Finite Group Databasebeta

← #37571829

Subgroups — #37571829

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#27133413C211yesmaximal, characteristic
2#27133069C212yesmaximal, complemented
3#27132404C212yesmaximal, complemented
4#27132403C211yesmaximal, characteristic
5#497×C191C211yesmaximal, characteristic
6#20259320—21no—
7#20259315—22no—
8#20259314C2212yes—
9#20259303—21no—
10#20259299—22no—
11#20259255—22no—
12#20259253—22no—
13#20259233C2212yes—
14C2×Q16×C191—21no—
15C2×SD16×C191—21no—
16C4⋊Q8×C191C2211yescharacteristic
17#124×C191C2211yescharacteristic
18#97×C191C2211yescharacteristic
19C2×Dic382—21no—
20C22×Dic191—21no—
21C2×C191⋊C8—22no—
22C4⋊Dic191D412yes—
23C4⋊Dic191D412yes—
24Dic191⋊C4—22no—
25Dic191⋊C4—22no—
26Dic191⋊C4—22no—
27C22⋊Dic191—22no—
28C22⋊Dic191—21no—
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
39Dic382—81no—
40C2×Dic191—22no—
41C2×Dic191—22no—
42C2×Dic191—81no—
43C191⋊C8—24no—
44C23×C191—21no—
45Q8×C191—41no—
46Q8×C191—41no—
47Q8×C191—41no—
48D4×C191—41no—
49C2×C4×C191—21no—
50C2×C4×C191—41no—
51C2×C4×C191C2×D412yes—
52C2×C4×C191C2×D411yescharacteristic
53C8×C191—22no—
54C22×C191—41no—
55C22×C191—42no—
56C22×C191#12011yescharacteristic
57C4×C191—21no—
58C4×C191—22no—
59C4×C191—41no—
60C4×C191—81no—
61Dic191—42no—
62Dic191—42no—
63C2×C191—81no—
64C2×C191#45311yescharacteristic
65C2×C191#45612yes—
66C191#166011yescharacteristic, complemented, Sylow
67#1660—1911nomaximal, Sylow
68#563—1911no—
69#501—1912no—
70#497D19111yescharacteristic
71#330—1912no—
72#329—1911no—
73C2×Q16—21no—
74C2×SD16—21no—
75C4⋊Q8—3821no—
76C4⋊Q8C2×D19111yescharacteristic
77#126—3822no—
78#125—1912no—
79#124C2×D19111yescharacteristic
80#123—3821no—
81C22⋊Q8—3822no—
82#107—3822no—
83#106—3822no—
84#97—1912no—
85#97C2×D19111yescharacteristic
86C2×Q8—21no—
87C2×Q8—3821no—
88C2×Q8C22⋊D19111yescharacteristic
89C2×D4C22⋊D19111yescharacteristic
90C22×C4—3821no—
91Q16—42no—
92SD16—42no—
93C2×C8—21no—
94C2×C8—3822no—
95C4⋊C4—22no—
96C4⋊C4—3822no—
97C4⋊C4—3822no—
98C4⋊C4—3822no—
99C4⋊C4—1912no—
100C4⋊C4—1912no—
101C22⋊C4—22no—
102C22⋊C4—3822no—
103C22⋊C4—3821no—
104C42—3821no—
105C42C22×D19111yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—41no—
109Q8—41no—
110Q8—15281no—
111D4—41no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—3822no—
115C2×C4—3822no—
116C2×C4—15281no—
117C2×C4D4×D19112yes—
118C2×C4C2×C22⋊D19111yescharacteristic
119C8—22no—
120C8—3824no—
121C22—41no—
122C22—42no—
123C22#2025929011yescharacteristic
124C4—21no—
125C4—22no—
126C4—41no—
127C4—81no—
128C4—7642no—
129C4—7642no—
130C2—81no—
131C2#2713283011yescharacteristic
132C2#2713284412yes—
subgroupG/H
centerC22#20259290
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#497×C191C2
socleC22×C191#120
prime radicalC2×D4×C191D4
hypercenter#497D191
hypocenterC191#1660
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. #27133069C2semidirect
3. #27132404C2semidirect
66. C191#1660semidirect

Maximal subgroups

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

subgroupindex
#271334132
#271330692
#271324042
#271324032
#497×C1912
#1660191