Finite Group Databasebeta

← C2×#31774639 #46850484

Subgroups — C2×#31774639 #46850484

Order 34688.

1408 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 469 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
1#31774639C2116yesmaximal, direct product factor
2C2×#22576149C212yesmaximal, complemented
3C2×#22576148C212yesmaximal, complemented
4C2×#22576146C214yesmaximal, complemented
5C2×#22576130C214yesmaximal, complemented
6C2×#22576128C212yesmaximal, complemented
7C2×#124×C271C211yesmaximal, characteristic, complemented
8C24×D271C2212yescomplemented
9C22×Dic542C2212yescomplemented
10C22×D1084C2212yes—
11C22×D1084C2212yes—
12C22×C22⋊D271C2214yes—
13#22576149C22116yescomplemented
14#22576148C22116yescomplemented
15#22576146C22132yescomplemented
16#22576130C22132yescomplemented
17#22576128C22116yescomplemented
18C2×C22⋊Dic271C2212yescomplemented
19C2×(C2×D271)⋊C4C2214yescomplemented
20C2×(C2×D271)⋊C4C2214yescomplemented
21C2×(C2×D271)⋊C4C2214yescomplemented
22C2×C4×Dic271C2212yescomplemented
23C22×Q8×C271C2211yescharacteristic, complemented
24C22×D4×C271C2211yescharacteristic
25#124×C271C2218yescomplemented
26C2×C22⋊C4×C271C2214yescomplemented
27C2×C42×C271C2211yescharacteristic, complemented
28C23×D271—44no—
29C23×D271—48no—
30C23×D271C2314yes—
31C23×D271C2318yescomplemented
32C2×Dic542—44no—
33C2×Dic542C2318yescomplemented
34C2×D1084—44no—
35C2×D1084—44no—
36C2×D1084C2318yes—
37C2×D1084C2318yes—
38C2×C22⋊D271—48no—
39C2×C22⋊D271C23116yes—
40C22×Dic271C2314yes—
41C22⋊Dic271C2318yescomplemented
42(C2×D271)⋊C4C23116yescomplemented
43(C2×D271)⋊C4C23116yescomplemented
44(C2×D271)⋊C4C23116yescomplemented
45C4×Dic271C2318yescomplemented
46C24×C271C2312yes—
47C2×Q8×C271—42no—
48C2×Q8×C271C2314yescomplemented
49C2×D4×C271—42no—
50C2×D4×C271C2314yes—
51C22×C4×C271C2312yes—
52C22×C4×C271C2312yes—
53C22×C4×C271C2311yescharacteristic
54C22⋊C4×C271C23116yescomplemented
55C42×C271C2314yescomplemented
56C22×D271—28no—
57C22×D271—28no—
58C22×D271—28no—
59C22×D271—416no—
60C22×D271—48no—
61C22×D271—432no—
62C22×D271C2418yes—
63Dic542—48no—
64D1084—48no—
65D1084—48no—
66C22⋊D271—416no—
67C2×Dic271—28no—
68C2×Dic271C2418yes—
69C23×C271—24no—
70C23×C271—28no—
71C23×C271C2414yes—
72C23×C271C2411yescharacteristic
73Q8×C271—44no—
74D4×C271—44no—
75C2×C4×C271—22no—
76C2×C4×C271—24no—
77C2×C4×C271—24no—
78C2×C4×C271C2414yes—
79C2×C4×C271C2412yes—
80C2×C4×C271C2414yes—
81C22×C271—24no—
82C22×C271—28no—
83C22×C271—48no—
84C22×C271H(5,𝔽2)12yes—
85C22×C271H(5,𝔽2)14yes—
86C22×C271C2511yescharacteristic
87C2×D271—28no—
88C2×D271—28no—
89C2×D271—28no—
90C2×D271—432no—
91C2×D271—416no—
92C4×C271—22no—
93C4×C271—24no—
94C4×C271—24no—
95Dic271—28no—
96C2×C271—44no—
97C2×C271H(3,𝔽4)14yescomplemented
98C2×C271C2×H(5,𝔽2)11yescharacteristic
99C2×C271C2×H(5,𝔽2)12yes—
100D271—48no—
101C271C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
102C2×H(3,𝔽4)—2711nomaximal, Sylow
103H(3,𝔽4)—27116no—
104C2×#124—2712no—
105C2×#124—2714no—
106C2×#124—2712no—
107C2×#124D27111yescharacteristic, complemented
108C2×#120—2712no—
109C2×#120—2714no—
110C25—2712no—
111C22×Q8—2712no—
112C22×Q8C2×D27111yescharacteristic, complemented
113C22×D4—2714no—
114C22×D4—2712no—
115C22×D4—2712no—
116C22×D4C2×D27111yescharacteristic
117#124—27116no—
118#124—27132no—
119#124—27116no—
120#124C2×D27118yescomplemented
121#120—27116no—
122#120—27132no—
123C2×C22⋊C4—2712no—
124C2×C22⋊C4—2714no—
125C2×C22⋊C4—2714no—
126C2×C22⋊C4—2714no—
127C2×C22⋊C4C2×D27114yescomplemented
128C2×C42—2712no—
129C2×C42C2×D27111yescharacteristic, complemented
130C24—2714no—
131C24—2718no—
132C24—10844no—
133C24—10848no—
134C24C22×D27112yes—
135C2×Q8—42no—
136C2×Q8—2718no—
137C2×Q8—10844no—
138C2×Q8C22×D27114yescomplemented
139C2×D4—42no—
140C2×D4—27116no—
141C2×D4—2718no—
142C2×D4—2718no—
143C2×D4—10844no—
144C2×D4—10844no—
145C2×D4—10848no—
146C2×D4C22×D27114yes—
147C22×C4—2714no—
148C22×C4C22×D27112yes—
149C22×C4C22×D27111yescharacteristic
150C22×C4C22×D27112yes—
151C22⋊C4—2718no—
152C22⋊C4—27116no—
153C22⋊C4—27116no—
154C22⋊C4—27116no—
155C22⋊C4C22×D271116yescomplemented
156C42—2718no—
157C42C22×D27114yescomplemented
158C23—24no—
159C23—28no—
160C23—2718no—
161C23—5428no—
162C23—5428no—
163C23—5428no—
164C23—10848no—
165C23—108416no—
166C23—108432no—
167C23C23×D27114yes—
168C23C23×D27111yescharacteristic
169Q8—44no—
170Q8—10848no—
171D4—44no—
172D4—10848no—
173D4—10848no—
174D4—108416no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—2718no—
179C2×C4—5428no—
180C2×C4C23×D27114yes—
181C2×C4C23×D27112yes—
182C2×C4C23×D27114yes—
183C22—24no—
184C22—28no—
185C22—48no—
186C22—5428no—
187C22—5428no—
188C22—5428no—
189C22—108432no—
190C22—108416no—
191C22#2257620512yes—
192C22#2257620614yes—
193C22C24×D27111yescharacteristic
194C4—22no—
195C4—24no—
196C4—24no—
197C4—5428no—
198C2—44no—
199C2—10848no—
200C2#3177463914yesdirect product factor
201C2C2×#2257620511yescharacteristic
202C2C2×#2257620612yes—
subgroupG/H
centerC23C23×D271
commutator, cosocleC22×C271C25
FrattiniC22C24×D271
FittingC2×#124×C271C2
socleC23×C271C24
hypercenterC2×#124D271
hypocenterC271C2×H(3,𝔽4)
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #31774639C2direct (N × H)
2. C2×#22576149C2semidirect
3. C2×#22576148C2semidirect
4. C2×#22576146C2semidirect
5. C2×#22576130C2semidirect
6. C2×#22576128C2semidirect
7. C2×#124×C271C2semidirect
8. C24×D271C22semidirect
9. C22×Dic542C22semidirect
13. #22576149C22semidirect
14. #22576148C22semidirect
15. #22576146C22semidirect
16. #22576130C22semidirect
17. #22576128C22semidirect
18. C2×C22⋊Dic271C22semidirect
19. C2×(C2×D271)⋊C4C22semidirect
20. C2×(C2×D271)⋊C4C22semidirect
21. C2×(C2×D271)⋊C4C22semidirect
22. C2×C4×Dic271C22semidirect
23. C22×Q8×C271C22semidirect
25. #124×C271C22semidirect
26. C2×C22⋊C4×C271C22semidirect
27. C2×C42×C271C22semidirect
31. C23×D271C23semidirect
33. C2×Dic542C23semidirect
41. C22⋊Dic271C23semidirect
42. (C2×D271)⋊C4C23semidirect
43. (C2×D271)⋊C4C23semidirect
44. (C2×D271)⋊C4C23semidirect
45. C4×Dic271C23semidirect
48. C2×Q8×C271C23semidirect
54. C22⋊C4×C271C23semidirect
55. C42×C271C23semidirect
97. C2×C271H(3,𝔽4)semidirect
101. C271C2×H(3,𝔽4)semidirect
107. C2×#124D271semidirect
112. C22×Q8C2×D271semidirect
120. #124C2×D271semidirect
127. C2×C22⋊C4C2×D271semidirect
129. C2×C42C2×D271semidirect
138. C2×Q8C22×D271semidirect
155. C22⋊C4C22×D271semidirect
157. C42C22×D271semidirect
200. C2#31774639direct (N × H)

Maximal subgroups

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

subgroupindex
#317746392
C2×#225761492
C2×#225761482
C2×#225761462
C2×#225761302
C2×#225761282
C2×#124×C2712
C2×H(3,𝔽4)271