Finite Group Databasebeta

← #30641229

Subgroups — #30641229

Order 16256.

1224 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 249 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
1C24×Dic127C211yesmaximal, characteristic, complemented
2C2×#16861178C211yesmaximal, characteristic
3C2×(C22⋊D127)⋊C4C213yesmaximal, complemented
4#22549753C213yesmaximal, complemented
5#22549748C213yesmaximal, complemented
6C22⋊C4×Dic127C213yesmaximal, complemented
7#379×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127C2213yes—
9C23×Dic127—212no—
10C23×Dic127C2213yes—
11C23×Dic127C2213yes—
12C22×C4×D127C2211yescharacteristic, complemented
13#16861178C418yescomplemented
14(C22⋊D127)⋊C4—212no—
15C2×Dic127⋊C4C2213yescomplemented
16C2×C22⋊Dic127C2213yes—
17C2×(C2×D127)⋊C4C2213yescomplemented
18C2×C4×Dic127C2213yescomplemented
19C127⋊#95C2216yescomplemented
20C25×C127C2211yescharacteristic
21C2×C22⋊C4×C127C2213yes—
22C2×C22⋊C4×C127C2213yes—
23C23×D127C2311yescharacteristic
24C2×C22⋊D127—26no—
25C2×C22⋊D127—26no—
26C2×C22⋊D127C2×C4112yescomplemented
27C22×Dic127—232no—
28C22×Dic127—212no—
29C22×Dic127—212no—
30C22×Dic127—26no—
31C22×Dic127D4112yescomplemented
32C22×Dic127C2313yes—
33C22×Dic127C2313yes—
34C22×Dic127C2311yescharacteristic
35C2×C4×D127—26no—
36Dic127⋊C4—26no—
37C22⋊Dic127C2×C4112yescomplemented
38(C2×D127)⋊C4—26no—
39C4×Dic127—26no—
40C24×C127—26no—
41C24×C127—26no—
42C24×C127C2×C414yescomplemented
43C24×C127C2313yes—
44C22×C4×C127C2311yescharacteristic
45C22×C4×C127C2313yes—
46C22⋊C4×C127—26no—
47C22⋊C4×C127—26no—
48C22×D127—23no—
49C22×D127—23no—
50C22×D127C22×C412yescomplemented
51C22⋊D127—224no—
52C2×Dic127—224no—
53C2×Dic127—23no—
54C2×Dic127—26no—
55C2×Dic127—23no—
56C2×Dic127—224no—
57C2×Dic127C22×C416yescomplemented
58C2×Dic127C2×D416yes—
59C2×Dic127C2×D416yes—
60C4×D127—44no—
61C23×C127—224no—
62C23×C127—28no—
63C23×C127—224no—
64C23×C127—26no—
65C23×C127—26no—
66C23×C127C22×C416yes—
67C23×C127C2×D416yes—
68C23×C127#4116yes—
69C23×C127C2411yescharacteristic
70C2×C4×C127—23no—
71C2×C4×C127—23no—
72C2×C4×C127—26no—
73C22×C127—232no—
74C22×C127—212no—
75C22×C127—26no—
76C22×C127—26no—
77C22×C127—212no—
78C22×C127C4×D4112yescomplemented
79C22×C127C23×C411yescharacteristic
80C22×C127C22×D413yes—
81C22×C127C2×#4113yes—
82C2×D127—26no—
83C2×D127—41no—
84C2×D127—43no—
85C4×C127—41no—
86C4×C127—43no—
87Dic127—26no—
88Dic127—26no—
89Dic127#12014yescomplemented
90C2×C127—212no—
91C2×C127C2×C4×D413yes—
92C2×C127C2×#12011yescharacteristic
93C2×C127#52513yes—
94D127—42no—
95C4×#120—1271nomaximal, Sylow
96C127C4×#12011yescharacteristic, complemented, Sylow
97C24×C4—1271no—
98C2×#120—1271no—
99C2×C4×D4—1273no—
100#386—1273no—
101#385—1273no—
102#379D12711yescharacteristic, complemented
103C4×C22⋊C4—1273no—
104C25C2×D12711yescharacteristic
105C22×D4—1273no—
106C23×C4—1273no—
107C23×C4—1273no—
108C23×C4—1271no—
109C23×C4—25412no—
110#120—1278no—
111C4×D4—25412no—
112C2×C4⋊C4—1273no—
113C2×C22⋊C4—1273no—
114C2×C22⋊C4—1273no—
115C2×C22⋊C4C2×D12713yes—
116C2×C22⋊C4C2×D12713yes—
117C2×C42—1273no—
118#95—1276no—
119C24—26no—
120C24—26no—
121C24—1271no—
122C24C4×D12714yescomplemented
123C24C22×D12713yes—
124C2×D4—12712no—
125C2×D4—2546no—
126C2×D4—2546no—
127C22×C4—1271no—
128C22×C4—12712no—
129C22×C4—1273no—
130C22×C4—1273no—
131C22×C4—2546no—
132C22×C4—25412no—
133C22×C4—25412no—
134C22×C4—25432no—
135C22×C4—2546no—
136C22×C4C22×D12713yes—
137C22×C4C22×D12711yescharacteristic
138C4⋊C4—2546no—
139C22⋊C4—26no—
140C22⋊C4—26no—
141C22⋊C4—12712no—
142C22⋊C4—2546no—
143C42—2546no—
144C23—28no—
145C23—224no—
146C23—26no—
147C23—26no—
148C23—224no—
149C23—1272no—
150C23—2543no—
151C23—2543no—
152C23C2×C4×D12716yes—
153C23D4×D12716yes—
154C23#1452324716yes—
155C23C23×D12711yescharacteristic
156D4—25424no—
157C2×C4—23no—
158C2×C4—23no—
159C2×C4—26no—
160C2×C4—1276no—
161C2×C4—1276no—
162C2×C4—1276no—
163C2×C4—5084no—
164C2×C4—25424no—
165C2×C4—25424no—
166C2×C4—2543no—
167C2×C4—2546no—
168C2×C4—2543no—
169C22—232no—
170C22—212no—
171C22—26no—
172C22—26no—
173C22—212no—
174C22—5081no—
175C22—5083no—
176C22—2546no—
177C22(C22⋊D127)⋊C4112yescomplemented
178C22C22×C4×D12711yescharacteristic
179C22C2×D4×D12713yes—
180C22C2×#1452324713yes—
181C4—41no—
182C4—43no—
183C4—1274no—
184C4—2546no—
185C4—2546no—
186C2—212no—
187C2—5082no—
188C2C2×(C22⋊D127)⋊C413yes—
189C2#120×D12711yescharacteristic
190C2#2255038213yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#379×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861178C2
hypercenter#379D127
hypocenterC127C4×#120
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C24×Dic127C2semidirect
3. C2×(C22⋊D127)⋊C4C2semidirect
4. #22549753C2semidirect
5. #22549748C2semidirect
6. C22⋊C4×Dic127C2semidirect
7. #379×C127C2semidirect
12. C22×C4×D127C22semidirect
13. #16861178C4semidirect
15. C2×Dic127⋊C4C22semidirect
17. C2×(C2×D127)⋊C4C22semidirect
18. C2×C4×Dic127C22semidirect
19. C127⋊#95C22semidirect
26. C2×C22⋊D127C2×C4semidirect
31. C22×Dic127D4semidirect
37. C22⋊Dic127C2×C4semidirect
42. C24×C127C2×C4semidirect
50. C22×D127C22×C4semidirect
57. C2×Dic127C22×C4semidirect
78. C22×C127C4×D4semidirect
89. Dic127#120semidirect
96. C127C4×#120semidirect
102. #379D127semidirect
122. C24C4×D127semidirect
177. C22(C22⋊D127)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C24×Dic1272
C2×#168611782
C2×(C22⋊D127)⋊C42
#225497532
#225497482
C22⋊C4×Dic1272
#379×C1272
C4×#120127