Finite Group Databasebeta

← #45663327

Subgroups — #45663327

Order 32896.

210 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 53 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#30657368C211yesmaximal, characteristic
2#30657067C212yesmaximal
3#30656473C212yesmaximal
4#30656463C211yesmaximal, characteristic
5#398×C257C211yesmaximal, characteristic
6#22551591—22no—
7#22551583—22no—
8#22551571—21no—
9#22551571C2212yes—
10#22551551—21no—
11#22551523—22no—
12#22551515—22no—
13C22⋊#14523339C2212yes—
14C2×C4⋊C4×C257—21no—
15C2×C4⋊C4×C257C2211yescharacteristic
16C2×C4⋊C4×C257C2211yescharacteristic
17#95×C257—21no—
18#95×C257C2211yescharacteristic
19C2×Dic514—21no—
20C2×Dic514D412yes—
21C2×#14523339—22no—
22C4⋊Dic257—21no—
23C4⋊Dic257—21no—
24C4⋊Dic257—21no—
25C4⋊Dic257D412yes—
26#16861597—22no—
27#16861597—22no—
28C22⋊Dic257—22no—
29C4×Dic257—22no—
30C22×C4×C257—21no—
31C22×C4×C257—21no—
32C22×C4×C257D411yescharacteristic
33C22×C4×C257D411yescharacteristic
34C22×C4×C257C2311yescharacteristic
35C4⋊C4×C257—22no—
36C4⋊C4×C257—22no—
37C4⋊C4×C257—42no—
38Dic514—42no—
39Dic514—81no—
40C2×Dic257—22no—
41C2×Dic257—22no—
42#14523339—42no—
43C23×C257C2×D411yescharacteristic
44C2×C4×C257—21no—
45C2×C4×C257—21no—
46C2×C4×C257—22no—
47C2×C4×C257—21no—
48C2×C4×C257—41no—
49C2×C4×C257—41no—
50C2×C4×C257—41no—
51C2×C4×C257—41no—
52C2×C4×C257—41no—
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—81no—
67Dic257—42no—
68Dic257—42no—
69C2×C257—21no—
70C2×C257—21no—
71C2×C257#44811yescharacteristic
72C2×C257#45611yescharacteristic
73C2×C257#45811yescharacteristic
74C257#161111yescharacteristic, complemented, Sylow
75#1611—2571nomaximal, Sylow
76#544—2571no—
77#483—2572no—
78#398D25711yescharacteristic
79#328—2572no—
80#324—2571no—
81C4⋊Q8—5142no—
82#125—5142no—
83C22⋊Q8—2572no—
84C22⋊Q8—5141no—
85#117—5141no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4C2×D25711yescharacteristic
88C2×C4⋊C4C2×D25711yescharacteristic
89#106—5142no—
90#103—5142no—
91C22⋊C8—2572no—
92#95—21no—
93#95C2×D25711yescharacteristic
94C2×Q8—2572no—
95C2×Q8—5141no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4C22⋊D25711yescharacteristic
99C22×C4C22⋊D25711yescharacteristic
100C22×C4C22×D25711yescharacteristic
101C2×C8—5142no—
102C4⋊C4—2572no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—5141no—
106C4⋊C4—5141no—
107C4⋊C4—5141no—
108C4⋊C4—5142no—
109C4⋊C4—5142no—
110C4⋊C4—42no—
111C22⋊C4—5142no—
112C42—5142no—
113C23C2×C22⋊D25711yescharacteristic
114Q8—10282no—
115Q8—20561no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—21no—
120C2×C4—5142no—
121C2×C4—5142no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4#1686160612yes—
130C2×C4D4×D25712yes—
131C8—10282no—
132C22—21no—
133C22—21no—
134C22#2255156011yescharacteristic
135C22C2×#1686160611yescharacteristic
136C22#2255162211yescharacteristic
137C4—22no—
138C4—22no—
139C4—42no—
140C4—41no—
141C4—10282no—
142C4—10282no—
143C4—81no—
144C2—21no—
145C2—21no—
146C2#3065687511yescharacteristic
147C2#3065691911yescharacteristic
148C2#3065692611yescharacteristic
subgroupG/H
centerC22#22551560
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#398×C257C2
socleC22×C257#120
prime radicalC23×C257C2×D4
hypercenter#398D257
hypocenterC257#1611
LCSG ▷ C22×C4×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

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
#306573682
#306570672
#306564732
#306564632
#398×C2572
#1611257