Finite Group Databasebeta

← #37571844

Subgroups — #37571844

Order 24448.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#27133411C211yesmaximal, characteristic
2#27133033C211yesmaximal, characteristic, complemented
3#27133023C212yesmaximal, complemented
4#27132402C212yesmaximal, complemented
5#330×C191C211yesmaximal, characteristic, complemented
6#20259322C2211yescharacteristic
7#20259310C2212yes—
8#20259307—22no—
9#20259300—22no—
10#20259297—22no—
11#20259247—22no—
12#20259245—22no—
13#20259244—22no—
14#20259233C2211yescharacteristic
15#125×C191C2211yescharacteristic
16#97×C191C2212yes—
17C2×Dic382—21no—
18C2×Dic382D412yes—
19C2×D764—22no—
20C2×D764D412yes—
21C2×C4×D191—22no—
22C2×C191⋊C8—21no—
23C4⋊Dic191—22no—
24Dic191⋊C4—22no—
25(C2×D191)⋊C4—22no—
26(C2×D191)⋊C4—22no—
27(C2×D191)⋊C4—22no—
28C2×C8×C191—22no—
29C4⋊C4×C191—22no—
30C4⋊C4×C191D412yes—
31C42×C191C2311yescharacteristic
32C22×D191—22no—
33Dic382—42no—
34Dic382—42no—
35D764—42no—
36D764—82no—
37C2×Dic191—22no—
38C2×Dic191—41no—
39C4×D191—82no—
40C191⋊C8—41no—
41C2×C4×C191—22no—
42C2×C4×C191C2×D412yes—
43C2×C4×C191C2×D411yescharacteristic
44C8×C191—42no—
45C22×C191#12011yescharacteristic
46C2×D191—42no—
47C2×D191—42no—
48C2×D191—42no—
49C4×C191—21no—
50C4×C191—22no—
51C4×C191—42no—
52Dic191—42no—
53Dic191—81no—
54C2×C191#45411yescharacteristic
55C2×C191#45512yes—
56D191—82no—
57C191#166211yescharacteristic, complemented, Sylow
58#1662—1911nomaximal, Sylow
59#562—1911no—
60#491—1911no—
61#489—1912no—
62#330D19111yescharacteristic, complemented
63#329—1912no—
64C4⋊Q8—1911no—
65#125C2×D19111yescharacteristic
66#124—1912no—
67#123—3822no—
68C22⋊Q8—3822no—
69#121—3822no—
70#103—3822no—
71#103—3822no—
72#102—3822no—
73#97—1911no—
74#97C2×D19112yes—
75C2×Q8—3821no—
76C2×Q8—1912no—
77C2×D4—3822no—
78C2×D4—1912no—
79C22×C4—3822no—
80C2×C8—22no—
81C2×C8—3821no—
82C4⋊C4—22no—
83C4⋊C4—3822no—
84C4⋊C4—3822no—
85C4⋊C4D76412yes—
86C22⋊C4—3822no—
87C22⋊C4—3822no—
88C22⋊C4—3822no—
89C42C22×D19111yescharacteristic
90C23—3822no—
91Q8—7642no—
92Q8—7642no—
93D4—15282no—
94D4—7642no—
95C2×C4—22no—
96C2×C4—3822no—
97C2×C4—15282no—
98C2×C4—7641no—
99C2×C4D4×D19112yes—
100C2×C4C2×D76411yescharacteristic
101C8—42no—
102C8—7641no—
103C22—7642no—
104C22—7642no—
105C22—7642no—
106C22#2025929211yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—15281no—
111C4—7642no—
112C2—15282no—
113C2#2713283811yescharacteristic
114C2#2713284212yes—
subgroupG/H
centerC22#20259292
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#330×C191C2
socleC22×C191#120
prime radical#27133411C2
hypercenter#330D191
hypocenterC191#1662
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #27133033C2semidirect
3. #27133023C2semidirect
4. #27132402C2semidirect
5. #330×C191C2semidirect
57. C191#1662semidirect
62. #330D191semidirect

Maximal subgroups

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

subgroupindex
#271334112
#271330332
#271330232
#271324022
#330×C1912
#1662191