Finite Group Databasebeta

← C2×#28357247 #42122599

Subgroups — C2×#28357247 #42122599

Order 27712.

298 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C2×#20294098C211yesmaximal, characteristic, complemented
2C2×D4×D433C211yesmaximal, characteristic, complemented
3#28357247C218yesmaximal, direct product factor
4C2×#20294084C211yesmaximal, characteristic, complemented
5C2×D3464C211yesmaximal, characteristic, complemented
6C2×#20294082C211yesmaximal, characteristic, complemented
7C2×#20294080C211yesmaximal, characteristic, complemented
8C2×SD16×C433C211yesmaximal, characteristic, complemented
9C23×D433—21no—
10#20294098—22no—
11#20294098C2214yescomplemented
12C2×D1732—21no—
13C2×D1732C2211yescharacteristic
14C2×D1732C2211yescharacteristic, complemented
15C2×C22⋊D433—21no—
16D4×D433—22no—
17D4×D433C2214yescomplemented
18C2×C4×D433—21no—
19C2×C4×D433C2211yescharacteristic
20#20294084C2214yescomplemented
21D3464C2214yescomplemented
22#20294082C2214yescomplemented
23#20294080C2214yescomplemented
24C2×#16812358C2211yescharacteristic, complemented
25C2×Q8×C433C2211yescharacteristic, complemented
26C2×D4×C433C2211yescharacteristic
27SD16×C433C2214yescomplemented
28C2×C8×C433C2211yescharacteristic, complemented
29C22×D433—21no—
30C22×D433—21no—
31C22×D433—24no—
32C22×D433—44no—
33C22×D433D411yescharacteristic
34D1732—24no—
35D1732—21no—
36D1732—21no—
37D1732C2312yes—
38D1732C2312yescomplemented
39C22⋊D433—24no—
40C2×Dic433D411yescharacteristic
41C4×D433—24no—
42C4×D433D412yes—
43C4×D433C2312yes—
44#16812358C2312yescomplemented
45C23×C433—21no—
46Q8×C433—21no—
47Q8×C433C2312yescomplemented
48D4×C433—21no—
49D4×C433C2312yes—
50C2×C4×C433—21no—
51C2×C4×C433C2311yescharacteristic
52C8×C433C2312yescomplemented
53C22×C433—22no—
54C22×C433—42no—
55C22×C433C2×D411yescharacteristic
56C2×D433—22no—
57C2×D433—22no—
58C2×D433—22no—
59C2×D433—48no—
60C2×D433—42no—
61C2×D433—42no—
62C2×D433C2×D412yes—
63C4×C433—22no—
64C4×C433C2×D411yescharacteristic
65C4×C433C2411yescharacteristic
66Dic433C2×D412yes—
67C2×C433—42no—
68C2×C433AGL(1,ℤ/8)12yescomplemented
69C2×C433C22×D411yescharacteristic
70D433—22no—
71D433—42no—
72D433—42no—
73C433C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
74C2×AGL(1,ℤ/8)—4331nomaximal, Sylow
75C2×#41—4331no—
76C22×D4—4331no—
77AGL(1,ℤ/8)—4338no—
78C2×SD16—4331no—
79C2×SD16D43311yescharacteristic, complemented
80C2×D8—4331no—
81C2×D8—4331no—
82C2×M16—4331no—
83C24—8661no—
84#41—8662no—
85#41—4334no—
86C2×Q8C2×D43311yescharacteristic, complemented
87C2×D4—8661no—
88C2×D4—8662no—
89C2×D4—8661no—
90C2×D4—4334no—
91C2×D4—4331no—
92C2×D4—4331no—
93C2×D4C2×D43311yescharacteristic
94C22×C4—8661no—
95C22×C4—4331no—
96SD16—4334no—
97SD16C2×D43314yescomplemented
98D8—4334no—
99D8—4334no—
100M16—4334no—
101C2×C8—4331no—
102C2×C8C2×D43311yescharacteristic, complemented
103C23—21no—
104C23—8664no—
105C23—8661no—
106C23—8661no—
107C23—4331no—
108C23—17324no—
109Q8—21no—
110Q8C22×D43312yescomplemented
111D4—21no—
112D4—8664no—
113D4—8664no—
114D4—8661no—
115D4—8661no—
116D4—4332no—
117D4—4332no—
118D4C22×D43312yes—
119C2×C4—21no—
120C2×C4—8664no—
121C2×C4—4331no—
122C2×C4—4332no—
123C2×C4—4332no—
124C2×C4C22×D43311yescharacteristic
125C8—4332no—
126C8C22×D43312yescomplemented
127C22—22no—
128C22—42no—
129C22—8662no—
130C22—8662no—
131C22—8662no—
132C22—4332no—
133C22—17328no—
134C22—17322no—
135C22—17322no—
136C22D4×D43311yescharacteristic
137C4—22no—
138C4—4332no—
139C4D4×D43311yescharacteristic
140C4C23×D43311yescharacteristic
141C2—42no—
142C2—8662no—
143C2—17322no—
144C2—17322no—
145C2#2835724712yesdirect product factor
146C2C2×D4×D43311yescharacteristic
subgroupG/H
centerC22D4×D433
commutator, cosocleC4×C433C24
FrattiniC4C23×D433
FittingC2×SD16×C433C2
socleC22×C433C2×D4
hypercenterC2×SD16D433
hypocenterC433C2×AGL(1,ℤ/8)
LCSG ▷ C4×C433 ▷ C2×C433 ▷ C433
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#20294098C2semidirect
2. C2×D4×D433C2semidirect
3. #28357247C2direct (N × H)
4. C2×#20294084C2semidirect
5. C2×D3464C2semidirect
6. C2×#20294082C2semidirect
7. C2×#20294080C2semidirect
8. C2×SD16×C433C2semidirect
11. #20294098C22semidirect
14. C2×D1732C22semidirect
17. D4×D433C22semidirect
20. #20294084C22semidirect
21. D3464C22semidirect
22. #20294082C22semidirect
23. #20294080C22semidirect
24. C2×#16812358C22semidirect
25. C2×Q8×C433C22semidirect
27. SD16×C433C22semidirect
28. C2×C8×C433C22semidirect
38. D1732C23semidirect
44. #16812358C23semidirect
47. Q8×C433C23semidirect
52. C8×C433C23semidirect
68. C2×C433AGL(1,ℤ/8)semidirect
73. C433C2×AGL(1,ℤ/8)semidirect
79. C2×SD16D433semidirect
86. C2×Q8C2×D433semidirect
97. SD16C2×D433semidirect
102. C2×C8C2×D433semidirect
110. Q8C22×D433semidirect
126. C8C22×D433semidirect
145. C2#28357247direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#202940982
C2×D4×D4332
#283572472
C2×#202940842
C2×D34642
C2×#202940822
C2×#202940802
C2×SD16×C4332
C2×AGL(1,ℤ/8)433