Finite Group Databasebeta

← #49130663

Subgroups — #49130663

Order 36224.

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

normal subgroups only

This group has 772 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#31827029C213yesmaximal, complemented
2#31826652C218yesmaximal, complemented
3#31826439C213yesmaximal, complemented
4#585×C283C211yesmaximal, characteristic, complemented
5C22×D1132C2211yescharacteristic, complemented
6#22583606—23no—
7#22583599—23no—
8#22583584—26no—
9#22583578C22112yescomplemented
10#22583569C2213yescomplemented
11#22583525C22112yescomplemented
122−1+4×C283—23no—
13H(5,𝔽2)×C283—23no—
14H(5,𝔽2)×C283C2214yes—
15C2×#41×C283—26no—
16C2×#41×C283C2213yes—
17C23×D283C2312yes—
18C2×D1132—23no—
19C2×D1132—23no—
20C2×C22⋊D283—212no—
21C4⋊Dic283—23no—
22C22⋊Dic283C2316yes—
23(C2×D283)⋊C4—212no—
24C4×Dic283—23no—
25#41×C283—24no—
26#41×C283—212no—
27#41×C283—26no—
28#41×C283—212no—
29#41×C283—26no—
30C2×Q8×C283—26no—
31C2×Q8×C283D413yescomplemented
32C2×D4×C283—26no—
33C2×D4×C283—212no—
34C2×D4×C283D413yescomplemented
35C2×D4×C283C2316yes—
36C22×C4×C283—24no—
37C22×C4×C283D416yescomplemented
38C22×C4×C283C2311yescharacteristic
39C22×D283—26no—
40C22×D283—82no—
41D1132—41no—
42D1132—43no—
43C22⋊D283—812no—
44C2×Dic283—26no—
45C23×C283—24no—
46C23×C283C2×D416yes—
47C23×C283C2411yescharacteristic
48Q8×C283—24no—
49Q8×C283—26no—
50D4×C283—212no—
51D4×C283—212no—
52D4×C283—26no—
53C2×C4×C283—212no—
54C2×C4×C283—26no—
55C2×C4×C283—26no—
56C2×C4×C283C2×D416yes—
57C2×C4×C283C2×D413yes—
58C2×C4×C283C2×D413yes—
59C22×C283—26no—
60C22×C283—41no—
61C22×C283—46no—
62C22×C283—84no—
63C22×C283C22×D413yes—
64C2×D283—412no—
65C2×D283—42no—
66C4×C283—26no—
67C4×C283#12011yescharacteristic
68C4×C283#12013yes—
69Dic283—46no—
70C2×C283—23no—
71C2×C283—46no—
72C2×C283C2×#12011yescharacteristic
73D283—82no—
74C283#302711yescharacteristic, complemented, Sylow
75#3027—2831nomaximal, Sylow
76#585D28311yescharacteristic, complemented
77#535—2833no—
78B(4,𝔽2)—2838no—
79#410—2833no—
802−1+4—23no—
81H(5,𝔽2)—23no—
82H(5,𝔽2)C2×D28314yes—
83C2×#41—26no—
84C2×#41C2×D28313yes—
85C22×D4—2831no—
86#127—5663no—
87#124—5663no—
88#121—5666no—
89#120—28312no—
90#117—2833no—
91#99—28312no—
92C24—2832no—
93#41—24no—
94#41—212no—
95#41—26no—
96#41—212no—
97#41—26no—
98C2×Q8—26no—
99C2×Q8C22⋊D28313yescomplemented
100C2×D4—26no—
101C2×D4—212no—
102C2×D4—56612no—
103C2×D4—5663no—
104C2×D4—5663no—
105C2×D4C22⋊D28313yescomplemented
106C2×D4C22×D28316yes—
107C22×C4—24no—
108C22×C4C22⋊D28316yescomplemented
109C22×C4C22×D28311yescharacteristic
110C4⋊C4—5663no—
111C22⋊C4—2836no—
112C22⋊C4—56612no—
113C42—5663no—
114C23—24no—
115C23—5666no—
116C23—22642no—
117C23C2×C22⋊D28316yes—
118C23C23×D28311yescharacteristic
119Q8—24no—
120Q8—26no—
121D4—212no—
122D4—212no—
123D4—26no—
124D4—11321no—
125D4—11323no—
126D4—226412no—
127C2×C4—212no—
128C2×C4—26no—
129C2×C4—26no—
130C2×C4—5666no—
131C2×C4C2×C22⋊D28316yes—
132C2×C4C2×C22⋊D28313yes—
133C2×C4C2×C22⋊D28313yes—
134C22—26no—
135C22—41no—
136C22—46no—
137C22—84no—
138C22—113212no—
139C22—11322no—
140C22C22×C22⋊D28313yes—
141C4—26no—
142C4—11326no—
143C4#2258357911yescharacteristic
144C4#2258357913yes—
145C2—23no—
146C2—46no—
147C2—22642no—
148C2C2×#2258357911yescharacteristic
subgroupG/H
centerC2C2×#22583579
commutator, cosocleC23×C283C24
FrattiniC23C23×D283
Fitting#585×C283C2
socleC2×C283C2×#120
hypercenter#585D283
hypocenterC283#3027
LCSG ▷ C23×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #31827029C2semidirect
2. #31826652C2semidirect
3. #31826439C2semidirect
4. #585×C283C2semidirect
5. C22×D1132C22semidirect
9. #22583578C22semidirect
10. #22583569C22semidirect
11. #22583525C22semidirect
31. C2×Q8×C283D4semidirect
34. C2×D4×C283D4semidirect
37. C22×C4×C283D4semidirect
74. C283#3027semidirect
76. #585D283semidirect
99. C2×Q8C22⋊D283semidirect
105. C2×D4C22⋊D283semidirect
108. C22×C4C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
#318270292
#318266522
#318264392
#585×C2832
#3027283