Finite Group Databasebeta

← #30648127

Subgroups — #30648127

Order 16256.

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

normal subgroups only

This group has 800 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#22550435C213yesmaximal, complemented
2#22550066C214yesmaximal, complemented
3#22550063C214yesmaximal, complemented
4C2×#16861124C213yesmaximal, complemented
5C2×H(5,𝔽2)×C127C211yesmaximal, characteristic, complemented
6C22×C22⋊D127C2211yescharacteristic, complemented
7#16861192—23no—
8#16861190C2216yescomplemented
9#16861181—26no—
10#16861178—23no—
11#16861177C2216yescomplemented
12C2×C22⋊Dic127C2213yescomplemented
13#16861124C22112yescomplemented
14H(5,𝔽2)×C127—26no—
15H(5,𝔽2)×C127C2214yes—
16C2×#41×C127—23no—
17C22×D4×C127—23no—
18C22×D4×C127C2213yes—
19C23×D127C2311yescharacteristic, complemented
20C2×C22⋊D127—23no—
21C2×C22⋊D127—23no—
22C2×C22⋊D127—26no—
23C22×Dic127—23no—
24C22×Dic127C2311yescharacteristic, complemented
25Dic127⋊C4—26no—
26C22⋊Dic127—26no—
27C22⋊Dic127—23no—
28C22⋊Dic127C2316yescomplemented
29(C2×D127)⋊C4—26no—
30C24×C127—21no—
31C24×C127D413yescomplemented
32C24×C127C2311yescharacteristic
33#41×C127—212no—
34#41×C127—212no—
35C2×Q8×C127—21no—
36C2×D4×C127—23no—
37C2×D4×C127—212no—
38C2×D4×C127—26no—
39C2×D4×C127—212no—
40C2×D4×C127—26no—
41C2×D4×C127D416yescomplemented
42C2×D4×C127C2316yes—
43C22×C4×C127—23no—
44C22×C4×C127D413yescomplemented
45C22×D127—23no—
46C22×D127—81no—
47C22⋊D127—41no—
48C22⋊D127—43no—
49C22⋊D127—86no—
50C2×Dic127—26no—
51C2×Dic127—23no—
52C2×Dic127—83no—
53C23×C127—24no—
54C23×C127—23no—
55C23×C127—23no—
56C23×C127—81no—
57C23×C127—83no—
58C23×C127—82no—
59C23×C127C2×D416yes—
60C23×C127C2×D413yes—
61C23×C127C2×D413yes—
62C23×C127C2411yescharacteristic
63Q8×C127—24no—
64D4×C127—212no—
65D4×C127—212no—
66D4×C127—212no—
67C2×C4×C127—212no—
68C2×C4×C127—23no—
69C2×C4×C127—23no—
70C2×C4×C127C2×D416yes—
71C22×C127—26no—
72C22×C127—43no—
73C22×C127—43no—
74C22×C127—41no—
75C22×C127—46no—
76C22×C127—46no—
77C22×C127—83no—
78C22×C127—84no—
79C22×C127#12011yescharacteristic
80C22×C127#12013yes—
81C22×C127C22×D413yes—
82C2×D127—46no—
83C2×D127—41no—
84C4×C127—26no—
85Dic127—46no—
86Dic127—41no—
87C2×C127—23no—
88C2×C127—21no—
89C2×C127—23no—
90C2×C127—46no—
91C2×C127C2×#12011yescharacteristic
92D127—81no—
93#3026—1271nomaximal, Sylow
94C127#302611yescharacteristic, complemented, Sylow
95C2×H(5,𝔽2)D12711yescharacteristic, complemented
96#534—1273no—
97#458—1274no—
98B(4,𝔽2)—1274no—
99C2×#99—1273no—
100H(5,𝔽2)—26no—
101H(5,𝔽2)C2×D12714yes—
102C2×#41—23no—
103C22×D4—23no—
104C22×D4—1271no—
105C22×D4C2×D12713yes—
106#123—1276no—
107#123—2543no—
108#121—2546no—
109#120—1276no—
110#120—2543no—
111C2×C22⋊C4—1273no—
112#99—12712no—
113C24—21no—
114C24—1271no—
115C24C22⋊D12713yescomplemented
116C24C22×D12711yescharacteristic
117#41—212no—
118#41—212no—
119C2×Q8—21no—
120C2×D4—23no—
121C2×D4—212no—
122C2×D4—26no—
123C2×D4—212no—
124C2×D4—26no—
125C2×D4—2546no—
126C2×D4—2543no—
127C2×D4—2543no—
128C2×D4C22⋊D12716yescomplemented
129C2×D4C22×D12716yes—
130C22×C4—23no—
131C22×C4—1271no—
132C22×C4—2543no—
133C22×C4C22⋊D12713yescomplemented
134C4⋊C4—2546no—
135C22⋊C4—1276no—
136C22⋊C4—2543no—
137C22⋊C4—2546no—
138C22⋊C4—2546no—
139C23—24no—
140C23—23no—
141C23—23no—
142C23—81no—
143C23—83no—
144C23—82no—
145C23—10161no—
146C23—2543no—
147C23C2×C22⋊D12716yes—
148C23C2×C22⋊D12713yes—
149C23C2×C22⋊D12713yes—
150C23C23×D12711yescharacteristic
151Q8—24no—
152D4—212no—
153D4—212no—
154D4—212no—
155D4—10166no—
156D4—5081no—
157D4—5083no—
158C2×C4—212no—
159C2×C4—23no—
160C2×C4—23no—
161C2×C4—10163no—
162C2×C4—2546no—
163C2×C4—2543no—
164C2×C4C2×C22⋊D12716yes—
165C22—26no—
166C22—43no—
167C22—43no—
168C22—41no—
169C22—46no—
170C22—46no—
171C22—83no—
172C22—84no—
173C22—5086no—
174C22—5081no—
175C22#1686117811yescharacteristic
176C22#1686117813yes—
177C22C22×C22⋊D12713yes—
178C4—26no—
179C4—5086no—
180C4—5081no—
181C2—23no—
182C2—21no—
183C2—23no—
184C2—46no—
185C2—10161no—
186C2C2×#1686117811yescharacteristic
subgroupG/H
centerC2C2×#16861178
commutator, cosocleC23×C127C24
FrattiniC23C23×D127
FittingC2×H(5,𝔽2)×C127C2
socleC2×C127C2×#120
hypercenterC2×H(5,𝔽2)D127
hypocenterC127#3026
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C24 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22550435C2semidirect
2. #22550066C2semidirect
3. #22550063C2semidirect
4. C2×#16861124C2semidirect
5. C2×H(5,𝔽2)×C127C2semidirect
6. C22×C22⋊D127C22semidirect
8. #16861190C22semidirect
11. #16861177C22semidirect
12. C2×C22⋊Dic127C22semidirect
13. #16861124C22semidirect
19. C23×D127C23semidirect
24. C22×Dic127C23semidirect
28. C22⋊Dic127C23semidirect
31. C24×C127D4semidirect
41. C2×D4×C127D4semidirect
44. C22×C4×C127D4semidirect
94. C127#3026semidirect
95. C2×H(5,𝔽2)D127semidirect
115. C24C22⋊D127semidirect
128. C2×D4C22⋊D127semidirect
133. C22×C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225504352
#225500662
#225500632
C2×#168611242
C2×H(5,𝔽2)×C1272
#3026127