Finite Group Databasebeta

← #36378597

Subgroups — #36378597

Order 22592.

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#26015039C211yesmaximal, characteristic, complemented
2#26015037C211yesmaximal, characteristic, complemented
3#26015025C211yesmaximal, characteristic, complemented
4#26015021C212yesmaximal, complemented
5#26015016C212yesmaximal, complemented
6#26015012C211yesmaximal, characteristic, complemented
7#26015011C212yesmaximal, complemented
8C2×#19150725C211yesmaximal, characteristic, complemented
9SD16×D353C212yesmaximal, complemented
10#26015000C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C353C211yesmaximal, characteristic, complemented
12#19150737—21no—
13#19150737C2211yescharacteristic
14#19150737C2211yescharacteristic
15#19150736—21no—
16#19150736—22no—
17#19150736C2212yes—
18#19150736C2212yescomplemented
19C2×Dic706—21no—
20C2×Dic706C2211yescharacteristic, complemented
21Q8×D353C2212yescomplemented
22C2×C22⋊D353—22no—
23D4×D353—21no—
24D4×D353C2212yes—
25Dic1412C2212yescomplemented
26#19150727C2211yescharacteristic, complemented
27#19150726C2212yescomplemented
28#19150725C2212yescomplemented
29#19150725C2212yescomplemented
30#19150725C2211yescharacteristic, complemented
31#19150724C2211yescharacteristic, complemented
32#19150722C2211yescharacteristic, complemented
33#19150720C2212yescomplemented
34C2×#15667725C2211yescharacteristic, complemented
35C8×D353C2212yescomplemented
36#41×C353C2211yescharacteristic, complemented
37C2×D4×C353C2211yescharacteristic
38SD16×C353C2212yescomplemented
39D8×C353C2212yescomplemented
40M16×C353C2211yescharacteristic, complemented
41C22×D353—22no—
42Dic706—21no—
43Dic706—22no—
44Dic706D411yescharacteristic
45Dic706C2312yes—
46Dic706C2311yescharacteristic
47D1412D411yescharacteristic
48D1412C2311yescharacteristic
49C22⋊D353—21no—
50C22⋊D353—22no—
51C22⋊D353—22no—
52C22⋊D353—22no—
53C22⋊D353D412yes—
54C2×Dic353—21no—
55C2×Dic353—22no—
56C2×Dic353—22no—
57C4×D353—22no—
58C4×D353—21no—
59C4×D353C2312yes—
60#15667725C2311yescharacteristic
61#15667725C2311yescharacteristic
62C23×C353—21no—
63Q8×C353C2311yescharacteristic
64D4×C353—21no—
65D4×C353—21no—
66D4×C353C2311yescharacteristic
67D4×C353C2312yes—
68C2×C4×C353—21no—
69C2×C4×C353C2311yescharacteristic
70C8×C353C2312yes—
71C22×C353—21no—
72C22×C353—22no—
73C22×C353—81no—
74C22×C353C2×D411yescharacteristic
75C2×D353—21no—
76C2×D353—82no—
77C2×D353C2×D412yes—
78C4×C353—21no—
79C4×C353C2×D411yescharacteristic
80C4×C353C2411yescharacteristic
81Dic353—21no—
82Dic353—22no—
83Dic353C2×D412yes—
84C2×C353—21no—
85C2×C353—42no—
86C2×C353—41no—
87C2×C353C22×D411yescharacteristic
88D353—22no—
89D353—41no—
90C353#57711yescharacteristic, complemented, Sylow
91#577—3531nomaximal, Sylow
922−1+4—3531no—
93H(5,𝔽2)—3531no—
94#137—3532no—
95#137—3531no—
96AGL(1,ℤ/8)—3532no—
97AGL(1,ℤ/8)D35311yescharacteristic, complemented
98#135—3531no—
99#135—3532no—
100C2×SD16—3531no—
101C2×SD16—3532no—
102#131—3531no—
103#41—3531no—
104#41—3532no—
105#41—3532no—
106#41—3531no—
107#41—7061no—
108#41—7062no—
109#41—7061no—
110#41C2×D35311yescharacteristic, complemented
111C2×Q8—3532no—
112C2×Q8—3531no—
113C2×Q8—7061no—
114C2×D4—3532no—
115C2×D4—7061no—
116C2×D4—7062no—
117C2×D4C2×D35311yescharacteristic
118Q16—3531no—
119Q16—3532no—
120SD16—3531no—
121SD16—3532no—
122SD16—3532no—
123SD16—3531no—
124SD16—3532no—
125SD16C2×D35312yescomplemented
126D8—3531no—
127D8C2×D35312yescomplemented
128M16—3532no—
129M16C2×D35311yescharacteristic, complemented
130C2×C8—3531no—
131C2×C8—3532no—
132C23—21no—
133C23—7062no—
134Q8—3531no—
135Q8—3532no—
136Q8—3531no—
137Q8—7062no—
138Q8—7061no—
139Q8C22×D35311yescharacteristic
140D4—21no—
141D4—21no—
142D4—3532no—
143D4—3531no—
144D4—3531no—
145D4—7062no—
146D4—7062no—
147D4—7062no—
148D4—7061no—
149D4C22×D35311yescharacteristic
150D4C22×D35312yes—
151C2×C4—21no—
152C2×C4—3532no—
153C2×C4—7062no—
154C2×C4—7062no—
155C2×C4—7062no—
156C2×C4—7061no—
157C2×C4—7061no—
158C2×C4C22×D35311yescharacteristic
159C8—3531no—
160C8—3531no—
161C8C22×D35312yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—28242no—
166C22—3532no—
167C22—7061no—
168C22D4×D35311yescharacteristic
169C4—21no—
170C4—3532no—
171C4—7061no—
172C4—7062no—
173C4D4×D35311yescharacteristic
174C4C23×D35311yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—14121no—
179C2—7062no—
180C2C2×D4×D35311yescharacteristic
subgroupG/H
centerC2C2×D4×D353
commutator, cosocleC4×C353C24
FrattiniC4C23×D353
FittingAGL(1,ℤ/8)×C353C2
socleC2×C353C22×D4
hypercenterAGL(1,ℤ/8)D353
hypocenterC353#577
LCSG ▷ C4×C353 ▷ C2×C353 ▷ C353
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #26015039C2semidirect
2. #26015037C2semidirect
3. #26015025C2semidirect
4. #26015021C2semidirect
5. #26015016C2semidirect
6. #26015012C2semidirect
7. #26015011C2semidirect
8. C2×#19150725C2semidirect
9. SD16×D353C2semidirect
10. #26015000C2semidirect
11. AGL(1,ℤ/8)×C353C2semidirect
18. #19150736C22semidirect
20. C2×Dic706C22semidirect
21. Q8×D353C22semidirect
25. Dic1412C22semidirect
26. #19150727C22semidirect
27. #19150726C22semidirect
28. #19150725C22semidirect
29. #19150725C22semidirect
30. #19150725C22semidirect
31. #19150724C22semidirect
32. #19150722C22semidirect
33. #19150720C22semidirect
34. C2×#15667725C22semidirect
35. C8×D353C22semidirect
36. #41×C353C22semidirect
38. SD16×C353C22semidirect
39. D8×C353C22semidirect
40. M16×C353C22semidirect
90. C353#577semidirect
97. AGL(1,ℤ/8)D353semidirect
110. #41C2×D353semidirect
125. SD16C2×D353semidirect
127. D8C2×D353semidirect
129. M16C2×D353semidirect

Maximal subgroups

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

subgroupindex
#260150392
#260150372
#260150252
#260150212
#260150162
#260150122
#260150112
C2×#191507252
SD16×D3532
#260150002
AGL(1,ℤ/8)×C3532
#577353