no—
50C2×C4×D173—21no—
51C2×C4×D173—21no—
52C2×C4×D173—22no—
53C2×C4×D173—22no—
54C2×C4×D173—44no—
55C2×C4×D173D411yescharacteristic
56C2×C4×D173D411yescharacteristic
57C2×C4×D173C2311yescharacteristic
58C2×C4×D173C2311yescharacteristic
59C2×#13063436C2312yes—
60C8×D173—24no—
61C4⋊Dic173—21no—
62C4⋊Dic173—21no—
63C4⋊Dic173C2312yes—
64C4⋊Dic173C2311yescharacteristic
65#14583805—22no—
66#14583805—22no—
67#14583805—21no—
68#14583805—21no—
69#14583805—21no—
70C22⋊Dic173—21no—
71#14583803—21no—
72#14583803—21no—
73C2×Q8×C173C2311yescharacteristic
74C22×C4×C173—21no—
75C22×C4×C173C2311yescharacteristic
76C2×C8×C173C2312yes—
77C4⋊C4×C173—21no—
78C4⋊C4×C173—21no—
79C4⋊C4×C173C2312yes—
80C4⋊C4×C173C2311yescharacteristic
81C22⋊C4×C173—21no—
82C22×D173—22no—
83C22×D173—22no—
84C22×D173SD1614yes—
85C22×D173C2×D411yescharacteristic
86C22×D173C2×D411yescharacteristic
87Dic346—22no—
88Dic346—42no—
89C2×Dic173—22no—
90C2×Dic173—22no—
91C2×Dic173—21no—
92C2×Dic173—21no—
93C2×Dic173—42no—
94C2×Dic173C2×D411yescharacteristic
95C2×Dic173C2×D411yescharacteristic
96C4×D173—22no—
97C4×D173—22no—
98C4×D173—22no—
99C4×D173—42no—
100C4×D173—44no—
101C4×D173—44no—
102C4×D173—42no—
103C4×D173#4114yes—
104#13063436—22no—
105C23×C173C2×D411yescharacteristic
106Q8×C173—22no—
107C2×C4×C173—22no—
108C2×C4×C173—22no—
109C2×C4×C173—21no—
110C2×C4×C173—21no—
111C2×C4×C173—42no—
112C2×C4×C173C2×D411yescharacteristic
113C2×C4×C173C2411yescharacteristic
114C8×C173—22no—
115C22×C173—21no—
116C22×C173—21no—
117C22×C173C2×SD1612yes—
118C22×C173C22×D411yescharacteristic
119C2×D173—21no—
120C2×D173—21no—
121C2×D173—28no—
122C2×D173#12312yes—
123C2×D173C2×SD1612yes—
124C2×D173C2×SD1612yes—
125C2×D173#13712yes—
126C4×C173—21no—
127C4×C173—41no—
128C4×C173—42no—
129C4×C173—41no—
130C4×C173C2×#4112yes—
131Dic173—21no—
132Dic173—41no—
133Dic173—42no—
134Dic173—41no—
135Dic173#12312yes—
136C2×C173—22no—
137C2×C173C2×#12311yescharacteristic
138C2×C173C22×SD1611yescharacteristic
139C2×C173C2×#13711yescharacteristic
140D173—22no—
141D173#48314yesdirect product factor
142C173C2×#48311yescharacteristic, complemented, Sylow
143C2×#483—1731nomaximal, Sylow
144C2×C22⋊Q8—1731no—
145C22×C4⋊C4—1731no—
146#483—1731no—
147#483—1731no—
148#483—1732no—
149#483—1732no—
150#483—1731no—
151#483D17311yescharacteristic, direct product factor
152C2×#106—1732no—
153C2×#103—1732no—
154C2×C22⋊C8—1731no—
155C22×Q8—1731no—
156C23×C4—3461no—
157C23×C4—1731no—
158C22×C8—1732no—
159C22⋊Q8—3461no—
160C22⋊Q8—3461no—
161C22⋊Q8—1731no—
162C22⋊Q8—1731no—
163C22⋊Q8—1731no—
164C22⋊Q8C2×D17311yescharacteristic
165C2×C4⋊C4—3461no—
166C2×C4⋊C4—3461no—
167C2×C4⋊C4—3461no—
168C2×C4⋊C4—3461no—
169C2×C4⋊C4—1731no—
170C2×C4⋊C4—1732no—
171C2×C4⋊C4—1732no—
172C2×C4⋊C4—1731no—
173C2×C4⋊C4C2×D17311yescharacteristic
174C2×C22⋊C4—3461no—
175#106—1732no—
176#106—1732no—
177#106—1732no—
178#106C2×D17312yescomplemented
179#103—1732no—
180#103—1732no—
181#103—1732no—
182#103C2×D17312yescomplemented
183C22⋊C8—1731no—
184C22⋊C8—1732no—
185C22⋊C8C2×D17311yescharacteristic
186C24—1731no—
187C2×Q8—3461no—
188C2×Q8—3464no—
189C2×Q8—1731no—
190C2×Q8C22×D17311yescharacteristic
191C22×C4—21no—
192C22×C4—3464no—
193C22×C4—3461no—
194C22×C4—3461no—
195C22×C4—3462no—
196C22×C4—3462no—
197C22×C4—3461no—
198C22×C4—1731no—
199C22×C4—1731no—
200C22×C4—1731no—
201C22×C4—1731no—
202C22×C4—1731no—
203C22×C4—6924no—
204C22×C4C22×D17311yescharacteristic
205C2×C8—3464no—
206C2×C8—1732no—
207C2×C8C22×D17312yes—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—3462no—
211C4⋊C4—3462no—
212C4⋊C4—3461no—
213C4⋊C4—3461no—
214C4⋊C4—3461no—
215C4⋊C4—3461no—
216C4⋊C4—3461no—
217C4⋊C4—1732no—
218C4⋊C4—1731no—
219C4⋊C4C22×D17312yes—
220C4⋊C4C22×D17311yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—3461no—
223C22⋊C4—3461no—
224C22⋊C4—3461no—
225C23—3462no—
226C23—3462no—
227C23—1734no—
228C23—1731no—
229C23—1731no—
230C23D4×D17311yescharacteristic
231Q8—22no—
232Q8—3462no—
233Q8—6922no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—3462no—
240C2×C4—3462no—
241C2×C4—3462no—
242C2×C4—3462no—
243C2×C4—3461no—
244C2×C4—3461no—
245C2×C4—3462no—
246C2×C4—1731no—
247C2×C4—1731no—
248C2×C4—1734no—
249C2×C4—6922no—
250C2×C4—6924no—
251C2×C4—6924no—
252C2×C4—6922no—
253C2×C4—6922no—
254C2×C4D4×D17311yescharacteristic
255C2×C4C23×D17311yescharacteristic
256C8—22no—
257C8—3462no—
258C22—21no—
259C22—21no—
260C22—3468no—
261C22—3461no—
262C22—3461no—
263C22—1732no—
264C22—1732no—
265C22—1732no—
266C22—1732no—
267C22SD16×D17312yes—
268C22C2×D4×D17311yescharacteristic
269C4—21no—
270C4—41no—
271C4—42no—
272C4—41no—
273C4—3461no—
274C4—1732no—
275C4—6921no—
276C4—6922no—
277C4—6921no—
278C4#41×D17312yes—
279C2—22no—
280C2—3462no—
281C2—1734no—
282C2#123×D17311yescharacteristic
283C2C2×SD16×D17311yescharacteristic
284C2#137×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#483×C173C2
socleC22×C173C22×D4
prime radicalC23×D173D4
hypercenter#483D173
hypocenterC173C2×#483
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #26011823C2semidirect
4. #26011822C2semidirect
5. #26011821C2semidirect
6. #26011820C2semidirect
7. #26011819C2semidirect
8. #106×D173C2semidirect
9. #103×D173C2semidirect
11. #483×C173C2semidirect
30. #19149611C22semidirect
31. #19149610C22semidirect
32. #19149610C22semidirect
33. #19149604C22semidirect
34. #19149603C22semidirect
35. #19149602C22semidirect
40. #106×C173C22semidirect
41. #103×C173C22semidirect
141. D173#483direct (N × H)
142. C173C2×#483semidirect
151. #483D173direct (N × H)
178. #106C2×D173semidirect
182. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1732
C2×C4⋊C4×D1732
#260118232
#260118222
#260118212
#260118202
#260118192
#106×D1732
#103×D1732
C22⋊C8×D1732
#483×C1732
C2×#483173