Finite Group Databasebeta

← #37571637

Subgroups — #37571637

Order 24448.

314 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 57 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#27133298C211yesmaximal, characteristic
2#27133035C211yesmaximal, characteristic, complemented
3#27132954C212yesmaximal, complemented
4#27132390C212yesmaximal, complemented
5#331×C191C211yesmaximal, characteristic, complemented
6C2×D4×D191—22no—
7C2×#15690275—22no—
8#20259322C2211yescharacteristic, complemented
9#20259293—22no—
10#20259290—22no—
11C4×D764C2212yes—
12#20259254—22no—
13#20259244—22no—
14C4×C191⋊C8C2211yescharacteristic, complemented
15#127×C191C2211yescharacteristic
16C4⋊C8×C191C2212yescomplemented
17C23×D191—22no—
18C2×Dic382—21no—
19C2×D764D412yes—
20C2×C22⋊D191—24no—
21D4×D191—44no—
22D4×D191—44no—
23C2×C4×D191—22no—
24#15690275—44no—
25C2×C191⋊C8—21no—
26C4⋊Dic191—22no—
27C4⋊Dic191D412yes—
28C22⋊Dic191—22no—
29(C2×D191)⋊C4—22no—
30C2×D4×C191—22no—
31C2×D4×C191D412yes—
32C2×C8×C191—22no—
33C42×C191C2311yescharacteristic
34C22×D191—22no—
35C22×D191—48no—
36C22×D191—82no—
37Dic382—42no—
38D764—22no—
39D764SD1614yescomplemented
40C22⋊D191—48no—
41C2×Dic191—22no—
42C2×Dic191—41no—
43C4×D191—42no—
44C4×D191—42no—
45C191⋊C8—22no—
46C23×C191—22no—
47D4×C191—42no—
48D4×C191—42no—
49D4×C191—42no—
50C2×C4×C191C2×D412yes—
51C2×C4×C191C2×D411yescharacteristic
52C8×C191—42no—
53C22×C191—44no—
54C22×C191—42no—
55C22×C191#12011yescharacteristic
56C2×D191—24no—
57C2×D191—42no—
58C2×D191—42no—
59C2×D191—88no—
60C4×C191—21no—
61C4×C191C2×SD1612yes—
62C4×C191AGL(1,ℤ/8)12yes—
63Dic191—42no—
64Dic191—81no—
65C2×C191—82no—
66C2×C191#45012yes—
67C2×C191#45311yescharacteristic
68D191—44no—
69C191#162911yescharacteristic, complemented, Sylow
70#1629—1911nomaximal, Sylow
71D42—1911no—
72#492—1911no—
73#476—1912no—
74#331D19111yescharacteristic, complemented
75D4⋊C8—1912no—
76C22×D4—3822no—
77C2×SD16—3822no—
78C4⋊Q8—1911no—
79#127C2×D19111yescharacteristic
80#121—3822no—
81#120—3822no—
82C4×D4—1912no—
83#106—3822no—
84C4⋊C8C2×D19112yescomplemented
85#102—3822no—
86C4×C8—1911no—
87C24—3822no—
88C2×Q8—3821no—
89C2×D4—22no—
90C2×D4—3824no—
91C2×D4—1912no—
92C2×D4—7644no—
93C2×D4—7644no—
94C2×D4D76412yes—
95C22×C4—3822no—
96SD16—7644no—
97C2×C8—22no—
98C2×C8—3821no—
99C4⋊C4—3822no—
100C4⋊C4—1912no—
101C22⋊C4—3822no—
102C22⋊C4—3822no—
103C42C22×D19111yescharacteristic
104C23—22no—
105C23—3822no—
106C23—15282no—
107C23—7648no—
108Q8—7642no—
109D4—42no—
110D4—42no—
111D4—42no—
112D4—3822no—
113D4—1914no—
114D4—7648no—
115C2×C4—3822no—
116C2×C4—7641no—
117C2×C4—7642no—
118C2×C4—7642no—
119C2×C4D4×D19112yes—
120C2×C4C2×D76411yescharacteristic
121C8—42no—
122C8—3822no—
123C22—44no—
124C22—42no—
125C22—3824no—
126C22—15288no—
127C22—7642no—
128C22—7642no—
129C22#2025929211yescharacteristic
130C4—21no—
131C4—15281no—
132C4—7642no—
133C4SD16×D19112yes—
134C4#2025934612yes—
135C2—82no—
136C2—7644no—
137C2#2713281512yes—
138C2#2713283311yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#331×C191C2
socleC22×C191#120
prime radical#27133298C2
hypercenter#331D191
hypocenterC191#1629
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #27133035C2semidirect
3. #27132954C2semidirect
4. #27132390C2semidirect
5. #331×C191C2semidirect
8. #20259322C22semidirect
14. C4×C191⋊C8C22semidirect
16. C4⋊C8×C191C22semidirect
39. D764SD16semidirect
69. C191#1629semidirect
74. #331D191semidirect
84. C4⋊C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271332982
#271330352
#271329542
#271323902
#331×C1912
#1629191