Finite Group Databasebeta

← #46844824

Subgroups — #46844824

Order 34688.

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

normal subgroups only

This group has 800 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
1C2×#22576205C211yesmaximal, characteristic
2#31774372C211yesmaximal, characteristic, complemented
3#31773870C214yesmaximal, complemented
4C2×#22576074C218yesmaximal, complemented
5#518×C271C211yesmaximal, characteristic, complemented
6#22576205—24no—
7#22576205—24no—
8C2×#16864654C2212yes—
9C2×#16864653C414yescomplemented
10C22×C22⋊D271C2214yes—
11C2×D4×D271C414yescomplemented
12D4×Dic271—24no—
13#22576119C2212yescomplemented
14C2×C22⋊Dic271C2214yescomplemented
15#22576074C22116yescomplemented
16C22×D4×C271C2211yescharacteristic
17C4×D4×C271—24no—
18#117×C271C2212yescomplemented
19C2×C22⋊C4×C271C2214yescomplemented
20C23×D271C2×C414yescomplemented
21#16864654—24no—
22#16864654—24no—
23#16864653—216no—
24C2×Dic542C2×C412yescomplemented
25C2×D1084C2×C412yescomplemented
26C2×C22⋊D271—28no—
27C2×C22⋊D271—28no—
28C2×C22⋊D271C2×C418yescomplemented
29C2×C22⋊D271C2314yes—
30D4×D271—216no—
31C22×Dic271—24no—
32C22×Dic271C2×C414yescomplemented
33C2×C4×D271C2×C414yescomplemented
34C4⋊Dic271—22no—
35C22⋊Dic271—24no—
36C22⋊Dic271C2314yes—
37C4×Dic271—22no—
38C24×C271C2312yes—
39C2×D4×C271C2×C414yes—
40C2×D4×C271D414yes—
41C2×D4×C271D414yes—
42C22×C4×C271—24no—
43C22×C4×C271C2311yescharacteristic
44C4⋊C4×C271—22no—
45C22⋊C4×C271—24no—
46C22⋊C4×C271C2314yes—
47C42×C271—22no—
48C22×D271—28no—
49C22×D271—84no—
50C22×D271C22×C414yes—
51Dic542—24no—
52D1084—24no—
53C22⋊D271—216no—
54C22⋊D271—24no—
55C22⋊D271—24no—
56C2×Dic271—28no—
57C2×Dic271—24no—
58C2×Dic271—84no—
59C2×Dic271C22×C414yes—
60C4×D271—28no—
61C23×C271—82no—
62C23×C271C22×C414yes—
63C23×C271C2×D414yes—
64C23×C271C2×D414yes—
65C23×C271C2411yescharacteristic
66D4×C271C22⋊C4116yes—
67C2×C4×C271—24no—
68C2×C4×C271—84no—
69C2×C4×C271C22×C412yes—
70C2×C4×C271C2×D412yes—
71C2×C4×C271C2×D412yes—
72C22×C271—44no—
73C22×C271—41no—
74C22×C271—44no—
75C22×C271—44no—
76C22×C271C2×C22⋊C418yes—
77C22×C271C23×C411yescharacteristic
78C22×C271C22×D411yescharacteristic
79C22×C271C22×D411yescharacteristic
80C2×D271—24no—
81C2×D271—44no—
82C2×D271—88no—
83C4×C271—44no—
84C4×C271C2×C22⋊C414yes—
85Dic271—24no—
86Dic271—44no—
87C2×C271—21no—
88C2×C271—21no—
89C2×C271—21no—
90C2×C271—28no—
91C2×C271C22×C22⋊C411yescharacteristic
92D271—44no—
93C271#288311yescharacteristic, complemented, Sylow
94#2883—2711nomaximal, Sylow
95C2×H(5,𝔽2)—2711no—
96#518—2711no—
97#518D27111yescharacteristic, complemented
98#410—2714no—
99C2×#99—2718no—
100H(5,𝔽2)—5424no—
101H(5,𝔽2)—5424no—
102C2×#41—2714no—
103C2×#41—2712no—
104C22×D4—2714no—
105C22×D4—2714no—
106C22×D4C2×D27111yescharacteristic
107C4×D4—24no—
108C4×D4—5424no—
109#117—2712no—
110#117C2×D27112yescomplemented
111C2×C22⋊C4—2714no—
112C2×C22⋊C4C2×D27114yescomplemented
113#99—27116no—
114C24—2714no—
115C24C22×D27112yes—
116#41—54216no—
117#41—5424no—
118#41—5424no—
119C2×Q8—2712no—
120C2×D4—2718no—
121C2×D4—2714no—
122C2×D4—2712no—
123C2×D4—5428no—
124C2×D4—5428no—
125C2×D4—54216no—
126C2×D4C4×D27114yes—
127C2×D4C22⋊D27114yes—
128C2×D4D108414yes—
129C22×C4—24no—
130C22×C4—2714no—
131C22×C4—2714no—
132C22×C4—5424no—
133C22×C4C22×D27111yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—5422no—
136C22⋊C4—24no—
137C22⋊C4—2714no—
138C22⋊C4—5424no—
139C22⋊C4C22×D27114yes—
140C42—22no—
141C42—5422no—
142C23—82no—
143C23—2714no—
144C23—5428no—
145C23—21684no—
146C23C2×C4×D27114yes—
147C23C2×C22⋊D27114yes—
148C23C2×D108414yes—
149C23C23×D27111yescharacteristic
150Q8—5424no—
151D4—5424no—
152D4—5424no—
153D4—54216no—
154D4—5424no—
155D4(C2×D271)⋊C4116yes—
156C2×C4—24no—
157C2×C4—84no—
158C2×C4—2714no—
159C2×C4—5428no—
160C2×C4—5428no—
161C2×C4—5424no—
162C2×C4—21684no—
163C2×C4C2×C4×D27112yes—
164C2×C4C2×C22⋊D27112yes—
165C2×C4C2×D108412yes—
166C22—44no—
167C22—44no—
168C22—41no—
169C22—44no—
170C22—5424no—
171C22—10844no—
172C22—21688no—
173C22C2×(C2×D271)⋊C418yes—
174C22C22×C4×D27111yescharacteristic
175C22C22×C22⋊D27111yescharacteristic
176C22C22×D108411yescharacteristic
177C4—44no—
178C4—5424no—
179C4—10844no—
180C4C2×(C2×D271)⋊C414yes—
181C2—21no—
182C2—21no—
183C2—21no—
184C2—28no—
185C2—10844no—
186C2C22×(C2×D271)⋊C411yescharacteristic
subgroupG/H
centerC2C22×(C2×D271)⋊C4
commutatorC22×C271C23×C4
FrattiniC23C23×D271
Fitting#518×C271C2
socleC2×C271C22×C22⋊C4
prime radicalC2×#22576205C2
cosocleC23×C271C24
hypercenter#518D271
hypocenterC271#2883
LCSG ▷ C22×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22×D4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #31774372C2semidirect
3. #31773870C2semidirect
4. C2×#22576074C2semidirect
5. #518×C271C2semidirect
9. C2×#16864653C4semidirect
11. C2×D4×D271C4semidirect
13. #22576119C22semidirect
14. C2×C22⋊Dic271C22semidirect
15. #22576074C22semidirect
18. #117×C271C22semidirect
19. C2×C22⋊C4×C271C22semidirect
20. C23×D271C2×C4semidirect
24. C2×Dic542C2×C4semidirect
25. C2×D1084C2×C4semidirect
28. C2×C22⋊D271C2×C4semidirect
32. C22×Dic271C2×C4semidirect
33. C2×C4×D271C2×C4semidirect
93. C271#2883semidirect
97. #518D271semidirect
110. #117C2×D271semidirect
112. C2×C22⋊C4C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225762052
#317743722
#317738702
C2×#225760742
#518×C2712
#2883271