Finite Group Databasebeta

← #44549375

Subgroups — #44549375

Order 32128.

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#30632318C211yesmaximal, characteristic
2#30632018C212yesmaximal
3#30631617C211yesmaximal, characteristic
4#30631425C212yesmaximal
5#324×C251C211yesmaximal, characteristic
6#22548714—22no—
7#22548706—22no—
8#22548694—21no—
9#22548694C2212yes—
10#22548674—21no—
11C2×C4⋊Dic251C2211yescharacteristic
12C2×Dic251⋊C4—21no—
13#22548647—22no—
14#22548640—22no—
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—
30Dic251⋊C4—42no—
31C22⋊Dic251—22no—
32C4×Dic251—22no—
33C22×C4×C251D411yescharacteristic
34C22×C4×C251D411yescharacteristic
35C22×C4×C251C2311yescharacteristic
36C2×C8×C251—22no—
37C4⋊C4×C251—22no—
38Dic502—42no—
39Dic502—81no—
40C2×Dic251—22no—
41C2×Dic251—22no—
42C2×Dic251—22no—
43C2×Dic251—41no—
44C2×Dic251—41no—
45C2×Dic251—41no—
46C2×Dic251—41no—
47C2×Dic251—41no—
48C23×C251C2×D411yescharacteristic
49C2×C4×C251—21no—
50C2×C4×C251—21no—
51C2×C4×C251—21no—
52C2×C4×C251—41no—
53C2×C4×C251—41no—
54C2×C4×C251SD1612yes—
55C2×C4×C251C2×D412yes—
56C8×C251—42no—
57C22×C251—21no—
58C22×C251—21no—
59C22×C251#12011yescharacteristic
60C22×C251C2×SD1611yescharacteristic
61C22×C251#13711yescharacteristic
62C4×C251—22no—
63C4×C251—22no—
64C4×C251—41no—
65Dic251—42no—
66Dic251—42no—
67Dic251—42no—
68Dic251—81no—
69C2×C251—21no—
70C2×C251—21no—
71C2×C251#44811yescharacteristic
72C2×C251#45611yescharacteristic
73C2×C251#45811yescharacteristic
74C251#161111yescharacteristic, complemented, Sylow
75#1611—2511nomaximal, Sylow
76#544—2511no—
77#483—2512no—
78#398—2511no—
79#328—2512no—
80#324D25111yescharacteristic
81C4⋊Q8—5022no—
82#125—5022no—
83C22⋊Q8—5021no—
84C22⋊Q8—2512no—
85#117—5021no—
86C2×C4⋊C4—5021no—
87C2×C4⋊C4—2511no—
88C2×C4⋊C4C2×D25111yescharacteristic
89#106—5022no—
90#103—5022no—
91C22⋊C8C2×D25112yes—
92#95—5021no—
93#95—2511no—
94C2×Q8—5021no—
95C2×Q8—2512no—
96C22×C4—5021no—
97C22×C4—5021no—
98C22×C4D100411yescharacteristic
99C22×C4D100411yescharacteristic
100C22×C4C22×D25111yescharacteristic
101C2×C8—22no—
102C4⋊C4—22no—
103C4⋊C4—10042no—
104C4⋊C4—5021no—
105C4⋊C4—5021no—
106C4⋊C4—5022no—
107C4⋊C4—5021no—
108C4⋊C4—5022no—
109C4⋊C4—5022no—
110C4⋊C4—2512no—
111C22⋊C4—5022no—
112C42—5022no—
113C23C2×D100411yescharacteristic
114Q8—20081no—
115Q8—10042no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—10041no—
122C2×C4—10041no—
123C2×C4—10041no—
124C2×C4—10041no—
125C2×C4—10041no—
126C2×C4—5022no—
127C2×C4—5022no—
128C2×C4—5022no—
129C2×C4#1686079912yes—
130C2×C4D4×D25112yes—
131C8—42no—
132C22—21no—
133C22—21no—
134C22#2254868511yescharacteristic
135C22C2×#1686079911yescharacteristic
136C22#2254874811yescharacteristic
137C4—22no—
138C4—22no—
139C4—41no—
140C4—20081no—
141C4—10042no—
142C4—10042no—
143C4—10042no—
144C2—21no—
145C2—21no—
146C2#3063182811yescharacteristic
147C2#3063187211yescharacteristic
148C2#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#1611
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
74. C251#1611semidirect

Maximal subgroups

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

subgroupindex
#306323182
#306320182
#306316172
#306314252
#324×C2512
#1611251