Finite Group Databasebeta

← C22×#22576134 #46849586

Subgroups — C22×#22576134 #46849586

Order 34688.

2528 conjugacy classes of subgroups; only the 725 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 725 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×D1084C211yesmaximal, characteristic, complemented
2C23×C22⋊D271C212yesmaximal, complemented
3C2×#22576134C2124yesmaximal, direct product factor
4C22×C4⋊Dic271C211yesmaximal, characteristic, complemented
5C22×(C2×D271)⋊C4C212yesmaximal, complemented
6C24×C4×C271C211yesmaximal, characteristic, complemented
7C24×D271C2212yes—
8C22×D1084—28no—
9C22×D1084C22112yescomplemented
10C22×C22⋊D271—216no—
11C22×C22⋊D271C22124yescomplemented
12C23×Dic271C2212yes—
13#22576134C22164yesdirect product factor
14C2×C4⋊Dic271C22112yescomplemented
15C2×(C2×D271)⋊C4C22124yescomplemented
16C25×C271C2211yescharacteristic
17C23×C4×C271—28no—
18C23×C4×C271C2211yescharacteristic
19C23×C4×C271C2211yescharacteristic
20C23×C4×C271C22112yescomplemented
21C23×D271—28no—
22C23×D271—28no—
23C23×D271—28no—
24C23×D271C23112yes—
25C2×D1084—248no—
26C2×D1084C23116yescomplemented
27C2×C22⋊D271—296no—
28C2×C22⋊D271C23132yescomplemented
29C22×Dic271—28no—
30C22×Dic271C23112yes—
31C4⋊Dic271C23116yescomplemented
32(C2×D271)⋊C4C23132yescomplemented
33C24×C271—24no—
34C24×C271—24no—
35C24×C271D418yescomplemented
36C24×C271C2316yes—
37C24×C271C2311yescharacteristic
38C22×C4×C271—24no—
39C22×C4×C271—248no—
40C22×C4×C271D418yescomplemented
41C22×C4×C271C2316yes—
42C22×C4×C271C2316yes—
43C22×C4×C271C23116yescomplemented
44C22×D271—224no—
45C22×D271—224no—
46C22×D271—224no—
47C22×D271—432no—
48C22×D271C2418yes—
49D1084—232no—
50C22⋊D271—264no—
51C2×Dic271—224no—
52C2×Dic271C2418yes—
53C23×C271—232no—
54C23×C271—212no—
55C23×C271—212no—
56C23×C271C2×D414yes—
57C23×C271C2×D414yes—
58C23×C271C2×D4124yescomplemented
59C23×C271#4114yes—
60C23×C271C2414yes—
61C23×C271C2413yes—
62C2×C4×C271—212no—
63C2×C4×C271—232no—
64C2×C4×C271C2×D4124yescomplemented
65C2×C4×C271C2414yes—
66C2×C4×C271C2414yes—
67C22×C271—248no—
68C22×C271—24no—
69C22×C271—24no—
70C22×C271#121116yescomplemented
71C22×C271C22×D416yes—
72C22×C271C22×D416yes—
73C22×C271C22×D418yescomplemented
74C22×C271C2×#4116yes—
75C22×C271C2511yescharacteristic
76C2×D271—28no—
77C2×D271—28no—
78C2×D271—28no—
79C2×D271—448no—
80C4×C271—24no—
81C4×C271C22×D418yescomplemented
82Dic271—28no—
83C2×C271—28no—
84C2×C271C2×#121112yescomplemented
85C2×C271C23×D411yescharacteristic
86C2×C271C23×D411yescharacteristic
87C2×C271C22×#4111yescharacteristic
88D271—48no—
89C271C22×#12111yescharacteristic, complemented, Sylow
90C22×#121—2711nomaximal, Sylow
91C23×D4—2712no—
92C23×D4—2711no—
93C24×C4D27111yescharacteristic, complemented
94C2×#121—27124no—
95C22×C4⋊C4—2711no—
96C22×C22⋊C4—2712no—
97C25—2712no—
98C25C2×D27111yescharacteristic
99C22×D4—27124no—
100C22×D4—27112no—
101C22×D4—54216no—
102C22×D4—5428no—
103C23×C4—28no—
104C23×C4—2712no—
105C23×C4C2×D27111yescharacteristic
106C23×C4C2×D27111yescharacteristic
107C23×C4C2×D271112yescomplemented
108#121—27164no—
109C2×C4⋊C4—27112no—
110C2×C22⋊C4—27124no—
111C24—24no—
112C24—24no—
113C24—27112no—
114C24—5428no—
115C24—5428no—
116C24—5428no—
117C24D108418yescomplemented
118C24C22×D27116yes—
119C24C22×D27111yescharacteristic
120C2×D4—27132no—
121C2×D4—27116no—
122C2×D4—54296no—
123C2×D4—54248no—
124C22×C4—24no—
125C22×C4—248no—
126C22×C4—27112no—
127C22×C4—5428no—
128C22×C4C22⋊D27118yescomplemented
129C22×C4C22×D27116yes—
130C22×C4C22×D27116yes—
131C22×C4C22×D271116yescomplemented
132C4⋊C4—27116no—
133C22⋊C4—27132no—
134C23—232no—
135C23—212no—
136C23—212no—
137C23—2718no—
138C23—54224no—
139C23—54224no—
140C23—54224no—
141C23—108432no—
142C23C2×C22⋊D27114yes—
143C23C2×D108414yes—
144C23C2×D1084124yescomplemented
145C23#1686465414yes—
146C23C23×D27114yes—
147C23C23×D27113yes—
148D4—54264no—
149D4—54232no—
150C2×C4—212no—
151C2×C4—232no—
152C2×C4—2718no—
153C2×C4—54224no—
154C2×C4C2×C22⋊D271124yescomplemented
155C2×C4C23×D27114yes—
156C2×C4C23×D27114yes—
157C22—24no—
158C22—248no—
159C22—24no—
160C22—5428no—
161C22—5428no—
162C22—5428no—
163C22—108448no—
164C22#22576134116yesdirect product factor
165C22C22×C22⋊D27116yes—
166C22C22×D108416yes—
167C22C22×D108418yescomplemented
168C22C2×#1686465416yes—
169C22C24×D27111yescharacteristic
170C4—24no—
171C4—5428no—
172C4C22×C22⋊D27118yescomplemented
173C2—28no—
174C2—10848no—
175C2C2×#22576134112yesdirect product factor
176C2C23×C22⋊D27111yescharacteristic
177C2C23×D108411yescharacteristic
178C2C22×#1686465411yescharacteristic
subgroupG/H
centerC24C22×D271
commutator, cosocleC22×C271C25
FrattiniC22C24×D271
FittingC24×C4×C271C2
socleC24×C271C23
hypercenterC24×C4D271
hypocenterC271C22×#121
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C24 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. C23×D1084C2semidirect
2. C23×C22⋊D271C2semidirect
3. C2×#22576134C2direct (N × H)
4. C22×C4⋊Dic271C2semidirect
5. C22×(C2×D271)⋊C4C2semidirect
6. C24×C4×C271C2semidirect
9. C22×D1084C22semidirect
11. C22×C22⋊D271C22semidirect
13. #22576134C22direct (N × H)
14. C2×C4⋊Dic271C22semidirect
15. C2×(C2×D271)⋊C4C22semidirect
20. C23×C4×C271C22semidirect
26. C2×D1084C23semidirect
28. C2×C22⋊D271C23semidirect
31. C4⋊Dic271C23semidirect
32. (C2×D271)⋊C4C23semidirect
35. C24×C271D4semidirect
40. C22×C4×C271D4semidirect
43. C22×C4×C271C23semidirect
58. C23×C271C2×D4semidirect
64. C2×C4×C271C2×D4semidirect
70. C22×C271#121semidirect
73. C22×C271C22×D4semidirect
81. C4×C271C22×D4semidirect
84. C2×C271C2×#121semidirect
89. C271C22×#121semidirect
93. C24×C4D271semidirect
107. C23×C4C2×D271semidirect
117. C24D1084semidirect
128. C22×C4C22⋊D271semidirect
131. C22×C4C22×D271semidirect
144. C23C2×D1084semidirect
154. C2×C4C2×C22⋊D271semidirect
164. C22#22576134direct (N × H)
167. C22C22×D1084semidirect
172. C4C22×C22⋊D271semidirect
175. C2C2×#22576134direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D10842
C23×C22⋊D2712
C2×#225761342
C22×C4⋊Dic2712
C22×(C2×D271)⋊C42
C24×C4×C2712
C22×#121271