Finite Group Databasebeta

← #49130633

Subgroups — #49130633

Order 36224.

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×#22583635C212yesmaximal, complemented
2#31827029C211yesmaximal, characteristic, complemented
3#31826638C214yesmaximal, complemented
4#31826633C214yesmaximal, complemented
5#31826482C212yesmaximal, complemented
6#31826449C211yesmaximal, characteristic, complemented
7#585×C283C211yesmaximal, characteristic, complemented
8C22×D1132C2211yescharacteristic, complemented
9#22583635—24no—
10#22583635C2218yescomplemented
11C2×#17985379—22no—
12C2×#17985377—22no—
13C2×#17985376C2212yescomplemented
14#22583606C2212yescomplemented
15#22583599C2212yescomplemented
16#22583584—22no—
17#22583578—22no—
18#22583569C2211yescharacteristic, complemented
19#22583537C2218yescomplemented
20#22583529C2212yescomplemented
21#22583527C2212yescomplemented
222−1+4×C283—22no—
232−1+4×C283C2212yescomplemented
24H(5,𝔽2)×C283—22no—
25H(5,𝔽2)×C283—22no—
26H(5,𝔽2)×C283C2212yescomplemented
27C2×#41×C283—24no—
28C2×#41×C283—22no—
29C2×#41×C283C2212yes—
30C2×#41×C283C2211yescharacteristic
31C23×D283—21no—
32C2×D1132—22no—
33C2×D1132—22no—
34C2×D1132—21no—
35C2×D1132C2312yes—
36C2×C22⋊D283—24no—
37#17985379—28no—
38#17985377—28no—
39#17985376—22no—
40#17985376C2314yes—
41C2×C283⋊C8—22no—
42C4⋊Dic283—21no—
43C22⋊Dic283—21no—
44(C2×D283)⋊C4—24no—
45C4×Dic283C2312yes—
46#41×C283—24no—
47#41×C283—28no—
48#41×C283—24no—
49#41×C283—24no—
50#41×C283—24no—
51#41×C283—22no—
52#41×C283—22no—
53#41×C283—24no—
54#41×C283—22no—
55#41×C283—22no—
56#41×C283D414yescomplemented
57#41×C283C2314yes—
58C2×Q8×C283—22no—
59C2×Q8×C283—24no—
60C2×Q8×C283D412yescomplemented
61C2×Q8×C283C2311yescharacteristic
62C2×D4×C283—28no—
63C2×D4×C283—22no—
64C2×D4×C283—24no—
65C2×D4×C283—22no—
66C2×D4×C283—24no—
67C2×D4×C283D412yes—
68C2×D4×C283D412yescomplemented
69C2×D4×C283C2311yescharacteristic
70C22×C4×C283—24no—
71C22×C4×C283—22no—
72C22×C4×C283D412yes—
73C22×C4×C283C2311yescharacteristic
74C22×D283—22no—
75C22×D283—42no—
76C22×D283—82no—
77D1132—24no—
78D1132—41no—
79D1132—41no—
80C22⋊D283—84no—
81C2×Dic283—22no—
82C283⋊C8—24no—
83C23×C283—24no—
84C23×C283—22no—
85C23×C283C2×D412yes—
86C23×C283C2×D411yescharacteristic
87Q8×C283—24no—
88Q8×C283—22no—
89Q8×C283—22no—
90Q8×C283C2×D414yes—
91D4×C283—28no—
92D4×C283—24no—
93D4×C283—24no—
94D4×C283—24no—
95D4×C283—24no—
96D4×C283—22no—
97D4×C283—22no—
98D4×C283C2×D414yes—
99C2×C4×C283—24no—
100C2×C4×C283—28no—
101C2×C4×C283—22no—
102C2×C4×C283—24no—
103C2×C4×C283—22no—
104C2×C4×C283—22no—
105C2×C4×C283—24no—
106C2×C4×C283C2×D412yes—
107C2×C4×C283C2×D411yescharacteristic
108C2×C4×C283C2×D412yes—
109C2×C4×C283C2×D412yes—
110C2×C4×C283C2411yescharacteristic
111C22×C283—22no—
112C22×C283—24no—
113C22×C283—41no—
114C22×C283—42no—
115C22×C283—82no—
116C22×C283—84no—
117C22×C283#12012yes—
118C22×C283C22×D411yescharacteristic
119C2×D283—44no—
120C2×D283—42no—
121C2×D283—84no—
122C4×C283—24no—
123C4×C283—22no—
124C4×C283#12011yescharacteristic
125C4×C283#12011yescharacteristic
126C4×C283C22×D412yes—
127Dic283—42no—
128C2×C283—21no—
129C2×C283—22no—
130C2×C283—42no—
131C2×C283—44no—
132C2×C283C2×#12011yescharacteristic
133D283—82no—
134C283#302111yescharacteristic, complemented, Sylow
135#3021—2831nomaximal, Sylow
136#585D28311yescharacteristic, complemented
137C2×AGL(1,ℤ/8)—2832no—
138#535—2831no—
139#454—2834no—
140#453—2834no—
141#421—2832no—
142#413—2831no—
1432−1+4—22no—
1442−1+4C2×D28312yescomplemented
145H(5,𝔽2)—22no—
146H(5,𝔽2)—22no—
147H(5,𝔽2)C2×D28312yescomplemented
148C2×#41—24no—
149C2×#41—22no—
150C2×#41C2×D28312yes—
151C2×#41C2×D28311yescharacteristic
152C22×D4—2831no—
153AGL(1,ℤ/8)—2838no—
154AGL(1,ℤ/8)—5664no—
155C2×SD16—5662no—
156C2×D8—5662no—
157C2×M16—2832no—
158#127—2832no—
159#124—2832no—
160#121—5662no—
161#120—5662no—
162#117—2831no—
163#104—2838no—
164#101—2832no—
165#100—2832no—
166C24—5661no—
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⋊D28314yescomplemented
178#41C22×D28314yes—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8C22⋊D28312yescomplemented
182C2×Q8C22×D28311yescharacteristic
183C2×D4—28no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4—22no—
187C2×D4—24no—
188C2×D4—2832no—
189C2×D4—5664no—
190C2×D4—5662no—
191C2×D4—5662no—
192C2×D4—5661no—
193C2×D4C22⋊D28312yes—
194C2×D4C22⋊D28312yescomplemented
195C2×D4C22×D28311yescharacteristic
196C22×C4—24no—
197C22×C4—22no—
198C22×C4C22⋊D28312yes—
199C22×C4C22×D28311yescharacteristic
200SD16—5668no—
201D8—5668no—
202M16—2834no—
203M16—5662no—
204C2×C8—5662no—
205C4⋊C4—5661no—
206C22⋊C4—5661no—
207C22⋊C4—5664no—
208C42—2832no—
209C23—24no—
210C23—22no—
211C23—5662no—
212C23—11322no—
213C23—22642no—
214C23C2×C22⋊D28312yes—
215C23C2×C22⋊D28311yescharacteristic
216Q8—24no—
217Q8—22no—
218Q8—22no—
219Q8C2×C22⋊D28314yes—
220D4—28no—
221D4—24no—
222D4—24no—
223D4—24no—
224D4—24no—
225D4—22no—
226D4—22no—
227D4—5664no—
228D4—11321no—
229D4—11321no—
230D4—22644no—
231D4C2×C22⋊D28314yes—
232C2×C4—24no—
233C2×C4—28no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—5662no—
240C2×C4C2×C22⋊D28312yes—
241C2×C4C2×C22⋊D28311yescharacteristic
242C2×C4C2×C22⋊D28312yes—
243C2×C4C2×C22⋊D28312yes—
244C2×C4C23×D28311yescharacteristic
245C8—5664no—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—42no—
250C22—82no—
251C22—84no—
252C22—11324no—
253C22—11322no—
254C22—22644no—
255C22#2258357912yes—
256C22C22×C22⋊D28311yescharacteristic
257C4—24no—
258C4—22no—
259C4—11322no—
260C4#2258357911yescharacteristic
261C4#2258357911yescharacteristic
262C4C22×C22⋊D28312yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—44no—
267C2—22642no—
268C2C2×#2258357911yescharacteristic
subgroupG/H
centerC2C2×#22583579
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#585×C283C2
socleC2×C283C2×#120
hypercenter#585D283
hypocenterC283#3021
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#22583635C2semidirect
2. #31827029C2semidirect
3. #31826638C2semidirect
4. #31826633C2semidirect
5. #31826482C2semidirect
6. #31826449C2semidirect
7. #585×C283C2semidirect
8. C22×D1132C22semidirect
10. #22583635C22semidirect
13. C2×#17985376C22semidirect
14. #22583606C22semidirect
15. #22583599C22semidirect
18. #22583569C22semidirect
19. #22583537C22semidirect
20. #22583529C22semidirect
21. #22583527C22semidirect
23. 2−1+4×C283C22semidirect
26. H(5,𝔽2)×C283C22semidirect
56. #41×C283D4semidirect
60. C2×Q8×C283D4semidirect
68. C2×D4×C283D4semidirect
134. C283#3021semidirect
136. #585D283semidirect
144. 2−1+4C2×D283semidirect
147. H(5,𝔽2)C2×D283semidirect
177. #41C22⋊D283semidirect
181. C2×Q8C22⋊D283semidirect
194. C2×D4C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836352
#318270292
#318266382
#318266332
#318264822
#318264492
#585×C2832
#3021283