Finite Group Databasebeta

← C2×#20297331 #28384368

Subgroups — C2×#20297331 #28384368

Order 14272.

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×#16813171C211yesmaximal, characteristic, complemented
2C2×D4×D223C211yesmaximal, characteristic, complemented
3#20297331C218yesmaximal, direct product factor
4C2×#16813157C211yesmaximal, characteristic, complemented
5C2×D1784C211yesmaximal, characteristic, complemented
6C2×#16813155C211yesmaximal, characteristic, complemented
7C2×#16813153C211yesmaximal, characteristic, complemented
8C2×SD16×C223C211yesmaximal, characteristic, complemented
9C23×D223—21no—
10#16813171—22no—
11#16813171C2214yescomplemented
12C2×D892—21no—
13C2×D892C2211yescharacteristic
14C2×D892C2211yescharacteristic, complemented
15C2×C22⋊D223—21no—
16D4×D223—22no—
17D4×D223C2214yescomplemented
18C2×C4×D223—21no—
19C2×C4×D223C2211yescharacteristic
20#16813157C2214yescomplemented
21D1784C2214yescomplemented
22#16813155C2214yescomplemented
23#16813153C2214yescomplemented
24C2×C223⋊C8C2211yescharacteristic, complemented
25C2×Q8×C223C2211yescharacteristic, complemented
26C2×D4×C223C2211yescharacteristic
27SD16×C223C2214yescomplemented
28C2×C8×C223C2211yescharacteristic, complemented
29C22×D223—21no—
30C22×D223—21no—
31C22×D223—24no—
32C22×D223—44no—
33C22×D223D411yescharacteristic
34D892—24no—
35D892—21no—
36D892—21no—
37D892C2312yes—
38D892C2312yescomplemented
39C22⋊D223—24no—
40C2×Dic223D411yescharacteristic
41C4×D223—24no—
42C4×D223D412yes—
43C4×D223C2312yes—
44C223⋊C8C2312yescomplemented
45C23×C223—21no—
46Q8×C223—21no—
47Q8×C223C2312yescomplemented
48D4×C223—21no—
49D4×C223C2312yes—
50C2×C4×C223—21no—
51C2×C4×C223C2311yescharacteristic
52C8×C223C2312yescomplemented
53C22×C223—22no—
54C22×C223—42no—
55C22×C223C2×D411yescharacteristic
56C2×D223—22no—
57C2×D223—22no—
58C2×D223—22no—
59C2×D223—48no—
60C2×D223—42no—
61C2×D223—42no—
62C2×D223C2×D412yes—
63C4×C223—22no—
64C4×C223C2×D411yescharacteristic
65C4×C223C2411yescharacteristic
66Dic223C2×D412yes—
67C2×C223—42no—
68C2×C223AGL(1,ℤ/8)12yescomplemented
69C2×C223C22×D411yescharacteristic
70D223—22no—
71D223—42no—
72D223—42no—
73C223C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
74C2×AGL(1,ℤ/8)—2231nomaximal, Sylow
75C2×#41—2231no—
76C22×D4—2231no—
77AGL(1,ℤ/8)—2238no—
78C2×SD16—2231no—
79C2×SD16D22311yescharacteristic, complemented
80C2×D8—2231no—
81C2×D8—2231no—
82C2×M16—2231no—
83C24—4461no—
84#41—4462no—
85#41—2234no—
86C2×Q8C2×D22311yescharacteristic, complemented
87C2×D4—4461no—
88C2×D4—4462no—
89C2×D4—4461no—
90C2×D4—2234no—
91C2×D4—2231no—
92C2×D4—2231no—
93C2×D4C2×D22311yescharacteristic
94C22×C4—4461no—
95C22×C4—2231no—
96SD16—2234no—
97SD16C2×D22314yescomplemented
98D8—2234no—
99D8—2234no—
100M16—2234no—
101C2×C8—2231no—
102C2×C8C2×D22311yescharacteristic, complemented
103C23—21no—
104C23—8924no—
105C23—4464no—
106C23—4461no—
107C23—4461no—
108C23—2231no—
109Q8—21no—
110Q8C22×D22312yescomplemented
111D4—21no—
112D4—4464no—
113D4—4464no—
114D4—4461no—
115D4—4461no—
116D4—2232no—
117D4—2232no—
118D4C22×D22312yes—
119C2×C4—21no—
120C2×C4—4464no—
121C2×C4—2231no—
122C2×C4—2232no—
123C2×C4—2232no—
124C2×C4C22×D22311yescharacteristic
125C8—2232no—
126C8C22×D22312yescomplemented
127C22—22no—
128C22—42no—
129C22—8928no—
130C22—8922no—
131C22—8922no—
132C22—4462no—
133C22—4462no—
134C22—4462no—
135C22—2232no—
136C22D4×D22311yescharacteristic
137C4—22no—
138C4—2232no—
139C4D4×D22311yescharacteristic
140C4C23×D22311yescharacteristic
141C2—42no—
142C2—8922no—
143C2—8922no—
144C2—4462no—
145C2#2029733112yesdirect product factor
146C2C2×D4×D22311yescharacteristic
subgroupG/H
centerC22D4×D223
commutator, cosocleC4×C223C24
FrattiniC4C23×D223
FittingC2×SD16×C223C2
socleC22×C223C2×D4
hypercenterC2×SD16D223
hypocenterC223C2×AGL(1,ℤ/8)
LCSG ▷ C4×C223 ▷ C2×C223 ▷ C223
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16813171C2semidirect
2. C2×D4×D223C2semidirect
3. #20297331C2direct (N × H)
4. C2×#16813157C2semidirect
5. C2×D1784C2semidirect
6. C2×#16813155C2semidirect
7. C2×#16813153C2semidirect
8. C2×SD16×C223C2semidirect
11. #16813171C22semidirect
14. C2×D892C22semidirect
17. D4×D223C22semidirect
20. #16813157C22semidirect
21. D1784C22semidirect
22. #16813155C22semidirect
23. #16813153C22semidirect
24. C2×C223⋊C8C22semidirect
25. C2×Q8×C223C22semidirect
27. SD16×C223C22semidirect
28. C2×C8×C223C22semidirect
38. D892C23semidirect
44. C223⋊C8C23semidirect
47. Q8×C223C23semidirect
52. C8×C223C23semidirect
68. C2×C223AGL(1,ℤ/8)semidirect
73. C223C2×AGL(1,ℤ/8)semidirect
79. C2×SD16D223semidirect
86. C2×Q8C2×D223semidirect
97. SD16C2×D223semidirect
102. C2×C8C2×D223semidirect
110. Q8C22×D223semidirect
126. C8C22×D223semidirect
145. C2#20297331direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168131712
C2×D4×D2232
#202973312
C2×#168131572
C2×D17842
C2×#168131552
C2×#168131532
C2×SD16×C2232
C2×AGL(1,ℤ/8)223