Finite Group Databasebeta

← #49130171

Subgroups — #49130171

Order 36224.

458 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 217 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#31827232C212yesmaximal, complemented
2#31826587C214yesmaximal, complemented
3#31826582C214yesmaximal, complemented
4#31826514C211yesmaximal, characteristic, complemented
5C2×#22583537C212yesmaximal, complemented
6M16×Dic283C211yesmaximal, characteristic, complemented
7#575×C283C211yesmaximal, characteristic
8#22583617—24no—
9#22583617C2218yescomplemented
10C2×#17985376—24no—
11C2×#17985376C2212yes—
12C2×#17985376C2211yescharacteristic
13C2×C4×Dic283C2211yescharacteristic
14#22583546C2214yescomplemented
15#22583537C2214yescomplemented
16#22583537C2214yescomplemented
17#22583520C2212yescomplemented
18C8×Dic283C2212yescomplemented
19C2×#41×C283C2212yes—
20#137×C283C414yescomplemented
21AGL(1,ℤ/8)×C283C414yescomplemented
22#135×C283C2214yes—
23C2×M16×C283C2211yescharacteristic
24C22×Dic283—21no—
25#17985376—28no—
26#17985376—22no—
27#17985376D414yes—
28#17985376C2314yes—
29C2×C283⋊C8—28no—
30C2×C283⋊C8—22no—
31C2×C283⋊C8C2312yes—
32C4×Dic283—21no—
33C4×Dic283C2311yescharacteristic
34C4×Dic283C2311yescharacteristic
35#41×C283—22no—
36#41×C283—22no—
37#41×C283C2×C414yes—
38#41×C283C2314yes—
39C2×Q8×C283C2×C412yes—
40C2×D4×C283—22no—
41C2×D4×C283C2×C412yes—
42C22×C4×C283—22no—
43C22×C4×C283C2311yescharacteristic
44Q16×C283C2×C414yescomplemented
45SD16×C283C2×C418yescomplemented
46D8×C283C2×C414yescomplemented
47M16×C283C2×C414yes—
48C2×C8×C283C2312yes—
49C2×Dic283—21no—
50C2×Dic283—21no—
51C2×Dic283—42no—
52C283⋊C8—24no—
53C283⋊C8C2×D414yes—
54C23×C283—22no—
55C23×C283#4111yescharacteristic
56Q8×C283—22no—
57Q8×C283C22×C414yes—
58D4×C283—24no—
59D4×C283—24no—
60D4×C283—22no—
61D4×C283C22×C414yes—
62C2×C4×C283—22no—
63C2×C4×C283—24no—
64C2×C4×C283—24no—
65C2×C4×C283C22×C412yes—
66C2×C4×C283C2×D412yes—
67C2×C4×C283#4111yescharacteristic
68C2×C4×C283C2411yescharacteristic
69C8×C283C22×C414yes—
70C22×C283—24no—
71C22×C283—41no—
72C22×C283—82no—
73C22×C283C4×D412yes—
74C22×C283C2×#4111yescharacteristic
75C4×C283—24no—
76C4×C283C4×D412yes—
77C4×C283C23×C411yescharacteristic
78C4×C283C22×D411yescharacteristic
79Dic283—22no—
80Dic283—41no—
81C2×C283—21no—
82C2×C283—22no—
83C2×C283—44no—
84C2×C283C2×C4×D411yescharacteristic
85C283#295611yescharacteristic, complemented, Sylow
86#2956—2831nomaximal, Sylow
87#575D28311yescharacteristic
88#568—2832no—
89#444—2834no—
90#443—2834no—
91#430—2831no—
92C2×#104—2832no—
93C4×M16—2831no—
94C2×#41C2×D28312yes—
95#137Dic28314yescomplemented
96AGL(1,ℤ/8)Dic28314yescomplemented
97#135C2×D28314yes—
98#131—2838no—
99#131—5664no—
100C2×M16—2831no—
101C2×M16—2832no—
102C2×M16—5664no—
103C2×M16C2×D28311yescharacteristic
104C2×C42—2831no—
105#108—2834no—
106#104—2834no—
107#104—2834no—
108#97—2832no—
109C4×C8—2832no—
110#41—22no—
111#41—22no—
112#41C2×Dic28314yes—
113#41C22×D28314yes—
114C2×Q8C2×Dic28312yes—
115C2×D4—22no—
116C2×D4C2×Dic28312yes—
117C22×C4—22no—
118C22×C4—5661no—
119C22×C4C22×D28311yescharacteristic
120Q16C2×Dic28314yescomplemented
121SD16C2×Dic28318yescomplemented
122D8C2×Dic28314yescomplemented
123M16—2834no—
124M16—2834no—
125M16—5668no—
126M16—5662no—
127M16C2×Dic28314yes—
128C2×C8—2832no—
129C2×C8—5668no—
130C2×C8—5662no—
131C2×C8C22×D28312yes—
132C42—2831no—
133C42—2831no—
134C42—5661no—
135C23—22no—
136C23#1798539111yescharacteristic
137Q8—22no—
138Q8C22×Dic28314yes—
139D4—24no—
140D4—24no—
141D4—22no—
142D4C22×Dic28314yes—
143C2×C4—22no—
144C2×C4—24no—
145C2×C4—24no—
146C2×C4—5661no—
147C2×C4—5661no—
148C2×C4—11322no—
149C2×C4C22×Dic28312yes—
150C2×C4D4×D28312yes—
151C2×C4#1798539111yescharacteristic
152C2×C4C23×D28311yescharacteristic
153C8—2834no—
154C8—5664no—
155C8C22×Dic28314yes—
156C22—24no—
157C22—41no—
158C22—82no—
159C22D4×Dic28312yes—
160C22C2×#1798539111yescharacteristic
161C4—24no—
162C4—5662no—
163C4—11321no—
164C4D4×Dic28312yes—
165C4C23×Dic28311yescharacteristic
166C4C2×D4×D28311yescharacteristic
167C2—21no—
168C2—22no—
169C2—44no—
170C2C2×D4×Dic28311yescharacteristic
subgroupG/H
centerC4C2×D4×D283
commutatorC4×C283C23×C4
FrattiniC2×C4C23×D283
Fitting, prime radical#575×C283C2
socleC2×C283C2×C4×D4
cosocleC2×C4×C283C24
hypercenter#575D283
hypocenterC283#2956
LCSG ▷ C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #31827232C2semidirect
2. #31826587C2semidirect
3. #31826582C2semidirect
4. #31826514C2semidirect
5. C2×#22583537C2semidirect
6. M16×Dic283C2semidirect
9. #22583617C22semidirect
14. #22583546C22semidirect
15. #22583537C22semidirect
16. #22583537C22semidirect
17. #22583520C22semidirect
18. C8×Dic283C22semidirect
20. #137×C283C4semidirect
21. AGL(1,ℤ/8)×C283C4semidirect
44. Q16×C283C2×C4semidirect
45. SD16×C283C2×C4semidirect
46. D8×C283C2×C4semidirect
85. C283#2956semidirect
95. #137Dic283semidirect
96. AGL(1,ℤ/8)Dic283semidirect
120. Q16C2×Dic283semidirect
121. SD16C2×Dic283semidirect
122. D8C2×Dic283semidirect

Maximal subgroups

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

subgroupindex
#318272322
#318265872
#318265822
#318265142
C2×#225835372
M16×Dic2832
#575×C2832
#2956283