Finite Group Databasebeta

← #49130618

Subgroups — #49130618

Order 36224.

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

normal subgroups only

This group has 762 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
1#31827263C212yesmaximal, complemented
2#31827032C211yesmaximal, characteristic, complemented
3#31826643C214yesmaximal, complemented
4#31826633C214yesmaximal, complemented
5#31826482C212yesmaximal, complemented
6C2×#22583527C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C283C211yesmaximal, characteristic, complemented
8C2×#17985392C2211yescharacteristic, complemented
9#22583641—22no—
10#22583640C2214yescomplemented
11#22583636—22no—
12#22583635C2214yescomplemented
13#22583630—24no—
14C2×#17985376C2212yescomplemented
15#22583606C2212yescomplemented
16#22583597C2212yescomplemented
17#22583581—22no—
18#22583579—22no—
19#22583569C2211yescharacteristic, complemented
20#22583537C2218yescomplemented
21#22583527C2214yescomplemented
22H(5,𝔽2)×C283—24no—
23H(5,𝔽2)×C283—22no—
24H(5,𝔽2)×C283C2214yes—
25C2×#41×C283—22no—
26C2×#41×C283C2212yes—
27C22×D4×C283—22no—
28C22×D4×C283—22no—
29C22×D4×C283C2211yescharacteristic
30#17985392—22no—
31#17985392—22no—
32C2×Dic566C2311yescharacteristic, complemented
33C2×D1132C2311yescharacteristic, complemented
34C2×C22⋊D283—21no—
35C2×C22⋊D283—24no—
36C2×C4×D283—21no—
37#17985382—24no—
38#17985380—24no—
39#17985379—24no—
40#17985377—24no—
41#17985376—22no—
42#17985376C2314yescomplemented
43C2×C283⋊C8—22no—
44C4⋊Dic283—21no—
45C22⋊Dic283—21no—
46C22⋊Dic283—24no—
47C4×Dic283C2312yescomplemented
48C24×C283—22no—
49C24×C283D412yes—
50#41×C283—28no—
51#41×C283—28no—
52#41×C283—22no—
53#41×C283—22no—
54#41×C283D414yescomplemented
55#41×C283C2314yes—
56C2×Q8×C283D412yescomplemented
57C2×D4×C283—24no—
58C2×D4×C283—28no—
59C2×D4×C283—28no—
60C2×D4×C283—22no—
61C2×D4×C283—28no—
62C2×D4×C283—28no—
63C2×D4×C283—22no—
64C2×D4×C283—22no—
65C2×D4×C283D412yes—
66C2×D4×C283D412yescomplemented
67C2×D4×C283C2312yes—
68C22×C4×C283—22no—
69C22×C4×C283—22no—
70C22×C4×C283C2311yescharacteristic
71C22×D283—21no—
72Dic566—22no—
73D1132—22no—
74C22⋊D283—41no—
75C22⋊D283—41no—
76C22⋊D283—84no—
77C2×Dic283—22no—
78C2×Dic283—21no—
79C4×D283—42no—
80C283⋊C8—24no—
81C23×C283—28no—
82C23×C283—22no—
83C23×C283—22no—
84C23×C283—22no—
85C23×C283—82no—
86C23×C283—84no—
87C23×C283C2×D414yes—
88C23×C283C2×D411yescharacteristic
89Q8×C283—22no—
90Q8×C283C2×D414yes—
91D4×C283—216no—
92D4×C283—24no—
93D4×C283—24no—
94D4×C283—24no—
95D4×C283—22no—
96D4×C283—24no—
97D4×C283C2×D414yes—
98C2×C4×C283—28no—
99C2×C4×C283—22no—
100C2×C4×C283—24no—
101C2×C4×C283—24no—
102C2×C4×C283C2×D411yescharacteristic
103C2×C4×C283C2×D412yes—
104C2×C4×C283C2×D412yes—
105C2×C4×C283C2411yescharacteristic
106C22×C283—24no—
107C22×C283—24no—
108C22×C283—41no—
109C22×C283—44no—
110C22×C283—44no—
111C22×C283—44no—
112C22×C283—82no—
113C22×C283—88no—
114C22×C283#12011yescharacteristic
115C22×C283#12011yescharacteristic
116C22×C283C22×D411yescharacteristic
117C2×D283—42no—
118C2×D283—41no—
119C4×C283—24no—
120C4×C283#12012yes—
121C4×C283C22×D412yes—
122Dic283—42no—
123Dic283—41no—
124C2×C283—21no—
125C2×C283—21no—
126C2×C283—21no—
127C2×C283—44no—
128C2×C283—44no—
129C2×C283C2×#12011yescharacteristic
130D283—81no—
131C283#301911yescharacteristic, complemented, Sylow
132#3019—2831nomaximal, Sylow
133C2×H(5,𝔽2)D28311yescharacteristic, complemented
134#575—2832no—
135#535—2831no—
136#455—2834no—
137#453—2834no—
138#421—2832no—
139C2×#100—2831no—
140H(5,𝔽2)—24no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D28314yes—
143C2×#41—22no—
144C2×#41—2831no—
145C2×#41C2×D28312yes—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C2×D28311yescharacteristic
149#137—2834no—
150#137—5662no—
151AGL(1,ℤ/8)—2834no—
152AGL(1,ℤ/8)—5662no—
153#135—5664no—
154C2×M16—2832no—
155#127—2832no—
156#124—2832no—
157#121—5662no—
158#120—5662no—
159#117—2831no—
160#104—2838no—
161#100—2834no—
162C24—22no—
163C24C22⋊D28312yes—
164#41—28no—
165#41—28no—
166#41—22no—
167#41—22no—
168#41—5662no—
169#41—5662no—
170#41C22⋊D28314yescomplemented
171#41C22×D28314yes—
172C2×Q8—2831no—
173C2×Q8C22⋊D28312yescomplemented
174C2×D4—24no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—2831no—
183C2×D4—5661no—
184C2×D4—5664no—
185C2×D4C22⋊D28312yes—
186C2×D4C22⋊D28312yescomplemented
187C2×D4C22×D28312yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—5661no—
191C22×C4C22×D28311yescharacteristic
192Q16—5664no—
193SD16—5664no—
194SD16—5664no—
195D8—5664no—
196M16—2834no—
197M16—5662no—
198C2×C8—5662no—
199C4⋊C4—5661no—
200C22⋊C4—5661no—
201C22⋊C4—5664no—
202C42—2832no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—82no—
208C23—84no—
209C23—5661no—
210C23C2×C22⋊D28314yes—
211C23C2×C22⋊D28311yescharacteristic
212Q8—22no—
213Q8—5662no—
214Q8C2×C22⋊D28314yes—
215D4—216no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—22no—
220D4—24no—
221D4—5662no—
222D4—11321no—
223D4—11321no—
224D4—22644no—
225D4C2×C22⋊D28314yes—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—5662no—
231C2×C4—5661no—
232C2×C4—11322no—
233C2×C4C2×C22⋊D28311yescharacteristic
234C2×C4C2×C22⋊D28312yes—
235C2×C4C2×C22⋊D28312yes—
236C2×C4C23×D28311yescharacteristic
237C8—5664no—
238C22—24no—
239C22—24no—
240C22—41no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—82no—
245C22—88no—
246C22—11322no—
247C22—11321no—
248C22#2258357911yescharacteristic
249C22#2258357911yescharacteristic
250C22C22×C22⋊D28311yescharacteristic
251C4—24no—
252C4—11322no—
253C4—11321no—
254C4#2258357912yes—
255C4C22×C22⋊D28312yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—22641no—
262C2C2×#2258357911yescharacteristic
subgroupG/H
centerC2C2×#22583579
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×H(5,𝔽2)×C283C2
socleC2×C283C2×#120
hypercenterC2×H(5,𝔽2)D283
hypocenterC283#3019
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #31827263C2semidirect
2. #31827032C2semidirect
3. #31826643C2semidirect
4. #31826633C2semidirect
5. #31826482C2semidirect
6. C2×#22583527C2semidirect
7. C2×H(5,𝔽2)×C283C2semidirect
8. C2×#17985392C22semidirect
10. #22583640C22semidirect
12. #22583635C22semidirect
14. C2×#17985376C22semidirect
15. #22583606C22semidirect
16. #22583597C22semidirect
19. #22583569C22semidirect
20. #22583537C22semidirect
21. #22583527C22semidirect
32. C2×Dic566C23semidirect
33. C2×D1132C23semidirect
42. #17985376C23semidirect
47. C4×Dic283C23semidirect
54. #41×C283D4semidirect
56. C2×Q8×C283D4semidirect
66. C2×D4×C283D4semidirect
131. C283#3019semidirect
133. C2×H(5,𝔽2)D283semidirect
170. #41C22⋊D283semidirect
173. C2×Q8C22⋊D283semidirect
186. C2×D4C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
#318272632
#318270322
#318266432
#318266332
#318264822
C2×#225835272
C2×H(5,𝔽2)×C2832
#3019283