Finite Group Databasebeta

← #30648136

Subgroups — #30648136

Order 16256.

716 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 161 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#22550447C213yesmaximal, complemented
2#22550066C218yesmaximal, complemented
3#22549850C213yesmaximal, complemented
4#585×C127C211yesmaximal, characteristic
5C22×Dic254C2211yescharacteristic, complemented
6#16861207—23no—
7#16861196—23no—
8#16861190C22112yescomplemented
9#16861188—26no—
10#16861168C2213yescomplemented
11#16861124C22112yescomplemented
122−1+4×C127—23no—
13H(5,𝔽2)×C127—23no—
14H(5,𝔽2)×C127C2214yes—
15C2×#41×C127—26no—
16C2×#41×C127C2213yes—
17C2×Dic254—23no—
18C2×Dic254—23no—
19C22×Dic127C2312yes—
20C4⋊Dic127—23no—
21Dic127⋊C4—212no—
22C22⋊Dic127—212no—
23C22⋊Dic127C2316yes—
24C4×Dic127—23no—
25#41×C127—24no—
26#41×C127—212no—
27#41×C127—26no—
28#41×C127—212no—
29#41×C127—26no—
30C2×Q8×C127—26no—
31C2×Q8×C127D413yes—
32C2×D4×C127—26no—
33C2×D4×C127—212no—
34C2×D4×C127D413yes—
35C2×D4×C127C2316yes—
36C22×C4×C127—24no—
37C22×C4×C127D416yes—
38C22×C4×C127C2311yescharacteristic
39Dic254—41no—
40Dic254—43no—
41C2×Dic127—26no—
42C2×Dic127—26no—
43C23×C127—24no—
44C23×C127C2×D416yes—
45C23×C127C2411yescharacteristic
46Q8×C127—24no—
47Q8×C127—26no—
48D4×C127—212no—
49D4×C127—212no—
50D4×C127—26no—
51C2×C4×C127—212no—
52C2×C4×C127—26no—
53C2×C4×C127—26no—
54C2×C4×C127C2×D416yes—
55C2×C4×C127C2×D413yes—
56C2×C4×C127C2×D413yes—
57C22×C127—26no—
58C22×C127—41no—
59C22×C127—46no—
60C22×C127—84no—
61C22×C127C22×D413yes—
62C4×C127—26no—
63C4×C127#12011yescharacteristic
64C4×C127#12013yes—
65Dic127—46no—
66Dic127—42no—
67C2×C127—23no—
68C2×C127—46no—
69C2×C127C2×#12011yescharacteristic
70#3028—1271nomaximal, Sylow
71C127#302811yescharacteristic, complemented, Sylow
72#585D12711yescharacteristic
73#536—1273no—
74#458—1278no—
75#410—1273no—
762−1+4—23no—
77H(5,𝔽2)—23no—
78H(5,𝔽2)C2×D12714yes—
79C2×#41—26no—
80C2×#41C2×D12713yes—
81C22×Q8—1271no—
82C4⋊Q8—2543no—
83#124—2543no—
84#123—12712no—
85C22⋊Q8—2546no—
86#117—1273no—
87#99—12712no—
88#41—24no—
89#41—212no—
90#41—26no—
91#41—212no—
92#41—26no—
93C2×Q8—26no—
94C2×Q8—2543no—
95C2×Q8—2543no—
96C2×Q8C22⋊D12713yes—
97C2×D4—26no—
98C2×D4—212no—
99C2×D4C22⋊D12713yes—
100C2×D4C22×D12716yes—
101C22×C4—24no—
102C22×C4—1272no—
103C22×C4C22⋊D12716yes—
104C22×C4C22×D12711yescharacteristic
105C4⋊C4—25412no—
106C4⋊C4—2543no—
107C22⋊C4—1276no—
108C22⋊C4—25412no—
109C42—2543no—
110C23—24no—
111C23C2×C22⋊D12716yes—
112C23C23×D12711yescharacteristic
113Q8—24no—
114Q8—26no—
115Q8—5081no—
116Q8—5083no—
117D4—212no—
118D4—212no—
119D4—26no—
120C2×C4—212no—
121C2×C4—26no—
122C2×C4—26no—
123C2×C4—2546no—
124C2×C4—2546no—
125C2×C4C2×C22⋊D12716yes—
126C2×C4C2×C22⋊D12713yes—
127C2×C4C2×C22⋊D12713yes—
128C22—26no—
129C22—41no—
130C22—46no—
131C22—84no—
132C22C22×C22⋊D12713yes—
133C4—26no—
134C4—5086no—
135C4—5082no—
136C4#1686117811yescharacteristic
137C4#1686117813yes—
138C2—23no—
139C2—46no—
140C2C2×#1686117811yescharacteristic
subgroupG/H
centerC2C2×#16861178
commutator, cosocleC23×C127C24
FrattiniC23C23×D127
Fitting, prime radical#585×C127C2
socleC2×C127C2×#120
hypercenter#585D127
hypocenterC127#3028
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #22550447C2semidirect
2. #22550066C2semidirect
3. #22549850C2semidirect
5. C22×Dic254C22semidirect
8. #16861190C22semidirect
10. #16861168C22semidirect
11. #16861124C22semidirect
71. C127#3028semidirect

Maximal subgroups

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

subgroupindex
#225504472
#225500662
#225498502
#585×C1272
#3028127