Finite Group Databasebeta

← #34060970

Subgroups — #34060970

Order 19072.

270 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 54 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#23750604C212yesmaximal, complemented
2#23750529C214yesmaximal, complemented
3#23750192C211yesmaximal, characteristic
4#18011990—22no—
5#18011970C2214yescomplemented
6#18011969—22no—
7C2×#14558736C2211yescharacteristic
8#18011907C2212yes—
9#18011898C412yescomplemented
10#18011882—22no—
11#18011880—22no—
12#18011870C411yescharacteristic, complemented
13C22×D4×C149C411yescharacteristic, complemented
14C4⋊#10597010—42no—
15C22⋊#10597010—24no—
16C4×#10597010—42no—
17#14558736—42no—
18C2×Dic298C2×C411yescharacteristic
19C2×D596C2311yescharacteristic
20C2×C22⋊D149—24no—
21C2×C22⋊D149D412yes—
22C2×C4×D149D412yes—
23C4⋊Dic149—21no—
24C22⋊Dic149—24no—
25C22⋊Dic149—21no—
26C4×Dic149C2×C412yes—
27C24×C149—21no—
28C2×D4×C149—21no—
29C2×D4×C149—42no—
30C2×D4×C149C2×C412yes—
31C22×C4×C149C2×C411yescharacteristic
32C2×#10597010—44no—
33C22×D149C2×D412yes—
34Dic298—41no—
35D596—41no—
36C22⋊D149—42no—
37C22⋊D149—84no—
38C2×Dic149—22no—
39C2×Dic149C22⋊C412yes—
40C4×D149—22no—
41C4×D149—22no—
42C23×C149—22no—
43C23×C149—42no—
44C23×C149—82no—
45C23×C149C22⋊C411yescharacteristic
46D4×C149—42no—
47D4×C149—42no—
48C2×C4×C149—22no—
49C2×C4×C149C22⋊C411yescharacteristic
50C2×C4×C149C22×C411yescharacteristic
51C22×C149—21no—
52C22×C149—41no—
53C22×C149—44no—
54C22×C149—42no—
55C22×C149—84no—
56C22×C149C2×C22⋊C411yescharacteristic
57C2×D149—22no—
58C2×D149—42no—
59#10597010—84no—
60C4×C149—21no—
61C4×C149#9912yes—
62Dic149—22no—
63Dic149—42no—
64C2×C149—21no—
65C2×C149—41no—
66C2×C149—82no—
67C2×C149C2×#9911yescharacteristic
68D149—42no—
69C149#212811yescharacteristic, complemented, Sylow
70#2128—1491nomaximal, Sylow
71#535—1491no—
72#410—1492no—
73#353—1494no—
74C2×#41—1491no—
75C22×D4#1059701011yescharacteristic, complemented
76#127—1492no—
77#124—1492no—
78#121—2982no—
79#120—2982no—
80#117—2982no—
81#117—1491no—
82#99—2982no—
83#99—1494no—
84C24—21no—
85#41—5962no—
86C2×Q8—1491no—
87C2×D4—21no—
88C2×D4—42no—
89C2×D4—2984no—
90C2×D4—1492no—
91C2×D4—1491no—
92C2×D4C2×#1059701012yes—
93C22×C4—1492no—
94C22×C4C2×#1059701011yescharacteristic
95C4⋊C4—2981no—
96C4⋊C4—5962no—
97C22⋊C4—2981no—
98C22⋊C4—2984no—
99C22⋊C4—2984no—
100C42—5962no—
101C42—1492no—
102C23—22no—
103C23—42no—
104C23—82no—
105C23—1492no—
106C23C22⋊#1059701011yescharacteristic
107Q8—5961no—
108D4—42no—
109D4—42no—
110D4—5961no—
111D4—5962no—
112D4—11924no—
113C2×C4—22no—
114C2×C4—2982no—
115C2×C4—2982no—
116C2×C4—2982no—
117C2×C4—5964no—
118C2×C4—1492no—
119C2×C4C22⋊#1059701011yescharacteristic
120C2×C4C22×#1059701011yescharacteristic
121C22—21no—
122C22—41no—
123C22—44no—
124C22—42no—
125C22—84no—
126C22—2982no—
127C22—5962no—
128C22C2×C22⋊#1059701011yescharacteristic
129C4—21no—
130C4—2982no—
131C4—5962no—
132C4—11924no—
133C4#1801196912yes—
134C2—21no—
135C2—41no—
136C2—82no—
137C2—5962no—
138C2C2×#1801196911yescharacteristic
subgroupG/H
centerC2C2×#18011969
commutatorC2×C4×C149C22×C4
FrattiniC2×C4C22×#10597010
FittingC22×D4×C149C4
socleC2×C149C2×#99
prime radical#23750192C2
cosocleC2×D596C23
hypercenterC22×D4#10597010
hypocenterC149#2128
LCSG ▷ C2×C4×C149 ▷ C22×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #23750604C2semidirect
2. #23750529C2semidirect
5. #18011970C22semidirect
9. #18011898C4semidirect
12. #18011870C4semidirect
13. C22×D4×C149C4semidirect
69. C149#2128semidirect
75. C22×D4#10597010semidirect

Maximal subgroups

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

subgroupindex
#237506042
#237505292
#237501922
#2128149