Finite Group Databasebeta

← #42122617

Subgroups — #42122617

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#28357268C211yesmaximal, characteristic, complemented
2#28357265C211yesmaximal, characteristic, complemented
3#28357251C211yesmaximal, characteristic, complemented
4#28357249C212yesmaximal, complemented
5#28357247C212yesmaximal, complemented
6#28357246C211yesmaximal, characteristic, complemented
7#28357242C211yesmaximal, characteristic, complemented
8#28357241C212yesmaximal, complemented
9SD16×D433C212yesmaximal, complemented
10#28357226C211yesmaximal, characteristic, complemented
11C2×SD16×C433C211yesmaximal, characteristic, complemented
12#20294098—21no—
13#20294098C2212yescomplemented
14#20294097—22no—
15#20294097C2211yescharacteristic
16#20294097C2211yescharacteristic
17#20294097C2211yescharacteristic, complemented
18#20294096—21no—
19#20294096C2212yes—
20Q8×D433—21no—
21Q8×D433C2212yescomplemented
22C2×C22⋊D433—22no—
23D4×D433—21no—
24D4×D433C2212yes—
25Dic1732C2211yescharacteristic, complemented
26#20294087C2212yescomplemented
27#20294086C2211yescharacteristic, complemented
28#20294086C2211yescharacteristic, complemented
29#20294085C2212yescomplemented
30#20294084C2212yescomplemented
31D3464C2211yescharacteristic, complemented
32#20294082C2212yescomplemented
33#20294081C2211yescharacteristic, complemented
34#20294080C2212yescomplemented
35C8×D433C2212yescomplemented
36C2×Q8×C433C2211yescharacteristic, complemented
37C2×D4×C433C2211yescharacteristic
38SD16×C433C2212yescomplemented
39SD16×C433C2212yescomplemented
40C2×C8×C433C2211yescharacteristic, complemented
41C22×D433—22no—
42Dic866—22no—
43Dic866D411yescharacteristic
44Dic866C2311yescharacteristic
45Dic866C2311yescharacteristic
46D1732—22no—
47D1732D411yescharacteristic
48D1732C2311yescharacteristic
49D1732C2311yescharacteristic
50C22⋊D433—21no—
51C22⋊D433—21no—
52C22⋊D433—22no—
53C22⋊D433—22no—
54C22⋊D433D412yes—
55C2×Dic433—22no—
56C4×D433—22no—
57C4×D433—21no—
58C4×D433—21no—
59C4×D433—22no—
60C4×D433C2312yes—
61#16812358C2312yes—
62C23×C433—21no—
63Q8×C433—21no—
64Q8×C433C2312yes—
65D4×C433—21no—
66D4×C433C2312yes—
67C2×C4×C433—21no—
68C2×C4×C433C2311yescharacteristic
69C8×C433C2311yescharacteristic
70C8×C433C2311yescharacteristic
71C22×C433—22no—
72C22×C433—81no—
73C22×C433C2×D411yescharacteristic
74C2×D433—21no—
75C2×D433—21no—
76C2×D433—82no—
77C2×D433C2×D412yes—
78C4×C433—22no—
79C4×C433C2×D411yescharacteristic
80C4×C433C2411yescharacteristic
81Dic433—21no—
82Dic433—21no—
83Dic433C2×D412yes—
84C2×C433—21no—
85C2×C433—42no—
86C2×C433C22×D411yescharacteristic
87D433—22no—
88D433—41no—
89D433—41no—
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#135—4331no—
100C2×SD16—4332no—
101C2×SD16D43311yescharacteristic, complemented
102#131—4331no—
103#41—8662no—
104#41—8661no—
105#41—8661no—
106#41—4332no—
107#41—4332no—
108#41—4331no—
109#41—4331no—
110#41—4331no—
111C2×Q8—8661no—
112C2×Q8—4332no—
113C2×Q8C2×D43311yescharacteristic, complemented
114C2×D4—8661no—
115C2×D4—8662no—
116C2×D4—4332no—
117C2×D4C2×D43311yescharacteristic
118Q16—4332no—
119Q16—4331no—
120SD16—4332no—
121SD16—4332no—
122SD16—4331no—
123SD16—4331no—
124SD16C2×D43312yescomplemented
125SD16C2×D43312yescomplemented
126D8—4332no—
127D8—4331no—
128M16—4331no—
129M16—4332no—
130C2×C8—4332no—
131C2×C8C2×D43311yescharacteristic, complemented
132C23—21no—
133C23—8662no—
134Q8—21no—
135Q8—8662no—
136Q8—4331no—
137Q8—4331no—
138Q8—4331no—
139Q8C22×D43312yes—
140D4—21no—
141D4—8662no—
142D4—8662no—
143D4—8661no—
144D4—8661no—
145D4—8662no—
146D4—4332no—
147D4—4331no—
148D4—4331no—
149D4—4331no—
150D4C22×D43312yes—
151C2×C4—21no—
152C2×C4—8662no—
153C2×C4—8662no—
154C2×C4—8662no—
155C2×C4—8661no—
156C2×C4—8661no—
157C2×C4—4332no—
158C2×C4C22×D43311yescharacteristic
159C8—4332no—
160C8C22×D43311yescharacteristic
161C8C22×D43311yescharacteristic
162C22—22no—
163C22—81no—
164C22—8661no—
165C22—8661no—
166C22—34642no—
167C22—4332no—
168C22D4×D43311yescharacteristic
169C4—22no—
170C4—8661no—
171C4—8661no—
172C4—4332no—
173C4D4×D43311yescharacteristic
174C4C23×D43311yescharacteristic
175C2—21no—
176C2—42no—
177C2—8662no—
178C2—17321no—
179C2—17321no—
180C2C2×D4×D43311yescharacteristic
subgroupG/H
centerC2C2×D4×D433
commutator, cosocleC4×C433C24
FrattiniC4C23×D433
FittingC2×SD16×C433C2
socleC2×C433C22×D4
hypercenterC2×SD16D433
hypocenterC433#577
LCSG ▷ C4×C433 ▷ C2×C433 ▷ C433
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. #28357268C2semidirect
2. #28357265C2semidirect
3. #28357251C2semidirect
4. #28357249C2semidirect
5. #28357247C2semidirect
6. #28357246C2semidirect
7. #28357242C2semidirect
8. #28357241C2semidirect
9. SD16×D433C2semidirect
10. #28357226C2semidirect
11. C2×SD16×C433C2semidirect
13. #20294098C22semidirect
17. #20294097C22semidirect
21. Q8×D433C22semidirect
25. Dic1732C22semidirect
26. #20294087C22semidirect
27. #20294086C22semidirect
28. #20294086C22semidirect
29. #20294085C22semidirect
30. #20294084C22semidirect
31. D3464C22semidirect
32. #20294082C22semidirect
33. #20294081C22semidirect
34. #20294080C22semidirect
35. C8×D433C22semidirect
36. C2×Q8×C433C22semidirect
38. SD16×C433C22semidirect
39. SD16×C433C22semidirect
40. C2×C8×C433C22semidirect
90. C433#577semidirect
101. C2×SD16D433semidirect
113. C2×Q8C2×D433semidirect
124. SD16C2×D433semidirect
125. SD16C2×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
#283572682
#283572652
#283572512
#283572492
#283572472
#283572462
#283572422
#283572412
SD16×D4332
#283572262
C2×SD16×C4332
#577433