Finite Group Databasebeta

← #17986809

Subgroups — #17986809

Order 4544.

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#14556175C211yesmaximal, characteristic, complemented
2#14556173C211yesmaximal, characteristic, complemented
3#14556161C211yesmaximal, characteristic, complemented
4#14556157C212yesmaximal, complemented
5#14556152C212yesmaximal, complemented
6#14556148C211yesmaximal, characteristic, complemented
7#14556147C212yesmaximal, complemented
8C2×#11765853C211yesmaximal, characteristic, complemented
9SD16×D71C212yesmaximal, complemented
10#14556136C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C71C211yesmaximal, characteristic, complemented
12#11765865—21no—
13#11765865C2211yescharacteristic
14#11765865C2211yescharacteristic
15#11765864—22no—
16#11765864—21no—
17#11765864C2212yes—
18#11765864C2212yescomplemented
19C2×Dic142—21no—
20C2×Dic142C2211yescharacteristic, complemented
21Q8×D71C2212yescomplemented
22C2×C22⋊D71—22no—
23D4×D71—21no—
24D4×D71C2212yes—
25Dic284C2212yescomplemented
26#11765855C2211yescharacteristic, complemented
27#11765854C2212yescomplemented
28#11765853C2212yescomplemented
29#11765853C2212yescomplemented
30#11765853C2211yescharacteristic, complemented
31#11765852C2211yescharacteristic, complemented
32#11765850C2211yescharacteristic, complemented
33#11765848C2212yescomplemented
34C2×C71⋊C8C2211yescharacteristic, complemented
35C8×D71C2212yescomplemented
36#41×C71C2211yescharacteristic, complemented
37C2×D4×C71C2211yescharacteristic
38SD16×C71C2212yescomplemented
39D8×C71C2212yescomplemented
40M16×C71C2211yescharacteristic, complemented
41C23×C71—21no—
42C22×D71—22no—
43Q8×C71C2311yescharacteristic
44D4×C71—21no—
45D4×C71—21no—
46D4×C71C2311yescharacteristic
47D4×C71C2312yes—
48C2×C4×C71—21no—
49C2×C4×C71C2311yescharacteristic
50C22⋊D71—21no—
51C22⋊D71—22no—
52C22⋊D71—22no—
53C22⋊D71—22no—
54C22⋊D71D412yes—
55C2×Dic71—21no—
56C2×Dic71—22no—
57C2×Dic71—22no—
58D284D411yescharacteristic
59D284C2311yescharacteristic
60C4×D71—22no—
61C4×D71—21no—
62C4×D71C2312yes—
63Dic142—21no—
64Dic142—22no—
65Dic142D411yescharacteristic
66Dic142C2312yes—
67Dic142C2311yescharacteristic
68C8×C71C2312yes—
69C71⋊C8C2311yescharacteristic
70C71⋊C8C2311yescharacteristic
71C22×C71—21no—
72C22×C71—22no—
73C22×C71—81no—
74C22×C71C2×D411yescharacteristic
75C2×D71—21no—
76C2×D71—82no—
77C2×D71C2×D412yes—
78C4×C71—21no—
79C4×C71C2×D411yescharacteristic
80C4×C71C2411yescharacteristic
81Dic71—22no—
82Dic71—21no—
83Dic71C2×D412yes—
84C2×C71—21no—
85C2×C71—42no—
86C2×C71—41no—
87C2×C71C22×D411yescharacteristic
88D71—22no—
89D71—41no—
90C71#57711yescharacteristic, complemented, Sylow
91#577—711nomaximal, Sylow
922−1+4—711no—
93H(5,𝔽2)—711no—
94#137—712no—
95#137—711no—
96AGL(1,ℤ/8)—712no—
97AGL(1,ℤ/8)D7111yescharacteristic, complemented
98#135—711no—
99#135—712no—
100C2×SD16—711no—
101C2×SD16—712no—
102#131—711no—
103#41—711no—
104#41—712no—
105#41—712no—
106#41—711no—
107#41—1421no—
108#41—1422no—
109#41—1421no—
110#41C2×D7111yescharacteristic, complemented
111C2×Q8—712no—
112C2×Q8—711no—
113C2×Q8—1421no—
114C2×D4—712no—
115C2×D4—1421no—
116C2×D4—1422no—
117C2×D4C2×D7111yescharacteristic
118Q16—711no—
119Q16—712no—
120SD16—711no—
121SD16—712no—
122SD16—712no—
123SD16—711no—
124SD16—712no—
125SD16C2×D7112yescomplemented
126D8—711no—
127D8C2×D7112yescomplemented
128M16—712no—
129M16C2×D7111yescharacteristic, complemented
130C2×C8—711no—
131C2×C8—712no—
132C23—21no—
133C23—1422no—
134Q8—711no—
135Q8—712no—
136Q8—711no—
137Q8—1422no—
138Q8—1421no—
139Q8C22×D7111yescharacteristic
140D4—21no—
141D4—21no—
142D4—712no—
143D4—711no—
144D4—711no—
145D4—1422no—
146D4—1422no—
147D4—1422no—
148D4—1421no—
149D4C22×D7111yescharacteristic
150D4C22×D7112yes—
151C2×C4—21no—
152C2×C4—712no—
153C2×C4—1422no—
154C2×C4—1422no—
155C2×C4—1422no—
156C2×C4—1421no—
157C2×C4—1421no—
158C2×C4C22×D7111yescharacteristic
159C8—711no—
160C8—711no—
161C8C22×D7112yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—5682no—
166C22—712no—
167C22—1421no—
168C22D4×D7111yescharacteristic
169C4—21no—
170C4—712no—
171C4—1421no—
172C4—1422no—
173C4D4×D7111yescharacteristic
174C4C23×D7111yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—2841no—
179C2—1422no—
180C2C2×D4×D7111yescharacteristic
subgroupG/H
centerC2C2×D4×D71
commutator, cosocleC4×C71C24
FrattiniC4C23×D71
FittingAGL(1,ℤ/8)×C71C2
socleC2×C71C22×D4
hypercenterAGL(1,ℤ/8)D71
hypocenterC71#577
LCSG ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #14556175C2semidirect
2. #14556173C2semidirect
3. #14556161C2semidirect
4. #14556157C2semidirect
5. #14556152C2semidirect
6. #14556148C2semidirect
7. #14556147C2semidirect
8. C2×#11765853C2semidirect
9. SD16×D71C2semidirect
10. #14556136C2semidirect
11. AGL(1,ℤ/8)×C71C2semidirect
18. #11765864C22semidirect
20. C2×Dic142C22semidirect
21. Q8×D71C22semidirect
25. Dic284C22semidirect
26. #11765855C22semidirect
27. #11765854C22semidirect
28. #11765853C22semidirect
29. #11765853C22semidirect
30. #11765853C22semidirect
31. #11765852C22semidirect
32. #11765850C22semidirect
33. #11765848C22semidirect
34. C2×C71⋊C8C22semidirect
35. C8×D71C22semidirect
36. #41×C71C22semidirect
38. SD16×C71C22semidirect
39. D8×C71C22semidirect
40. M16×C71C22semidirect
90. C71#577semidirect
97. AGL(1,ℤ/8)D71semidirect
110. #41C2×D71semidirect
125. SD16C2×D71semidirect
127. D8C2×D71semidirect
129. M16C2×D71semidirect

Maximal subgroups

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

subgroupindex
#145561752
#145561732
#145561612
#145561572
#145561522
#145561482
#145561472
C2×#117658532
SD16×D712
#145561362
AGL(1,ℤ/8)×C712
#57771