Finite Group Databasebeta

← #44549024

Subgroups — #44549024

Order 32128.

246 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 67 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#30632235C211yesmaximal, characteristic
2#30631739C212yesmaximal, complemented
3#30631435C212yesmaximal, complemented
4#30631431C211yesmaximal, characteristic, complemented
5#331×C251C211yesmaximal, characteristic, complemented
6C2×#16860809C412yes—
7#22548715C2211yescharacteristic
8#22548702C412yes—
9#22548686—22no—
10D4×Dic251—22no—
11C42×D251C2211yescharacteristic
12C4⋊(C251⋊C8)C2212yes—
13(C2×D251)⋊C8—22no—
14C8×Dic251—22no—
15#127×C251C2211yescharacteristic
16C4⋊C8×C251C2212yes—
17#16860809—44no—
18C2×Dic502C2×C412yes—
19C2×C22⋊D251—22no—
20C22×Dic251—22no—
21C2×C4×D251C2×C411yescharacteristic
22C2×C4×D251D412yes—
23C2×C251⋊C8—22no—
24C4⋊Dic251—22no—
25C22⋊Dic251—24no—
26C4×Dic251C2×C411yescharacteristic
27C4×Dic251D412yes—
28C2×D4×C251—22no—
29C2×D4×C251C2×C412yes—
30C2×C8×C251—22no—
31C42×C251C2311yescharacteristic
32C22×D251C22⋊C411yescharacteristic
33Dic502—42no—
34C22⋊D251—44no—
35C2×Dic251—22no—
36C2×Dic251—44no—
37C2×Dic251C22⋊C412yes—
38C2×Dic251C22⋊C411yescharacteristic
39C4×D251—22no—
40C4×D251—22no—
41C4×D251—22no—
42C251⋊C8—42no—
43C23×C251—22no—
44D4×C251—42no—
45D4×C251—42no—
46D4×C251—42no—
47C2×C4×C251C22×C411yescharacteristic
48C2×C4×C251C2×D412yes—
49C8×C251—42no—
50C22×C251—42no—
51C22×C251—44no—
52C22×C251C2×C22⋊C411yescharacteristic
53C2×D251—21no—
54C2×D251—22no—
55C4×C251—21no—
56C4×C251#13512yes—
57C4×C251#13512yes—
58Dic251—22no—
59Dic251—42no—
60Dic251#10012yes—
61C2×C251—82no—
62C2×C251C2×#10011yescharacteristic
63C2×C251#41612yes—
64D251—41no—
65C251#154311yescharacteristic, complemented, Sylow
66#1543—2511nomaximal, Sylow
67#532—2511no—
68#431—2512no—
69#333—2512no—
70#331—2511no—
71#331D25111yescharacteristic, complemented
72C2×#41—2512no—
73C4⋊Q8—2511no—
74#127C2×D25111yescharacteristic
75#124—2512no—
76#121—5022no—
77C4×D4—5022no—
78C2×C42—2511no—
79C4⋊C8—2512no—
80C4⋊C8C2×D25112yes—
81C22⋊C8—5022no—
82C4×C8—5022no—
83#41—10044no—
84C2×Q8—2512no—
85C2×D4—22no—
86C2×D4—5022no—
87C2×D4C4×D25112yes—
88C22×C4—5022no—
89C22×C4—2511no—
90C22×C4—2512no—
91C2×C8—22no—
92C2×C8—5022no—
93C4⋊C4—5022no—
94C22⋊C4—5024no—
95C42—2511no—
96C42—2512no—
97C42C22×D25111yescharacteristic
98C23—22no—
99C23—2511no—
100Q8—10042no—
101D4—42no—
102D4—42no—
103D4—42no—
104D4—10044no—
105C2×C4—10044no—
106C2×C4—5022no—
107C2×C4—5022no—
108C2×C4—5022no—
109C2×C4—5022no—
110C2×C4—2512no—
111C2×C4—2511no—
112C2×C4C2×C4×D25111yescharacteristic
113C2×C4D4×D25112yes—
114C8—42no—
115C8—10042no—
116C22—44no—
117C22—42no—
118C22—5022no—
119C22—5021no—
120C22C22⋊C4×D25111yescharacteristic
121C4—21no—
122C4—10042no—
123C4—5022no—
124C4—2512no—
125C4#2254873412yes—
126C4#2254873612yes—
127C2—82no—
128C2—10041no—
129C2#100×D25111yescharacteristic
130C2#3063168012yes—
subgroupG/H
centerC22C22⋊C4×D251
commutatorC2×C4×C251C22×C4
FrattiniC42C22×D251
Fitting#331×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632235C2
cosocleC42×C251C23
hypercenter#331D251
hypocenterC251#1543
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #30631739C2semidirect
3. #30631435C2semidirect
4. #30631431C2semidirect
5. #331×C251C2semidirect
65. C251#1543semidirect
71. #331D251semidirect

Maximal subgroups

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

subgroupindex
#306322352
#306317392
#306314352
#306314312
#331×C2512
#1543251