Finite Group Databasebeta

← #44549409

Subgroups — #44549409

Order 32128.

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#30632318C211yesmaximal, characteristic
2#30632023C212yesmaximal
3#30631626C211yesmaximal, characteristic
4#30631425C212yesmaximal
5#324×C251C211yesmaximal, characteristic
6#22548714—22no—
7#22548706—22no—
8#22548694—21no—
9#22548694C2212yes—
10#22548674—21no—
11C2×C4⋊Dic251C2211yescharacteristic
12#22548649—22no—
13#22548640—22no—
14C251⋊#95—21no—
15C251⋊#95—21no—
16C251⋊#95C2211yescharacteristic
17C2×C4⋊C4×C251C2211yescharacteristic
18C22⋊C8×C251C2212yes—
19C2×Dic502—21no—
20C2×Dic502D412yes—
21C22×Dic251—21no—
22C22×Dic251—21no—
23C4⋊Dic251—21no—
24C4⋊Dic251—21no—
25C4⋊Dic251—21no—
26C4⋊Dic251—22no—
27C4⋊Dic251D412yes—
28Dic251⋊C4—22no—
29Dic251⋊C4—22no—
30C22⋊Dic251—22no—
31C4×Dic251—22no—
32C22×C4×C251D411yescharacteristic
33C22×C4×C251D411yescharacteristic
34C22×C4×C251C2311yescharacteristic
35C2×C8×C251—22no—
36C4⋊C4×C251—22no—
37Dic502—42no—
38Dic502—81no—
39C2×Dic251—22no—
40C2×Dic251—22no—
41C2×Dic251—41no—
42C2×Dic251—41no—
43C2×Dic251—42no—
44C2×Dic251—41no—
45C2×Dic251—41no—
46C23×C251C2×D411yescharacteristic
47C2×C4×C251—21no—
48C2×C4×C251—21no—
49C2×C4×C251—21no—
50C2×C4×C251—41no—
51C2×C4×C251—41no—
52C2×C4×C251Q1612yes—
53C2×C4×C251C2×D412yes—
54C8×C251—42no—
55C22×C251—21no—
56C22×C251—21no—
57C22×C251#12011yescharacteristic
58C22×C251C2×Q1611yescharacteristic
59C22×C251#13711yescharacteristic
60C4×C251—22no—
61C4×C251—22no—
62C4×C251—41no—
63Dic251—42no—
64Dic251—42no—
65Dic251—81no—
66Dic251—81no—
67C2×C251—21no—
68C2×C251—21no—
69C2×C251#45111yescharacteristic
70C2×C251#45411yescharacteristic
71C2×C251#45811yescharacteristic
72C251#161711yescharacteristic, complemented, Sylow
73#1617—2511nomaximal, Sylow
74#544—2511no—
75#484—2512no—
76#400—2511no—
77#328—2512no—
78#324D25111yescharacteristic
79C4⋊Q8—5022no—
80#125—5022no—
81C22⋊Q8—5021no—
82C22⋊Q8—2512no—
83#117—5021no—
84C2×C4⋊C4—2511no—
85C2×C4⋊C4C2×D25111yescharacteristic
86#107—5022no—
87#103—5022no—
88C22⋊C8C2×D25112yes—
89#95—5021no—
90#95—5021no—
91#95—2511no—
92C2×Q8—5021no—
93C2×Q8—2512no—
94C22×C4—5021no—
95C22×C4—5021no—
96C22×C4D100411yescharacteristic
97C22×C4D100411yescharacteristic
98C22×C4C22×D25111yescharacteristic
99C2×C8—22no—
100C4⋊C4—22no—
101C4⋊C4—5021no—
102C4⋊C4—5021no—
103C4⋊C4—5022no—
104C4⋊C4—5021no—
105C4⋊C4—5022no—
106C4⋊C4—5022no—
107C4⋊C4—2512no—
108C22⋊C4—5022no—
109C42—5022no—
110C23C2×D100411yescharacteristic
111Q8—20081no—
112Q8—10042no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—10041no—
119C2×C4—10041no—
120C2×C4—10042no—
121C2×C4—10041no—
122C2×C4—10041no—
123C2×C4—5022no—
124C2×C4—5022no—
125C2×C4Dic100412yes—
126C2×C4D4×D25112yes—
127C8—42no—
128C22—21no—
129C22—21no—
130C22#2254868511yescharacteristic
131C22C2×Dic100411yescharacteristic
132C22#2254874811yescharacteristic
133C4—22no—
134C4—22no—
135C4—41no—
136C4—20081no—
137C4—20081no—
138C4—10042no—
139C4—10042no—
140C2—21no—
141C2—21no—
142C2#3063184511yescharacteristic
143C2#3063186211yescharacteristic
144C2#3063187811yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#324×C251C2
socleC22×C251#120
prime radicalC23×C251C2×D4
hypercenter#324D251
hypocenterC251#1617
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
72. C251#1617semidirect

Maximal subgroups

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

subgroupindex
#306323182
#306320232
#306316262
#306314252
#324×C2512
#1617251