Finite Group Databasebeta

← #31774463

Subgroups — #31774463

Order 17344.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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×#16864654C211yesmaximal, characteristic, complemented
2C2×D4×D271C211yesmaximal, characteristic, complemented
3#22576156C212yesmaximal, complemented
4#22576146C212yesmaximal, complemented
5#22576135C211yesmaximal, characteristic, complemented
6#22576134C211yesmaximal, characteristic, complemented
7#22576131C211yesmaximal, characteristic, complemented
8#22576130C212yesmaximal, complemented
9#22576128C212yesmaximal, complemented
10#22576118C211yesmaximal, characteristic, complemented
11#121×C271C211yesmaximal, characteristic, complemented
12C23×D271C2212yes—
13#16864654—24no—
14C2×Dic542C2211yescharacteristic, complemented
15C2×D1084C2211yescharacteristic
16C2×D1084C2211yescharacteristic
17C2×D1084C2212yescomplemented
18C2×C22⋊D271C2211yescharacteristic
19C2×C22⋊D271C2212yes—
20C2×C22⋊D271C2211yescharacteristic, complemented
21C2×C22⋊D271C2212yescomplemented
22C2×C22⋊D271C2212yescomplemented
23D4×D271—24no—
24C2×C4×D271C2211yescharacteristic
25C2×C4×D271C2211yescharacteristic, complemented
26C4⋊Dic271C2211yescharacteristic, complemented
27C22⋊Dic271C2212yescomplemented
28(C2×D271)⋊C4C2212yes—
29(C2×D271)⋊C4C2212yescomplemented
30(C2×D271)⋊C4C2212yescomplemented
31C4×Dic271C2212yescomplemented
32C2×D4×C271C2211yescharacteristic
33C2×D4×C271C2212yescomplemented
34C22×C4×C271C2211yescharacteristic, complemented
35C4⋊C4×C271C2211yescharacteristic, complemented
36C22⋊C4×C271C2212yescomplemented
37C22×D271—24no—
38C22×D271—42no—
39C22×D271—42no—
40C22×D271C2312yes—
41C22×D271C2311yescharacteristic
42C22×D271C2311yescharacteristic
43Dic542—22no—
44D1084—22no—
45D1084—22no—
46D1084—42no—
47C22⋊D271—22no—
48C22⋊D271—24no—
49C22⋊D271—22no—
50C22⋊D271—42no—
51C22⋊D271—42no—
52C2×Dic271C2312yes—
53C2×Dic271C2311yescharacteristic
54C2×Dic271C2311yescharacteristic
55C4×D271—22no—
56C4×D271D414yescomplemented
57C23×C271C2312yes—
58C23×C271C2311yescharacteristic
59D4×C271—22no—
60D4×C271—42no—
61C2×C4×C271—22no—
62C2×C4×C271C2311yescharacteristic
63C2×C4×C271C2312yes—
64C2×C4×C271C2311yescharacteristic
65C22×C271—21no—
66C22×C271—22no—
67C22×C271—21no—
68C22×C271—22no—
69C22×C271—21no—
70C22×C271—22no—
71C22×C271C2411yescharacteristic
72C2×D271—21no—
73C2×D271—21no—
74C2×D271—21no—
75C2×D271—22no—
76C2×D271—21no—
77C2×D271—22no—
78C2×D271—21no—
79C2×D271—22no—
80C2×D271—48no—
81C2×D271C2×D412yes—
82C4×C271—21no—
83C4×C271—22no—
84C4×C271C2×D412yes—
85Dic271—21no—
86Dic271—22no—
87Dic271C2×D412yes—
88C2×C271—42no—
89C2×C271—41no—
90C2×C271C22×D411yescharacteristic
91C2×C271H(5,𝔽2)11yescharacteristic
92C2×C271H(5,𝔽2)11yescharacteristic
93D271—22no—
94D271—41no—
95D271—42no—
96C271#53511yescharacteristic, complemented, Sylow
97#535—2711nomaximal, Sylow
98C2×#41—2711no—
99C22×D4—2711no—
100#127—2712no—
101#124—2712no—
102#121—2711no—
103#121—2711no—
104#121—2711no—
105#121D27111yescharacteristic, complemented
106#120—2712no—
107#120—2712no—
108#117—2711no—
109C24—2712no—
110#41—5424no—
111C2×Q8—2711no—
112C2×D4—2711no—
113C2×D4—2712no—
114C2×D4—2711no—
115C2×D4—2712no—
116C2×D4—2712no—
117C2×D4—2711no—
118C2×D4—2711no—
119C2×D4—2712no—
120C2×D4—5424no—
121C2×D4C2×D27111yescharacteristic
122C2×D4C2×D27112yescomplemented
123C22×C4—2711no—
124C22×C4—2711no—
125C22×C4C2×D27111yescharacteristic, complemented
126C4⋊C4—2711no—
127C4⋊C4C2×D27111yescharacteristic, complemented
128C22⋊C4—2712no—
129C22⋊C4—2712no—
130C22⋊C4—2712no—
131C22⋊C4—2712no—
132C22⋊C4C2×D27112yescomplemented
133C42—2712no—
134C23—2711no—
135C23—2712no—
136C23—2711no—
137C23—5424no—
138C23—10842no—
139C23—10842no—
140C23C22×D27112yes—
141C23C22×D27111yescharacteristic
142Q8—5422no—
143D4—22no—
144D4—42no—
145D4—5422no—
146D4—5424no—
147D4—5422no—
148D4—5422no—
149D4—5422no—
150D4—10842no—
151D4—10842no—
152D4—10842no—
153C2×C4—22no—
154C2×C4—2711no—
155C2×C4—2714no—
156C2×C4—2712no—
157C2×C4—2711no—
158C2×C4—5422no—
159C2×C4C22×D27112yes—
160C2×C4C22×D27111yescharacteristic
161C2×C4C22×D27111yescharacteristic
162C22—21no—
163C22—22no—
164C22—21no—
165C22—22no—
166C22—21no—
167C22—22no—
168C22—2712no—
169C22—5421no—
170C22—5421no—
171C22—5421no—
172C22—5422no—
173C22—5421no—
174C22—5422no—
175C22—5421no—
176C22—5422no—
177C22—10848no—
178C22C23×D27111yescharacteristic
179C4—21no—
180C4—22no—
181C4—2712no—
182C4—5421no—
183C4—5422no—
184C4D4×D27112yes—
185C2—42no—
186C2—41no—
187C2—5422no—
188C2—10842no—
189C2—10841no—
190C2C2×D4×D27111yescharacteristic
191C2#2257620511yescharacteristic
192C2#2257620611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D271
commutator, socle, cosocleC22×C271C24
Fitting#121×C271C2
hypercenter#121D271
hypocenterC271#535
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#16864654C2semidirect
2. C2×D4×D271C2semidirect
3. #22576156C2semidirect
4. #22576146C2semidirect
5. #22576135C2semidirect
6. #22576134C2semidirect
7. #22576131C2semidirect
8. #22576130C2semidirect
9. #22576128C2semidirect
10. #22576118C2semidirect
11. #121×C271C2semidirect
14. C2×Dic542C22semidirect
17. C2×D1084C22semidirect
20. C2×C22⋊D271C22semidirect
21. C2×C22⋊D271C22semidirect
22. C2×C22⋊D271C22semidirect
25. C2×C4×D271C22semidirect
26. C4⋊Dic271C22semidirect
27. C22⋊Dic271C22semidirect
29. (C2×D271)⋊C4C22semidirect
30. (C2×D271)⋊C4C22semidirect
31. C4×Dic271C22semidirect
33. C2×D4×C271C22semidirect
34. C22×C4×C271C22semidirect
35. C4⋊C4×C271C22semidirect
36. C22⋊C4×C271C22semidirect
56. C4×D271D4semidirect
96. C271#535semidirect
105. #121D271semidirect
122. C2×D4C2×D271semidirect
125. C22×C4C2×D271semidirect
127. C4⋊C4C2×D271semidirect
132. C22⋊C4C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#168646542
C2×D4×D2712
#225761562
#225761462
#225761352
#225761342
#225761312
#225761302
#225761282
#225761182
#121×C2712
#535271