Finite Group Databasebeta

← #1557×C251 #44546144

Subgroups — #1557×C251 #44546144

Order 32128.

246 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 92 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#532×C251C211yesmaximal, characteristic
2#432×C251C211yesmaximal, characteristic, complemented
3#423×C251C211yesmaximal, characteristic, complemented
4#332×C251C212yesmaximal, complemented
5#331×C251C212yesmaximal, complemented
6C2×#41×C251—21no—
7C2×M16×C251—21no—
8C4⋊Q8×C251C2211yescharacteristic
9#127×C251C2211yescharacteristic
10#124×C251—21no—
11#121×C251—21no—
12#121×C251C412yes—
13C4×D4×C251—21no—
14C4×D4×C251C412yes—
15C2×C42×C251C2211yescharacteristic
16C4⋊C8×C251—21no—
17C4⋊C8×C251C2212yes—
18C4⋊C8×C251C2212yes—
19C22⋊C8×C251—21no—
20#97×C251—21no—
21#41×C251—44no—
22C2×Q8×C251—21no—
23C2×D4×C251—22no—
24C2×D4×C251—21no—
25C2×D4×C251—21no—
26C2×D4×C251C2×C412yes—
27C22×C4×C251—22no—
28C22×C4×C251C2×C411yescharacteristic
29C22×C4×C251D411yescharacteristic
30C22×C4×C251D411yescharacteristic
31M16×C251—42no—
32C2×C8×C251—22no—
33C2×C8×C251—22no—
34C4⋊C4×C251—21no—
35C4⋊C4×C251C2×C412yes—
36C22⋊C4×C251—22no—
37C22⋊C4×C251—22no—
38C42×C251C2×C411yescharacteristic
39C42×C251D411yescharacteristic
40C42×C251D411yescharacteristic
41C42×C251C2311yescharacteristic
42C23×C251—22no—
43C23×C251C22⋊C411yescharacteristic
44Q8×C251—42no—
45D4×C251—42no—
46D4×C251—44no—
47D4×C251—41no—
48D4×C251—42no—
49D4×C251—41no—
50C2×C4×C251—21no—
51C2×C4×C251—22no—
52C2×C4×C251—21no—
53C2×C4×C251—22no—
54C2×C4×C251—41no—
55C2×C4×C251—44no—
56C2×C4×C251C22⋊C411yescharacteristic
57C2×C4×C251C22⋊C411yescharacteristic
58C2×C4×C251C22⋊C411yescharacteristic
59C2×C4×C251C22×C411yescharacteristic
60C2×C4×C251C2×D411yescharacteristic
61C2×C4×C251C2×D411yescharacteristic
62C8×C251—42no—
63C8×C251—42no—
64C22×C251—21no—
65C22×C251—21no—
66C22×C251—21no—
67C22×C251—42no—
68C22×C251—42no—
69C22×C251—42no—
70C22×C251C2×C22⋊C411yescharacteristic
71C4×C251—21no—
72C4×C251—21no—
73C4×C251—21no—
74C4×C251—21no—
75C4×C251—42no—
76C4×C251AGL(1,ℤ/8)11yescharacteristic
77C4×C251AGL(1,ℤ/8)12yes—
78C4×C251#13711yescharacteristic
79C2×C251—41no—
80C2×C251—82no—
81C2×C251#41311yescharacteristic
82C2×C251#41711yescharacteristic
83C2×C251#41811yescharacteristic
84C251#155711yescharacteristic, direct product factor, Sylow
85#1557C25111yesmaximal, characteristic, direct product factor, Sylow
86#532C2×C25111yescharacteristic
87#432C2×C25111yescharacteristic, complemented
88#423C2×C25111yescharacteristic, complemented
89#332C2×C25112yescomplemented
90#331C2×C25112yescomplemented
91C2×#41—21no—
92C2×M16—21no—
93C4⋊Q8C22×C25111yescharacteristic
94#127C22×C25111yescharacteristic
95#124—21no—
96#121—21no—
97#121C4×C25112yes—
98C4×D4—21no—
99C4×D4C4×C25112yes—
100C2×C42C22×C25111yescharacteristic
101C4⋊C8—21no—
102C4⋊C8C22×C25112yes—
103C4⋊C8C22×C25112yes—
104C22⋊C8—21no—
105#97—21no—
106#41—44no—
107C2×Q8—21no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4C2×C4×C25112yes—
112C22×C4—22no—
113C22×C4C2×C4×C25111yescharacteristic
114C22×C4D4×C25111yescharacteristic
115C22×C4D4×C25111yescharacteristic
116M16—42no—
117C2×C8—22no—
118C2×C8—22no—
119C4⋊C4—21no—
120C4⋊C4C2×C4×C25112yes—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C42C2×C4×C25111yescharacteristic
124C42D4×C25111yescharacteristic
125C42D4×C25111yescharacteristic
126C42C23×C25111yescharacteristic
127C23—22no—
128C23C22⋊C4×C25111yescharacteristic
129Q8—42no—
130D4—42no—
131D4—44no—
132D4—42no—
133D4—41no—
134D4—41no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—41no—
140C2×C4—44no—
141C2×C4C22⋊C4×C25111yescharacteristic
142C2×C4C22⋊C4×C25111yescharacteristic
143C2×C4C22⋊C4×C25111yescharacteristic
144C2×C4C22×C4×C25111yescharacteristic
145C2×C4C2×D4×C25111yescharacteristic
146C2×C4C2×D4×C25111yescharacteristic
147C8—42no—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—42no—
153C22—42no—
154C22—42no—
155C22C2×C22⋊C4×C25111yescharacteristic
156C4—21no—
157C4—21no—
158C4—21no—
159C4—21no—
160C4—42no—
161C4AGL(1,ℤ/8)×C25111yescharacteristic
162C4AGL(1,ℤ/8)×C25112yes—
163C4#137×C25111yescharacteristic
164C2—41no—
165C2—82no—
166C2#413×C25111yescharacteristic
167C2#417×C25111yescharacteristic
168C2#418×C25111yescharacteristic
subgroupG/H
center, socleC22×C251C2×C22⋊C4
commutatorC2×C4C22×C4×C251
Frattini, cosocleC42C23×C251
prime radical#532×C251C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C251 ◁ C2×C42×C251 ◁ G

Complements

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

NH ≅ G/Ntype
2. #432×C251C2semidirect
3. #423×C251C2semidirect
4. #332×C251C2semidirect
5. #331×C251C2semidirect
84. C251#1557direct (N × H)
85. #1557C251direct (N × H)
87. #432C2×C251semidirect
88. #423C2×C251semidirect
89. #332C2×C251semidirect
90. #331C2×C251semidirect

Maximal subgroups

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

subgroupindex
#532×C2512
#432×C2512
#423×C2512
#332×C2512
#331×C2512
#1557251