Finite Group Databasebeta

← #37571824

Subgroups — #37571824

Order 24448.

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

normal subgroups only

This group has 228 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#27133405C211yesmaximal, characteristic
2#27133025C212yesmaximal, complemented
3#27132403C211yesmaximal, characteristic, complemented
4#27132402C212yesmaximal, complemented
5#497×C191C211yesmaximal, characteristic, complemented
6#20259311—21no—
7#20259310C2212yes—
8#20259308—22no—
9#20259292—22no—
10#20259290—21no—
11#20259245—22no—
12#20259242—22no—
13#20259233C2212yescomplemented
14C2×Q16×C191—21no—
15C2×SD16×C191—21no—
16C4⋊Q8×C191C2211yescharacteristic, complemented
17#124×C191C2211yescharacteristic
18#97×C191C2211yescharacteristic, complemented
19C23×D191—21no—
20C2×Dic382D412yes—
21C2×D764—21no—
22C2×D764D412yes—
23C2×C22⋊D191—22no—
24C2×C191⋊C8—22no—
25C22⋊Dic191—21no—
26(C2×D191)⋊C4—22no—
27(C2×D191)⋊C4—22no—
28(C2×D191)⋊C4—22no—
29C4×Dic191—21no—
30C2×Q8×C191—21no—
31C2×Q8×C191D411yescharacteristic
32C2×D4×C191D411yescharacteristic
33Q16×C191—42no—
34SD16×C191—42no—
35C2×C8×C191—21no—
36C4⋊C4×C191—22no—
37C22⋊C4×C191—22no—
38C42×C191C2311yescharacteristic
39C22×D191—22no—
40C22×D191—81no—
41C22×D191—82no—
42Dic382—42no—
43D764—42no—
44D764—81no—
45C22⋊D191—82no—
46C2×Dic191—22no—
47C191⋊C8—42no—
48C23×C191—21no—
49Q8×C191—41no—
50Q8×C191—41no—
51Q8×C191—41no—
52D4×C191—41no—
53C2×C4×C191—21no—
54C2×C4×C191—41no—
55C2×C4×C191C2×D412yes—
56C2×C4×C191C2×D411yescharacteristic
57C8×C191—22no—
58C22×C191—41no—
59C22×C191—42no—
60C22×C191#12011yescharacteristic
61C2×D191—42no—
62C2×D191—42no—
63C2×D191—42no—
64C2×D191—84no—
65C4×C191—21no—
66C4×C191—22no—
67C4×C191—41no—
68C4×C191—81no—
69Dic191—42no—
70C2×C191—81no—
71C2×C191#45311yescharacteristic
72C2×C191#45512yes—
73D191—82no—
74C191#165911yescharacteristic, complemented, Sylow
75#1659—1911nomaximal, Sylow
76H(3,𝔽4)—1911no—
77#497D19111yescharacteristic, complemented
78#489—1912no—
79#329—1911no—
80#329—1912no—
81C2×Q16—21no—
82C2×SD16—21no—
83C4⋊Q8C2×D19111yescharacteristic, complemented
84#124—3822no—
85#124—3821no—
86#124—1912no—
87#124C2×D19111yescharacteristic
88#120—3822no—
89#120—3821no—
90#103—3822no—
91#102—3822no—
92#97—1912no—
93#97C2×D19111yescharacteristic, complemented
94C24—3821no—
95C2×Q8—21no—
96C2×Q8—1912no—
97C2×Q8C22⋊D19111yescharacteristic
98C2×D4—3821no—
99C2×D4—3822no—
100C2×D4—1912no—
101C2×D4C22⋊D19111yescharacteristic
102Q16—42no—
103SD16—42no—
104C2×C8—21no—
105C2×C8—3822no—
106C4⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—3822no—
109C22⋊C4—3821no—
110C22⋊C4—3822no—
111C22⋊C4—3822no—
112C42—3821no—
113C42C22×D19111yescharacteristic
114C23—21no—
115C23—3822no—
116C23—15281no—
117C23—15282no—
118Q8—41no—
119Q8—41no—
120Q8—41no—
121Q8—7642no—
122D4—41no—
123D4—15281no—
124D4—15282no—
125D4—7642no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—3822no—
129C2×C4D4×D19112yes—
130C2×C4C2×C22⋊D19111yescharacteristic
131C8—22no—
132C8—7642no—
133C22—41no—
134C22—42no—
135C22—15284no—
136C22—7642no—
137C22—7642no—
138C22—7642no—
139C22#2025929011yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—81no—
144C4—7642no—
145C2—81no—
146C2—15282no—
147C2#2713283011yescharacteristic
148C2#2713283912yes—
subgroupG/H
centerC22#20259290
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#497×C191C2
socleC22×C191#120
prime radical#27133405C2
hypercenter#497D191
hypocenterC191#1659
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #27133025C2semidirect
3. #27132403C2semidirect
4. #27132402C2semidirect
5. #497×C191C2semidirect
13. #20259233C22semidirect
16. C4⋊Q8×C191C22semidirect
18. #97×C191C22semidirect
74. C191#1659semidirect
77. #497D191semidirect
83. C4⋊Q8C2×D191semidirect
93. #97C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271334052
#271330252
#271324032
#271324022
#497×C1912
#1659191