Finite Group Databasebeta

← #49130640

Subgroups — #49130640

Order 36224.

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

normal subgroups only

This group has 704 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×#22583640C212yesmaximal, complemented
2#31827036C211yesmaximal, characteristic, complemented
3#31826648C214yesmaximal, complemented
4#31826643C214yesmaximal, complemented
5#31826482C212yesmaximal, complemented
6#31826449C211yesmaximal, characteristic, complemented
7#585×C283C211yesmaximal, characteristic
8C22×Dic566C2211yescharacteristic, complemented
9#22583640—24no—
10#22583640C2218yescomplemented
11C2×#17985382—22no—
12C2×#17985380—22no—
13C2×#17985376C2212yescomplemented
14#22583608C2212yescomplemented
15#22583597C2212yescomplemented
16#22583591—22no—
17#22583589—22no—
18#22583569C2211yescharacteristic, complemented
19#22583537C2218yescomplemented
20#22583529C2212yescomplemented
21#22583527C2212yescomplemented
222−1+4×C283—22no—
232−1+4×C283C2212yes—
24H(5,𝔽2)×C283—22no—
25H(5,𝔽2)×C283—22no—
26H(5,𝔽2)×C283C2212yes—
27C2×#41×C283—24no—
28C2×#41×C283—22no—
29C2×#41×C283C2212yes—
30C2×#41×C283C2211yescharacteristic
31C2×Dic566—21no—
32C2×Dic566—22no—
33C2×Dic566—22no—
34C2×Dic566C2312yes—
35C22×Dic283—21no—
36#17985382—28no—
37#17985380—28no—
38#17985376—22no—
39#17985376C2314yes—
40C2×C283⋊C8—22no—
41C4⋊Dic283—21no—
42Dic283⋊C4—24no—
43C22⋊Dic283—21no—
44C22⋊Dic283—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×C283D414yes—
57#41×C283C2314yes—
58C2×Q8×C283—22no—
59C2×Q8×C283—24no—
60C2×Q8×C283D412yes—
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×C283D412yes—
69C2×D4×C283C2311yescharacteristic
70C22×C4×C283—24no—
71C22×C4×C283—22no—
72C22×C4×C283D412yes—
73C22×C4×C283C2311yescharacteristic
74Dic566—24no—
75Dic566—41no—
76Dic566—41no—
77C2×Dic283—22no—
78C2×Dic283—22no—
79C2×Dic283—42no—
80C283⋊C8—24no—
81C23×C283—24no—
82C23×C283—22no—
83C23×C283C2×D412yes—
84C23×C283C2×D411yescharacteristic
85Q8×C283—24no—
86Q8×C283—22no—
87Q8×C283—22no—
88Q8×C283C2×D414yes—
89D4×C283—28no—
90D4×C283—24no—
91D4×C283—24no—
92D4×C283—24no—
93D4×C283—24no—
94D4×C283—22no—
95D4×C283—22no—
96D4×C283C2×D414yes—
97C2×C4×C283—24no—
98C2×C4×C283—28no—
99C2×C4×C283—22no—
100C2×C4×C283—24no—
101C2×C4×C283—22no—
102C2×C4×C283—22no—
103C2×C4×C283—24no—
104C2×C4×C283C2×D412yes—
105C2×C4×C283C2×D411yescharacteristic
106C2×C4×C283C2×D412yes—
107C2×C4×C283C2×D412yes—
108C2×C4×C283C2411yescharacteristic
109C22×C283—22no—
110C22×C283—24no—
111C22×C283—41no—
112C22×C283—42no—
113C22×C283—82no—
114C22×C283—84no—
115C22×C283#12012yes—
116C22×C283C22×D411yescharacteristic
117C4×C283—24no—
118C4×C283—22no—
119C4×C283#12011yescharacteristic
120C4×C283#12011yescharacteristic
121C4×C283C22×D412yes—
122Dic283—42no—
123Dic283—42no—
124C2×C283—21no—
125C2×C283—22no—
126C2×C283—42no—
127C2×C283—44no—
128C2×C283C2×#12011yescharacteristic
129C283#302211yescharacteristic, complemented, Sylow
130#3022—2831nomaximal, Sylow
131#585D28311yescharacteristic
132C2×#137—2832no—
133#536—2831no—
134#456—2834no—
135#455—2834no—
136#421—2832no—
137#413—2831no—
1382−1+4—22no—
1392−1+4C2×D28312yes—
140H(5,𝔽2)—22no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D28312yes—
143C2×#41—24no—
144C2×#41—22no—
145C2×#41C2×D28312yes—
146C2×#41C2×D28311yescharacteristic
147C22×Q8—2831no—
148#137—2838no—
149#137—5664no—
150C2×Q16—5662no—
151C2×SD16—5662no—
152C2×M16—2832no—
153C4⋊Q8—2832no—
154#124—2832no—
155#123—5662no—
156C22⋊Q8—5662no—
157#117—2831no—
158#104—2838no—
159#101—2832no—
160#100—2832no—
161#41—24no—
162#41—28no—
163#41—24no—
164#41—24no—
165#41—24no—
166#41—22no—
167#41—22no—
168#41—24no—
169#41—22no—
170#41—22no—
171#41C22⋊D28314yes—
172#41C22×D28314yes—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—2832no—
176C2×Q8—5661no—
177C2×Q8—5662no—
178C2×Q8—5662no—
179C2×Q8C22⋊D28312yes—
180C2×Q8C22×D28311yescharacteristic
181C2×D4—28no—
182C2×D4—22no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4C22⋊D28312yes—
187C2×D4C22⋊D28312yes—
188C2×D4C22×D28311yescharacteristic
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—5661no—
192C22×C4C22⋊D28312yes—
193C22×C4C22×D28311yescharacteristic
194Q16—5668no—
195SD16—5668no—
196M16—2834no—
197M16—5662no—
198C2×C8—5662no—
199C4⋊C4—5664no—
200C4⋊C4—5661no—
201C22⋊C4—5661no—
202C22⋊C4—5664no—
203C42—2832no—
204C23—24no—
205C23—22no—
206C23C2×C22⋊D28312yes—
207C23C2×C22⋊D28311yescharacteristic
208Q8—24no—
209Q8—22no—
210Q8—22no—
211Q8—5664no—
212Q8—11321no—
213Q8—11321no—
214Q8C2×C22⋊D28314yes—
215D4—28no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4C2×C22⋊D28314yes—
223C2×C4—24no—
224C2×C4—28no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—5662no—
231C2×C4—5662no—
232C2×C4—11322no—
233C2×C4C2×C22⋊D28312yes—
234C2×C4C2×C22⋊D28311yescharacteristic
235C2×C4C2×C22⋊D28312yes—
236C2×C4C2×C22⋊D28312yes—
237C2×C4C23×D28311yescharacteristic
238C8—5664no—
239C22—22no—
240C22—24no—
241C22—41no—
242C22—42no—
243C22—82no—
244C22—84no—
245C22#2258357912yes—
246C22C22×C22⋊D28311yescharacteristic
247C4—24no—
248C4—22no—
249C4—11322no—
250C4—11322no—
251C4#2258357911yescharacteristic
252C4#2258357911yescharacteristic
253C4C22×C22⋊D28312yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—44no—
258C2C2×#2258357911yescharacteristic
subgroupG/H
centerC2C2×#22583579
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting, prime radical#585×C283C2
socleC2×C283C2×#120
hypercenter#585D283
hypocenterC283#3022
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×#22583640C2semidirect
2. #31827036C2semidirect
3. #31826648C2semidirect
4. #31826643C2semidirect
5. #31826482C2semidirect
6. #31826449C2semidirect
8. C22×Dic566C22semidirect
10. #22583640C22semidirect
13. C2×#17985376C22semidirect
14. #22583608C22semidirect
15. #22583597C22semidirect
18. #22583569C22semidirect
19. #22583537C22semidirect
20. #22583529C22semidirect
21. #22583527C22semidirect
129. C283#3022semidirect

Maximal subgroups

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

subgroupindex
C2×#225836402
#318270362
#318266482
#318266432
#318264822
#318264492
#585×C2832
#3022283