Finite Group Databasebeta

← #36378596

Subgroups — #36378596

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#26015040C211yesmaximal, characteristic, complemented
2#26015038C211yesmaximal, characteristic, complemented
3#26015024C212yesmaximal, complemented
4#26015020C211yesmaximal, characteristic, complemented
5#26015019C212yesmaximal, complemented
6#26015015C212yesmaximal, complemented
7#26015012C211yesmaximal, characteristic, complemented
8C2×#19150724C211yesmaximal, characteristic, complemented
9SD16×D353C212yesmaximal, complemented
10#26015000C211yesmaximal, characteristic, complemented
11#137×C353C211yesmaximal, characteristic, complemented
12#19150738—21no—
13#19150738C2212yes—
14#19150738C2212yescomplemented
15#19150737—21no—
16#19150737—22no—
17#19150737C2211yescharacteristic
18#19150737C2211yescharacteristic, complemented
19Q8×D353—21no—
20Q8×D353C2212yescomplemented
21C2×D1412—21no—
22C2×D1412C2211yescharacteristic
23D4×D353—22no—
24D4×D353C2212yes—
25#19150727C2211yescharacteristic, complemented
26#19150726C2212yescomplemented
27#19150725C2211yescharacteristic, complemented
28#19150724C2212yescomplemented
29#19150724C2212yescomplemented
30#19150724C2211yescharacteristic, complemented
31D2824C2212yescomplemented
32#19150722C2211yescharacteristic, complemented
33#19150720C2212yescomplemented
34C2×#15667725C2211yescharacteristic, complemented
35C8×D353C2212yescomplemented
36#41×C353C2211yescharacteristic
37C2×Q8×C353C2211yescharacteristic, complemented
38Q16×C353C2212yescomplemented
39SD16×C353C2212yescomplemented
40M16×C353C2211yescharacteristic, complemented
41C22×D353—21no—
42C22×D353—22no—
43Dic706—22no—
44Dic706D411yescharacteristic
45Dic706C2311yescharacteristic
46D1412—22no—
47D1412—22no—
48D1412—21no—
49D1412D411yescharacteristic
50D1412C2311yescharacteristic
51D1412C2312yes—
52C22⋊D353—21no—
53C22⋊D353—22no—
54C22⋊D353D412yes—
55C4×D353—22no—
56C4×D353—21no—
57C4×D353—22no—
58C4×D353—22no—
59C4×D353C2312yes—
60#15667725C2311yescharacteristic
61#15667725C2311yescharacteristic
62Q8×C353—21no—
63Q8×C353C2312yes—
64Q8×C353C2311yescharacteristic
65D4×C353—21no—
66D4×C353C2311yescharacteristic
67C2×C4×C353—21no—
68C2×C4×C353—21no—
69C2×C4×C353C2311yescharacteristic
70C8×C353C2312yes—
71C22×C353—21no—
72C22×C353C2×D411yescharacteristic
73C2×D353—21no—
74C2×D353—22no—
75C2×D353—82no—
76C2×D353—81no—
77C2×D353C2×D412yes—
78C4×C353—21no—
79C4×C353—22no—
80C4×C353C2×D411yescharacteristic
81C4×C353C2411yescharacteristic
82Dic353—21no—
83Dic353C2×D412yes—
84C2×C353—21no—
85C2×C353—41no—
86C2×C353C22×D411yescharacteristic
87D353—22no—
88D353—41no—
89D353—42no—
90C353#57711yescharacteristic, complemented, Sylow
91#577—3531nomaximal, Sylow
922−1+4—3531no—
93H(5,𝔽2)—3531no—
94#137—3532no—
95#137D35311yescharacteristic, complemented
96AGL(1,ℤ/8)—3532no—
97AGL(1,ℤ/8)—3531no—
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—7062no—
108#41—7061no—
109#41—7061no—
110#41C2×D35311yescharacteristic
111C2×Q8—3532no—
112C2×Q8—7061no—
113C2×Q8C2×D35311yescharacteristic, complemented
114C2×D4—3532no—
115C2×D4—3531no—
116C2×D4—7061no—
117C2×D4—7062no—
118Q16—3531no—
119Q16C2×D35312yescomplemented
120SD16—3532no—
121SD16—3532no—
122SD16—3531no—
123SD16—3531no—
124SD16—3532no—
125SD16C2×D35312yescomplemented
126D8—3531no—
127D8—3532no—
128M16—3532no—
129M16C2×D35311yescharacteristic, complemented
130C2×C8—3531no—
131C2×C8—3532no—
132C23—7062no—
133C23—7061no—
134Q8—21no—
135Q8—3531no—
136Q8—3531no—
137Q8—7062no—
138Q8C22×D35312yes—
139Q8C22×D35311yescharacteristic
140D4—21no—
141D4—3532no—
142D4—3531no—
143D4—3531no—
144D4—3532no—
145D4—7062no—
146D4—7062no—
147D4—7062no—
148D4—7061no—
149D4—7061no—
150D4C22×D35311yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—3532no—
154C2×C4—7062no—
155C2×C4—7062no—
156C2×C4—7062no—
157C2×C4—7061no—
158C2×C4C22×D35311yescharacteristic
159C8—3531no—
160C8—3531no—
161C8C22×D35312yes—
162C22—21no—
163C22—28241no—
164C22—28242no—
165C22—3532no—
166C22—7061no—
167C22—7062no—
168C22D4×D35311yescharacteristic
169C4—21no—
170C4—22no—
171C4—3532no—
172C4—7061no—
173C4D4×D35311yescharacteristic
174C4C23×D35311yescharacteristic
175C2—21no—
176C2—41no—
177C2—14122no—
178C2—14121no—
179C2—7062no—
180C2C2×D4×D35311yescharacteristic
subgroupG/H
centerC2C2×D4×D353
commutator, cosocleC4×C353C24
FrattiniC4C23×D353
Fitting#137×C353C2
socleC2×C353C22×D4
hypercenter#137D353
hypocenterC353#577
LCSG ▷ C4×C353 ▷ C2×C353 ▷ C353
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #26015040C2semidirect
2. #26015038C2semidirect
3. #26015024C2semidirect
4. #26015020C2semidirect
5. #26015019C2semidirect
6. #26015015C2semidirect
7. #26015012C2semidirect
8. C2×#19150724C2semidirect
9. SD16×D353C2semidirect
10. #26015000C2semidirect
11. #137×C353C2semidirect
14. #19150738C22semidirect
18. #19150737C22semidirect
20. Q8×D353C22semidirect
25. #19150727C22semidirect
26. #19150726C22semidirect
27. #19150725C22semidirect
28. #19150724C22semidirect
29. #19150724C22semidirect
30. #19150724C22semidirect
31. D2824C22semidirect
32. #19150722C22semidirect
33. #19150720C22semidirect
34. C2×#15667725C22semidirect
35. C8×D353C22semidirect
37. C2×Q8×C353C22semidirect
38. Q16×C353C22semidirect
39. SD16×C353C22semidirect
40. M16×C353C22semidirect
90. C353#577semidirect
95. #137D353semidirect
113. C2×Q8C2×D353semidirect
119. Q16C2×D353semidirect
125. SD16C2×D353semidirect
129. M16C2×D353semidirect

Maximal subgroups

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

subgroupindex
#260150402
#260150382
#260150242
#260150202
#260150192
#260150152
#260150122
C2×#191507242
SD16×D3532
#260150002
#137×C3532
#577353