Finite Group Databasebeta

← #30648132

Subgroups — #30648132

Order 16256.

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

normal subgroups only

This group has 772 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#22550440C213yesmaximal, complemented
2#22550063C218yesmaximal, complemented
3#22549850C213yesmaximal, complemented
4#585×C127C211yesmaximal, characteristic, complemented
5C22×D508C2211yescharacteristic, complemented
6#16861205—23no—
7#16861198—23no—
8#16861183—26no—
9#16861177C22112yescomplemented
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—
17C23×D127C2312yes—
18C2×D508—23no—
19C2×D508—23no—
20C2×C22⋊D127—212no—
21C4⋊Dic127—23no—
22C22⋊Dic127C2316yes—
23(C2×D127)⋊C4—212no—
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×C127D413yescomplemented
32C2×D4×C127—26no—
33C2×D4×C127—212no—
34C2×D4×C127D413yescomplemented
35C2×D4×C127C2316yes—
36C22×C4×C127—24no—
37C22×C4×C127D416yescomplemented
38C22×C4×C127C2311yescharacteristic
39C22×D127—26no—
40C22×D127—82no—
41D508—41no—
42D508—43no—
43C22⋊D127—812no—
44C2×Dic127—26no—
45C23×C127—24no—
46C23×C127C2×D416yes—
47C23×C127C2411yescharacteristic
48Q8×C127—24no—
49Q8×C127—26no—
50D4×C127—212no—
51D4×C127—212no—
52D4×C127—26no—
53C2×C4×C127—212no—
54C2×C4×C127—26no—
55C2×C4×C127—26no—
56C2×C4×C127C2×D416yes—
57C2×C4×C127C2×D413yes—
58C2×C4×C127C2×D413yes—
59C22×C127—26no—
60C22×C127—41no—
61C22×C127—46no—
62C22×C127—84no—
63C22×C127C22×D413yes—
64C2×D127—412no—
65C2×D127—42no—
66C4×C127—26no—
67C4×C127#12011yescharacteristic
68C4×C127#12013yes—
69Dic127—46no—
70C2×C127—23no—
71C2×C127—46no—
72C2×C127C2×#12011yescharacteristic
73D127—82no—
74#3027—1271nomaximal, Sylow
75C127#302711yescharacteristic, complemented, Sylow
76#585D12711yescharacteristic, complemented
77#535—1273no—
78B(4,𝔽2)—1278no—
79#410—1273no—
802−1+4—23no—
81H(5,𝔽2)—23no—
82H(5,𝔽2)C2×D12714yes—
83C2×#41—26no—
84C2×#41C2×D12713yes—
85C22×D4—1271no—
86#127—2543no—
87#124—2543no—
88#121—2546no—
89#120—12712no—
90#117—1273no—
91#99—12712no—
92C24—1272no—
93#41—24no—
94#41—212no—
95#41—26no—
96#41—212no—
97#41—26no—
98C2×Q8—26no—
99C2×Q8C22⋊D12713yescomplemented
100C2×D4—26no—
101C2×D4—212no—
102C2×D4—25412no—
103C2×D4—2543no—
104C2×D4—2543no—
105C2×D4C22⋊D12713yescomplemented
106C2×D4C22×D12716yes—
107C22×C4—24no—
108C22×C4C22⋊D12716yescomplemented
109C22×C4C22×D12711yescharacteristic
110C4⋊C4—2543no—
111C22⋊C4—1276no—
112C22⋊C4—25412no—
113C42—2543no—
114C23—24no—
115C23—10162no—
116C23—2546no—
117C23C2×C22⋊D12716yes—
118C23C23×D12711yescharacteristic
119Q8—24no—
120Q8—26no—
121D4—212no—
122D4—212no—
123D4—26no—
124D4—101612no—
125D4—5081no—
126D4—5083no—
127C2×C4—212no—
128C2×C4—26no—
129C2×C4—26no—
130C2×C4—2546no—
131C2×C4C2×C22⋊D12716yes—
132C2×C4C2×C22⋊D12713yes—
133C2×C4C2×C22⋊D12713yes—
134C22—26no—
135C22—41no—
136C22—46no—
137C22—84no—
138C22—50812no—
139C22—5082no—
140C22C22×C22⋊D12713yes—
141C4—26no—
142C4—5086no—
143C4#1686117811yescharacteristic
144C4#1686117813yes—
145C2—23no—
146C2—46no—
147C2—10162no—
148C2C2×#1686117811yescharacteristic
subgroupG/H
centerC2C2×#16861178
commutator, cosocleC23×C127C24
FrattiniC23C23×D127
Fitting#585×C127C2
socleC2×C127C2×#120
hypercenter#585D127
hypocenterC127#3027
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. #22550440C2semidirect
2. #22550063C2semidirect
3. #22549850C2semidirect
4. #585×C127C2semidirect
5. C22×D508C22semidirect
9. #16861177C22semidirect
10. #16861168C22semidirect
11. #16861124C22semidirect
31. C2×Q8×C127D4semidirect
34. C2×D4×C127D4semidirect
37. C22×C4×C127D4semidirect
75. C127#3027semidirect
76. #585D127semidirect
99. C2×Q8C22⋊D127semidirect
105. C2×D4C22⋊D127semidirect
108. C22×C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225504402
#225500632
#225498502
#585×C1272
#3027127