Finite Group Databasebeta

← #2135×C251 #44546722

Subgroups — #2135×C251 #44546722

Order 32128.

230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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#536×C251C211yesmaximal, characteristic
2#413×C251C212yesmaximal
3#356×C251C214yesmaximal, complemented
4C2×#41×C251C2211yescharacteristic
5C22×Q8×C251C411yescharacteristic
6C2×M16×C251—22no—
7C4⋊Q8×C251C2212yes—
8#124×C251C412yes—
9#123×C251—22no—
10C22⋊Q8×C251—22no—
11#117×C251C411yescharacteristic
12#101×C251C2214yes—
13#100×C251—22no—
14#41×C251—42no—
15C2×Q8×C251—21no—
16C2×Q8×C251—42no—
17C2×Q8×C251C2×C412yes—
18C2×Q8×C251C2311yescharacteristic
19C2×D4×C251C2×C411yescharacteristic
20C2×D4×C251D412yes—
21C22×C4×C251—21no—
22C22×C4×C251C2×C411yescharacteristic
23C22×C4×C251D412yes—
24M16×C251—24no—
25M16×C251—42no—
26C2×C8×C251—42no—
27C4⋊C4×C251—21no—
28C4⋊C4×C251—24no—
29C22⋊C4×C251—21no—
30C22⋊C4×C251—24no—
31C42×C251C2×C412yes—
32C23×C251C22⋊C412yes—
33C23×C251C22⋊C411yescharacteristic
34Q8×C251—41no—
35Q8×C251—42no—
36Q8×C251—42no—
37D4×C251—41no—
38D4×C251—42no—
39C2×C4×C251—22no—
40C2×C4×C251—22no—
41C2×C4×C251—22no—
42C2×C4×C251—22no—
43C2×C4×C251—22no—
44C2×C4×C251—42no—
45C2×C4×C251C22⋊C411yescharacteristic
46C2×C4×C251C22×C411yescharacteristic
47C2×C4×C251C2×D412yes—
48C8×C251—44no—
49C22×C251—21no—
50C22×C251—22no—
51C22×C251—42no—
52C22×C251—41no—
53C22×C251C2×C22⋊C411yescharacteristic
54C4×C251—22no—
55C4×C251—21no—
56C4×C251—42no—
57C4×C251—42no—
58C4×C251#9912yes—
59C2×C251—21no—
60C2×C251—41no—
61C2×C251—42no—
62C2×C251C2×#9911yescharacteristic
63C251#213511yescharacteristic, direct product factor, Sylow
64#2135C25111yesmaximal, characteristic, direct product factor, Sylow
65#536C2×C25111yescharacteristic
66#413C2×C25112yes—
67#356C2×C25114yescomplemented
68C2×#41C22×C25111yescharacteristic
69C22×Q8C4×C25111yescharacteristic
70C2×M16—22no—
71C4⋊Q8C22×C25112yes—
72#124C4×C25112yes—
73#123—22no—
74C22⋊Q8—22no—
75#117C4×C25111yescharacteristic
76#101C22×C25114yes—
77#100—22no—
78#41—42no—
79C2×Q8—21no—
80C2×Q8—42no—
81C2×Q8C2×C4×C25112yes—
82C2×Q8C23×C25111yescharacteristic
83C2×D4C2×C4×C25111yescharacteristic
84C2×D4D4×C25112yes—
85C22×C4—21no—
86C22×C4C2×C4×C25111yescharacteristic
87C22×C4D4×C25112yes—
88M16—24no—
89M16—42no—
90C2×C8—42no—
91C4⋊C4—21no—
92C4⋊C4—24no—
93C22⋊C4—21no—
94C22⋊C4—24no—
95C42C2×C4×C25112yes—
96C23C22⋊C4×C25112yes—
97C23C22⋊C4×C25111yescharacteristic
98Q8—41no—
99Q8—42no—
100Q8—42no—
101D4—41no—
102D4—42no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—42no—
109C2×C4C22⋊C4×C25111yescharacteristic
110C2×C4C22×C4×C25111yescharacteristic
111C2×C4C2×D4×C25112yes—
112C8—44no—
113C22—21no—
114C22—22no—
115C22—42no—
116C22—41no—
117C22C2×C22⋊C4×C25111yescharacteristic
118C4—22no—
119C4—21no—
120C4—42no—
121C4—42no—
122C4#99×C25112yes—
123C2—21no—
124C2—41no—
125C2—42no—
126C2C2×#99×C25111yescharacteristic
subgroupG/H
center, socleC2×C251C2×#99
commutatorC2×C4C22×C4×C251
Frattini, cosocleC2×Q8C23×C251
prime radicalC2×#41×C251C22
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C251 ◁ C2×C4×C251 ◁ C2×#41×C251 ◁ G

Complements

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

NH ≅ G/Ntype
3. #356×C251C2semidirect
63. C251#2135direct (N × H)
64. #2135C251direct (N × H)
67. #356C2×C251semidirect

Maximal subgroups

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

subgroupindex
#536×C2512
#413×C2512
#356×C2512
#2135251