Finite Group Databasebeta

← #26012295

Subgroups — #26012295

Order 11072.

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

normal subgroups only

This group has 268 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#19149725C211yesmaximal, characteristic, complemented
2#19149724C211yesmaximal, characteristic, complemented
3#19149707C211yesmaximal, characteristic, complemented
4#19149706C212yesmaximal, complemented
5#19149703C212yesmaximal, complemented
6#19149700C212yesmaximal, complemented
7#19149699C211yesmaximal, characteristic, complemented
8C2×#14583811C211yesmaximal, characteristic, complemented
9D8×D173C212yesmaximal, complemented
10#19149687C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C173C211yesmaximal, characteristic, complemented
12#14583827C2212yescomplemented
13#14583826—21no—
14#14583826C2211yescharacteristic
15#14583826C2211yescharacteristic, complemented
16#14583825—21no—
17#14583825C2212yescomplemented
18C2×D692—21no—
19C2×D692C2211yescharacteristic, complemented
20C2×C22⋊D173—22no—
21D4×D173—21no—
22D4×D173—22no—
23D4×D173C2212yescomplemented
24D4×D173C2212yescomplemented
25#14583816C2211yescharacteristic, complemented
26#14583815C2212yescomplemented
27#14583814C2211yescharacteristic, complemented
28#14583813C2211yescharacteristic, complemented
29D1384C2212yescomplemented
30#14583811C2211yescharacteristic, complemented
31#14583811C2212yescomplemented
32#14583811C2212yescomplemented
33#14583809C2212yescomplemented
34C2×#13063436C2211yescharacteristic, complemented
35C8×D173C2212yescomplemented
36#41×C173C2211yescharacteristic, complemented
37C2×D4×C173C2211yescharacteristic, complemented
38SD16×C173C2212yescomplemented
39D8×C173C2212yescomplemented
40M16×C173C2211yescharacteristic, complemented
41C22×D173—21no—
42C22×D173—22no—
43C22×D173—22no—
44Dic346D411yescharacteristic
45Dic346C2311yescharacteristic
46D692—22no—
47D692—21no—
48D692D411yescharacteristic
49D692C2312yes—
50D692C2311yescharacteristic
51C22⋊D173—21no—
52C22⋊D173—22no—
53C22⋊D173—22no—
54C22⋊D173—22no—
55C22⋊D173D412yes—
56C2×Dic173—22no—
57C4×D173—21no—
58C4×D173—22no—
59C4×D173C2312yes—
60#13063436C2311yescharacteristic
61#13063436C2311yescharacteristic
62C23×C173—21no—
63Q8×C173C2311yescharacteristic
64D4×C173—21no—
65D4×C173—21no—
66D4×C173C2312yes—
67D4×C173C2311yescharacteristic
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—22no—
77C2×D173—82no—
78C2×D173—82no—
79C2×D173—81no—
80C2×D173C2×D412yes—
81C4×C173—21no—
82C4×C173C2×D411yescharacteristic
83C4×C173C2411yescharacteristic
84Dic173—21no—
85Dic173C2×D412yes—
86C2×C173—21no—
87C2×C173—41no—
88C2×C173—42no—
89C2×C173C22×D411yescharacteristic
90D173—22no—
91D173—41no—
92D173—42no—
93C173#57611yescharacteristic, complemented, Sylow
94#576—1731nomaximal, Sylow
95H(5,𝔽2)—1731no—
96H(5,𝔽2)—1731no—
97AGL(1,ℤ/8)—1732no—
98AGL(1,ℤ/8)—1732no—
99AGL(1,ℤ/8)—1731no—
100AGL(1,ℤ/8)D17311yescharacteristic, complemented
101#135—1731no—
102#135—1732no—
103C2×D8—1731no—
104C2×D8—1732no—
105#131—1731no—
106#41—3461no—
107#41—3461no—
108#41—1732no—
109#41—1732no—
110#41—1731no—
111#41—1731no—
112#41C2×D17311yescharacteristic, complemented
113C2×D4—3461no—
114C2×D4—3462no—
115C2×D4—3461no—
116C2×D4—3462no—
117C2×D4—1732no—
118C2×D4—1732no—
119C2×D4—1731no—
120C2×D4C2×D17311yescharacteristic, complemented
121Q16—1731no—
122SD16—1731no—
123SD16—1731no—
124SD16—1732no—
125SD16C2×D17312yescomplemented
126D8—1732no—
127D8—1731no—
128D8—1732no—
129D8—1732no—
130D8C2×D17312yescomplemented
131M16—1732no—
132M16C2×D17311yescharacteristic, complemented
133C2×C8—1731no—
134C2×C8—1732no—
135C23—21no—
136C23—3462no—
137C23—3462no—
138C23—3461no—
139Q8—1731no—
140Q8—1731no—
141Q8C22×D17311yescharacteristic
142D4—21no—
143D4—21no—
144D4—3462no—
145D4—3462no—
146D4—3461no—
147D4—3462no—
148D4—3462no—
149D4—3461no—
150D4—1732no—
151D4—1731no—
152D4—1732no—
153D4—1731no—
154D4C22×D17312yes—
155D4C22×D17311yescharacteristic
156C2×C4—21no—
157C2×C4—3462no—
158C2×C4—3462no—
159C2×C4—3461no—
160C2×C4—1732no—
161C2×C4C22×D17311yescharacteristic
162C8—1731no—
163C8—1731no—
164C8C22×D17312yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—3462no—
169C22—3461no—
170C22—13841no—
171C22—13842no—
172C22—13842no—
173C22—1732no—
174C22D4×D17311yescharacteristic
175C4—21no—
176C4—3461no—
177C4—1732no—
178C4D4×D17311yescharacteristic
179C4C23×D17311yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—3462no—
184C2—6922no—
185C2—6921no—
186C2C2×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#576
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. #19149725C2semidirect
2. #19149724C2semidirect
3. #19149707C2semidirect
4. #19149706C2semidirect
5. #19149703C2semidirect
6. #19149700C2semidirect
7. #19149699C2semidirect
8. C2×#14583811C2semidirect
9. D8×D173C2semidirect
10. #19149687C2semidirect
11. AGL(1,ℤ/8)×C173C2semidirect
12. #14583827C22semidirect
15. #14583826C22semidirect
17. #14583825C22semidirect
19. C2×D692C22semidirect
23. D4×D173C22semidirect
24. D4×D173C22semidirect
25. #14583816C22semidirect
26. #14583815C22semidirect
27. #14583814C22semidirect
28. #14583813C22semidirect
29. D1384C22semidirect
30. #14583811C22semidirect
31. #14583811C22semidirect
32. #14583811C22semidirect
33. #14583809C22semidirect
34. C2×#13063436C22semidirect
35. C8×D173C22semidirect
36. #41×C173C22semidirect
37. C2×D4×C173C22semidirect
38. SD16×C173C22semidirect
39. D8×C173C22semidirect
40. M16×C173C22semidirect
93. C173#576semidirect
100. AGL(1,ℤ/8)D173semidirect
112. #41C2×D173semidirect
120. C2×D4C2×D173semidirect
125. SD16C2×D173semidirect
130. D8C2×D173semidirect
132. M16C2×D173semidirect

Maximal subgroups

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

subgroupindex
#191497252
#191497242
#191497072
#191497062
#191497032
#191497002
#191496992
C2×#145838112
D8×D1732
#191496872
AGL(1,ℤ/8)×C1732
#576173