Finite Group Databasebeta

← #36362935

Subgroups — #36362935

Order 22144.

736 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 161 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×#19149704C212yesmaximal, complemented
2#26012055C211yesmaximal, characteristic, complemented
3#26011664C214yesmaximal, complemented
4#26011659C214yesmaximal, complemented
5#26011508C212yesmaximal, complemented
6#26011475C211yesmaximal, characteristic, complemented
7#585×C173C211yesmaximal, characteristic, complemented
8C22×D692C2211yescharacteristic, complemented
9#19149704—24no—
10#19149704C2218yescomplemented
11C2×#14583813—22no—
12C2×#14583811—22no—
13C2×#14583810C2212yescomplemented
14#19149675C2212yescomplemented
15#19149668C2212yescomplemented
16#19149653—22no—
17#19149647—22no—
18#19149638C2211yescharacteristic, complemented
19#19149606C2218yescomplemented
20#19149598C2212yescomplemented
21#19149596C2212yescomplemented
222−1+4×C173—22no—
232−1+4×C173C2212yescomplemented
24H(5,𝔽2)×C173—22no—
25H(5,𝔽2)×C173—22no—
26H(5,𝔽2)×C173C2212yescomplemented
27C2×#41×C173—24no—
28C2×#41×C173—22no—
29C2×#41×C173C2212yes—
30C2×#41×C173C2211yescharacteristic
31C23×D173—21no—
32C2×D692—22no—
33C2×D692—22no—
34C2×D692—21no—
35C2×D692C2312yes—
36C2×C22⋊D173—24no—
37#14583813—28no—
38#14583811—28no—
39#14583810—22no—
40#14583810C2314yes—
41C2×#13063436—22no—
42C4⋊Dic173—21no—
43C22⋊Dic173—21no—
44#14583803—24no—
45C4×Dic173C2312yes—
46#41×C173—24no—
47#41×C173—28no—
48#41×C173—24no—
49#41×C173—24no—
50#41×C173—24no—
51#41×C173—22no—
52#41×C173—22no—
53#41×C173—24no—
54#41×C173—22no—
55#41×C173—22no—
56#41×C173D414yescomplemented
57#41×C173C2314yes—
58C2×Q8×C173—22no—
59C2×Q8×C173—24no—
60C2×Q8×C173D412yescomplemented
61C2×Q8×C173C2311yescharacteristic
62C2×D4×C173—28no—
63C2×D4×C173—22no—
64C2×D4×C173—24no—
65C2×D4×C173—22no—
66C2×D4×C173—24no—
67C2×D4×C173D412yes—
68C2×D4×C173D412yescomplemented
69C2×D4×C173C2311yescharacteristic
70C22×C4×C173—24no—
71C22×C4×C173—22no—
72C22×C4×C173D412yes—
73C22×C4×C173C2311yescharacteristic
74C22×D173—22no—
75C22×D173—42no—
76C22×D173—82no—
77D692—24no—
78D692—41no—
79D692—41no—
80C22⋊D173—84no—
81C2×Dic173—22no—
82#13063436—24no—
83C23×C173—24no—
84C23×C173—22no—
85C23×C173C2×D412yes—
86C23×C173C2×D411yescharacteristic
87Q8×C173—24no—
88Q8×C173—22no—
89Q8×C173—22no—
90Q8×C173C2×D414yes—
91D4×C173—28no—
92D4×C173—24no—
93D4×C173—24no—
94D4×C173—24no—
95D4×C173—24no—
96D4×C173—22no—
97D4×C173—22no—
98D4×C173C2×D414yes—
99C2×C4×C173—24no—
100C2×C4×C173—28no—
101C2×C4×C173—22no—
102C2×C4×C173—24no—
103C2×C4×C173—22no—
104C2×C4×C173—22no—
105C2×C4×C173—24no—
106C2×C4×C173C2×D412yes—
107C2×C4×C173C2×D411yescharacteristic
108C2×C4×C173C2×D412yes—
109C2×C4×C173C2×D412yes—
110C2×C4×C173C2411yescharacteristic
111C22×C173—22no—
112C22×C173—24no—
113C22×C173—41no—
114C22×C173—42no—
115C22×C173—82no—
116C22×C173—84no—
117C22×C173#12012yes—
118C22×C173C22×D411yescharacteristic
119C2×D173—44no—
120C2×D173—42no—
121C2×D173—84no—
122C4×C173—24no—
123C4×C173—22no—
124C4×C173#12011yescharacteristic
125C4×C173#12011yescharacteristic
126C4×C173C22×D412yes—
127Dic173—42no—
128C2×C173—21no—
129C2×C173—22no—
130C2×C173—42no—
131C2×C173—44no—
132C2×C173C2×#12011yescharacteristic
133D173—82no—
134C173#302111yescharacteristic, complemented, Sylow
135#3021—1731nomaximal, Sylow
136#585D17311yescharacteristic, complemented
137C2×AGL(1,ℤ/8)—1732no—
138#535—1731no—
139#454—1734no—
140#453—1734no—
141#421—1732no—
142#413—1731no—
1432−1+4—22no—
1442−1+4C2×D17312yescomplemented
145H(5,𝔽2)—22no—
146H(5,𝔽2)—22no—
147H(5,𝔽2)C2×D17312yescomplemented
148C2×#41—24no—
149C2×#41—22no—
150C2×#41C2×D17312yes—
151C2×#41C2×D17311yescharacteristic
152C22×D4—1731no—
153AGL(1,ℤ/8)—3464no—
154AGL(1,ℤ/8)—1738no—
155C2×SD16—3462no—
156C2×D8—3462no—
157C2×M16—1732no—
158#127—1732no—
159#124—1732no—
160#121—3462no—
161#120—3462no—
162#117—1731no—
163#104—1738no—
164#101—1732no—
165#100—1732no—
166C24—3461no—
167#41—24no—
168#41—28no—
169#41—24no—
170#41—24no—
171#41—24no—
172#41—22no—
173#41—22no—
174#41—24no—
175#41—22no—
176#41—22no—
177#41C22⋊D17314yescomplemented
178#41C22×D17314yes—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8C22⋊D17312yescomplemented
182C2×Q8C22×D17311yescharacteristic
183C2×D4—28no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4—22no—
187C2×D4—24no—
188C2×D4—3464no—
189C2×D4—3462no—
190C2×D4—3462no—
191C2×D4—3461no—
192C2×D4—1732no—
193C2×D4C22⋊D17312yes—
194C2×D4C22⋊D17312yescomplemented
195C2×D4C22×D17311yescharacteristic
196C22×C4—24no—
197C22×C4—22no—
198C22×C4C22⋊D17312yes—
199C22×C4C22×D17311yescharacteristic
200SD16—3468no—
201D8—3468no—
202M16—3462no—
203M16—1734no—
204C2×C8—3462no—
205C4⋊C4—3461no—
206C22⋊C4—3461no—
207C22⋊C4—3464no—
208C42—1732no—
209C23—24no—
210C23—22no—
211C23—3462no—
212C23—13842no—
213C23—6922no—
214C23C2×C22⋊D17312yes—
215C23C2×C22⋊D17311yescharacteristic
216Q8—24no—
217Q8—22no—
218Q8—22no—
219Q8C2×C22⋊D17314yes—
220D4—28no—
221D4—24no—
222D4—24no—
223D4—24no—
224D4—24no—
225D4—22no—
226D4—22no—
227D4—3464no—
228D4—13844no—
229D4—6921no—
230D4—6921no—
231D4C2×C22⋊D17314yes—
232C2×C4—24no—
233C2×C4—28no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—3462no—
240C2×C4C2×C22⋊D17312yes—
241C2×C4C2×C22⋊D17311yescharacteristic
242C2×C4C2×C22⋊D17312yes—
243C2×C4C2×C22⋊D17312yes—
244C2×C4C23×D17311yescharacteristic
245C8—3464no—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—42no—
250C22—82no—
251C22—84no—
252C22—13844no—
253C22—6924no—
254C22—6922no—
255C22#1914964812yes—
256C22C22×C22⋊D17311yescharacteristic
257C4—24no—
258C4—22no—
259C4—6922no—
260C4#1914964811yescharacteristic
261C4#1914964811yescharacteristic
262C4C22×C22⋊D17312yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—44no—
267C2—13842no—
268C2C2×#1914964811yescharacteristic
subgroupG/H
centerC2C2×#19149648
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#585×C173C2
socleC2×C173C2×#120
hypercenter#585D173
hypocenterC173#3021
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#19149704C2semidirect
2. #26012055C2semidirect
3. #26011664C2semidirect
4. #26011659C2semidirect
5. #26011508C2semidirect
6. #26011475C2semidirect
7. #585×C173C2semidirect
8. C22×D692C22semidirect
10. #19149704C22semidirect
13. C2×#14583810C22semidirect
14. #19149675C22semidirect
15. #19149668C22semidirect
18. #19149638C22semidirect
19. #19149606C22semidirect
20. #19149598C22semidirect
21. #19149596C22semidirect
23. 2−1+4×C173C22semidirect
26. H(5,𝔽2)×C173C22semidirect
56. #41×C173D4semidirect
60. C2×Q8×C173D4semidirect
68. C2×D4×C173D4semidirect
134. C173#3021semidirect
136. #585D173semidirect
144. 2−1+4C2×D173semidirect
147. H(5,𝔽2)C2×D173semidirect
177. #41C22⋊D173semidirect
181. C2×Q8C22⋊D173semidirect
194. C2×D4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497042
#260120552
#260116642
#260116592
#260115082
#260114752
#585×C1732
#3021173