Finite Group Databasebeta

← #26012298

Subgroups — #26012298

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#19149725C211yesmaximal, characteristic, complemented
3#19149711C212yesmaximal, complemented
4#19149709C211yesmaximal, characteristic, complemented
5#19149704C211yesmaximal, characteristic, complemented
6#19149703C212yesmaximal, complemented
7#19149700C212yesmaximal, complemented
8C2×#14583815C211yesmaximal, characteristic, complemented
9SD16×D173C212yesmaximal, complemented
10#19149685C211yesmaximal, characteristic, complemented
11#135×C173C211yesmaximal, characteristic, complemented
12#14583827C2212yes—
13#14583826—21no—
14#14583826—21no—
15#14583826C2211yescharacteristic
16#14583825—22no—
17#14583825C2212yescomplemented
18C2×Dic346—21no—
19C2×Dic346C2211yescharacteristic, complemented
20Q8×D173C2212yescomplemented
21C2×D692—21no—
22C2×D692C2211yescharacteristic
23D4×D173—22no—
24D4×D173C2212yes—
25#14583816C2212yescomplemented
26#14583815C2212yescomplemented
27#14583815C2212yescomplemented
28#14583814C2212yescomplemented
29#14583813C2212yescomplemented
30#14583811C2212yescomplemented
31#14583810C2211yescharacteristic, complemented
32#14583809C2212yescomplemented
33C8×D173C2212yescomplemented
34#41×C173C2211yescharacteristic
35#41×C173C2211yescharacteristic, complemented
36Q16×C173C2211yescharacteristic, complemented
37SD16×C173C2211yescharacteristic, complemented
38SD16×C173C2211yescharacteristic, complemented
39D8×C173C2211yescharacteristic, complemented
40C2×C8×C173C2211yescharacteristic, complemented
41C22×D173—21no—
42C22×D173—22no—
43Dic346—21no—
44Dic346—22no—
45Dic346D411yescharacteristic
46Dic346C2312yes—
47D692—22no—
48D692—21no—
49D692D411yescharacteristic
50D692C2312yes—
51C22⋊D173—22no—
52C22⋊D173—22no—
53C22⋊D173D412yes—
54C2×Dic173—21no—
55C2×Dic173—22no—
56C4×D173—22no—
57C4×D173—22no—
58C4×D173C2312yes—
59#13063436C2312yes—
60Q8×C173C2311yescharacteristic
61Q8×C173C2311yescharacteristic
62D4×C173—21no—
63D4×C173—21no—
64D4×C173C2311yescharacteristic
65D4×C173C2311yescharacteristic
66C2×C4×C173—21no—
67C2×C4×C173—21no—
68C2×C4×C173C2311yescharacteristic
69C8×C173C2311yescharacteristic
70C8×C173C2311yescharacteristic
71C22×C173—21no—
72C22×C173—21no—
73C22×C173C2×D411yescharacteristic
74C2×D173—22no—
75C2×D173—82no—
76C2×D173—81no—
77C2×D173C2×D412yes—
78C4×C173—21no—
79C4×C173—21no—
80C4×C173C2×D411yescharacteristic
81C4×C173C2411yescharacteristic
82Dic173—22no—
83Dic173C2×D412yes—
84C2×C173—21no—
85C2×C173—41no—
86C2×C173—41no—
87C2×C173C22×D411yescharacteristic
88D173—22no—
89D173—42no—
90C173#57711yescharacteristic, complemented, Sylow
91#577—1731nomaximal, Sylow
922−1+4—1731no—
93H(5,𝔽2)—1731no—
94#137—1731no—
95#137—1732no—
96AGL(1,ℤ/8)—1731no—
97AGL(1,ℤ/8)—1732no—
98#135—1732no—
99#135D17311yescharacteristic, complemented
100C2×SD16—1732no—
101C2×SD16—1731no—
102#131—1731no—
103#41—3461no—
104#41—3461no—
105#41—3462no—
106#41—1732no—
107#41—1732no—
108#41—1731no—
109#41C2×D17311yescharacteristic
110#41C2×D17311yescharacteristic, complemented
111C2×Q8—3461no—
112C2×Q8—1732no—
113C2×Q8—1731no—
114C2×D4—3461no—
115C2×D4—3462no—
116C2×D4—1732no—
117C2×D4—1731no—
118Q16—1732no—
119Q16C2×D17311yescharacteristic, complemented
120SD16—1732no—
121SD16—1732no—
122SD16—1732no—
123SD16—1732no—
124SD16C2×D17311yescharacteristic, complemented
125SD16C2×D17311yescharacteristic, complemented
126D8—1732no—
127D8C2×D17311yescharacteristic, complemented
128M16—1731no—
129M16—1732no—
130C2×C8—1732no—
131C2×C8C2×D17311yescharacteristic, complemented
132C23—3462no—
133C23—3461no—
134Q8—3462no—
135Q8—3461no—
136Q8—1732no—
137Q8—1731no—
138Q8C22×D17311yescharacteristic
139Q8C22×D17311yescharacteristic
140D4—21no—
141D4—21no—
142D4—3462no—
143D4—3462no—
144D4—3462no—
145D4—3461no—
146D4—1732no—
147D4—1731no—
148D4—1732no—
149D4C22×D17311yescharacteristic
150D4C22×D17311yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—3462no—
154C2×C4—3462no—
155C2×C4—3461no—
156C2×C4—3462no—
157C2×C4—1732no—
158C2×C4C22×D17311yescharacteristic
159C8—1732no—
160C8C22×D17311yescharacteristic
161C8C22×D17311yescharacteristic
162C22—21no—
163C22—21no—
164C22—3462no—
165C22—13841no—
166C22—13842no—
167C22—1732no—
168C22D4×D17311yescharacteristic
169C4—21no—
170C4—21no—
171C4—3462no—
172C4—1732no—
173C4D4×D17311yescharacteristic
174C4C23×D17311yescharacteristic
175C2—21no—
176C2—41no—
177C2—41no—
178C2—3462no—
179C2—6922no—
180C2C2×D4×D17311yescharacteristic
subgroupG/H
centerC2C2×D4×D173
commutator, cosocleC4×C173C24
FrattiniC4C23×D173
Fitting#135×C173C2
socleC2×C173C22×D4
hypercenter#135D173
hypocenterC173#577
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #19149726C2semidirect
2. #19149725C2semidirect
3. #19149711C2semidirect
4. #19149709C2semidirect
5. #19149704C2semidirect
6. #19149703C2semidirect
7. #19149700C2semidirect
8. C2×#14583815C2semidirect
9. SD16×D173C2semidirect
10. #19149685C2semidirect
11. #135×C173C2semidirect
17. #14583825C22semidirect
19. C2×Dic346C22semidirect
20. Q8×D173C22semidirect
25. #14583816C22semidirect
26. #14583815C22semidirect
27. #14583815C22semidirect
28. #14583814C22semidirect
29. #14583813C22semidirect
30. #14583811C22semidirect
31. #14583810C22semidirect
32. #14583809C22semidirect
33. C8×D173C22semidirect
35. #41×C173C22semidirect
36. Q16×C173C22semidirect
37. SD16×C173C22semidirect
38. SD16×C173C22semidirect
39. D8×C173C22semidirect
40. C2×C8×C173C22semidirect
90. C173#577semidirect
99. #135D173semidirect
110. #41C2×D173semidirect
119. Q16C2×D173semidirect
124. SD16C2×D173semidirect
125. SD16C2×D173semidirect
127. D8C2×D173semidirect
131. C2×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#191497262
#191497252
#191497112
#191497092
#191497042
#191497032
#191497002
C2×#145838152
SD16×D1732
#191496852
#135×C1732
#577173