Finite Group Databasebeta

← #49130639

Subgroups — #49130639

Order 36224.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#31827301C211yesmaximal, characteristic
2C2×#22583641C212yesmaximal, complemented
3#31826647C214yesmaximal, complemented
4#31826642C214yesmaximal, complemented
5#31826483C212yesmaximal, complemented
6#31826449C211yesmaximal, characteristic, complemented
7#536×C283C211yesmaximal, characteristic, complemented
8#22583658C2212yes—
9#22583657—22no—
10#22583656—22no—
11#22583655—22no—
12#22583655C2212yes—
13C2×#17985392—22no—
14C2×#17985392C2212yes—
15#41×D283—24no—
16#22583641—24no—
17#22583641C2218yescomplemented
18C2×#17985382—22no—
19C2×#17985379—22no—
20C2×#17985376C2212yescomplemented
21#22583538C2218yescomplemented
22#22583529C2212yescomplemented
23#22583527C2212yescomplemented
24C2×#41×C283C2211yescharacteristic
25C22×Q8×C283C2211yescharacteristic, complemented
26C4⋊Q8×C283C2212yescomplemented
27#124×C283C2212yescomplemented
28#123×C283—22no—
29C22⋊Q8×C283—22no—
30#117×C283C2211yescharacteristic, complemented
31#17985393—24no—
32#17985392—24no—
33#17985392—24no—
34#17985392—22no—
35#17985392—24no—
36#17985392—22no—
37#17985392D414yes—
38#17985392C2314yes—
39#17985391—28no—
40#17985391—24no—
41C2×Dic566—22no—
42C2×Dic566D412yes—
43Q8×D283—24no—
44C2×D1132—22no—
45C2×D1132D412yes—
46C2×C22⋊D283—24no—
47C2×C22⋊D283—22no—
48D4×D283—28no—
49D4×D283—24no—
50C2×C4×D283—24no—
51C2×C4×D283—22no—
52#17985382—28no—
53#17985379—28no—
54#17985376—22no—
55#17985376C2314yes—
56C2×C283⋊C8—22no—
57#41×C283—22no—
58#41×C283—22no—
59C2×Q8×C283—21no—
60C2×Q8×C283—22no—
61C2×Q8×C283—22no—
62C2×Q8×C283C2312yes—
63C2×Q8×C283C2311yescharacteristic
64C2×D4×C283D412yes—
65C2×D4×C283C2311yescharacteristic
66C22×C4×C283—21no—
67C22×C4×C283D412yes—
68C22×C4×C283C2311yescharacteristic
69C4⋊C4×C283—21no—
70C4⋊C4×C283—24no—
71C22⋊C4×C283—21no—
72C22⋊C4×C283—24no—
73C42×C283C2312yes—
74C22×D283—24no—
75C22×D283—22no—
76Dic566—24no—
77Dic566—22no—
78Dic566C2×D414yes—
79D1132—24no—
80D1132—22no—
81D1132C2×D414yes—
82C22⋊D283—28no—
83C22⋊D283—24no—
84C22⋊D283—24no—
85C2×Dic283—24no—
86C2×Dic283—22no—
87C4×D283—28no—
88C4×D283—24no—
89C4×D283—24no—
90C283⋊C8—24no—
91C23×C283C2×D412yes—
92C23×C283C2×D411yescharacteristic
93Q8×C283—22no—
94Q8×C283—24no—
95Q8×C283—41no—
96Q8×C283—41no—
97D4×C283—24no—
98D4×C283—22no—
99C2×C4×C283—22no—
100C2×C4×C283—22no—
101C2×C4×C283—22no—
102C2×C4×C283—22no—
103C2×C4×C283—42no—
104C2×C4×C283C2×D412yes—
105C2×C4×C283C2×D412yes—
106C2×C4×C283C2×D412yes—
107C2×C4×C283C2×D411yescharacteristic
108C2×C4×C283C2411yescharacteristic
109C22×C283—22no—
110C22×C283—41no—
111C22×C283—42no—
112C22×C283#12012yes—
113C22×C283C22×D411yescharacteristic
114C2×D283—24no—
115C2×D283—82no—
116C2×D283—84no—
117C4×C283—22no—
118C4×C283—42no—
119C4×C283—42no—
120C4×C283#12011yescharacteristic
121C4×C283#12011yescharacteristic
122C4×C283C22×D412yes—
123Dic283—24no—
124C2×C283—21no—
125C2×C283—22no—
126C2×C283—42no—
127C2×C283C2×#12011yescharacteristic
128D283—44no—
129C283#302211yescharacteristic, complemented, Sylow
130#3022—2831nomaximal, Sylow
131#585—2831no—
132C2×#137—2832no—
133#536D28311yescharacteristic, complemented
134#456—2834no—
135#455—2834no—
136#421—2832no—
137#413—2831no—
1382−1+4—2832no—
1392−1+4—5662no—
140H(5,𝔽2)—2832no—
141H(5,𝔽2)—5662no—
142H(5,𝔽2)—5662no—
143C2×#41—2832no—
144C2×#41—5664no—
145C2×#41—5662no—
146C2×#41C2×D28311yescharacteristic
147C22×Q8C2×D28311yescharacteristic, complemented
148#137—2838no—
149#137—5664no—
150C2×Q16—5662no—
151C2×SD16—5662no—
152C2×M16—2832no—
153C4⋊Q8C2×D28312yescomplemented
154#124C2×D28312yescomplemented
155#123—22no—
156C22⋊Q8—22no—
157#117C2×D28311yescharacteristic, complemented
158#104—2838no—
159#101—2832no—
160#100—2832no—
161#41—22no—
162#41—22no—
163#41—2834no—
164#41—2834no—
165#41—5664no—
166#41—5664no—
167#41—5664no—
168#41—5668no—
169#41—5664no—
170#41—5662no—
171#41—5662no—
172#41—5664no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—2832no—
177C2×Q8—5662no—
178C2×Q8—5664no—
179C2×Q8C22×D28312yes—
180C2×Q8C22×D28311yescharacteristic
181C2×D4—2832no—
182C2×D4—5662no—
183C2×D4—5664no—
184C2×D4—5668no—
185C2×D4—5662no—
186C2×D4—5664no—
187C2×D4C22⋊D28312yes—
188C2×D4C22×D28311yescharacteristic
189C22×C4—21no—
190C22×C4—5664no—
191C22×C4—5662no—
192C22×C4C22⋊D28312yes—
193C22×C4C22×D28311yescharacteristic
194Q16—5668no—
195SD16—5668no—
196M16—2834no—
197M16—5662no—
198C2×C8—5662no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C22×D28312yes—
204C23—5664no—
205C23—5662no—
206C23C2×C22⋊D28312yes—
207C23C2×C22⋊D28311yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—41no—
211Q8—41no—
212Q8—2834no—
213Q8—5664no—
214Q8—5662no—
215D4—24no—
216D4—22no—
217D4—2834no—
218D4—5664no—
219D4—5668no—
220D4—5664no—
221D4—5664no—
222D4—5662no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—5668no—
229C2×C4—5664no—
230C2×C4—5662no—
231C2×C4—5664no—
232C2×C4—5664no—
233C2×C4D4×D28312yes—
234C2×C4D4×D28312yes—
235C2×C4C2×C22⋊D28312yes—
236C2×C4C2×C22⋊D28311yescharacteristic
237C2×C4C23×D28311yescharacteristic
238C8—5664no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—5664no—
243C22—22642no—
244C22—22644no—
245C22#2258357812yes—
246C22C22×C22⋊D28311yescharacteristic
247C4—22no—
248C4—42no—
249C4—42no—
250C4—5664no—
251C4#2258357811yescharacteristic
252C4#2258357811yescharacteristic
253C4C2×D4×D28312yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—11324no—
258C2C2×#2258357811yescharacteristic
subgroupG/H
centerC2C2×#22583578
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#536×C283C2
socleC2×C283C2×#120
prime radical#31827301C2
hypercenter#536D283
hypocenterC283#3022
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#22583641C2semidirect
3. #31826647C2semidirect
4. #31826642C2semidirect
5. #31826483C2semidirect
6. #31826449C2semidirect
7. #536×C283C2semidirect
17. #22583641C22semidirect
20. C2×#17985376C22semidirect
21. #22583538C22semidirect
22. #22583529C22semidirect
23. #22583527C22semidirect
25. C22×Q8×C283C22semidirect
26. C4⋊Q8×C283C22semidirect
27. #124×C283C22semidirect
30. #117×C283C22semidirect
129. C283#3022semidirect
133. #536D283semidirect
147. C22×Q8C2×D283semidirect
153. C4⋊Q8C2×D283semidirect
154. #124C2×D283semidirect
157. #117C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318273012
C2×#225836412
#318266472
#318266422
#318264832
#318264492
#536×C2832
#3022283