Finite Group Databasebeta

← #45663296

Subgroups — #45663296

Order 32896.

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

normal subgroups only

This group has 238 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#30656895C212yesmaximal
3#30656657C211yesmaximal, characteristic
4#30656475C212yesmaximal
5#324×C257C211yesmaximal, characteristic
6C22×Dic514C2211yescharacteristic
7C2×Dic1028—22no—
8#22551591—22no—
9#22551583—22no—
10#22551571—21no—
11#22551571C2212yes—
12#22551551—21no—
13C2×#16861597—21no—
14#22551517—22no—
15#22551499—21no—
16#22551499C2211yescharacteristic
17C2×C4⋊C4×C257C2211yescharacteristic
18C22⋊C8×C257C2212yes—
19C2×Dic514—21no—
20C2×Dic514—22no—
21C2×Dic514—42no—
22C2×Dic514D412yes—
23C22×Dic257—21no—
24C22×Dic257—21no—
25Dic1028—44no—
26C4⋊Dic257—21no—
27C4⋊Dic257—21no—
28C4⋊Dic257—21no—
29C4⋊Dic257D412yes—
30#16861597—22no—
31#16861597—22no—
32#16861597—42no—
33C22⋊Dic257—22no—
34C4×Dic257—22no—
35C22×C4×C257D411yescharacteristic
36C22×C4×C257D411yescharacteristic
37C22×C4×C257C2311yescharacteristic
38C2×C8×C257—22no—
39C4⋊C4×C257—22no—
40Dic514—24no—
41Dic514—42no—
42Dic514—42no—
43Dic514—81no—
44C2×Dic257—22no—
45C2×Dic257—22no—
46C2×Dic257—22no—
47C2×Dic257—22no—
48C2×Dic257—41no—
49C2×Dic257—41no—
50C2×Dic257—41no—
51C2×Dic257—41no—
52C23×C257C2×D411yescharacteristic
53C2×C4×C257—21no—
54C2×C4×C257—21no—
55C2×C4×C257—21no—
56C2×C4×C257—41no—
57C2×C4×C257—41no—
58C2×C4×C257Q1612yescomplemented
59C2×C4×C257C2×D412yes—
60C8×C257—42no—
61C22×C257—21no—
62C22×C257—21no—
63C22×C257#12011yescharacteristic
64C22×C257C2×Q1611yescharacteristic
65C22×C257#13711yescharacteristic
66C4×C257—22no—
67C4×C257—22no—
68C4×C257—41no—
69Dic257—42no—
70Dic257—42no—
71Dic257—42no—
72Dic257—42no—
73C2×C257—21no—
74C2×C257—21no—
75C2×C257#45111yescharacteristic
76C2×C257#45611yescharacteristic
77C2×C257B(4,𝔽2)11yescharacteristic
78C257#160511yescharacteristic, complemented, Sylow
79#1605—2571nomaximal, Sylow
80#544—2571no—
81#451—2572no—
82#395—2571no—
83#328—2572no—
84#324D25711yescharacteristic
85C22×Q8—2571no—
86C2×Q16—5142no—
87C4⋊Q8—5142no—
88#125—5142no—
89C22⋊Q8—2572no—
90C22⋊Q8—5141no—
91#117—5141no—
92C2×C4⋊C4—5141no—
93C2×C4⋊C4C2×D25711yescharacteristic
94#103—5142no—
95C22⋊C8C2×D25712yes—
96#95—2571no—
97#95—5141no—
98C2×Q8—2572no—
99C2×Q8—5141no—
100C2×Q8—5142no—
101C2×Q8—10282no—
102C22×C4—5141no—
103C22×C4—5141no—
104C22×C4D102811yescharacteristic
105C22×C4D102811yescharacteristic
106C22×C4C22×D25711yescharacteristic
107Q16—10284no—
108C2×C8—22no—
109C4⋊C4—2572no—
110C4⋊C4—22no—
111C4⋊C4—5141no—
112C4⋊C4—5141no—
113C4⋊C4—5141no—
114C4⋊C4—5142no—
115C4⋊C4—5142no—
116C4⋊C4—10282no—
117C22⋊C4—5142no—
118C42—5142no—
119C23C2×D102811yescharacteristic
120Q8—5144no—
121Q8—10282no—
122Q8—10282no—
123Q8—20561no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—5142no—
128C2×C4—5142no—
129C2×C4—5142no—
130C2×C4—5142no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—10281no—
134C2×C4—10281no—
135C2×C4—10281no—
136C2×C4—10281no—
137C2×C4Dic102812yescomplemented
138C2×C4D4×D25712yes—
139C8—42no—
140C22—21no—
141C22—21no—
142C22#2255156211yescharacteristic
143C22C2×Dic102811yescharacteristic
144C22#2255162511yescharacteristic
145C4—22no—
146C4—22no—
147C4—41no—
148C4—10282no—
149C4—10282no—
150C4—10282no—
151C4—10282no—
152C2—21no—
153C2—21no—
154C2#3065689511yescharacteristic
155C2#3065692211yescharacteristic
156C2#3065692511yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#324×C257C2
socleC22×C257#120
prime radicalC23×C257C2×D4
hypercenter#324D257
hypocenterC257#1605
LCSG ▷ C22×C4×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
58. C2×C4×C257Q16semidirect
78. C257#1605semidirect
137. C2×C4Dic1028semidirect

Maximal subgroups

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

subgroupindex
#306573682
#306568952
#306566572
#306564752
#324×C2572
#1605257