Finite Group Databasebeta

← #49130621

Subgroups — #49130621

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#31827265C212yesmaximal, complemented
3#31827032C211yesmaximal, characteristic, complemented
4#31826645C214yesmaximal, complemented
5#31826635C214yesmaximal, complemented
6#31826481C212yesmaximal, complemented
7C2×#100×C283C211yesmaximal, characteristic, complemented
8#22583655—24no—
9#22583655—22no—
10#22583655C2214yes—
11C2×#17985392C2212yes—
12C2×#17985392C2211yescharacteristic, complemented
13C2×#17985391—22no—
14C22×C22⋊D283—22no—
15C2×D4×D283—22no—
16#22583643—22no—
17#22583643C2214yescomplemented
18#22583638—22no—
19#22583638C2214yescomplemented
20#22583632—24no—
21#22583606C2212yescomplemented
22#22583597C2212yescomplemented
23#22583581—22no—
24#22583579—22no—
25#22583569C2211yescharacteristic, complemented
26#22583536C2218yescomplemented
27C22×D4×C283C2211yescharacteristic
28C2×M16×C283C2212yescomplemented
29#100×C283C2214yescomplemented
30C23×D283—22no—
31#17985392—22no—
32#17985392—22no—
33#17985392—22no—
34#17985392—22no—
35#17985392D414yescomplemented
36#17985392C2314yes—
37#17985391—28no—
38#17985391—28no—
39C2×Dic566D412yescomplemented
40C2×Dic566C2311yescharacteristic, complemented
41C2×D1132D412yescomplemented
42C2×D1132C2311yescharacteristic, complemented
43C2×C22⋊D283—24no—
44C2×C22⋊D283—28no—
45C2×C22⋊D283—22no—
46C2×C22⋊D283—28no—
47C2×C22⋊D283—21no—
48C2×C22⋊D283—24no—
49D4×D283—28no—
50D4×D283—28no—
51C22×Dic283—22no—
52C2×C4×D283—22no—
53C2×C4×D283—21no—
54Dic1132—24no—
55#17985381—24no—
56#17985381—24no—
57D2264—24no—
58C4⋊Dic283—21no—
59C22⋊Dic283—21no—
60C22⋊Dic283—24no—
61C4×Dic283C2312yescomplemented
62C24×C283D412yes—
63C2×D4×C283—22no—
64C2×D4×C283—22no—
65C2×D4×C283D412yes—
66C2×D4×C283C2312yes—
67C22×C4×C283C2311yescharacteristic
68M16×C283—22no—
69M16×C283C2314yescomplemented
70C2×C8×C283—22no—
71C22×D283—22no—
72C22×D283—28no—
73C22×D283—21no—
74C22×D283—84no—
75Dic566—22no—
76Dic566—22no—
77Dic566C2×D414yes—
78D1132—22no—
79D1132—22no—
80D1132C2×D414yes—
81C22⋊D283—216no—
82C22⋊D283—24no—
83C22⋊D283—24no—
84C22⋊D283—41no—
85C22⋊D283—41no—
86C22⋊D283—84no—
87C2×Dic283—28no—
88C2×Dic283—22no—
89C2×Dic283—22no—
90C2×Dic283—21no—
91C4×D283—24no—
92C4×D283—24no—
93C4×D283—42no—
94C23×C283—22no—
95C23×C283—22no—
96C23×C283—82no—
97C23×C283C2×D414yes—
98C23×C283C2×D411yescharacteristic
99D4×C283—24no—
100D4×C283—24no—
101C2×C4×C283C2×D412yes—
102C2×C4×C283C2×D412yes—
103C2×C4×C283C2×D411yescharacteristic
104C2×C4×C283C2411yescharacteristic
105C8×C283—24no—
106C22×C283—24no—
107C22×C283—41no—
108C22×C283—44no—
109C22×C283—44no—
110C22×C283—82no—
111C22×C283#12011yescharacteristic
112C22×C283#12011yescharacteristic
113C22×C283C22×D411yescharacteristic
114C2×D283—24no—
115C2×D283—44no—
116C2×D283—42no—
117C2×D283—41no—
118C2×D283—88no—
119C4×C283#12012yes—
120C4×C283C22×D412yes—
121Dic283—24no—
122Dic283—42no—
123Dic283—41no—
124C2×C283—21no—
125C2×C283—21no—
126C2×C283—21no—
127C2×C283—44no—
128C2×C283C2×#12011yescharacteristic
129D283—44no—
130D283—81no—
131C283#301911yescharacteristic, complemented, Sylow
132#3019—2831nomaximal, Sylow
133C2×H(5,𝔽2)—2831no—
134#575—2832no—
135#535—2831no—
136#455—2834no—
137#453—2834no—
138#421—2832no—
139C2×#100D28311yescharacteristic, complemented
140H(5,𝔽2)—2834no—
141H(5,𝔽2)—5664no—
142H(5,𝔽2)—5662no—
143C2×#41—2832no—
144C2×#41—2831no—
145C2×#41—5662no—
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×M16C2×D28312yescomplemented
155#127—2832no—
156#124—2832no—
157#121—5662no—
158#120—5662no—
159#117—2831no—
160#104—2838no—
161#100C2×D28314yescomplemented
162C24—5662no—
163C24D113212yes—
164#41—2834no—
165#41—2834no—
166#41—5668no—
167#41—5668no—
168#41—5662no—
169#41—5662no—
170#41—5662no—
171#41—5662no—
172C2×Q8—2832no—
173C2×Q8—2831no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—2832no—
177C2×D4—2831no—
178C2×D4—5668no—
179C2×D4—5668no—
180C2×D4—5664no—
181C2×D4—5668no—
182C2×D4—5662no—
183C2×D4—5668no—
184C2×D4—5661no—
185C2×D4—5664no—
186C2×D4D113212yes—
187C2×D4C22×D28312yes—
188C22×C4—5662no—
189C22×C4—5662no—
190C22×C4—5661no—
191C22×C4C22×D28311yescharacteristic
192Q16—5664no—
193SD16—5664no—
194SD16—5664no—
195D8—5664no—
196M16—22no—
197M16C22×D28314yescomplemented
198C2×C8—22no—
199C4⋊C4—5661no—
200C22⋊C4—5661no—
201C22⋊C4—5664no—
202C42—2832no—
203C23—22no—
204C23—22no—
205C23—82no—
206C23—5668no—
207C23—5662no—
208C23—5661no—
209C23—22644no—
210C23C2×D113214yes—
211C23C2×D113211yescharacteristic
212Q8—2834no—
213Q8—5662no—
214Q8—5662no—
215D4—24no—
216D4—24no—
217D4—2834no—
218D4—56616no—
219D4—5664no—
220D4—5664no—
221D4—5662no—
222D4—5662no—
223D4—11321no—
224D4—11321no—
225D4—22644no—
226C2×C4—5668no—
227C2×C4—5662no—
228C2×C4—5664no—
229C2×C4—5664no—
230C2×C4—5662no—
231C2×C4—5661no—
232C2×C4—11322no—
233C2×C4D4×D28312yes—
234C2×C4D4×D28312yes—
235C2×C4C2×D113211yescharacteristic
236C2×C4C23×D28311yescharacteristic
237C8—24no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—82no—
243C22—5664no—
244C22—11324no—
245C22—11322no—
246C22—11321no—
247C22—22648no—
248C22#2258358011yescharacteristic
249C22#2258358011yescharacteristic
250C22C22×D113211yescharacteristic
251C4—5664no—
252C4—11322no—
253C4—11321no—
254C4#2258358012yes—
255C4C2×D4×D28312yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—11324no—
261C2—22641no—
262C2C2×#2258358011yescharacteristic
subgroupG/H
centerC2C2×#22583580
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×#100×C283C2
socleC2×C283C2×#120
hypercenterC2×#100D283
hypocenterC283#3019
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×#22583655C2semidirect
2. #31827265C2semidirect
3. #31827032C2semidirect
4. #31826645C2semidirect
5. #31826635C2semidirect
6. #31826481C2semidirect
7. C2×#100×C283C2semidirect
12. C2×#17985392C22semidirect
17. #22583643C22semidirect
19. #22583638C22semidirect
21. #22583606C22semidirect
22. #22583597C22semidirect
25. #22583569C22semidirect
26. #22583536C22semidirect
28. C2×M16×C283C22semidirect
29. #100×C283C22semidirect
35. #17985392D4semidirect
39. C2×Dic566D4semidirect
40. C2×Dic566C23semidirect
41. C2×D1132D4semidirect
42. C2×D1132C23semidirect
61. C4×Dic283C23semidirect
69. M16×C283C23semidirect
131. C283#3019semidirect
139. C2×#100D283semidirect
154. C2×M16C2×D283semidirect
161. #100C2×D283semidirect
197. M16C22×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836552
#318272652
#318270322
#318266452
#318266352
#318264812
C2×#100×C2832
#3019283