Finite Group Databasebeta

← #45663613

Subgroups — #45663613

Order 32896.

230 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 49 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#30657127C212yesmaximal, complemented
2#30657103C211yesmaximal, characteristic, complemented
3#30656479C211yesmaximal, characteristic, complemented
4#30656477C212yesmaximal, complemented
5#562×C257C211yesmaximal, characteristic, complemented
6C2×#16861605—22no—
7C2×#16861603—22no—
8#22551589C2211yescharacteristic
9#22551512—22no—
10#22551503C2211yescharacteristic
11#22551503C2212yes—
12#125×C257C2211yescharacteristic
13#124×C257C2212yes—
14#123×C257—22no—
15C22⋊Q8×C257—22no—
16#121×C257—22no—
17C2×D1028—22no—
18C2×D1028—21no—
19#16861605—44no—
20#16861603—44no—
21C2×#14523339—22no—
22C2×#14523339—21no—
23C2×Q8×C257D412yes—
24C2×D4×C257—22no—
25C2×D4×C257D412yes—
26C22×C4×C257—22no—
27C4⋊C4×C257—22no—
28C4⋊C4×C257—22no—
29C4⋊C4×C257D412yes—
30C22⋊C4×C257—22no—
31C22⋊C4×C257—22no—
32C22⋊C4×C257—22no—
33C42×C257C2311yescharacteristic
34C22×D257—41no—
35D1028—42no—
36D1028—42no—
37D1028—42no—
38#14523339—24no—
39#14523339—41no—
40C23×C257—22no—
41Q8×C257—42no—
42D4×C257—42no—
43D4×C257—82no—
44C2×C4×C257—22no—
45C2×C4×C257—22no—
46C2×C4×C257—82no—
47C2×C4×C257C2×D411yescharacteristic
48C2×C4×C257C2×D412yes—
49C22×C257—42no—
50C22×C257—42no—
51C22×C257—42no—
52C22×C257#12011yescharacteristic
53C2×D257—82no—
54C2×D257—81no—
55C4×C257—22no—
56C4×C257—21no—
57C4×C257—42no—
58C4×C257—42no—
59C2×C257—82no—
60C2×C257#45312yes—
61C2×C257#45411yescharacteristic
62D257—161no—
63C257#165611yescharacteristic, complemented, Sylow
64#1656—2571nomaximal, Sylow
65#562D25711yescharacteristic, complemented
66#496—2572no—
67#490—2571no—
68#330—2571no—
69#329—2572no—
70C2×SD16—5142no—
71C2×D8—5142no—
72#127—2571no—
73#125C2×D25711yescharacteristic
74#124C2×D25712yes—
75#123—22no—
76C22⋊Q8—22no—
77#121—22no—
78#102—5142no—
79#97—2571no—
80#97—2572no—
81C2×Q8C22⋊D25712yes—
82C2×D4—22no—
83C2×D4—5141no—
84C2×D4—5142no—
85C2×D4C22⋊D25712yes—
86C22×C4—22no—
87SD16—10284no—
88D8—10284no—
89C2×C8—5142no—
90C2×C8—5141no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4C22⋊D25712yes—
94C22⋊C4—22no—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C42C22×D25711yescharacteristic
98C23—22no—
99C23—10281no—
100Q8—42no—
101D4—42no—
102D4—10282no—
103D4—10282no—
104D4—10282no—
105D4—82no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—82no—
109C2×C4C2×C22⋊D25711yescharacteristic
110C2×C4C2×C22⋊D25712yes—
111C8—5144no—
112C8—10281no—
113C22—42no—
114C22—42no—
115C22—42no—
116C22—20562no—
117C22—20561no—
118C22#2255156111yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—42no—
123C2—82no—
124C2—41121no—
125C2#3065690512yes—
126C2#3065691011yescharacteristic
subgroupG/H
centerC22#22551561
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#562×C257C2
socleC22×C257#120
hypercenter#562D257
hypocenterC257#1656
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #30657127C2semidirect
2. #30657103C2semidirect
3. #30656479C2semidirect
4. #30656477C2semidirect
5. #562×C257C2semidirect
63. C257#1656semidirect
65. #562D257semidirect

Maximal subgroups

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

subgroupindex
#306571272
#306571032
#306564792
#306564772
#562×C2572
#1656257