Finite Group Databasebeta

← #45663364

Subgroups — #45663364

Order 32896.

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

normal subgroups only

This group has 202 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#30657073C212yesmaximal
3#30656676C211yesmaximal, characteristic
4#30656475C212yesmaximal
5#324×C257C211yesmaximal, characteristic
6#22551591—22no—
7#22551583—22no—
8#22551571—21no—
9#22551571C2212yes—
10#22551551—21no—
11C2×C4⋊Dic257C2211yescharacteristic
12#22551526—22no—
13#22551517—22no—
14#22551499—21no—
15#22551499—21no—
16#22551499C2211yescharacteristic
17C2×C4⋊C4×C257C2211yescharacteristic
18C22⋊C8×C257C2212yes—
19C2×Dic514—21no—
20C2×Dic514D412yes—
21C22×Dic257—21no—
22C22×Dic257—21no—
23C4⋊Dic257—21no—
24C4⋊Dic257—21no—
25C4⋊Dic257—21no—
26C4⋊Dic257—22no—
27C4⋊Dic257D412yes—
28#16861597—22no—
29#16861597—22no—
30C22⋊Dic257—22no—
31C4×Dic257—22no—
32C22×C4×C257D411yescharacteristic
33C22×C4×C257D411yescharacteristic
34C22×C4×C257C2311yescharacteristic
35C2×C8×C257—22no—
36C4⋊C4×C257—22no—
37Dic514—42no—
38Dic514—81no—
39C2×Dic257—22no—
40C2×Dic257—22no—
41C2×Dic257—41no—
42C2×Dic257—41no—
43C2×Dic257—42no—
44C2×Dic257—41no—
45C2×Dic257—41no—
46C23×C257C2×D411yescharacteristic
47C2×C4×C257—21no—
48C2×C4×C257—21no—
49C2×C4×C257—21no—
50C2×C4×C257—41no—
51C2×C4×C257—41no—
52C2×C4×C257Q1612yes—
53C2×C4×C257C2×D412yes—
54C8×C257—42no—
55C22×C257—21no—
56C22×C257—21no—
57C22×C257#12011yescharacteristic
58C22×C257C2×Q1611yescharacteristic
59C22×C257#13711yescharacteristic
60C4×C257—22no—
61C4×C257—22no—
62C4×C257—41no—
63Dic257—42no—
64Dic257—42no—
65Dic257—81no—
66Dic257—81no—
67C2×C257—21no—
68C2×C257—21no—
69C2×C257#45111yescharacteristic
70C2×C257#45411yescharacteristic
71C2×C257#45811yescharacteristic
72C257#161711yescharacteristic, complemented, Sylow
73#1617—2571nomaximal, Sylow
74#544—2571no—
75#484—2572no—
76#400—2571no—
77#328—2572no—
78#324D25711yescharacteristic
79C4⋊Q8—5142no—
80#125—5142no—
81C22⋊Q8—2572no—
82C22⋊Q8—5141no—
83#117—5141no—
84C2×C4⋊C4—2571no—
85C2×C4⋊C4C2×D25711yescharacteristic
86#107—5142no—
87#103—5142no—
88C22⋊C8C2×D25712yes—
89#95—2571no—
90#95—5141no—
91#95—5141no—
92C2×Q8—2572no—
93C2×Q8—5141no—
94C22×C4—5141no—
95C22×C4—5141no—
96C22×C4D102811yescharacteristic
97C22×C4D102811yescharacteristic
98C22×C4C22×D25711yescharacteristic
99C2×C8—22no—
100C4⋊C4—2572no—
101C4⋊C4—22no—
102C4⋊C4—5141no—
103C4⋊C4—5141no—
104C4⋊C4—5141no—
105C4⋊C4—5142no—
106C4⋊C4—5142no—
107C4⋊C4—5142no—
108C22⋊C4—5142no—
109C42—5142no—
110C23C2×D102811yescharacteristic
111Q8—10282no—
112Q8—20561no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—5142no—
117C2×C4—5142no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—10281no—
121C2×C4—10281no—
122C2×C4—10282no—
123C2×C4—10281no—
124C2×C4—10281no—
125C2×C4Dic102812yes—
126C2×C4D4×D25712yes—
127C8—42no—
128C22—21no—
129C22—21no—
130C22#2255156211yescharacteristic
131C22C2×Dic102811yescharacteristic
132C22#2255162511yescharacteristic
133C4—22no—
134C4—22no—
135C4—41no—
136C4—10282no—
137C4—10282no—
138C4—20561no—
139C4—20561no—
140C2—21no—
141C2—21no—
142C2#3065689511yescharacteristic
143C2#3065691211yescharacteristic
144C2#3065692811yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#324×C257C2
socleC22×C257#120
prime radicalC23×C257C2×D4
hypercenter#324D257
hypocenterC257#1617
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
72. C257#1617semidirect

Maximal subgroups

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

subgroupindex
#306573682
#306570732
#306566762
#306564752
#324×C2572
#1617257