Finite Group Databasebeta

← C2×#20259346 #27133448

Subgroups — C2×#20259346 #27133448

Order 12224.

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×#15690286C211yesmaximal, characteristic, complemented
2C2×D4×D191C211yesmaximal, characteristic, complemented
3#20259346C218yesmaximal, direct product factor
4C2×#15690276C211yesmaximal, characteristic, complemented
5C2×#15690275C211yesmaximal, characteristic, complemented
6C2×#15690272C211yesmaximal, characteristic, complemented
7C2×#15690270C211yesmaximal, characteristic, complemented
8C2×D8×C191C211yesmaximal, characteristic, complemented
9C23×D191—21no—
10#15690286—22no—
11#15690286C2214yescomplemented
12C2×Dic382C2211yescharacteristic, complemented
13C2×D764C2211yescharacteristic
14C2×C22⋊D191—21no—
15C2×C22⋊D191—21no—
16D4×D191—22no—
17D4×D191C2214yescomplemented
18C22×Dic191—21no—
19C2×C4×D191C2211yescharacteristic
20#15690276C2214yescomplemented
21#15690275C2214yescomplemented
22#15690272C2214yescomplemented
23#15690270C2214yescomplemented
24C2×C191⋊C8C2211yescharacteristic, complemented
25C2×D4×C191C2211yescharacteristic
26C2×D4×C191C2211yescharacteristic, complemented
27D8×C191C2214yescomplemented
28C2×C8×C191C2211yescharacteristic, complemented
29C22×D191—21no—
30C22×D191—24no—
31C22×D191—44no—
32C22×D191D411yescharacteristic
33Dic382—21no—
34Dic382C2312yescomplemented
35D764—21no—
36D764C2312yes—
37C22⋊D191—24no—
38C22⋊D191—24no—
39C2×Dic191—21no—
40C2×Dic191—24no—
41C2×Dic191D411yescharacteristic
42C4×D191D412yes—
43C4×D191C2312yes—
44C191⋊C8C2312yescomplemented
45C23×C191—21no—
46C23×C191—21no—
47D4×C191—21no—
48D4×C191—21no—
49D4×C191C2312yes—
50D4×C191C2312yescomplemented
51C2×C4×C191C2311yescharacteristic
52C8×C191C2312yescomplemented
53C22×C191—22no—
54C22×C191—22no—
55C22×C191—42no—
56C22×C191—42no—
57C22×C191C2×D411yescharacteristic
58C2×D191—22no—
59C2×D191—22no—
60C2×D191—48no—
61C2×D191—42no—
62C2×D191C2×D412yes—
63C4×C191C2×D411yescharacteristic
64C4×C191C2411yescharacteristic
65Dic191—22no—
66Dic191C2×D412yes—
67C2×C191—42no—
68C2×C191—42no—
69C2×C191AGL(1,ℤ/8)12yescomplemented
70C2×C191C22×D411yescharacteristic
71D191—22no—
72D191—42no—
73C191C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
74C2×AGL(1,ℤ/8)—1911nomaximal, Sylow
75C2×#41—1911no—
76C22×D4—1911no—
77AGL(1,ℤ/8)—1918no—
78C2×SD16—1911no—
79C2×SD16—1911no—
80C2×D8—1911no—
81C2×D8D19111yescharacteristic, complemented
82C2×M16—1911no—
83C24—3821no—
84#41—3822no—
85#41—1914no—
86C2×Q8—1911no—
87C2×D4—3821no—
88C2×D4—3822no—
89C2×D4—3821no—
90C2×D4—1914no—
91C2×D4—1911no—
92C2×D4C2×D19111yescharacteristic
93C2×D4C2×D19111yescharacteristic, complemented
94C22×C4—3821no—
95C22×C4—1911no—
96SD16—1914no—
97SD16—1914no—
98D8—1914no—
99D8C2×D19114yescomplemented
100M16—1914no—
101C2×C8—1911no—
102C2×C8C2×D19111yescharacteristic, complemented
103C23—21no—
104C23—21no—
105C23—3824no—
106C23—3821no—
107C23—1911no—
108C23—7644no—
109Q8—3821no—
110Q8—1912no—
111D4—21no—
112D4—21no—
113D4—3824no—
114D4—3824no—
115D4—3821no—
116D4—1912no—
117D4C22×D19112yes—
118D4C22×D19112yescomplemented
119C2×C4—3821no—
120C2×C4—3824no—
121C2×C4—1911no—
122C2×C4—1912no—
123C2×C4—1912no—
124C2×C4C22×D19111yescharacteristic
125C8—1912no—
126C8C22×D19112yescomplemented
127C22—22no—
128C22—22no—
129C22—42no—
130C22—42no—
131C22—3822no—
132C22—3822no—
133C22—1912no—
134C22—7648no—
135C22—7642no—
136C22D4×D19111yescharacteristic
137C4—3822no—
138C4—1912no—
139C4D4×D19111yescharacteristic
140C4C23×D19111yescharacteristic
141C2—42no—
142C2—42no—
143C2—3822no—
144C2—7642no—
145C2#2025934612yesdirect product factor
146C2C2×D4×D19111yescharacteristic
subgroupG/H
centerC22D4×D191
commutator, cosocleC4×C191C24
FrattiniC4C23×D191
FittingC2×D8×C191C2
socleC22×C191C2×D4
hypercenterC2×D8D191
hypocenterC191C2×AGL(1,ℤ/8)
LCSG ▷ C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. C2×#15690286C2semidirect
2. C2×D4×D191C2semidirect
3. #20259346C2direct (N × H)
4. C2×#15690276C2semidirect
5. C2×#15690275C2semidirect
6. C2×#15690272C2semidirect
7. C2×#15690270C2semidirect
8. C2×D8×C191C2semidirect
11. #15690286C22semidirect
12. C2×Dic382C22semidirect
17. D4×D191C22semidirect
20. #15690276C22semidirect
21. #15690275C22semidirect
22. #15690272C22semidirect
23. #15690270C22semidirect
24. C2×C191⋊C8C22semidirect
26. C2×D4×C191C22semidirect
27. D8×C191C22semidirect
28. C2×C8×C191C22semidirect
34. Dic382C23semidirect
44. C191⋊C8C23semidirect
50. D4×C191C23semidirect
52. C8×C191C23semidirect
69. C2×C191AGL(1,ℤ/8)semidirect
73. C191C2×AGL(1,ℤ/8)semidirect
81. C2×D8D191semidirect
93. C2×D4C2×D191semidirect
99. D8C2×D191semidirect
102. C2×C8C2×D191semidirect
118. D4C22×D191semidirect
126. C8C22×D191semidirect
145. C2#20259346direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156902862
C2×D4×D1912
#202593462
C2×#156902762
C2×#156902752
C2×#156902722
C2×#156902702
C2×D8×C1912
C2×AGL(1,ℤ/8)191