Finite Group Databasebeta

← #31774707

Subgroups — #31774707

Order 17344.

258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 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#22576207C211yesmaximal, characteristic, complemented
2#22576205C211yesmaximal, characteristic, complemented
3#22576193C211yesmaximal, characteristic, complemented
4#22576189C212yesmaximal, complemented
5#22576184C212yesmaximal, complemented
6#22576180C211yesmaximal, characteristic, complemented
7#22576179C212yesmaximal, complemented
8C2×#16864642C211yesmaximal, characteristic, complemented
9SD16×D271C212yesmaximal, complemented
10#22576168C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C271C211yesmaximal, characteristic, complemented
12#16864654—21no—
13#16864654C2211yescharacteristic
14#16864654C2211yescharacteristic
15#16864653—21no—
16#16864653—22no—
17#16864653C2212yes—
18#16864653C2212yescomplemented
19C2×Dic542—21no—
20C2×Dic542C2211yescharacteristic, complemented
21Q8×D271C2212yescomplemented
22C2×C22⋊D271—22no—
23D4×D271—21no—
24D4×D271C2212yes—
25Dic1084C2212yescomplemented
26#16864644C2211yescharacteristic, complemented
27#16864643C2212yescomplemented
28#16864642C2212yescomplemented
29#16864642C2212yescomplemented
30#16864642C2211yescharacteristic, complemented
31#16864641C2211yescharacteristic, complemented
32#16864639C2211yescharacteristic, complemented
33#16864637C2212yescomplemented
34C2×C271⋊C8C2211yescharacteristic, complemented
35C8×D271C2212yescomplemented
36#41×C271C2211yescharacteristic, complemented
37C2×D4×C271C2211yescharacteristic
38SD16×C271C2212yescomplemented
39D8×C271C2212yescomplemented
40M16×C271C2211yescharacteristic, complemented
41C22×D271—22no—
42Dic542—21no—
43Dic542—22no—
44Dic542D411yescharacteristic
45Dic542C2312yes—
46Dic542C2311yescharacteristic
47D1084D411yescharacteristic
48D1084C2311yescharacteristic
49C22⋊D271—21no—
50C22⋊D271—22no—
51C22⋊D271—22no—
52C22⋊D271—22no—
53C22⋊D271D412yes—
54C2×Dic271—21no—
55C2×Dic271—22no—
56C2×Dic271—22no—
57C4×D271—22no—
58C4×D271—21no—
59C4×D271C2312yes—
60C271⋊C8C2311yescharacteristic
61C271⋊C8C2311yescharacteristic
62C23×C271—21no—
63Q8×C271C2311yescharacteristic
64D4×C271—21no—
65D4×C271—21no—
66D4×C271C2311yescharacteristic
67D4×C271C2312yes—
68C2×C4×C271—21no—
69C2×C4×C271C2311yescharacteristic
70C8×C271C2312yes—
71C22×C271—21no—
72C22×C271—22no—
73C22×C271—81no—
74C22×C271C2×D411yescharacteristic
75C2×D271—21no—
76C2×D271—82no—
77C2×D271C2×D412yes—
78C4×C271—21no—
79C4×C271C2×D411yescharacteristic
80C4×C271C2411yescharacteristic
81Dic271—21no—
82Dic271—22no—
83Dic271C2×D412yes—
84C2×C271—21no—
85C2×C271—42no—
86C2×C271—41no—
87C2×C271C22×D411yescharacteristic
88D271—22no—
89D271—41no—
90C271#57711yescharacteristic, complemented, Sylow
91#577—2711nomaximal, Sylow
922−1+4—2711no—
93H(5,𝔽2)—2711no—
94#137—2712no—
95#137—2711no—
96AGL(1,ℤ/8)—2712no—
97AGL(1,ℤ/8)D27111yescharacteristic, complemented
98#135—2711no—
99#135—2712no—
100C2×SD16—2711no—
101C2×SD16—2712no—
102#131—2711no—
103#41—2711no—
104#41—2712no—
105#41—2712no—
106#41—2711no—
107#41—5421no—
108#41—5422no—
109#41—5421no—
110#41C2×D27111yescharacteristic, complemented
111C2×Q8—2712no—
112C2×Q8—2711no—
113C2×Q8—5421no—
114C2×D4—2712no—
115C2×D4—5421no—
116C2×D4—5422no—
117C2×D4C2×D27111yescharacteristic
118Q16—2711no—
119Q16—2712no—
120SD16—2711no—
121SD16—2712no—
122SD16—2712no—
123SD16—2711no—
124SD16—2712no—
125SD16C2×D27112yescomplemented
126D8—2711no—
127D8C2×D27112yescomplemented
128M16—2712no—
129M16C2×D27111yescharacteristic, complemented
130C2×C8—2711no—
131C2×C8—2712no—
132C23—21no—
133C23—5422no—
134Q8—2711no—
135Q8—2712no—
136Q8—2711no—
137Q8—5422no—
138Q8—5421no—
139Q8C22×D27111yescharacteristic
140D4—21no—
141D4—21no—
142D4—2712no—
143D4—2711no—
144D4—2711no—
145D4—5422no—
146D4—5422no—
147D4—5422no—
148D4—5421no—
149D4C22×D27111yescharacteristic
150D4C22×D27112yes—
151C2×C4—21no—
152C2×C4—2712no—
153C2×C4—5422no—
154C2×C4—5422no—
155C2×C4—5422no—
156C2×C4—5421no—
157C2×C4—5421no—
158C2×C4C22×D27111yescharacteristic
159C8—2711no—
160C8—2711no—
161C8C22×D27112yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—2712no—
166C22—5421no—
167C22—21682no—
168C22D4×D27111yescharacteristic
169C4—21no—
170C4—2712no—
171C4—5421no—
172C4—5422no—
173C4D4×D27111yescharacteristic
174C4C23×D27111yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—5422no—
179C2—10841no—
180C2C2×D4×D27111yescharacteristic
subgroupG/H
centerC2C2×D4×D271
commutator, cosocleC4×C271C24
FrattiniC4C23×D271
FittingAGL(1,ℤ/8)×C271C2
socleC2×C271C22×D4
hypercenterAGL(1,ℤ/8)D271
hypocenterC271#577
LCSG ▷ C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #22576207C2semidirect
2. #22576205C2semidirect
3. #22576193C2semidirect
4. #22576189C2semidirect
5. #22576184C2semidirect
6. #22576180C2semidirect
7. #22576179C2semidirect
8. C2×#16864642C2semidirect
9. SD16×D271C2semidirect
10. #22576168C2semidirect
11. AGL(1,ℤ/8)×C271C2semidirect
18. #16864653C22semidirect
20. C2×Dic542C22semidirect
21. Q8×D271C22semidirect
25. Dic1084C22semidirect
26. #16864644C22semidirect
27. #16864643C22semidirect
28. #16864642C22semidirect
29. #16864642C22semidirect
30. #16864642C22semidirect
31. #16864641C22semidirect
32. #16864639C22semidirect
33. #16864637C22semidirect
34. C2×C271⋊C8C22semidirect
35. C8×D271C22semidirect
36. #41×C271C22semidirect
38. SD16×C271C22semidirect
39. D8×C271C22semidirect
40. M16×C271C22semidirect
90. C271#577semidirect
97. AGL(1,ℤ/8)D271semidirect
110. #41C2×D271semidirect
125. SD16C2×D271semidirect
127. D8C2×D271semidirect
129. M16C2×D271semidirect

Maximal subgroups

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

subgroupindex
#225762072
#225762052
#225761932
#225761892
#225761842
#225761802
#225761792
C2×#168646422
SD16×D2712
#225761682
AGL(1,ℤ/8)×C2712
#577271