Finite Group Databasebeta

← #42119866

Subgroups — #42119866

Order 27584.

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#28352236C211yesmaximal, characteristic, complemented
2#28352235C211yesmaximal, characteristic, complemented
3#28352221C212yesmaximal, complemented
4#28352219C211yesmaximal, characteristic, complemented
5#28352214C211yesmaximal, characteristic, complemented
6#28352213C212yesmaximal, complemented
7#28352210C212yesmaximal, complemented
8C2×#20293789C211yesmaximal, characteristic, complemented
9SD16×D431C212yesmaximal, complemented
10#28352195C211yesmaximal, characteristic, complemented
11#135×C431C211yesmaximal, characteristic, complemented
12#20293801C2212yes—
13#20293800—21no—
14#20293800—21no—
15#20293800C2211yescharacteristic
16#20293799—22no—
17#20293799C2212yescomplemented
18C2×Dic862—21no—
19C2×Dic862C2211yescharacteristic, complemented
20Q8×D431C2212yescomplemented
21C2×D1724—21no—
22C2×D1724C2211yescharacteristic
23D4×D431—22no—
24D4×D431C2212yes—
25#20293790C2212yescomplemented
26#20293789C2212yescomplemented
27#20293789C2212yescomplemented
28#20293788C2212yescomplemented
29#20293787C2212yescomplemented
30#20293785C2212yescomplemented
31#20293784C2211yescharacteristic, complemented
32#20293783C2212yescomplemented
33C8×D431C2212yescomplemented
34#41×C431C2211yescharacteristic
35#41×C431C2211yescharacteristic, complemented
36Q16×C431C2211yescharacteristic, complemented
37SD16×C431C2211yescharacteristic, complemented
38SD16×C431C2211yescharacteristic, complemented
39D8×C431C2211yescharacteristic, complemented
40C2×C8×C431C2211yescharacteristic, complemented
41C22×D431—21no—
42C22×D431—22no—
43Dic862—21no—
44Dic862—22no—
45Dic862D411yescharacteristic
46Dic862C2312yes—
47D1724—22no—
48D1724—21no—
49D1724D411yescharacteristic
50D1724C2312yes—
51C22⋊D431—22no—
52C22⋊D431—22no—
53C22⋊D431D412yes—
54C2×Dic431—21no—
55C2×Dic431—22no—
56C4×D431—22no—
57C4×D431—22no—
58C4×D431C2312yes—
59C431⋊C8C2312yes—
60Q8×C431C2311yescharacteristic
61Q8×C431C2311yescharacteristic
62D4×C431—21no—
63D4×C431—21no—
64D4×C431C2311yescharacteristic
65D4×C431C2311yescharacteristic
66C2×C4×C431—21no—
67C2×C4×C431—21no—
68C2×C4×C431C2311yescharacteristic
69C8×C431C2311yescharacteristic
70C8×C431C2311yescharacteristic
71C22×C431—21no—
72C22×C431—21no—
73C22×C431C2×D411yescharacteristic
74C2×D431—22no—
75C2×D431—82no—
76C2×D431—81no—
77C2×D431C2×D412yes—
78C4×C431—21no—
79C4×C431—21no—
80C4×C431C2×D411yescharacteristic
81C4×C431C2411yescharacteristic
82Dic431—22no—
83Dic431C2×D412yes—
84C2×C431—21no—
85C2×C431—41no—
86C2×C431—41no—
87C2×C431C22×D411yescharacteristic
88D431—22no—
89D431—42no—
90C431#57711yescharacteristic, complemented, Sylow
91#577—4311nomaximal, Sylow
922−1+4—4311no—
93H(5,𝔽2)—4311no—
94#137—4311no—
95#137—4312no—
96AGL(1,ℤ/8)—4311no—
97AGL(1,ℤ/8)—4312no—
98#135—4312no—
99#135D43111yescharacteristic, complemented
100C2×SD16—4312no—
101C2×SD16—4311no—
102#131—4311no—
103#41—8621no—
104#41—8621no—
105#41—8622no—
106#41—4312no—
107#41—4312no—
108#41—4311no—
109#41C2×D43111yescharacteristic
110#41C2×D43111yescharacteristic, complemented
111C2×Q8—8621no—
112C2×Q8—4312no—
113C2×Q8—4311no—
114C2×D4—8621no—
115C2×D4—8622no—
116C2×D4—4312no—
117C2×D4—4311no—
118Q16—4312no—
119Q16C2×D43111yescharacteristic, complemented
120SD16—4312no—
121SD16—4312no—
122SD16—4312no—
123SD16—4312no—
124SD16C2×D43111yescharacteristic, complemented
125SD16C2×D43111yescharacteristic, complemented
126D8—4312no—
127D8C2×D43111yescharacteristic, complemented
128M16—4311no—
129M16—4312no—
130C2×C8—4312no—
131C2×C8C2×D43111yescharacteristic, complemented
132C23—8622no—
133C23—8621no—
134Q8—8622no—
135Q8—8621no—
136Q8—4312no—
137Q8—4311no—
138Q8C22×D43111yescharacteristic
139Q8C22×D43111yescharacteristic
140D4—21no—
141D4—21no—
142D4—8622no—
143D4—8622no—
144D4—8622no—
145D4—8621no—
146D4—4312no—
147D4—4311no—
148D4—4312no—
149D4C22×D43111yescharacteristic
150D4C22×D43111yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—8622no—
154C2×C4—8622no—
155C2×C4—8621no—
156C2×C4—8622no—
157C2×C4—4312no—
158C2×C4C22×D43111yescharacteristic
159C8—4312no—
160C8C22×D43111yescharacteristic
161C8C22×D43111yescharacteristic
162C22—21no—
163C22—21no—
164C22—8622no—
165C22—34481no—
166C22—34482no—
167C22—4312no—
168C22D4×D43111yescharacteristic
169C4—21no—
170C4—21no—
171C4—8622no—
172C4—4312no—
173C4D4×D43111yescharacteristic
174C4C23×D43111yescharacteristic
175C2—21no—
176C2—41no—
177C2—41no—
178C2—8622no—
179C2—17242no—
180C2C2×D4×D43111yescharacteristic
subgroupG/H
centerC2C2×D4×D431
commutator, cosocleC4×C431C24
FrattiniC4C23×D431
Fitting#135×C431C2
socleC2×C431C22×D4
hypercenter#135D431
hypocenterC431#577
LCSG ▷ C4×C431 ▷ C2×C431 ▷ C431
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #28352236C2semidirect
2. #28352235C2semidirect
3. #28352221C2semidirect
4. #28352219C2semidirect
5. #28352214C2semidirect
6. #28352213C2semidirect
7. #28352210C2semidirect
8. C2×#20293789C2semidirect
9. SD16×D431C2semidirect
10. #28352195C2semidirect
11. #135×C431C2semidirect
17. #20293799C22semidirect
19. C2×Dic862C22semidirect
20. Q8×D431C22semidirect
25. #20293790C22semidirect
26. #20293789C22semidirect
27. #20293789C22semidirect
28. #20293788C22semidirect
29. #20293787C22semidirect
30. #20293785C22semidirect
31. #20293784C22semidirect
32. #20293783C22semidirect
33. C8×D431C22semidirect
35. #41×C431C22semidirect
36. Q16×C431C22semidirect
37. SD16×C431C22semidirect
38. SD16×C431C22semidirect
39. D8×C431C22semidirect
40. C2×C8×C431C22semidirect
90. C431#577semidirect
99. #135D431semidirect
110. #41C2×D431semidirect
119. Q16C2×D431semidirect
124. SD16C2×D431semidirect
125. SD16C2×D431semidirect
127. D8C2×D431semidirect
131. C2×C8C2×D431semidirect

Maximal subgroups

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

subgroupindex
#283522362
#283522352
#283522212
#283522192
#283522142
#283522132
#283522102
C2×#202937892
SD16×D4312
#283521952
#135×C4312
#577431