Finite Group Databasebeta

← #27132829

Subgroups — #27132829

Order 12224.

168 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 39 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#20259368C211yesmaximal, characteristic, complemented
2#20259348C211yesmaximal, characteristic, complemented
3#20259318C211yesmaximal, characteristic, complemented
4#20259249C211yesmaximal, characteristic, complemented
5#20259248C211yesmaximal, characteristic, complemented
6#20259238C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C191C211yesmaximal, characteristic, complemented
8#15690288—21no—
9#15690287—21no—
10#15690287C2211yescharacteristic
11C2×D764—21no—
12C2×D764C2211yescharacteristic
13C2×C22⋊D191—22no—
14D4×D191—22no—
15D4×D191—21no—
16#15690275—21no—
17#15690272—21no—
18#15690271C2211yescharacteristic, complemented
19C4×Dic191C2211yescharacteristic, complemented
20#41×C191C2211yescharacteristic
21C2×D4×C191C2211yescharacteristic, complemented
22SD16×C191—21no—
23D8×C191—21no—
24M16×C191C2211yescharacteristic, complemented
25C22×D191—22no—
26C22×D191D412yes—
27Dic382D411yescharacteristic, complemented
28D764—22no—
29D764—21no—
30D764—21no—
31D764D411yescharacteristic, complemented
32C22⋊D191—22no—
33C22⋊D191—21no—
34C22⋊D191—44no—
35C2×Dic191—21no—
36C4×D191—22no—
37C4×D191—21no—
38C191⋊C8—21no—
39C23×C191—21no—
40Q8×C191D411yescharacteristic, complemented
41D4×C191—21no—
42D4×C191—21no—
43D4×C191—21no—
44D4×C191D411yescharacteristic, complemented
45C2×C4×C191—21no—
46C2×C4×C191C2311yescharacteristic
47C8×C191—21no—
48C22×C191—21no—
49C22×C191—42no—
50C22×C191—41no—
51C22×C191C2×D411yescharacteristic
52C2×D191—22no—
53C2×D191—21no—
54C2×D191—42no—
55C2×D191—82no—
56C4×C191—21no—
57C4×C191C2×D411yescharacteristic
58C4×C191C2×D411yescharacteristic
59Dic191—22no—
60Dic191—21no—
61C2×C191—21no—
62C2×C191—41no—
63C2×C191—81no—
64C2×C191#12011yescharacteristic
65D191—42no—
66D191—41no—
67C191#45311yescharacteristic, complemented, Sylow
68#453—1911nomaximal, Sylow
69H(5,𝔽2)—1911no—
70AGL(1,ℤ/8)—1911no—
71AGL(1,ℤ/8)D19111yescharacteristic, complemented
72#127—1911no—
73#104—1911no—
74#104—1911no—
75#100—1911no—
76#41—3821no—
77#41—3821no—
78#41—1911no—
79#41C2×D19111yescharacteristic
80C2×D4—3821no—
81C2×D4—3822no—
82C2×D4—3821no—
83C2×D4—3822no—
84C2×D4—1911no—
85C2×D4C2×D19111yescharacteristic, complemented
86SD16—21no—
87SD16—3821no—
88D8—21no—
89D8—3821no—
90M16—1911no—
91M16C2×D19111yescharacteristic, complemented
92C42—1911no—
93C23—21no—
94C23—3822no—
95C23—1912no—
96Q8—1911no—
97Q8C22⋊D19111yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—3822no—
102D4—3822no—
103D4—3821no—
104D4—3821no—
105D4—3821no—
106D4—1911no—
107D4—7644no—
108D4C22⋊D19111yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—3822no—
111C2×C4—3821no—
112C2×C4—3821no—
113C2×C4C22×D19111yescharacteristic
114C8—21no—
115C8—3821no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—3822no—
120C22—3821no—
121C22—15282no—
122C22—7642no—
123C22D4×D19111yescharacteristic
124C4—21no—
125C4—3822no—
126C4—3821no—
127C4D4×D19111yescharacteristic
128C4C2×C22⋊D19111yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—7641no—
133C2—7642no—
134C2#2025929011yescharacteristic
subgroupG/H
centerC2#20259290
commutator, cosocleC2×C4×C191C23
FrattiniC2×C4C22×D191
FittingAGL(1,ℤ/8)×C191C2
socleC2×C191#120
hypercenterAGL(1,ℤ/8)D191
hypocenterC191#453
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #20259368C2semidirect
2. #20259348C2semidirect
3. #20259318C2semidirect
4. #20259249C2semidirect
5. #20259248C2semidirect
6. #20259238C2semidirect
7. AGL(1,ℤ/8)×C191C2semidirect
18. #15690271C22semidirect
19. C4×Dic191C22semidirect
21. C2×D4×C191C22semidirect
24. M16×C191C22semidirect
27. Dic382D4semidirect
31. D764D4semidirect
40. Q8×C191D4semidirect
44. D4×C191D4semidirect
67. C191#453semidirect
71. AGL(1,ℤ/8)D191semidirect
85. C2×D4C2×D191semidirect
91. M16C2×D191semidirect
97. Q8C22⋊D191semidirect
108. D4C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
#202593682
#202593482
#202593182
#202592492
#202592482
#202592382
AGL(1,ℤ/8)×C1912
#453191