Finite Group Databasebeta

← #120×Dic127 #30641233

Subgroups — #120×Dic127 #30641233

Order 16256.

1224 conjugacy classes of subgroups; only the 289 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 289 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×D4×Dic127C213yesmaximal, complemented
3#22549756C213yesmaximal, complemented
4#22549751C213yesmaximal, complemented
5#22549732C211yesmaximal, characteristic, complemented
6C22⋊C4×Dic127C213yesmaximal, complemented
7C2×#120×C127C211yesmaximal, characteristic
8C23×Dic127—212no—
9C23×Dic127C2213yes—
10C23×Dic127C2213yes—
11C23×Dic127C2211yescharacteristic, complemented
12D4×Dic127—212no—
13C2×C4⋊Dic127C2213yescomplemented
14C2×C22⋊Dic127C2213yes—
15C2×C22⋊Dic127C2213yes—
16C2×C22⋊Dic127C2213yescomplemented
17C2×C4×Dic127C2213yescomplemented
18C127⋊#95C2216yescomplemented
19C25×C127C2211yescharacteristic
20C22×D4×C127C2213yes—
21#120×C127C418yescomplemented
22C2×C22⋊C4×C127C2213yes—
23C22×Dic127—232no—
24C22×Dic127—212no—
25C22×Dic127—212no—
26C22×Dic127—26no—
27C22×Dic127—26no—
28C22×Dic127D4112yescomplemented
29C22×Dic127C2311yescharacteristic
30C22×Dic127C2313yes—
31C22×Dic127C2313yes—
32C22×Dic127C2311yescharacteristic
33C4⋊Dic127—26no—
34C22⋊Dic127—26no—
35C22⋊Dic127—26no—
36C22⋊Dic127—26no—
37C4×Dic127—26no—
38C24×C127—26no—
39C24×C127—26no—
40C24×C127C2×C414yescomplemented
41C24×C127C2311yescharacteristic
42C24×C127C2313yes—
43C2×D4×C127—26no—
44C2×D4×C127—26no—
45C2×D4×C127C2×C4112yescomplemented
46C22×C4×C127C2313yes—
47C22⋊C4×C127C2×C4112yescomplemented
48C2×Dic127—224no—
49C2×Dic127—26no—
50C2×Dic127—224no—
51C2×Dic127—23no—
52C2×Dic127—23no—
53C2×Dic127—26no—
54C2×Dic127—44no—
55C2×Dic127C2×D416yes—
56C2×Dic127C2×D416yes—
57C23×C127—224no—
58C23×C127—28no—
59C23×C127—224no—
60C23×C127—26no—
61C23×C127—26no—
62C23×C127—23no—
63C23×C127—23no—
64C23×C127C22×C416yes—
65C23×C127C22×C412yescomplemented
66C23×C127C2×D416yes—
67C23×C127#4116yes—
68C23×C127C2411yescharacteristic
69D4×C127—224no—
70C2×C4×C127—23no—
71C2×C4×C127—23no—
72C2×C4×C127C22×C416yescomplemented
73C22×C127—232no—
74C22×C127—212no—
75C22×C127—26no—
76C22×C127—26no—
77C22×C127—212no—
78C22×C127—26no—
79C22×C127—41no—
80C22×C127—43no—
81C22×C127C4×D4112yescomplemented
82C22×C127C23×C411yescharacteristic
83C22×C127C22×D413yes—
84C22×C127C2×#4113yes—
85C4×C127—26no—
86Dic127—26no—
87Dic127—41no—
88Dic127—43no—
89Dic127#12014yesdirect product factor
90C2×C127—212no—
91C2×C127—42no—
92C2×C127C2×C4×D413yes—
93C2×C127C2×#12011yescharacteristic
94C2×C127#52513yes—
95C4×#120—1271nomaximal, Sylow
96C127C4×#12011yescharacteristic, complemented, Sylow
97C24×C4—1271no—
98C2×#120D12711yescharacteristic
99C2×C4×D4—1273no—
100#386—1273no—
101#385—1273no—
102#379—1271no—
103C4×C22⋊C4—1273no—
104C25C2×D12711yescharacteristic
105C22×D4C2×D12713yes—
106C23×C4—1273no—
107C23×C4—1271no—
108C23×C4—1273no—
109C23×C4—25412no—
110#120Dic12718yesdirect product factor
111C4×D4—25412no—
112C2×C4⋊C4—1273no—
113C2×C22⋊C4—1273no—
114C2×C22⋊C4—1273no—
115C2×C22⋊C4—1273no—
116C2×C22⋊C4C2×D12713yes—
117C2×C42—1273no—
118#95—1276no—
119C24—26no—
120C24—26no—
121C24C2×Dic12714yescomplemented
122C24C22×D12711yescharacteristic
123C24C22×D12713yes—
124C2×D4—26no—
125C2×D4—26no—
126C2×D4C2×Dic127112yescomplemented
127C22×C4—1271no—
128C22×C4—12712no—
129C22×C4—1273no—
130C22×C4—1273no—
131C22×C4—1271no—
132C22×C4—2546no—
133C22×C4—25412no—
134C22×C4—25412no—
135C22×C4—25432no—
136C22×C4—2546no—
137C22×C4C22×D12713yes—
138C4⋊C4—2546no—
139C22⋊C4—2546no—
140C22⋊C4—2546no—
141C22⋊C4—2546no—
142C22⋊C4C2×Dic127112yescomplemented
143C42—2546no—
144C23—28no—
145C23—224no—
146C23—26no—
147C23—26no—
148C23—224no—
149C23—23no—
150C23—23no—
151C23C22×Dic12716yes—
152C23C22×Dic12712yescomplemented
153C23D4×D12716yes—
154C23#1452324716yes—
155C23C23×D12711yescharacteristic
156D4—224no—
157C2×C4—23no—
158C2×C4—23no—
159C2×C4—1276no—
160C2×C4—1276no—
161C2×C4—5084no—
162C2×C4—25424no—
163C2×C4—2543no—
164C2×C4—2543no—
165C2×C4—2546no—
166C2×C4—25424no—
167C2×C4—2546no—
168C2×C4C22×Dic12716yescomplemented
169C22—232no—
170C22—212no—
171C22—26no—
172C22—26no—
173C22—212no—
174C22—26no—
175C22—41no—
176C22—43no—
177C22D4×Dic127112yescomplemented
178C22C23×Dic12711yescharacteristic
179C22C2×D4×D12713yes—
180C22C2×#1452324713yes—
181C4—26no—
182C4—1274no—
183C4—5081no—
184C4—5083no—
185C4—2546no—
186C2—212no—
187C2—42no—
188C2C2×D4×Dic12713yes—
189C2#120×D12711yescharacteristic
190C2#2255038213yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting, prime radicalC2×#120×C127C2
socle, cosocleC23×C127C24
hypercenterC2×#120D127
hypocenterC127C4×#120
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C24×Dic127C2semidirect
2. C2×D4×Dic127C2semidirect
3. #22549756C2semidirect
4. #22549751C2semidirect
5. #22549732C2semidirect
6. C22⋊C4×Dic127C2semidirect
11. C23×Dic127C22semidirect
13. C2×C4⋊Dic127C22semidirect
16. C2×C22⋊Dic127C22semidirect
17. C2×C4×Dic127C22semidirect
18. C127⋊#95C22semidirect
21. #120×C127C4semidirect
28. C22×Dic127D4semidirect
40. C24×C127C2×C4semidirect
45. C2×D4×C127C2×C4semidirect
47. C22⋊C4×C127C2×C4semidirect
65. C23×C127C22×C4semidirect
72. C2×C4×C127C22×C4semidirect
81. C22×C127C4×D4semidirect
89. Dic127#120direct (N × H)
96. C127C4×#120semidirect
110. #120Dic127direct (N × H)
121. C24C2×Dic127semidirect
126. C2×D4C2×Dic127semidirect
142. C22⋊C4C2×Dic127semidirect
152. C23C22×Dic127semidirect
168. C2×C4C22×Dic127semidirect
177. C22D4×Dic127semidirect

Maximal subgroups

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

subgroupindex
C24×Dic1272
C2×D4×Dic1272
#225497562
#225497512
#225497322
C22⋊C4×Dic1272
C2×#120×C1272
C4×#120127