Finite Group Databasebeta

← C22×#16861180 #30651949

Subgroups — C22×#16861180 #30651949

Order 16256.

2528 conjugacy classes of subgroups; only the 685 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 685 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
1C23×C22⋊D127C212yesmaximal, complemented
2C23×C4×D127C211yesmaximal, characteristic, complemented
3C2×#16861180C2124yesmaximal, direct product factor
4C22×C4⋊Dic127C211yesmaximal, characteristic, complemented
5C22×C22⋊Dic127C212yesmaximal, complemented
6C23×D4×C127C211yesmaximal, characteristic, complemented
7C24×D127C2211yescharacteristic
8C22×C22⋊D127—216no—
9C22×C22⋊D127C22124yescomplemented
10C23×Dic127C2211yescharacteristic
11C23×Dic127C2212yes—
12C22×C4×D127—28no—
13C22×C4×D127C22112yescomplemented
14#16861180C22164yesdirect product factor
15C2×C4⋊Dic127C22112yescomplemented
16C2×C22⋊Dic127C22124yescomplemented
17C25×C127C2212yes—
18C22×D4×C127—28no—
19C22×D4×C127C22112yescomplemented
20C23×C4×C127C2211yescharacteristic
21C23×D127—24no—
22C23×D127—24no—
23C23×D127D418yescomplemented
24C23×D127C2316yes—
25C2×C22⋊D127—296no—
26C2×C22⋊D127C23132yescomplemented
27C22×Dic127—24no—
28C22×Dic127—28no—
29C22×Dic127C2316yes—
30C22×Dic127C23112yes—
31C2×C4×D127—248no—
32C2×C4×D127C23116yescomplemented
33C4⋊Dic127C23116yescomplemented
34C22⋊Dic127C23132yescomplemented
35C24×C127—28no—
36C24×C127—28no—
37C24×C127—28no—
38C24×C127C23112yes—
39C24×C127C2311yescharacteristic
40C2×D4×C127—248no—
41C2×D4×C127C23116yescomplemented
42C22×C4×C127D418yescomplemented
43C22×C4×C127C2316yes—
44C22×D127—232no—
45C22×D127—212no—
46C22×D127—212no—
47C22×D127C2×D4124yescomplemented
48C22×D127C2414yes—
49C22⋊D127—264no—
50C2×Dic127—212no—
51C2×Dic127—224no—
52C2×Dic127C2414yes—
53C2×Dic127C2418yes—
54C4×D127—232no—
55C23×C127—224no—
56C23×C127—224no—
57C23×C127—224no—
58C23×C127—432no—
59C23×C127C2×D414yes—
60C23×C127C2×D414yes—
61C23×C127#4114yes—
62C23×C127C2418yes—
63C23×C127C2413yes—
64D4×C127—232no—
65C2×C4×C127C2×D4124yescomplemented
66C2×C4×C127C2414yes—
67C22×C127—28no—
68C22×C127—28no—
69C22×C127—28no—
70C22×C127—448no—
71C22×C127#121116yescomplemented
72C22×C127C22×D416yes—
73C22×C127C22×D416yes—
74C22×C127C2×#4116yes—
75C22×C127C2511yescharacteristic
76C2×D127—248no—
77C2×D127—24no—
78C2×D127—24no—
79C2×D127C22×D418yescomplemented
80C4×C127C22×D418yescomplemented
81Dic127—24no—
82Dic127—28no—
83C2×C127—48no—
84C2×C127C2×#121112yescomplemented
85C2×C127C23×D411yescharacteristic
86C2×C127C23×D411yescharacteristic
87C2×C127C22×#4111yescharacteristic
88D127—28no—
89C22×#121—1271nomaximal, Sylow
90C127C22×#12111yescharacteristic, complemented, Sylow
91C23×D4—1272no—
92C23×D4D12711yescharacteristic, complemented
93C24×C4—1271no—
94C2×#121—12724no—
95C22×C4⋊C4—1271no—
96C22×C22⋊C4—1272no—
97C25—1271no—
98C25C2×D12712yes—
99C22×D4—28no—
100C22×D4—12724no—
101C22×D4—25416no—
102C22×D4C2×D127112yescomplemented
103C23×C4—1271no—
104C23×C4—1272no—
105C23×C4—12712no—
106C23×C4—2548no—
107C23×C4C2×D12711yescharacteristic
108#121—12764no—
109C2×C4⋊C4—12712no—
110C2×C22⋊C4—12724no—
111C24—28no—
112C24—28no—
113C24—28no—
114C24—1278no—
115C24—1276no—
116C24—2544no—
117C24—2544no—
118C24C22×D127112yes—
119C24C22×D12711yescharacteristic
120C2×D4—248no—
121C2×D4—12732no—
122C2×D4—25496no—
123C2×D4C22×D127116yescomplemented
124C22×C4—1276no—
125C22×C4—12712no—
126C22×C4—12716no—
127C22×C4—2548no—
128C22×C4—2544no—
129C22×C4—25448no—
130C22×C4C22⋊D12718yescomplemented
131C22×C4C22×D12716yes—
132C4⋊C4—12716no—
133C22⋊C4—12732no—
134C23—224no—
135C23—224no—
136C23—224no—
137C23—432no—
138C23—12724no—
139C23—1274no—
140C23—25412no—
141C23—25412no—
142C23—25432no—
143C23D4×D12714yes—
144C23C2×C22⋊D12714yes—
145C23#1452324714yes—
146C23C23×D12718yes—
147C23C23×D12713yes—
148D4—232no—
149D4—25464no—
150C2×C4—1274no—
151C2×C4—1278no—
152C2×C4—25424no—
153C2×C4—25412no—
154C2×C4—25432no—
155C2×C4C2×C22⋊D127124yescomplemented
156C2×C4C23×D12714yes—
157C22—28no—
158C22—28no—
159C22—28no—
160C22—448no—
161C22—1278no—
162C22—25448no—
163C22—2544no—
164C22—2544no—
165C22#16861180116yesdirect product factor
166C22C2×D4×D12716yes—
167C22C22×C22⋊D12716yes—
168C22C2×#1452324716yes—
169C22C24×D12711yescharacteristic
170C4—2548no—
171C4—2544no—
172C4C22×C22⋊D12718yescomplemented
173C2—48no—
174C2—2548no—
175C2C2×#16861180112yesdirect product factor
176C2C22×D4×D12711yescharacteristic
177C2C23×C22⋊D12711yescharacteristic
178C2C22×#1452324711yescharacteristic
subgroupG/H
centerC24C22×D127
commutator, cosocleC22×C127C25
FrattiniC22C24×D127
FittingC23×D4×C127C2
socleC24×C127C23
hypercenterC23×D4D127
hypocenterC127C22×#121
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D127C2semidirect
2. C23×C4×D127C2semidirect
3. C2×#16861180C2direct (N × H)
4. C22×C4⋊Dic127C2semidirect
5. C22×C22⋊Dic127C2semidirect
6. C23×D4×C127C2semidirect
9. C22×C22⋊D127C22semidirect
13. C22×C4×D127C22semidirect
14. #16861180C22direct (N × H)
15. C2×C4⋊Dic127C22semidirect
16. C2×C22⋊Dic127C22semidirect
19. C22×D4×C127C22semidirect
23. C23×D127D4semidirect
26. C2×C22⋊D127C23semidirect
32. C2×C4×D127C23semidirect
33. C4⋊Dic127C23semidirect
34. C22⋊Dic127C23semidirect
41. C2×D4×C127C23semidirect
42. C22×C4×C127D4semidirect
47. C22×D127C2×D4semidirect
65. C2×C4×C127C2×D4semidirect
71. C22×C127#121semidirect
79. C2×D127C22×D4semidirect
80. C4×C127C22×D4semidirect
84. C2×C127C2×#121semidirect
90. C127C22×#121semidirect
92. C23×D4D127semidirect
102. C22×D4C2×D127semidirect
123. C2×D4C22×D127semidirect
130. C22×C4C22⋊D127semidirect
155. C2×C4C2×C22⋊D127semidirect
165. C22#16861180direct (N × H)
172. C4C22×C22⋊D127semidirect
175. C2C2×#16861180direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D1272
C23×C4×D1272
C2×#168611802
C22×C4⋊Dic1272
C22×C22⋊Dic1272
C23×D4×C1272
C22×#121127