Finite Group Databasebeta

← #26012301

Subgroups — #26012301

Order 11072.

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#19149726C211yesmaximal, characteristic, complemented
2#19149724C211yesmaximal, characteristic, complemented
3#19149712C211yesmaximal, characteristic, complemented
4#19149708C212yesmaximal, complemented
5#19149703C212yesmaximal, complemented
6#19149699C211yesmaximal, characteristic, complemented
7#19149698C212yesmaximal, complemented
8C2×#14583814C211yesmaximal, characteristic, complemented
9SD16×D173C212yesmaximal, complemented
10#19149687C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C173C211yesmaximal, characteristic, complemented
12#14583826—21no—
13#14583826C2211yescharacteristic
14#14583826C2211yescharacteristic
15#14583825—21no—
16#14583825—22no—
17#14583825C2212yes—
18#14583825C2212yescomplemented
19C2×Dic346—21no—
20C2×Dic346C2211yescharacteristic, complemented
21Q8×D173C2212yescomplemented
22C2×C22⋊D173—22no—
23D4×D173—21no—
24D4×D173C2212yes—
25Dic692C2212yescomplemented
26#14583816C2211yescharacteristic, complemented
27#14583815C2212yescomplemented
28#14583814C2212yescomplemented
29#14583814C2212yescomplemented
30#14583814C2211yescharacteristic, complemented
31#14583813C2211yescharacteristic, complemented
32#14583811C2211yescharacteristic, complemented
33#14583809C2212yescomplemented
34C2×#13063436C2211yescharacteristic, complemented
35C8×D173C2212yescomplemented
36#41×C173C2211yescharacteristic, complemented
37C2×D4×C173C2211yescharacteristic
38SD16×C173C2212yescomplemented
39D8×C173C2212yescomplemented
40M16×C173C2211yescharacteristic, complemented
41C22×D173—22no—
42Dic346—21no—
43Dic346—22no—
44Dic346D411yescharacteristic
45Dic346C2312yes—
46Dic346C2311yescharacteristic
47D692D411yescharacteristic
48D692C2311yescharacteristic
49C22⋊D173—21no—
50C22⋊D173—22no—
51C22⋊D173—22no—
52C22⋊D173—22no—
53C22⋊D173D412yes—
54C2×Dic173—21no—
55C2×Dic173—22no—
56C2×Dic173—22no—
57C4×D173—22no—
58C4×D173—21no—
59C4×D173C2312yes—
60#13063436C2311yescharacteristic
61#13063436C2311yescharacteristic
62C23×C173—21no—
63Q8×C173C2311yescharacteristic
64D4×C173—21no—
65D4×C173—21no—
66D4×C173C2311yescharacteristic
67D4×C173C2312yes—
68C2×C4×C173—21no—
69C2×C4×C173C2311yescharacteristic
70C8×C173C2312yes—
71C22×C173—21no—
72C22×C173—22no—
73C22×C173—81no—
74C22×C173C2×D411yescharacteristic
75C2×D173—21no—
76C2×D173—82no—
77C2×D173C2×D412yes—
78C4×C173—21no—
79C4×C173C2×D411yescharacteristic
80C4×C173C2411yescharacteristic
81Dic173—21no—
82Dic173—22no—
83Dic173C2×D412yes—
84C2×C173—21no—
85C2×C173—42no—
86C2×C173—41no—
87C2×C173C22×D411yescharacteristic
88D173—22no—
89D173—41no—
90C173#57711yescharacteristic, complemented, Sylow
91#577—1731nomaximal, Sylow
922−1+4—1731no—
93H(5,𝔽2)—1731no—
94#137—1732no—
95#137—1731no—
96AGL(1,ℤ/8)—1732no—
97AGL(1,ℤ/8)D17311yescharacteristic, complemented
98#135—1731no—
99#135—1732no—
100C2×SD16—1731no—
101C2×SD16—1732no—
102#131—1731no—
103#41—3461no—
104#41—3461no—
105#41—3462no—
106#41—1732no—
107#41—1732no—
108#41—1731no—
109#41—1731no—
110#41C2×D17311yescharacteristic, complemented
111C2×Q8—3461no—
112C2×Q8—1732no—
113C2×Q8—1731no—
114C2×D4—3461no—
115C2×D4—3462no—
116C2×D4—1732no—
117C2×D4C2×D17311yescharacteristic
118Q16—1731no—
119Q16—1732no—
120SD16—1731no—
121SD16—1732no—
122SD16—1732no—
123SD16—1731no—
124SD16—1732no—
125SD16C2×D17312yescomplemented
126D8—1731no—
127D8C2×D17312yescomplemented
128M16—1732no—
129M16C2×D17311yescharacteristic, complemented
130C2×C8—1731no—
131C2×C8—1732no—
132C23—21no—
133C23—3462no—
134Q8—3462no—
135Q8—3461no—
136Q8—1732no—
137Q8—1731no—
138Q8—1731no—
139Q8C22×D17311yescharacteristic
140D4—21no—
141D4—21no—
142D4—3462no—
143D4—3462no—
144D4—3461no—
145D4—3462no—
146D4—1732no—
147D4—1731no—
148D4—1731no—
149D4C22×D17311yescharacteristic
150D4C22×D17312yes—
151C2×C4—21no—
152C2×C4—3462no—
153C2×C4—3461no—
154C2×C4—3462no—
155C2×C4—3462no—
156C2×C4—3461no—
157C2×C4—1732no—
158C2×C4C22×D17311yescharacteristic
159C8—1731no—
160C8—1731no—
161C8C22×D17312yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—3461no—
166C22—13842no—
167C22—1732no—
168C22D4×D17311yescharacteristic
169C4—21no—
170C4—3461no—
171C4—3462no—
172C4—1732no—
173C4D4×D17311yescharacteristic
174C4C23×D17311yescharacteristic
175C2—21no—
176C2—41no—
177C2—42no—
178C2—3462no—
179C2—6921no—
180C2C2×D4×D17311yescharacteristic
subgroupG/H
centerC2C2×D4×D173
commutator, cosocleC4×C173C24
FrattiniC4C23×D173
FittingAGL(1,ℤ/8)×C173C2
socleC2×C173C22×D4
hypercenterAGL(1,ℤ/8)D173
hypocenterC173#577
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #19149726C2semidirect
2. #19149724C2semidirect
3. #19149712C2semidirect
4. #19149708C2semidirect
5. #19149703C2semidirect
6. #19149699C2semidirect
7. #19149698C2semidirect
8. C2×#14583814C2semidirect
9. SD16×D173C2semidirect
10. #19149687C2semidirect
11. AGL(1,ℤ/8)×C173C2semidirect
18. #14583825C22semidirect
20. C2×Dic346C22semidirect
21. Q8×D173C22semidirect
25. Dic692C22semidirect
26. #14583816C22semidirect
27. #14583815C22semidirect
28. #14583814C22semidirect
29. #14583814C22semidirect
30. #14583814C22semidirect
31. #14583813C22semidirect
32. #14583811C22semidirect
33. #14583809C22semidirect
34. C2×#13063436C22semidirect
35. C8×D173C22semidirect
36. #41×C173C22semidirect
38. SD16×C173C22semidirect
39. D8×C173C22semidirect
40. M16×C173C22semidirect
90. C173#577semidirect
97. AGL(1,ℤ/8)D173semidirect
110. #41C2×D173semidirect
125. SD16C2×D173semidirect
127. D8C2×D173semidirect
129. M16C2×D173semidirect

Maximal subgroups

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

subgroupindex
#191497262
#191497242
#191497122
#191497082
#191497032
#191496992
#191496982
C2×#145838142
SD16×D1732
#191496872
AGL(1,ℤ/8)×C1732
#577173