Finite Group Databasebeta

← #30648122

Subgroups — #30648122

Order 16256.

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

normal subgroups only

This group has 766 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#22550383C213yesmaximal, complemented
2#22550066C214yesmaximal, complemented
3#22550063C214yesmaximal, complemented
4#22549850C213yesmaximal, complemented
5#585×C127C211yesmaximal, characteristic, complemented
6C22×C4×D127C2211yescharacteristic, complemented
7#16861190C2216yescomplemented
8#16861185—23no—
9#16861180—23no—
10#16861177C2216yescomplemented
11C4×C22⋊D127—26no—
12#16861168C2213yescomplemented
13#16861124C22112yescomplemented
142−1+4×C127—23no—
15H(5,𝔽2)×C127—23no—
16H(5,𝔽2)×C127C2214yes—
17C2×#41×C127—26no—
18C2×#41×C127C2213yes—
19C23×D127C2311yescharacteristic
20C2×C22⋊D127—26no—
21C22×Dic127C2311yescharacteristic
22C2×C4×D127—23no—
23C2×C4×D127—23no—
24C4⋊Dic127—23no—
25Dic127⋊C4—26no—
26C22⋊Dic127—26no—
27C22⋊Dic127C2316yes—
28(C2×D127)⋊C4—26no—
29C4×Dic127—23no—
30#41×C127—24no—
31#41×C127—212no—
32#41×C127—26no—
33#41×C127—212no—
34#41×C127—26no—
35C2×Q8×C127—26no—
36C2×Q8×C127D413yescomplemented
37C2×D4×C127—26no—
38C2×D4×C127—212no—
39C2×D4×C127D413yescomplemented
40C2×D4×C127C2316yes—
41C22×C4×C127—24no—
42C22×C4×C127D416yescomplemented
43C22×C4×C127C2311yescharacteristic
44C22×D127—23no—
45C22×D127—42no—
46C22⋊D127—412no—
47C2×Dic127—26no—
48C2×Dic127—23no—
49C4×D127—43no—
50C4×D127—41no—
51C23×C127—24no—
52C23×C127C2×D416yes—
53C23×C127C2411yescharacteristic
54Q8×C127—24no—
55Q8×C127—26no—
56D4×C127—212no—
57D4×C127—212no—
58D4×C127—26no—
59C2×C4×C127—212no—
60C2×C4×C127—26no—
61C2×C4×C127—26no—
62C2×C4×C127C2×D416yes—
63C2×C4×C127C2×D413yes—
64C2×C4×C127C2×D413yes—
65C22×C127—26no—
66C22×C127—41no—
67C22×C127—46no—
68C22×C127—84no—
69C22×C127C22×D413yes—
70C2×D127—26no—
71C2×D127—43no—
72C2×D127—41no—
73C4×C127—26no—
74C4×C127#12013yes—
75C4×C127#12011yescharacteristic
76Dic127—46no—
77Dic127—41no—
78C2×C127—23no—
79C2×C127—46no—
80C2×C127C2×#12011yescharacteristic
81D127—42no—
82#3025—1271nomaximal, Sylow
83C127#302511yescharacteristic, complemented, Sylow
84#585D12711yescharacteristic, complemented
85#525—1273no—
86#458—1274no—
87B(4,𝔽2)—1274no—
88#410—1273no—
892−1+4—23no—
90H(5,𝔽2)—23no—
91H(5,𝔽2)C2×D12714yes—
92C2×#41—26no—
93C2×#41C2×D12713yes—
94C23×C4—1271no—
95#123—1276no—
96C22⋊Q8—2543no—
97#121—2543no—
98#120—1276no—
99C4×D4—2546no—
100#117—1273no—
101#99—12712no—
102C24—1271no—
103#41—24no—
104#41—212no—
105#41—26no—
106#41—212no—
107#41—26no—
108C2×Q8—26no—
109C2×Q8C22⋊D12713yescomplemented
110C2×D4—26no—
111C2×D4—212no—
112C2×D4—2546no—
113C2×D4C22⋊D12713yescomplemented
114C2×D4C22×D12716yes—
115C22×C4—24no—
116C22×C4—1271no—
117C22×C4—2543no—
118C22×C4—2543no—
119C22×C4C22⋊D12716yescomplemented
120C22×C4C22×D12711yescharacteristic
121C4⋊C4—2546no—
122C4⋊C4—2543no—
123C22⋊C4—1276no—
124C22⋊C4—2546no—
125C22⋊C4—2546no—
126C42—2543no—
127C23—24no—
128C23—5082no—
129C23—2543no—
130C23C2×C22⋊D12716yes—
131C23C23×D12711yescharacteristic
132Q8—24no—
133Q8—26no—
134D4—212no—
135D4—212no—
136D4—26no—
137D4—50812no—
138C2×C4—212no—
139C2×C4—26no—
140C2×C4—26no—
141C2×C4—5083no—
142C2×C4—5081no—
143C2×C4—2546no—
144C2×C4—2543no—
145C2×C4C2×C22⋊D12716yes—
146C2×C4C2×C22⋊D12713yes—
147C2×C4C2×C22⋊D12713yes—
148C22—26no—
149C22—41no—
150C22—46no—
151C22—84no—
152C22—5083no—
153C22—5081no—
154C22—2546no—
155C22C22×C22⋊D12713yes—
156C4—26no—
157C4—5086no—
158C4—5081no—
159C4#1686117813yes—
160C4#1686117811yescharacteristic
161C2—23no—
162C2—46no—
163C2—5082no—
164C2C2×#1686117811yescharacteristic
subgroupG/H
centerC4#16861178
commutator, cosocleC23×C127C24
FrattiniC23C23×D127
Fitting#585×C127C2
socleC2×C127C2×#120
hypercenter#585D127
hypocenterC127#3025
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #22550383C2semidirect
2. #22550066C2semidirect
3. #22550063C2semidirect
4. #22549850C2semidirect
5. #585×C127C2semidirect
6. C22×C4×D127C22semidirect
7. #16861190C22semidirect
10. #16861177C22semidirect
12. #16861168C22semidirect
13. #16861124C22semidirect
36. C2×Q8×C127D4semidirect
39. C2×D4×C127D4semidirect
42. C22×C4×C127D4semidirect
83. C127#3025semidirect
84. #585D127semidirect
109. C2×Q8C22⋊D127semidirect
113. C2×D4C22⋊D127semidirect
119. C22×C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225503832
#225500662
#225500632
#225498502
#585×C1272
#3025127