Finite Group Databasebeta

← #42122618

Subgroups — #42122618

Order 27712.

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#28357267C211yesmaximal, characteristic, complemented
2#28357266C211yesmaximal, characteristic, complemented
3#28357252C212yesmaximal, complemented
4#28357250C211yesmaximal, characteristic, complemented
5#28357245C211yesmaximal, characteristic, complemented
6#28357244C212yesmaximal, complemented
7#28357241C212yesmaximal, complemented
8C2×#20294086C211yesmaximal, characteristic, complemented
9SD16×D433C212yesmaximal, complemented
10#28357226C211yesmaximal, characteristic, complemented
11#135×C433C211yesmaximal, characteristic, complemented
12#20294098C2212yes—
13#20294097—21no—
14#20294097—21no—
15#20294097C2211yescharacteristic
16#20294096—22no—
17#20294096C2212yescomplemented
18C2×Dic866—21no—
19C2×Dic866C2211yescharacteristic, complemented
20Q8×D433C2212yescomplemented
21C2×D1732—21no—
22C2×D1732C2211yescharacteristic
23D4×D433—22no—
24D4×D433C2212yes—
25#20294087C2212yescomplemented
26#20294086C2212yescomplemented
27#20294086C2212yescomplemented
28#20294085C2212yescomplemented
29#20294084C2212yescomplemented
30#20294082C2212yescomplemented
31#20294081C2211yescharacteristic, complemented
32#20294080C2212yescomplemented
33C8×D433C2212yescomplemented
34#41×C433C2211yescharacteristic
35#41×C433C2211yescharacteristic, complemented
36Q16×C433C2211yescharacteristic, complemented
37SD16×C433C2211yescharacteristic, complemented
38SD16×C433C2211yescharacteristic, complemented
39D8×C433C2211yescharacteristic, complemented
40C2×C8×C433C2211yescharacteristic, complemented
41C22×D433—21no—
42C22×D433—22no—
43Dic866—21no—
44Dic866—22no—
45Dic866D411yescharacteristic
46Dic866C2312yes—
47D1732—22no—
48D1732—21no—
49D1732D411yescharacteristic
50D1732C2312yes—
51C22⋊D433—22no—
52C22⋊D433—22no—
53C22⋊D433D412yes—
54C2×Dic433—21no—
55C2×Dic433—22no—
56C4×D433—22no—
57C4×D433—22no—
58C4×D433C2312yes—
59#16812358C2312yes—
60Q8×C433C2311yescharacteristic
61Q8×C433C2311yescharacteristic
62D4×C433—21no—
63D4×C433—21no—
64D4×C433C2311yescharacteristic
65D4×C433C2311yescharacteristic
66C2×C4×C433—21no—
67C2×C4×C433—21no—
68C2×C4×C433C2311yescharacteristic
69C8×C433C2311yescharacteristic
70C8×C433C2311yescharacteristic
71C22×C433—21no—
72C22×C433—21no—
73C22×C433C2×D411yescharacteristic
74C2×D433—22no—
75C2×D433—82no—
76C2×D433—81no—
77C2×D433C2×D412yes—
78C4×C433—21no—
79C4×C433—21no—
80C4×C433C2×D411yescharacteristic
81C4×C433C2411yescharacteristic
82Dic433—22no—
83Dic433C2×D412yes—
84C2×C433—21no—
85C2×C433—41no—
86C2×C433—41no—
87C2×C433C22×D411yescharacteristic
88D433—22no—
89D433—42no—
90C433#57711yescharacteristic, complemented, Sylow
91#577—4331nomaximal, Sylow
922−1+4—4331no—
93H(5,𝔽2)—4331no—
94#137—4331no—
95#137—4332no—
96AGL(1,ℤ/8)—4331no—
97AGL(1,ℤ/8)—4332no—
98#135—4332no—
99#135D43311yescharacteristic, complemented
100C2×SD16—4332no—
101C2×SD16—4331no—
102#131—4331no—
103#41—8661no—
104#41—8661no—
105#41—8662no—
106#41—4332no—
107#41—4332no—
108#41—4331no—
109#41C2×D43311yescharacteristic
110#41C2×D43311yescharacteristic, complemented
111C2×Q8—8661no—
112C2×Q8—4332no—
113C2×Q8—4331no—
114C2×D4—8661no—
115C2×D4—8662no—
116C2×D4—4332no—
117C2×D4—4331no—
118Q16—4332no—
119Q16C2×D43311yescharacteristic, complemented
120SD16—4332no—
121SD16—4332no—
122SD16—4332no—
123SD16—4332no—
124SD16C2×D43311yescharacteristic, complemented
125SD16C2×D43311yescharacteristic, complemented
126D8—4332no—
127D8C2×D43311yescharacteristic, complemented
128M16—4331no—
129M16—4332no—
130C2×C8—4332no—
131C2×C8C2×D43311yescharacteristic, complemented
132C23—8662no—
133C23—8661no—
134Q8—8662no—
135Q8—8661no—
136Q8—4332no—
137Q8—4331no—
138Q8C22×D43311yescharacteristic
139Q8C22×D43311yescharacteristic
140D4—21no—
141D4—21no—
142D4—8662no—
143D4—8662no—
144D4—8662no—
145D4—8661no—
146D4—4332no—
147D4—4331no—
148D4—4332no—
149D4C22×D43311yescharacteristic
150D4C22×D43311yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—8662no—
154C2×C4—8662no—
155C2×C4—8661no—
156C2×C4—8662no—
157C2×C4—4332no—
158C2×C4C22×D43311yescharacteristic
159C8—4332no—
160C8C22×D43311yescharacteristic
161C8C22×D43311yescharacteristic
162C22—21no—
163C22—21no—
164C22—8662no—
165C22—34641no—
166C22—34642no—
167C22—4332no—
168C22D4×D43311yescharacteristic
169C4—21no—
170C4—21no—
171C4—8662no—
172C4—4332no—
173C4D4×D43311yescharacteristic
174C4C23×D43311yescharacteristic
175C2—21no—
176C2—41no—
177C2—41no—
178C2—8662no—
179C2—17322no—
180C2C2×D4×D43311yescharacteristic
subgroupG/H
centerC2C2×D4×D433
commutator, cosocleC4×C433C24
FrattiniC4C23×D433
Fitting#135×C433C2
socleC2×C433C22×D4
hypercenter#135D433
hypocenterC433#577
LCSG ▷ C4×C433 ▷ C2×C433 ▷ C433
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #28357267C2semidirect
2. #28357266C2semidirect
3. #28357252C2semidirect
4. #28357250C2semidirect
5. #28357245C2semidirect
6. #28357244C2semidirect
7. #28357241C2semidirect
8. C2×#20294086C2semidirect
9. SD16×D433C2semidirect
10. #28357226C2semidirect
11. #135×C433C2semidirect
17. #20294096C22semidirect
19. C2×Dic866C22semidirect
20. Q8×D433C22semidirect
25. #20294087C22semidirect
26. #20294086C22semidirect
27. #20294086C22semidirect
28. #20294085C22semidirect
29. #20294084C22semidirect
30. #20294082C22semidirect
31. #20294081C22semidirect
32. #20294080C22semidirect
33. C8×D433C22semidirect
35. #41×C433C22semidirect
36. Q16×C433C22semidirect
37. SD16×C433C22semidirect
38. SD16×C433C22semidirect
39. D8×C433C22semidirect
40. C2×C8×C433C22semidirect
90. C433#577semidirect
99. #135D433semidirect
110. #41C2×D433semidirect
119. Q16C2×D433semidirect
124. SD16C2×D433semidirect
125. SD16C2×D433semidirect
127. D8C2×D433semidirect
131. C2×C8C2×D433semidirect

Maximal subgroups

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

subgroupindex
#283572672
#283572662
#283572522
#283572502
#283572452
#283572442
#283572412
C2×#202940862
SD16×D4332
#283572262
#135×C4332
#577433