Finite Group Databasebeta

← #45663618

Subgroups — #45663618

Order 32896.

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

normal subgroups only

This group has 206 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#30657127C211yesmaximal, characteristic
2#30657103C212yesmaximal, complemented
3#30656479C212yesmaximal, complemented
4#30656477C211yesmaximal, characteristic, complemented
5#563×C257C211yesmaximal, characteristic, complemented
6C2×#16861605—21no—
7C2×#16861603—21no—
8#22551589C2211yescharacteristic
9#22551512—22no—
10#22551512—22no—
11#22551503C2212yes—
12#22551503C2211yescharacteristic
13C4⋊Q8×C257—21no—
14#126×C257—22no—
15#125×C257C2212yes—
16#124×C257C2211yescharacteristic
17#123×C257—21no—
18C22⋊Q8×C257—22no—
19C2×D1028—21no—
20C2×D1028—22no—
21#16861605—42no—
22#16861603—42no—
23C2×#14523339—21no—
24C2×#14523339—22no—
25C2×Q8×C257—21no—
26C2×Q8×C257D411yescharacteristic
27C2×D4×C257D411yescharacteristic
28C22×C4×C257—21no—
29C4⋊C4×C257—22no—
30C4⋊C4×C257—22no—
31C4⋊C4×C257—22no—
32C4⋊C4×C257D412yes—
33C4⋊C4×C257D412yes—
34C22⋊C4×C257—22no—
35C22⋊C4×C257—21no—
36C22⋊C4×C257—22no—
37C42×C257—21no—
38C42×C257C2311yescharacteristic
39C22×D257—41no—
40D1028—41no—
41D1028—41no—
42D1028—44no—
43#14523339—22no—
44#14523339—42no—
45C23×C257—21no—
46Q8×C257—41no—
47Q8×C257—81no—
48D4×C257—41no—
49C2×C4×C257—22no—
50C2×C4×C257—21no—
51C2×C4×C257—22no—
52C2×C4×C257—81no—
53C2×C4×C257C2×D412yes—
54C2×C4×C257C2×D411yescharacteristic
55C22×C257—41no—
56C22×C257—42no—
57C22×C257#12011yescharacteristic
58C2×D257—81no—
59C2×D257—82no—
60C4×C257—21no—
61C4×C257—22no—
62C4×C257—42no—
63C4×C257—41no—
64C4×C257—42no—
65C2×C257—81no—
66C2×C257#45311yescharacteristic
67C2×C257#45412yes—
68D257—161no—
69C257#165711yescharacteristic, complemented, Sylow
70#1657—2571nomaximal, Sylow
71#563D25711yescharacteristic, complemented
72#496—2571no—
73#490—2572no—
74#330—2572no—
75#329—2571no—
76C2×SD16—5141no—
77C2×D8—5141no—
78C4⋊Q8—21no—
79#127—2571no—
80#126—22no—
81#125C2×D25712yes—
82#124C2×D25711yescharacteristic
83#123—21no—
84C22⋊Q8—22no—
85#102—5142no—
86#102—5142no—
87#97—2572no—
88#97—2571no—
89C2×Q8—21no—
90C2×Q8C22⋊D25711yescharacteristic
91C2×D4—5142no—
92C2×D4—5141no—
93C2×D4C22⋊D25711yescharacteristic
94C22×C4—21no—
95SD16—10282no—
96D8—10282no—
97C2×C8—5141no—
98C2×C8—5142no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4C22⋊D25712yes—
103C4⋊C4C22⋊D25712yes—
104C22⋊C4—22no—
105C22⋊C4—21no—
106C22⋊C4—22no—
107C42—21no—
108C42C22×D25711yescharacteristic
109C23—21no—
110C23—10281no—
111Q8—41no—
112Q8—81no—
113D4—41no—
114D4—10281no—
115D4—10281no—
116D4—10284no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—81no—
121C2×C4C2×C22⋊D25712yes—
122C2×C4C2×C22⋊D25711yescharacteristic
123C8—5142no—
124C8—10282no—
125C22—41no—
126C22—42no—
127C22—20561no—
128C22—20562no—
129C22#2255156111yescharacteristic
130C4—21no—
131C4—22no—
132C4—42no—
133C4—41no—
134C4—42no—
135C2—81no—
136C2—41121no—
137C2#3065690511yescharacteristic
138C2#3065691012yes—
subgroupG/H
centerC22#22551561
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#563×C257C2
socleC22×C257#120
prime radical#30657127C2
hypercenter#563D257
hypocenterC257#1657
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
2. #30657103C2semidirect
3. #30656479C2semidirect
4. #30656477C2semidirect
5. #563×C257C2semidirect
69. C257#1657semidirect
71. #563D257semidirect

Maximal subgroups

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

subgroupindex
#306571272
#306571032
#306564792
#306564772
#563×C2572
#1657257