Finite Group Databasebeta

← #470×D173 #36363161

Subgroups — #470×D173 #36363161

Order 22144.

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

normal subgroups only

This group has 492 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
1C2×Q16×D173C211yesmaximal, characteristic, complemented
2C2×M16×D173C211yesmaximal, characteristic
3C22⋊Q8×D173C212yesmaximal
4#26011750C211yesmaximal, characteristic, complemented
5#26011749C211yesmaximal, characteristic, complemented
6#26011748C212yesmaximal, complemented
7#26011747C212yesmaximal, complemented
8#26011746C211yesmaximal, characteristic, complemented
9#106×D173C211yesmaximal, characteristic, complemented
10#103×D173C212yesmaximal, complemented
11#470×C173C211yesmaximal, characteristic, complemented
12C2×Q8×D173C2212yes—
13C22×C4×D173C2211yescharacteristic
14C2×Dic692C2211yescharacteristic, complemented
15C2×#14583816C2212yescomplemented
16Q16×D173—24no—
17C2×#14583810C2211yescharacteristic
18C2×#14583809C2211yescharacteristic
19C2×#14583809C2211yescharacteristic
20M16×D173—24no—
21C2×C8×D173C2211yescharacteristic
22C2×C8×D173C2211yescharacteristic
23#19149659C2212yes—
24#19149658C2212yes—
25#19149657—22no—
26#19149656—22no—
27#19149655C2212yes—
28C4⋊C4×D173—22no—
29C4⋊C4×D173C2212yes—
30C22⋊C4×D173—22no—
31#19149611C2211yescharacteristic, complemented
32#19149610C2212yescomplemented
33#19149604C2212yescomplemented
34#19149603C2212yescomplemented
35#19149602C2212yescomplemented
36C2×Q16×C173C2211yescharacteristic, complemented
37C2×M16×C173C2211yescharacteristic
38C22⋊Q8×C173C2212yes—
39#106×C173C2211yescharacteristic, complemented
40#103×C173C2212yescomplemented
41C23×D173D411yescharacteristic
42C2×Dic346—22no—
43C2×Dic346C2312yes—
44Q8×D173—24no—
45Q8×D173—24no—
46C22×Dic173D411yescharacteristic
47C2×C4×D173—22no—
48C2×C4×D173—22no—
49C2×C4×D173—22no—
50C2×C4×D173—22no—
51C2×C4×D173D411yescharacteristic
52C2×C4×D173D411yescharacteristic
53C2×C4×D173C2311yescharacteristic
54C2×C4×D173C2311yescharacteristic
55Dic692—22no—
56#14583816—24no—
57#14583810—22no—
58#14583809—22no—
59#14583809—22no—
60C2×#13063436C2311yescharacteristic
61C2×#13063436C2311yescharacteristic
62C8×D173—22no—
63C8×D173D414yes—
64C4⋊Dic173—22no—
65C4⋊Dic173C2312yes—
66#14583805—22no—
67#14583805—22no—
68#14583805—22no—
69C22⋊Dic173—22no—
70#14583803—22no—
71#14583803—22no—
72C2×Q8×C173C2312yes—
73C22×C4×C173C2311yescharacteristic
74Q16×C173—22no—
75M16×C173—22no—
76C2×C8×C173C2311yescharacteristic
77C2×C8×C173C2311yescharacteristic
78C4⋊C4×C173—22no—
79C4⋊C4×C173C2312yes—
80C22⋊C4×C173—22no—
81C22×D173—22no—
82C22×D173—24no—
83C22×D173C2×D411yescharacteristic
84C22×D173C2×D411yescharacteristic
85Dic346—22no—
86Dic346—22no—
87Dic346—44no—
88C2×Dic173—22no—
89C2×Dic173—22no—
90C2×Dic173—22no—
91C2×Dic173C2×D411yescharacteristic
92C2×Dic173C2×D411yescharacteristic
93C4×D173—22no—
94C4×D173—21no—
95C4×D173—21no—
96C4×D173—22no—
97C4×D173—44no—
98C4×D173—44no—
99C4×D173C2×D412yes—
100C4×D173#4112yes—
101#13063436—21no—
102#13063436C2×D412yes—
103C23×C173C2×D411yescharacteristic
104Q8×C173—22no—
105Q8×C173—22no—
106C2×C4×C173—21no—
107C2×C4×C173—21no—
108C2×C4×C173—22no—
109C2×C4×C173—22no—
110C2×C4×C173C2×D411yescharacteristic
111C2×C4×C173C2411yescharacteristic
112C8×C173—21no—
113C8×C173C2×D412yes—
114C22×C173—21no—
115C22×C173—22no—
116C22×C173C22×D411yescharacteristic
117C2×D173—21no—
118C2×D173—22no—
119C2×D173—44no—
120C2×D173#12112yes—
121C2×D173#13714yes—
122C4×C173—21no—
123C4×C173—42no—
124C4×C173—42no—
125C4×C173C22×D411yescharacteristic
126C4×C173C2×#4111yescharacteristic
127Dic173—21no—
128Dic173—42no—
129Dic173—42no—
130Dic173#12112yes—
131C2×C173—41no—
132C2×C173C2×#12111yescharacteristic
133C2×C173C2×#13712yes—
134D173—41no—
135D173#47014yesdirect product factor
136C173C2×#47011yescharacteristic, complemented, Sylow
137C2×#470—1731nomaximal, Sylow
138C22×Q16—1731no—
139C22×M16—1731no—
140C2×C22⋊Q8—1732no—
141#470—1731no—
142#470—1731no—
143#470—1732no—
144#470—1731no—
145#470—1732no—
146#470D17311yescharacteristic, direct product factor
147C2×#106—1731no—
148C2×#103—1732no—
149C22×Q8—1732no—
150C23×C4—1731no—
151C2×Q16—3464no—
152C2×Q16—1732no—
153C2×Q16—1731no—
154C2×Q16C2×D17311yescharacteristic, complemented
155C2×M16—3464no—
156C2×M16—1731no—
157C2×M16—1731no—
158C2×M16—1731no—
159C2×M16C2×D17311yescharacteristic
160C22×C8—1731no—
161C22×C8—1731no—
162C22⋊Q8—3462no—
163C22⋊Q8—3462no—
164C22⋊Q8—1732no—
165C22⋊Q8—1732no—
166C22⋊Q8—1732no—
167C22⋊Q8C2×D17312yes—
168C2×C4⋊C4—3462no—
169C2×C4⋊C4—1732no—
170C2×C22⋊C4—3462no—
171#106—1731no—
172#106—1732no—
173#106C2×D17311yescharacteristic, complemented
174#103—1732no—
175#103—1732no—
176#103—1732no—
177#103C2×D17312yescomplemented
178C24—1731no—
179C2×Q8—3462no—
180C2×Q8—3464no—
181C2×Q8—3464no—
182C2×Q8—1732no—
183C2×Q8C22×D17312yes—
184C22×C4—3462no—
185C22×C4—3462no—
186C22×C4—3462no—
187C22×C4—3462no—
188C22×C4—1731no—
189C22×C4—1731no—
190C22×C4—1731no—
191C22×C4—1731no—
192C22×C4—1731no—
193C22×C4C22×D17311yescharacteristic
194Q16—22no—
195Q16—3464no—
196Q16—3462no—
197M16—22no—
198M16—3462no—
199M16—3462no—
200M16—3462no—
201C2×C8—3462no—
202C2×C8—1731no—
203C2×C8—1731no—
204C2×C8—1734no—
205C2×C8C22×D17311yescharacteristic
206C2×C8C22×D17311yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—3462no—
209C4⋊C4—3462no—
210C4⋊C4—3462no—
211C4⋊C4—3462no—
212C4⋊C4—1732no—
213C4⋊C4C22×D17312yes—
214C22⋊C4—22no—
215C22⋊C4—3462no—
216C22⋊C4—3462no—
217C22⋊C4—3462no—
218C23—3462no—
219C23—3464no—
220C23—1731no—
221C23—1731no—
222C23D4×D17311yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—3462no—
226Q8—3462no—
227Q8—6924no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—3462no—
233C2×C4—3462no—
234C2×C4—3462no—
235C2×C4—3461no—
236C2×C4—3461no—
237C2×C4—3462no—
238C2×C4—3462no—
239C2×C4—1731no—
240C2×C4—1731no—
241C2×C4—1732no—
242C2×C4—1732no—
243C2×C4—6924no—
244C2×C4—6924no—
245C2×C4D4×D17311yescharacteristic
246C2×C4C23×D17311yescharacteristic
247C8—21no—
248C8—3461no—
249C8—1732no—
250C8D4×D17312yes—
251C22—21no—
252C22—22no—
253C22—3461no—
254C22—3462no—
255C22—1732no—
256C22—1734no—
257C22—6924no—
258C22C2×D4×D17311yescharacteristic
259C4—21no—
260C4—42no—
261C4—42no—
262C4—3461no—
263C4—1732no—
264C4—6922no—
265C4—6922no—
266C4C2×D4×D17311yescharacteristic
267C4#41×D17311yescharacteristic
268C2—41no—
269C2—1734no—
270C2—6921no—
271C2#121×D17311yescharacteristic
272C2#137×D17312yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#470×C173C2
socleC22×C173C22×D4
prime radicalC23×D173D4
hypercenter#470D173
hypocenterC173C2×#470
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D173C2semidirect
4. #26011750C2semidirect
5. #26011749C2semidirect
6. #26011748C2semidirect
7. #26011747C2semidirect
8. #26011746C2semidirect
9. #106×D173C2semidirect
10. #103×D173C2semidirect
11. #470×C173C2semidirect
14. C2×Dic692C22semidirect
15. C2×#14583816C22semidirect
31. #19149611C22semidirect
32. #19149610C22semidirect
33. #19149604C22semidirect
34. #19149603C22semidirect
35. #19149602C22semidirect
36. C2×Q16×C173C22semidirect
39. #106×C173C22semidirect
40. #103×C173C22semidirect
135. D173#470direct (N × H)
136. C173C2×#470semidirect
146. #470D173direct (N × H)
154. C2×Q16C2×D173semidirect
173. #106C2×D173semidirect
177. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1732
C2×M16×D1732
C22⋊Q8×D1732
#260117502
#260117492
#260117482
#260117472
#260117462
#106×D1732
#103×D1732
#470×C1732
C2×#470173