Finite Group Databasebeta

← C2×#31774459 #46849667

Subgroups — C2×#31774459 #46849667

Order 34688.

1984 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1984 conjugacy classes of subgroups — too many to draw. Only its 565 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⋊D271C212yesmaximal, complemented
2#31774459C2116yesmaximal, direct product factor
3C2×#22576144C214yesmaximal, complemented
4C2×#22576134C214yesmaximal, complemented
5C2×#22576130C214yesmaximal, complemented
6C22×C22⋊C4×C271C211yesmaximal, characteristic, complemented
7C24×D271C2212yescomplemented
8C22×D1084C2214yescomplemented
9C22×C22⋊D271—216no—
10C22×C22⋊D271C2214yes—
11C22×C22⋊D271C2214yescomplemented
12C22×C22⋊D271C22116yescomplemented
13C23×Dic271C2212yescomplemented
14#22576144C22132yescomplemented
15#22576134C22132yescomplemented
16#22576130C22132yescomplemented
17C2×C4⋊Dic271C2214yescomplemented
18C2×(C2×D271)⋊C4C2218yescomplemented
19C25×C271C2211yescharacteristic
20C23×C4×C271C2212yescomplemented
21C2×C22⋊C4×C271C2214yescomplemented
22C2×C22⋊C4×C271C2218yescomplemented
23C23×D271—28no—
24C23×D271—48no—
25C23×D271C2314yes—
26C23×D271C2318yescomplemented
27C2×D1084—48no—
28C2×D1084C23116yescomplemented
29C2×C22⋊D271—216no—
30C2×C22⋊D271—216no—
31C2×C22⋊D271—264no—
32C2×C22⋊D271C23116yes—
33C2×C22⋊D271C23116yescomplemented
34C22×Dic271—28no—
35C22×Dic271C2314yes—
36C22×Dic271C2318yescomplemented
37C4⋊Dic271C23116yescomplemented
38(C2×D271)⋊C4C23132yescomplemented
39C24×C271—44no—
40C24×C271D418yescomplemented
41C24×C271C2312yes—
42C24×C271C2311yescharacteristic
43C24×C271C2314yes—
44C22×C4×C271—44no—
45C22×C4×C271C2314yes—
46C22×C4×C271C2318yescomplemented
47C22⋊C4×C271C23116yescomplemented
48C22×D271—28no—
49C22×D271—216no—
50C22×D271—216no—
51C22×D271—416no—
52C22×D271—432no—
53C22×D271C2418yes—
54D1084—416no—
55C22⋊D271—232no—
56C22⋊D271—232no—
57C2×Dic271—28no—
58C2×Dic271—216no—
59C2×Dic271C2418yes—
60C23×C271—24no—
61C23×C271—28no—
62C23×C271—416no—
63C23×C271—48no—
64C23×C271C2×D418yes—
65C23×C271C2×D414yes—
66C23×C271C2×D4116yescomplemented
67C23×C271C2414yes—
68C23×C271C2412yes—
69C23×C271C2411yescharacteristic
70C2×C4×C271—28no—
71C2×C4×C271—48no—
72C2×C4×C271C2418yes—
73C22×C271—28no—
74C22×C271—216no—
75C22×C271—24no—
76C22×C271—28no—
77C22×C271—416no—
78C22×C271C22×D418yes—
79C22×C271C22×D414yes—
80C22×C271C22×D412yes—
81C22×C271H(5,𝔽2)14yes—
82C22×C271C2511yescharacteristic
83C2×D271—28no—
84C2×D271—216no—
85C2×D271—432no—
86C2×D271—416no—
87C4×C271—28no—
88Dic271—28no—
89C2×C271—28no—
90C2×C271—24no—
91C2×C271#53414yescomplemented
92C2×C271C23×D411yescharacteristic
93C2×C271C2×H(5,𝔽2)12yes—
94D271—48no—
95C271C2×#53411yescharacteristic, complemented, Sylow
96C2×#534—2711nomaximal, Sylow
97C23×D4—2712no—
98#534—27116no—
99C2×#123—2714no—
100C2×#121—2714no—
101C2×#120—2714no—
102C22×C22⋊C4D27111yescharacteristic, complemented
103C25—2712no—
104C25C2×D27111yescharacteristic
105C22×D4—2714no—
106C22×D4—2714no—
107C22×D4—27116no—
108C22×D4—2714no—
109C22×D4—54216no—
110C23×C4—2712no—
111C23×C4C2×D27112yescomplemented
112#123—27132no—
113#121—27132no—
114#120—27132no—
115C2×C4⋊C4—2714no—
116C2×C22⋊C4—2718no—
117C2×C22⋊C4C2×D27114yescomplemented
118C2×C22⋊C4C2×D27118yescomplemented
119C24—44no—
120C24—2714no—
121C24—2718no—
122C24—5428no—
123C24—10848no—
124C24D108418yescomplemented
125C24C22×D27112yes—
126C24C22×D27111yescharacteristic
127C24C22×D27114yes—
128C2×D4—27116no—
129C2×D4—27116no—
130C2×D4—27116no—
131C2×D4—54216no—
132C2×D4—54216no—
133C2×D4—54264no—
134C2×D4—10848no—
135C22×C4—44no—
136C22×C4—2718no—
137C22×C4—2714no—
138C22×C4—5428no—
139C22×C4C22×D27114yes—
140C22×C4C22×D27118yescomplemented
141C4⋊C4—27116no—
142C22⋊C4—27132no—
143C22⋊C4C22×D271116yescomplemented
144C23—24no—
145C23—28no—
146C23—416no—
147C23—48no—
148C23—2718no—
149C23—54216no—
150C23—5428no—
151C23—54216no—
152C23—108416no—
153C23—108432no—
154C23C2×D108418yes—
155C23C2×D108414yes—
156C23C2×D1084116yescomplemented
157C23C23×D27114yes—
158C23C23×D27112yes—
159C23C23×D27111yescharacteristic
160D4—54232no—
161D4—54232no—
162D4—108416no—
163C2×C4—28no—
164C2×C4—48no—
165C2×C4—2718no—
166C2×C4—54216no—
167C2×C4—5428no—
168C2×C4C23×D27118yes—
169C22—24no—
170C22—28no—
171C22—216no—
172C22—28no—
173C22—416no—
174C22—54216no—
175C22—5428no—
176C22—108432no—
177C22—108416no—
178C22C22×D108418yes—
179C22C22×D108414yes—
180C22C22×D108412yes—
181C22#2257620514yes—
182C22C24×D27111yescharacteristic
183C4—28no—
184C4—5428no—
185C2—28no—
186C2—24no—
187C2—10848no—
188C2#3177445914yesdirect product factor
189C2C23×D108411yescharacteristic
190C2C2×#2257620512yes—
subgroupG/H
centerC23C23×D271
commutator, cosocleC22×C271C25
FrattiniC22C24×D271
FittingC22×C22⋊C4×C271C2
socleC23×C271C24
hypercenterC22×C22⋊C4D271
hypocenterC271C2×#534
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D271C2semidirect
2. #31774459C2direct (N × H)
3. C2×#22576144C2semidirect
4. C2×#22576134C2semidirect
5. C2×#22576130C2semidirect
6. C22×C22⋊C4×C271C2semidirect
7. C24×D271C22semidirect
8. C22×D1084C22semidirect
11. C22×C22⋊D271C22semidirect
12. C22×C22⋊D271C22semidirect
13. C23×Dic271C22semidirect
14. #22576144C22semidirect
15. #22576134C22semidirect
16. #22576130C22semidirect
17. C2×C4⋊Dic271C22semidirect
18. C2×(C2×D271)⋊C4C22semidirect
20. C23×C4×C271C22semidirect
21. C2×C22⋊C4×C271C22semidirect
22. C2×C22⋊C4×C271C22semidirect
26. C23×D271C23semidirect
28. C2×D1084C23semidirect
33. C2×C22⋊D271C23semidirect
36. C22×Dic271C23semidirect
37. C4⋊Dic271C23semidirect
38. (C2×D271)⋊C4C23semidirect
40. C24×C271D4semidirect
46. C22×C4×C271C23semidirect
47. C22⋊C4×C271C23semidirect
66. C23×C271C2×D4semidirect
91. C2×C271#534semidirect
95. C271C2×#534semidirect
102. C22×C22⋊C4D271semidirect
111. C23×C4C2×D271semidirect
117. C2×C22⋊C4C2×D271semidirect
118. C2×C22⋊C4C2×D271semidirect
124. C24D1084semidirect
140. C22×C4C22×D271semidirect
143. C22⋊C4C22×D271semidirect
156. C23C2×D1084semidirect
188. C2#31774459direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D2712
#317744592
C2×#225761442
C2×#225761342
C2×#225761302
C22×C22⋊C4×C2712
C2×#534271