Finite Group Databasebeta

← #45662345

Subgroups — #45662345

Order 32896.

154 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 35 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#30656673C211yesmaximal, characteristic
2#30656504C211yesmaximal, characteristic
3#323×C257C211yesmaximal, characteristic
4C2×C4⋊Dic257C411yescharacteristic, complemented
5C2×#16861597—21no—
6C2×C22⋊Dic257—21no—
7C2×C22⋊Dic257—21no—
8#22551499C411yescharacteristic, complemented
9C2×C22⋊C4×C257C2211yescharacteristic
10C22⋊C8×C257—21no—
11C22×Dic257—21no—
12C22×Dic257—21no—
13C22×Dic257—41no—
14C4⋊Dic257—41no—
15#16861597—42no—
16C22⋊Dic257—22no—
17C22⋊Dic257—42no—
18C22⋊Dic257—41no—
19C24×C257D411yescharacteristic
20C22×C4×C257C2×C411yescharacteristic
21C22×C4×C257D411yescharacteristic
22C2×C8×C257—41no—
23C22⋊C4×C257—41no—
24C2×Dic257—22no—
25C2×Dic257—41no—
26C2×Dic257—41no—
27C2×Dic257—42no—
28C2×Dic257—41no—
29C2×Dic257—41no—
30C2×Dic257—41no—
31C2×Dic257—81no—
32C2×Dic257—81no—
33C23×C257—21no—
34C23×C257—21no—
35C23×C257—41no—
36C23×C257—41no—
37C23×C257C22⋊C411yescharacteristic
38C23×C257SD1611yescharacteristic
39C23×C257Q1611yescharacteristic
40C2×C4×C257—21no—
41C2×C4×C257—21no—
42C2×C4×C257—41no—
43C2×C4×C257—41no—
44C8×C257—81no—
45C22×C257—21no—
46C22×C257—21no—
47C22×C257—21no—
48C22×C257—21no—
49C22×C257—41no—
50C22×C257—41no—
51C22×C257—41no—
52C22×C257—41no—
53C22×C257—81no—
54C22×C257#9911yescharacteristic
55C22×C257#10311yescharacteristic
56C22×C257#10411yescharacteristic
57C4×C257—41no—
58C4×C257—41no—
59Dic257—42no—
60Dic257—82no—
61Dic257—81no—
62C2×C257—21no—
63C2×C257—21no—
64C2×C257—41no—
65C2×C257—41no—
66C2×C257#32811yescharacteristic
67C2×C257#35211yescharacteristic
68C2×C257#35311yescharacteristic
69C257#135211yescharacteristic, complemented, Sylow
70#1352—2571nomaximal, Sylow
71#399—2571no—
72#342—2571no—
73#323D25711yescharacteristic
74C2×C4⋊C4—2571no—
75C2×C4⋊C4—5141no—
76C2×C22⋊C4—5141no—
77C2×C22⋊C4—5141no—
78C2×C22⋊C4C2×D25711yescharacteristic
79C22⋊C8—21no—
80#95—2571no—
81C24D102811yescharacteristic
82C22×C4—5141no—
83C22×C4—5141no—
84C22×C4—10281no—
85C22×C4C4×D25711yescharacteristic
86C22×C4C22⋊D25711yescharacteristic
87C2×C8—41no—
88C4⋊C4—10281no—
89C4⋊C4—10282no—
90C22⋊C4—5142no—
91C22⋊C4—41no—
92C22⋊C4—10282no—
93C22⋊C4—10281no—
94C23—21no—
95C23—21no—
96C23—41no—
97C23—41no—
98C23#1686159511yescharacteristic
99C23#1686160711yescharacteristic
100C23Dic102811yescharacteristic
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—5142no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—10281no—
107C2×C4—10281no—
108C2×C4—10282no—
109C2×C4—10281no—
110C2×C4—10281no—
111C2×C4—10281no—
112C2×C4—20561no—
113C2×C4—20561no—
114C8—81no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—21no—
119C22—41no—
120C22—41no—
121C22—41no—
122C22—41no—
123C22—81no—
124C22#2255150611yescharacteristic
125C22#2255151711yescharacteristic
126C22#2255151811yescharacteristic
127C4—41no—
128C4—41no—
129C4—10282no—
130C4—20562no—
131C4—20561no—
132C2—21no—
133C2—21no—
134C2—41no—
135C2—41no—
136C2#3065647511yescharacteristic
137C2#3065652811yescharacteristic
138C2#3065652911yescharacteristic
subgroupG/H
centerC22#22551506
commutatorC22×C4×C257C2×C4
FrattiniC2×C22⋊C4C2×D257
Fitting#323×C257C2
socleC22×C257#99
prime radicalC24×C257D4
cosocleC2×C22⋊C4×C257C22
hypercenter#323D257
hypocenterC257#1352
LCSG ▷ C22×C4×C257 ▷ C23×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #323

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊Dic257C4semidirect
8. #22551499C4semidirect
69. C257#1352semidirect

Maximal subgroups

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

subgroupindex
#306566732
#306565042
#323×C2572
#1352257