Finite Group Databasebeta

← #525×C257 #30656394

Subgroups — #525×C257 #30656394

Order 16448.

330 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 170 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
1C2×#41×C257C211yesmaximal, characteristic, complemented
2C23×C4×C257C211yesmaximal, characteristic, complemented
3#123×C257C212yesmaximal, complemented
4C22⋊Q8×C257C212yesmaximal, complemented
5#121×C257C212yesmaximal, complemented
6#120×C257C212yesmaximal, complemented
7C4×D4×C257C214yesmaximal, complemented
8#117×C257C211yesmaximal, characteristic, complemented
9C24×C257C2211yescharacteristic
10#41×C257—24no—
11C2×Q8×C257C2211yescharacteristic, complemented
12C2×D4×C257C2214yes—
13C2×D4×C257C2212yes—
14C2×D4×C257C2211yescharacteristic, complemented
15C22×C4×C257—24no—
16C22×C4×C257C2212yes—
17C22×C4×C257C2211yescharacteristic
18C22×C4×C257C2212yes—
19C22×C4×C257C2211yescharacteristic
20C22×C4×C257C2212yescomplemented
21C4⋊C4×C257C2212yescomplemented
22C4⋊C4×C257C2214yescomplemented
23C22⋊C4×C257C2212yes—
24C22⋊C4×C257C2214yes—
25C22⋊C4×C257C2214yescomplemented
26C42×C257C2212yescomplemented
27C23×C257—22no—
28C23×C257—24no—
29C23×C257C2312yes—
30C23×C257C2311yescharacteristic
31C23×C257C2312yes—
32Q8×C257—22no—
33D4×C257—24no—
34D4×C257—28no—
35D4×C257—22no—
36C2×C4×C257—22no—
37C2×C4×C257—24no—
38C2×C4×C257—24no—
39C2×C4×C257—24no—
40C2×C4×C257D414yescomplemented
41C2×C4×C257C2314yes—
42C2×C4×C257C2312yes—
43C2×C4×C257C2312yes—
44C2×C4×C257C2311yescharacteristic
45C2×C4×C257C2311yescharacteristic
46C22×C257—28no—
47C22×C257—22no—
48C22×C257—22no—
49C22×C257—22no—
50C22×C257—22no—
51C22×C257—24no—
52C22×C257C2×D412yes—
53C22×C257#4114yescomplemented
54C22×C257C2411yescharacteristic
55C4×C257—22no—
56C4×C257—22no—
57C4×C257—24no—
58C4×C257C2×D412yes—
59C4×C257C2×D412yes—
60C2×C257—22no—
61C2×C257—24no—
62C2×C257—42no—
63C2×C257C22×D411yescharacteristic
64C2×C257C2×#4112yes—
65C257#52511yescharacteristic, direct product factor, Sylow
66#525C25711yesmaximal, characteristic, direct product factor, Sylow
67C2×#41C2×C25711yescharacteristic, complemented
68C23×C4C2×C25711yescharacteristic, complemented
69#123C2×C25712yescomplemented
70C22⋊Q8C2×C25712yescomplemented
71#121C2×C25712yescomplemented
72#120C2×C25712yescomplemented
73C4×D4C2×C25714yescomplemented
74#117C2×C25711yescharacteristic, complemented
75C24C22×C25711yescharacteristic
76#41—24no—
77C2×Q8C22×C25711yescharacteristic, complemented
78C2×D4C22×C25714yes—
79C2×D4C22×C25712yes—
80C2×D4C22×C25711yescharacteristic, complemented
81C22×C4—24no—
82C22×C4C22×C25712yes—
83C22×C4C22×C25712yes—
84C22×C4C22×C25711yescharacteristic
85C22×C4C22×C25711yescharacteristic
86C22×C4C22×C25712yescomplemented
87C4⋊C4C22×C25712yescomplemented
88C4⋊C4C22×C25714yescomplemented
89C22⋊C4C22×C25712yes—
90C22⋊C4C22×C25714yes—
91C22⋊C4C22×C25714yescomplemented
92C42C22×C25712yescomplemented
93C23—22no—
94C23—24no—
95C23C23×C25712yes—
96C23C23×C25711yescharacteristic
97C23C23×C25712yes—
98Q8—22no—
99D4—24no—
100D4—28no—
101D4—22no—
102C2×C4—22no—
103C2×C4—24no—
104C2×C4—24no—
105C2×C4—24no—
106C2×C4D4×C25714yescomplemented
107C2×C4C23×C25714yes—
108C2×C4C23×C25712yes—
109C2×C4C23×C25712yes—
110C2×C4C23×C25711yescharacteristic
111C2×C4C23×C25711yescharacteristic
112C22—28no—
113C22—22no—
114C22—22no—
115C22—22no—
116C22—22no—
117C22—24no—
118C22C2×D4×C25712yes—
119C22#41×C25714yescomplemented
120C22C24×C25711yescharacteristic
121C4—22no—
122C4—22no—
123C4—24no—
124C4C2×D4×C25712yes—
125C4C2×D4×C25712yes—
126C2—22no—
127C2—24no—
128C2—42no—
129C2C22×D4×C25711yescharacteristic
130C2C2×#41×C25712yes—
subgroupG/H
centerC2×C4×C257C23
commutator, Frattini, cosocleC22C24×C257
socleC22×C257C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C257 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#41×C257C2semidirect
2. C23×C4×C257C2semidirect
3. #123×C257C2semidirect
4. C22⋊Q8×C257C2semidirect
5. #121×C257C2semidirect
6. #120×C257C2semidirect
7. C4×D4×C257C2semidirect
8. #117×C257C2semidirect
11. C2×Q8×C257C22semidirect
14. C2×D4×C257C22semidirect
20. C22×C4×C257C22semidirect
21. C4⋊C4×C257C22semidirect
22. C4⋊C4×C257C22semidirect
25. C22⋊C4×C257C22semidirect
26. C42×C257C22semidirect
40. C2×C4×C257D4semidirect
53. C22×C257#41semidirect
65. C257#525direct (N × H)
66. #525C257direct (N × H)
67. C2×#41C2×C257semidirect
68. C23×C4C2×C257semidirect
69. #123C2×C257semidirect
70. C22⋊Q8C2×C257semidirect
71. #121C2×C257semidirect
72. #120C2×C257semidirect
73. C4×D4C2×C257semidirect
74. #117C2×C257semidirect
77. C2×Q8C22×C257semidirect
80. C2×D4C22×C257semidirect
86. C22×C4C22×C257semidirect
87. C4⋊C4C22×C257semidirect
88. C4⋊C4C22×C257semidirect
91. C22⋊C4C22×C257semidirect
92. C42C22×C257semidirect
106. C2×C4D4×C257semidirect
119. C22#41×C257semidirect

Maximal subgroups

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

subgroupindex
C2×#41×C2572
C23×C4×C2572
#123×C2572
C22⋊Q8×C2572
#121×C2572
#120×C2572
C4×D4×C2572
#117×C2572
#525257