Finite Group Databasebeta

← #45663328

Subgroups — #45663328

Order 32896.

210 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 57 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#30657065C212yesmaximal
2#30656667C211yesmaximal, characteristic
3#30656474C212yesmaximal
4#30656463C211yesmaximal, characteristic
5#544×C257C211yesmaximal, characteristic
6C2×C4⋊Dic257C2211yescharacteristic
7C2×#16861597—21no—
8#22551523—22no—
9#22551516—22no—
10C22⋊#14523339C2212yes—
11#22551499—21no—
12#22551499C2211yescharacteristic
13C4⋊Q8×C257—22no—
14#125×C257—22no—
15C22⋊Q8×C257—21no—
16C22⋊Q8×C257C2212yes—
17#117×C257—21no—
18C2×C4⋊C4×C257C2211yescharacteristic
19C22×Dic257—21no—
20C22×Dic257—21no—
21C2×#14523339—22no—
22C4⋊Dic257—22no—
23#16861597—42no—
24C2×Q8×C257—21no—
25C2×Q8×C257D412yes—
26C22×C4×C257D411yescharacteristic
27C22×C4×C257D411yescharacteristic
28C22×C4×C257C2311yescharacteristic
29C4⋊C4×C257—21no—
30C4⋊C4×C257—21no—
31C4⋊C4×C257—21no—
32C4⋊C4×C257—22no—
33C4⋊C4×C257—22no—
34C4⋊C4×C257—22no—
35C4⋊C4×C257D412yes—
36C22⋊C4×C257—22no—
37C42×C257—22no—
38C2×Dic257—22no—
39C2×Dic257—41no—
40C2×Dic257—41no—
41C2×Dic257—41no—
42C2×Dic257—41no—
43C2×Dic257—41no—
44#14523339—42no—
45C23×C257C2×D411yescharacteristic
46Q8×C257—42no—
47Q8×C257—81no—
48C2×C4×C257—21no—
49C2×C4×C257—21no—
50C2×C4×C257—22no—
51C2×C4×C257—22no—
52C2×C4×C257—21no—
53C2×C4×C257—41no—
54C2×C4×C257—41no—
55C2×C4×C257SD1612yes—
56C2×C4×C257C2×D412yes—
57C22×C257—21no—
58C22×C257—21no—
59C22×C257#12011yescharacteristic
60C22×C257C2×SD1611yescharacteristic
61C22×C257#13711yescharacteristic
62C4×C257—22no—
63C4×C257—22no—
64C4×C257—42no—
65C4×C257—41no—
66C4×C257—42no—
67Dic257—42no—
68Dic257—81no—
69C2×C257—21no—
70C2×C257—21no—
71C2×C257#44811yescharacteristic
72C2×C257#45611yescharacteristic
73C2×C257#45811yescharacteristic
74C257#161111yescharacteristic, complemented, Sylow
75#1611—2571nomaximal, Sylow
76#544D25711yescharacteristic
77#483—2572no—
78#398—2571no—
79#328—2572no—
80#324—2571no—
81C4⋊Q8—22no—
82#125—22no—
83C22⋊Q8—21no—
84C22⋊Q8C2×D25712yes—
85#117—21no—
86C2×C4⋊C4—2571no—
87C2×C4⋊C4—5141no—
88C2×C4⋊C4C2×D25711yescharacteristic
89#106—5142no—
90#103—5142no—
91C22⋊C8—2572no—
92#95—2571no—
93#95—5141no—
94C2×Q8—21no—
95C2×Q8C22⋊D25712yes—
96C22×C4—5141no—
97C22×C4—5141no—
98C22×C4C22⋊D25711yescharacteristic
99C22×C4C22⋊D25711yescharacteristic
100C22×C4C22×D25711yescharacteristic
101C2×C8—5142no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—5142no—
109C4⋊C4—10282no—
110C4⋊C4C22⋊D25712yes—
111C22⋊C4—22no—
112C42—22no—
113C23C2×C22⋊D25711yescharacteristic
114Q8—42no—
115Q8—81no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—21no—
121C2×C4—5142no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—10281no—
125C2×C4—10281no—
126C2×C4—10281no—
127C2×C4—10281no—
128C2×C4—10281no—
129C2×C4#1686160512yes—
130C2×C4C2×C22⋊D25712yes—
131C8—10282no—
132C22—21no—
133C22—21no—
134C22#2255156111yescharacteristic
135C22C2×#1686160511yescharacteristic
136C22#2255162311yescharacteristic
137C4—22no—
138C4—22no—
139C4—42no—
140C4—41no—
141C4—42no—
142C4—10282no—
143C4—20561no—
144C2—21no—
145C2—21no—
146C2#3065687611yescharacteristic
147C2#3065692011yescharacteristic
148C2#3065692711yescharacteristic
subgroupG/H
centerC22#22551561
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#544×C257C2
socleC22×C257#120
prime radicalC23×C257C2×D4
hypercenter#544D257
hypocenterC257#1611
LCSG ▷ C22×C4×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
74. C257#1611semidirect

Maximal subgroups

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

subgroupindex
#306570652
#306566672
#306564742
#306564632
#544×C2572
#1611257