Finite Group Databasebeta

← #1657×C251 #44546244

Subgroups — #1657×C251 #44546244

Order 32128.

206 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 60 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#563×C251C211yesmaximal, characteristic, complemented
2#496×C251C211yesmaximal, characteristic
3#490×C251C212yesmaximal, complemented
4#330×C251C212yesmaximal, complemented
5#329×C251C211yesmaximal, characteristic, complemented
6C2×SD16×C251—21no—
7C2×D8×C251—21no—
8C4⋊Q8×C251—21no—
9#127×C251C2211yescharacteristic
10#126×C251—22no—
11#125×C251C2212yes—
12#124×C251C2211yescharacteristic
13#123×C251—21no—
14C22⋊Q8×C251—22no—
15#102×C251—22no—
16#102×C251—22no—
17#97×C251C2212yes—
18#97×C251C2211yescharacteristic
19C2×Q8×C251—21no—
20C2×Q8×C251D411yescharacteristic
21C2×D4×C251—22no—
22C2×D4×C251—21no—
23C2×D4×C251D411yescharacteristic
24C22×C4×C251—21no—
25SD16×C251—42no—
26D8×C251—42no—
27C2×C8×C251—21no—
28C2×C8×C251—22no—
29C4⋊C4×C251—22no—
30C4⋊C4×C251—22no—
31C4⋊C4×C251—22no—
32C4⋊C4×C251D412yes—
33C4⋊C4×C251D412yes—
34C22⋊C4×C251—22no—
35C22⋊C4×C251—21no—
36C22⋊C4×C251—22no—
37C42×C251—21no—
38C42×C251C2311yescharacteristic
39C23×C251—21no—
40C23×C251—41no—
41Q8×C251—41no—
42Q8×C251—81no—
43D4×C251—41no—
44D4×C251—41no—
45D4×C251—41no—
46D4×C251—44no—
47C2×C4×C251—22no—
48C2×C4×C251—21no—
49C2×C4×C251—22no—
50C2×C4×C251—81no—
51C2×C4×C251C2×D412yes—
52C2×C4×C251C2×D411yescharacteristic
53C8×C251—22no—
54C8×C251—42no—
55C22×C251—41no—
56C22×C251—42no—
57C22×C251—81no—
58C22×C251—82no—
59C22×C251#12011yescharacteristic
60C4×C251—21no—
61C4×C251—22no—
62C4×C251—42no—
63C4×C251—41no—
64C4×C251—42no—
65C2×C251—81no—
66C2×C251—161no—
67C2×C251#45311yescharacteristic
68C2×C251#45412yes—
69C251#165711yescharacteristic, direct product factor, Sylow
70#1657C25111yesmaximal, characteristic, direct product factor, Sylow
71#563C2×C25111yescharacteristic, complemented
72#496C2×C25111yescharacteristic
73#490C2×C25112yescomplemented
74#330C2×C25112yescomplemented
75#329C2×C25111yescharacteristic, complemented
76C2×SD16—21no—
77C2×D8—21no—
78C4⋊Q8—21no—
79#127C22×C25111yescharacteristic
80#126—22no—
81#125C22×C25112yes—
82#124C22×C25111yescharacteristic
83#123—21no—
84C22⋊Q8—22no—
85#102—22no—
86#102—22no—
87#97C22×C25112yes—
88#97C22×C25111yescharacteristic
89C2×Q8—21no—
90C2×Q8D4×C25111yescharacteristic
91C2×D4—22no—
92C2×D4—21no—
93C2×D4D4×C25111yescharacteristic
94C22×C4—21no—
95SD16—42no—
96D8—42no—
97C2×C8—21no—
98C2×C8—22no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4D4×C25112yes—
103C4⋊C4D4×C25112yes—
104C22⋊C4—22no—
105C22⋊C4—21no—
106C22⋊C4—22no—
107C42—21no—
108C42C23×C25111yescharacteristic
109C23—21no—
110C23—41no—
111Q8—41no—
112Q8—81no—
113D4—41no—
114D4—41no—
115D4—41no—
116D4—44no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—81no—
121C2×C4C2×D4×C25112yes—
122C2×C4C2×D4×C25111yescharacteristic
123C8—22no—
124C8—42no—
125C22—41no—
126C22—42no—
127C22—81no—
128C22—82no—
129C22#120×C25111yescharacteristic
130C4—21no—
131C4—22no—
132C4—42no—
133C4—41no—
134C4—42no—
135C2—81no—
136C2—161no—
137C2#453×C25111yescharacteristic
138C2#454×C25112yes—
subgroupG/H
center, socleC22×C251#120
commutator, Frattini, cosocleC42C23×C251
prime radical#496×C251C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C251 ◁ C42×C251 ◁ G

Complements

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

NH ≅ G/Ntype
1. #563×C251C2semidirect
3. #490×C251C2semidirect
4. #330×C251C2semidirect
5. #329×C251C2semidirect
69. C251#1657direct (N × H)
70. #1657C251direct (N × H)
71. #563C2×C251semidirect
73. #490C2×C251semidirect
74. #330C2×C251semidirect
75. #329C2×C251semidirect

Maximal subgroups

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

subgroupindex
#563×C2512
#496×C2512
#490×C2512
#330×C2512
#329×C2512
#1657251