Finite Group Databasebeta

← #49130617

Subgroups — #49130617

Order 36224.

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

normal subgroups only

This group has 762 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
1C2×#22583655C211yesmaximal, characteristic, complemented
2#31827263C212yesmaximal, complemented
3#31826642C214yesmaximal, complemented
4#31826632C214yesmaximal, complemented
5#31826483C212yesmaximal, complemented
6C2×#22583527C211yesmaximal, characteristic, complemented
7#535×C283C211yesmaximal, characteristic, complemented
8#22583655—24no—
9#22583655—22no—
10#22583655C2214yes—
11C2×#17985392C2212yes—
12C2×#17985391—22no—
13C22×C22⋊D283—22no—
14C2×D4×D283—22no—
15#22583641C2214yescomplemented
16#22583640—22no—
17#22583636C2214yescomplemented
18#22583635—22no—
19#22583630—24no—
20C2×#17985376C2212yescomplemented
21#22583538C2218yescomplemented
22#22583527C2214yescomplemented
23C2×#41×C283C2211yescharacteristic, complemented
24C22×D4×C283C2211yescharacteristic
25#127×C283C2212yescomplemented
26#124×C283C2212yescomplemented
27#121×C283—22no—
28#120×C283—22no—
29#117×C283C2211yescharacteristic, complemented
30C23×D283—22no—
31#17985392—22no—
32#17985392—22no—
33#17985392D414yescomplemented
34#17985392C2314yes—
35#17985391—28no—
36#17985391—28no—
37C2×Dic566D412yescomplemented
38C2×D1132D412yescomplemented
39C2×C22⋊D283—24no—
40C2×C22⋊D283—28no—
41C2×C22⋊D283—22no—
42C2×C22⋊D283—28no—
43D4×D283—28no—
44D4×D283—28no—
45C22×Dic283—22no—
46C2×C4×D283—22no—
47#17985382—24no—
48#17985380—24no—
49#17985379—24no—
50#17985377—24no—
51#17985376—22no—
52#17985376C2314yescomplemented
53C2×C283⋊C8—22no—
54C24×C283D412yes—
55#41×C283—22no—
56#41×C283—22no—
57C2×Q8×C283C2311yescharacteristic, complemented
58C2×D4×C283—21no—
59C2×D4×C283—24no—
60C2×D4×C283—22no—
61C2×D4×C283—22no—
62C2×D4×C283D412yes—
63C2×D4×C283C2312yes—
64C2×D4×C283C2311yescharacteristic, complemented
65C22×C4×C283—21no—
66C22×C4×C283C2311yescharacteristic
67C4⋊C4×C283—21no—
68C22⋊C4×C283—21no—
69C22⋊C4×C283—24no—
70C42×C283C2312yescomplemented
71C22×D283—22no—
72C22×D283—28no—
73C22×D283—84no—
74Dic566—22no—
75Dic566C2×D414yes—
76D1132—22no—
77D1132C2×D414yes—
78C22⋊D283—216no—
79C22⋊D283—24no—
80C22⋊D283—24no—
81C2×Dic283—28no—
82C2×Dic283—22no—
83C4×D283—24no—
84C4×D283—24no—
85C283⋊C8—24no—
86C23×C283—22no—
87C23×C283—22no—
88C23×C283—21no—
89C23×C283—82no—
90C23×C283C2×D414yes—
91C23×C283C2×D411yescharacteristic
92Q8×C283—22no—
93D4×C283—24no—
94D4×C283—24no—
95D4×C283—22no—
96D4×C283—41no—
97D4×C283—41no—
98D4×C283—84no—
99C2×C4×C283—22no—
100C2×C4×C283—21no—
101C2×C4×C283—42no—
102C2×C4×C283C2×D412yes—
103C2×C4×C283C2×D412yes—
104C2×C4×C283C2×D411yescharacteristic
105C2×C4×C283C2411yescharacteristic
106C22×C283—24no—
107C22×C283—41no—
108C22×C283—44no—
109C22×C283—44no—
110C22×C283—42no—
111C22×C283—41no—
112C22×C283—82no—
113C22×C283#12011yescharacteristic
114C22×C283#12011yescharacteristic
115C22×C283C22×D411yescharacteristic
116C2×D283—24no—
117C2×D283—44no—
118C2×D283—88no—
119C4×C283—42no—
120C4×C283—41no—
121C4×C283#12012yes—
122C4×C283C22×D412yes—
123Dic283—24no—
124C2×C283—21no—
125C2×C283—21no—
126C2×C283—21no—
127C2×C283—44no—
128C2×C283—81no—
129C2×C283C2×#12011yescharacteristic
130D283—44no—
131C283#301911yescharacteristic, complemented, Sylow
132#3019—2831nomaximal, Sylow
133C2×H(5,𝔽2)—2831no—
134#575—2832no—
135#535D28311yescharacteristic, complemented
136#455—2834no—
137#453—2834no—
138#421—2832no—
139C2×#100—2831no—
140H(5,𝔽2)—2834no—
141H(5,𝔽2)—5664no—
142H(5,𝔽2)—5662no—
143C2×#41—2832no—
144C2×#41—5662no—
145C2×#41C2×D28311yescharacteristic, complemented
146C22×D4—5662no—
147C22×D4—5662no—
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#127C2×D28312yescomplemented
156#124C2×D28312yescomplemented
157#121—22no—
158#120—22no—
159#117C2×D28311yescharacteristic, complemented
160#104—2838no—
161#100—2834no—
162C24—5662no—
163C24C22⋊D28312yes—
164#41—22no—
165#41—22no—
166#41—2834no—
167#41—2834no—
168#41—5668no—
169#41—5668no—
170#41—5662no—
171#41—5662no—
172C2×Q8—2832no—
173C2×Q8C22×D28311yescharacteristic, complemented
174C2×D4—21no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—2832no—
179C2×D4—5668no—
180C2×D4—5668no—
181C2×D4—5664no—
182C2×D4—5668no—
183C2×D4—5662no—
184C2×D4—5668no—
185C2×D4C22⋊D28312yes—
186C2×D4C22×D28312yes—
187C2×D4C22×D28311yescharacteristic, complemented
188C22×C4—21no—
189C22×C4—5662no—
190C22×C4—5662no—
191C22×C4C22×D28311yescharacteristic
192Q16—5664no—
193SD16—5664no—
194SD16—5664no—
195D8—5664no—
196M16—2834no—
197M16—5662no—
198C2×C8—5662no—
199C4⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—24no—
202C42C22×D28312yescomplemented
203C23—22no—
204C23—22no—
205C23—21no—
206C23—82no—
207C23—5668no—
208C23—5662no—
209C23—22644no—
210C23C2×C22⋊D28314yes—
211C23C2×C22⋊D28311yescharacteristic
212Q8—22no—
213Q8—2834no—
214Q8—5662no—
215D4—24no—
216D4—24no—
217D4—22no—
218D4—41no—
219D4—41no—
220D4—84no—
221D4—2834no—
222D4—56616no—
223D4—5664no—
224D4—5664no—
225D4—5662no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—5668no—
230C2×C4—5662no—
231C2×C4—5664no—
232C2×C4—5664no—
233C2×C4D4×D28312yes—
234C2×C4D4×D28312yes—
235C2×C4C2×C22⋊D28311yescharacteristic
236C2×C4C23×D28311yescharacteristic
237C8—5664no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—42no—
243C22—41no—
244C22—82no—
245C22—5664no—
246C22—11324no—
247C22—22648no—
248C22#2258357811yescharacteristic
249C22#2258357811yescharacteristic
250C22C22×C22⋊D28311yescharacteristic
251C4—42no—
252C4—41no—
253C4—5664no—
254C4#2258357812yes—
255C4C2×D4×D28312yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—81no—
261C2—11324no—
262C2C2×#2258357811yescharacteristic
subgroupG/H
centerC2C2×#22583578
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#535×C283C2
socleC2×C283C2×#120
hypercenter#535D283
hypocenterC283#3019
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#22583655C2semidirect
2. #31827263C2semidirect
3. #31826642C2semidirect
4. #31826632C2semidirect
5. #31826483C2semidirect
6. C2×#22583527C2semidirect
7. #535×C283C2semidirect
15. #22583641C22semidirect
17. #22583636C22semidirect
20. C2×#17985376C22semidirect
21. #22583538C22semidirect
22. #22583527C22semidirect
23. C2×#41×C283C22semidirect
25. #127×C283C22semidirect
26. #124×C283C22semidirect
29. #117×C283C22semidirect
33. #17985392D4semidirect
37. C2×Dic566D4semidirect
38. C2×D1132D4semidirect
52. #17985376C23semidirect
57. C2×Q8×C283C23semidirect
64. C2×D4×C283C23semidirect
70. C42×C283C23semidirect
131. C283#3019semidirect
135. #535D283semidirect
145. C2×#41C2×D283semidirect
155. #127C2×D283semidirect
156. #124C2×D283semidirect
159. #117C2×D283semidirect
173. C2×Q8C22×D283semidirect
187. C2×D4C22×D283semidirect
202. C42C22×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836552
#318272632
#318266422
#318266322
#318264832
C2×#225835272
#535×C2832
#3019283