Finite Group Databasebeta

← #34044002

Subgroups — #34044002

Order 19072.

196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 49 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#23750035C212yesmaximal, complemented
2#23750020C211yesmaximal, characteristic
3#23749367C212yesmaximal, complemented
4#23749365C211yesmaximal, characteristic
5#563×C149C211yesmaximal, characteristic
6C2×#14558726—21no—
7C2×#14558724—21no—
8#18011911C2211yescharacteristic
9#18011844—22no—
10#18011842—22no—
11#18011822C2211yescharacteristic
12#18011822C2212yes—
13C4⋊Q8×C149—21no—
14#126×C149—22no—
15#125×C149C2212yes—
16#124×C149C2211yescharacteristic
17#123×C149—21no—
18C22⋊Q8×C149—22no—
19C2×Dic298—21no—
20#14558726—42no—
21#14558724—42no—
22C2×#11924746—21no—
23C2×#11924746—22no—
24C4⋊Dic149—22no—
25C2×Q8×C149—21no—
26C2×Q8×C149D411yescharacteristic
27C2×D4×C149D411yescharacteristic
28C22×C4×C149—21no—
29C4⋊C4×C149—22no—
30C4⋊C4×C149—22no—
31C4⋊C4×C149—22no—
32C4⋊C4×C149D412yes—
33C4⋊C4×C149D412yes—
34C22⋊C4×C149—22no—
35C22⋊C4×C149—21no—
36C22⋊C4×C149—22no—
37C42×C149—21no—
38C42×C149C2311yescharacteristic
39Dic298—41no—
40Dic298—41no—
41C2×Dic149—41no—
42#11924746—22no—
43#11924746—24no—
44C23×C149—21no—
45Q8×C149—41no—
46Q8×C149—81no—
47D4×C149—41no—
48C2×C4×C149—22no—
49C2×C4×C149—21no—
50C2×C4×C149—22no—
51C2×C4×C149—81no—
52C2×C4×C149C2×D411yescharacteristic
53C2×C4×C149C2×D412yes—
54C22×C149—41no—
55C22×C149—42no—
56C22×C149#12011yescharacteristic
57C4×C149—21no—
58C4×C149—22no—
59C4×C149—42no—
60C4×C149—41no—
61C4×C149—42no—
62Dic149—81no—
63C2×C149—81no—
64C2×C149#45311yescharacteristic
65C2×C149#45612yes—
66C149#166011yescharacteristic, complemented, Sylow
67#1660—1491nomaximal, Sylow
68#563D14911yescharacteristic
69#501—1492no—
70#497—1491no—
71#330—1492no—
72#329—1491no—
73C2×Q16—2981no—
74C2×SD16—2981no—
75C4⋊Q8—21no—
76C4⋊Q8—1491no—
77#126—22no—
78#125C2×D14912yes—
79#124C2×D14911yescharacteristic
80#123—21no—
81C22⋊Q8—22no—
82#107—2982no—
83#106—2982no—
84#97—1492no—
85#97—1491no—
86C2×Q8—21no—
87C2×Q8—2981no—
88C2×Q8C22⋊D14911yescharacteristic
89C2×D4C22⋊D14911yescharacteristic
90C22×C4—21no—
91Q16—5962no—
92SD16—5962no—
93C2×C8—2982no—
94C2×C8—2981no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—2982no—
99C4⋊C4C22⋊D14912yes—
100C4⋊C4C22⋊D14912yes—
101C22⋊C4—22no—
102C22⋊C4—21no—
103C22⋊C4—22no—
104C42—21no—
105C42C22×D14911yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—81no—
109Q8—5961no—
110Q8—5961no—
111D4—41no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—81no—
116C2×C4—5961no—
117C2×C4C2×C22⋊D14912yes—
118C2×C4C2×C22⋊D14911yescharacteristic
119C8—2984no—
120C8—2982no—
121C22—41no—
122C22—42no—
123C22#1801188011yescharacteristic
124C4—21no—
125C4—22no—
126C4—42no—
127C4—41no—
128C4—42no—
129C4—11921no—
130C2—81no—
131C2#2374979311yescharacteristic
132C2#2374980812yes—
subgroupG/H
centerC22#18011880
commutator, cosocleC42×C149C23
FrattiniC42C22×D149
Fitting#563×C149C2
socleC22×C149#120
prime radicalC2×D4×C149D4
hypercenter#563D149
hypocenterC149#1660
LCSG ▷ C42×C149 ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #23750035C2semidirect
3. #23749367C2semidirect
66. C149#1660semidirect

Maximal subgroups

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

subgroupindex
#237500352
#237500202
#237493672
#237493652
#563×C1492
#1660149