Finite Group Databasebeta

← #30647256

Subgroups — #30647256

Order 16256.

778 conjugacy classes of subgroups; only the 293 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 293 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×#16861216C211yesmaximal, characteristic, complemented
2#22550359C211yesmaximal, characteristic, complemented
3#22549881C213yesmaximal, complemented
4#22549877C213yesmaximal, complemented
5#22549872C213yesmaximal, complemented
6C2×#16861130C213yesmaximal, complemented
7C2×H(5,𝔽2)×C127C211yesmaximal, characteristic
8#16861216—24no—
9C2×#14523232C2213yescomplemented
10C22×C127⋊C8C2213yescomplemented
11Q8×Dic127—21no—
12D4×Dic127—23no—
13#16861168C2213yescomplemented
14C2×C4⋊Dic127C2213yescomplemented
15#16861133C2211yescharacteristic, complemented
16#16861133C2213yescomplemented
17#16861130C2213yescomplemented
18#16861130C2216yescomplemented
19#16861130C2213yescomplemented
20H(5,𝔽2)×C127—24no—
21H(5,𝔽2)×C127C418yescomplemented
22C2×#41×C127—21no—
23C2×#41×C127C2213yes—
24C2×#41×C127C2211yescharacteristic
25C22×D4×C127—23no—
26C22×D4×C127C2213yes—
27C22×Dic127—23no—
28#14523232—26no—
29C2×C127⋊C8—26no—
30C2×C127⋊C8C2311yescharacteristic
31C2×C127⋊C8C2313yes—
32C4⋊Dic127—23no—
33C4⋊Dic127C2313yes—
34C4⋊Dic127C2311yescharacteristic
35C22⋊Dic127—23no—
36C4×Dic127—23no—
37C24×C127—23no—
38#41×C127—28no—
39#41×C127—26no—
40#41×C127C2×C414yes—
41#41×C127C2×C4112yescomplemented
42#41×C127D414yes—
43C2×Q8×C127D411yescharacteristic
44C2×Q8×C127C2311yescharacteristic
45C2×D4×C127—23no—
46C2×D4×C127—224no—
47C2×D4×C127—23no—
48C2×D4×C127—26no—
49C2×D4×C127C2×C4112yescomplemented
50C2×D4×C127D413yes—
51C2×D4×C127C2313yes—
52C22×C4×C127—23no—
53C22×C4×C127C2313yes—
54C2×Dic127—21no—
55C2×Dic127—23no—
56C2×Dic127—83no—
57C127⋊C8—21no—
58C127⋊C8—23no—
59C23×C127—23no—
60C23×C127—212no—
61C23×C127—83no—
62C23×C127—83no—
63C23×C127C2×D413yes—
64Q8×C127—21no—
65Q8×C127C22⋊C414yes—
66Q8×C127C22×C412yes—
67D4×C127—212no—
68D4×C127—212no—
69D4×C127—23no—
70D4×C127C22⋊C4112yes—
71D4×C127C22×C416yes—
72C2×C4×C127—21no—
73C2×C4×C127—212no—
74C2×C4×C127C22×C416yes—
75C2×C4×C127C2×D413yes—
76C2×C4×C127C2×D416yes—
77C2×C4×C127C2411yescharacteristic
78C22×C127—23no—
79C22×C127—23no—
80C22×C127—26no—
81C22×C127—43no—
82C22×C127—43no—
83C22×C127—812no—
84C22×C127C2×C22⋊C416yes—
85C22×C127C22×D411yescharacteristic
86C4×C127—22no—
87C4×C127C2×C22⋊C416yes—
88C4×C127C23×C411yescharacteristic
89C4×C127C22×D411yescharacteristic
90Dic127—43no—
91Dic127—41no—
92C2×C127—26no—
93C2×C127—46no—
94C2×C127C22×C22⋊C411yescharacteristic
95C2×C127#57611yescharacteristic
96C2×C127#57711yescharacteristic
97#2897—1271nomaximal, Sylow
98C127#289711yescharacteristic, complemented, Sylow
99C2×H(5,𝔽2)D12711yescharacteristic
100C2×#131—1271no—
101#520—1271no—
102#418—1273no—
103#417—1273no—
104#416—1273no—
105C2×#102—1273no—
106H(5,𝔽2)—24no—
107H(5,𝔽2)Dic12718yescomplemented
108C2×#41—21no—
109C2×#41C2×D12713yes—
110C2×#41C2×D12711yescharacteristic
111C22×D4—23no—
112C22×D4C2×D12713yes—
113#131—2544no—
114C2×M16—1273no—
115C22×C8—1273no—
116C4×Q8—2541no—
117C4×D4—2543no—
118#117—1273no—
119C2×C4⋊C4—1273no—
120#103—1271no—
121#103—1273no—
122#102—1273no—
123#102—1276no—
124#102—1273no—
125C24—23no—
126#41—28no—
127#41—26no—
128#41C2×Dic12714yes—
129#41C2×Dic127112yescomplemented
130#41C22⋊D12714yes—
131C2×Q8C22⋊D12711yescharacteristic
132C2×Q8C22×D12711yescharacteristic
133C2×D4—23no—
134C2×D4—224no—
135C2×D4—23no—
136C2×D4—26no—
137C2×D4C2×Dic127112yescomplemented
138C2×D4C22⋊D12713yes—
139C2×D4C22×D12713yes—
140C22×C4—23no—
141C22×C4—2543no—
142C22×C4C22×D12713yes—
143M16—2546no—
144C2×C8—1271no—
145C2×C8—1273no—
146C2×C8—2546no—
147C4⋊C4—1273no—
148C4⋊C4—1271no—
149C4⋊C4—2543no—
150C22⋊C4—2543no—
151C42—2543no—
152C23—23no—
153C23—212no—
154C23—83no—
155C23—83no—
156C23C2×C22⋊D12713yes—
157Q8—21no—
158Q8C22⋊Dic12714yes—
159Q8C22×Dic12712yes—
160D4—212no—
161D4—23no—
162D4—212no—
163D4C22⋊Dic127112yes—
164D4C22×Dic12716yes—
165C2×C4—21no—
166C2×C4—212no—
167C2×C4—10163no—
168C2×C4—2541no—
169C2×C4—2543no—
170C2×C4C22×Dic12716yes—
171C2×C4C2×C22⋊D12716yes—
172C2×C4C2×C22⋊D12713yes—
173C2×C4C23×D12711yescharacteristic
174C8—2541no—
175C8—2543no—
176C22—23no—
177C22—23no—
178C22—26no—
179C22—43no—
180C22—43no—
181C22—812no—
182C22C2×C22⋊Dic12716yes—
183C22C22×C22⋊D12711yescharacteristic
184C4—22no—
185C4—5083no—
186C4—5081no—
187C4C2×C22⋊Dic12716yes—
188C4C23×Dic12711yescharacteristic
189C4C22×C22⋊D12711yescharacteristic
190C2—26no—
191C2—46no—
192C2C22×C22⋊Dic12711yescharacteristic
193C2#2255067911yescharacteristic
194C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutatorC4×C127C23×C4
FrattiniC2×C4C23×D127
Fitting, prime radicalC2×H(5,𝔽2)×C127C2
socleC22×C127C22×D4
cosocleC2×C4×C127C24
hypercenterC2×H(5,𝔽2)D127
hypocenterC127#2897
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#16861216C2semidirect
2. #22550359C2semidirect
3. #22549881C2semidirect
4. #22549877C2semidirect
5. #22549872C2semidirect
6. C2×#16861130C2semidirect
9. C2×#14523232C22semidirect
10. C22×C127⋊C8C22semidirect
13. #16861168C22semidirect
14. C2×C4⋊Dic127C22semidirect
15. #16861133C22semidirect
16. #16861133C22semidirect
17. #16861130C22semidirect
18. #16861130C22semidirect
19. #16861130C22semidirect
21. H(5,𝔽2)×C127C4semidirect
41. #41×C127C2×C4semidirect
49. C2×D4×C127C2×C4semidirect
98. C127#2897semidirect
107. H(5,𝔽2)Dic127semidirect
129. #41C2×Dic127semidirect
137. C2×D4C2×Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612162
#225503592
#225498812
#225498772
#225498722
C2×#168611302
C2×H(5,𝔽2)×C1272
#2897127