Finite Group Databasebeta

← #34176116

Subgroups — #34176116

Order 20224.

138 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 25 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#23762721C211yesmaximal, characteristic
2#23761573C211yesmaximal, characteristic, complemented
3C79×#1356C211yesmaximal, characteristic, complemented
4#19099311C411yescharacteristic
5#19098925—21no—
6#19098362—21no—
7#19098305C411yescharacteristic
8#19098302—21no—
9#491×C79C2211yescharacteristic
10#347×C79—21no—
11#14579149—21no—
12#14579146—41no—
13#14579139—41no—
14#14579136—41no—
15#14579084—41no—
16#14579081—41no—
17#14579080—41no—
18#14579072—21no—
19C4⋊Q8×C79D411yescharacteristic
20#125×C79C2×C411yescharacteristic
21M32×C79—41no—
22#103×C79—41no—
23#97×C79D411yescharacteristic
24C2×Dic158—21no—
25C2×D316—21no—
26C2×D316—41no—
27C2×C4×D79—41no—
28#11929608—81no—
29C2×C79⋊C8—41no—
30Dic79⋊C4—41no—
31(C2×D79)⋊C4—41no—
32(C2×D79)⋊C4—41no—
33(C2×D79)⋊C4—41no—
34C2×Q8×C79—21no—
35C2×C8×C79—21no—
36C4⋊C4×C79—21no—
37C4⋊C4×C79—41no—
38C4⋊C4×C79—41no—
39C42×C79C22⋊C411yescharacteristic
40C16×C79—42no—
41C22×D79—41no—
42Q8×C79—81no—
43C2×C4×C79—21no—
44C2×C4×C79—41no—
45C2×C4×C79—41no—
46C2×C4×C79#9911yescharacteristic
47C2×Dic79—41no—
48D316—81no—
49D316—161no—
50C4×D79—161no—
51Dic158—81no—
52C8×C79—41no—
53C79⋊C8—81no—
54C79⋊C8—161no—
55C22×C79AΣL(1,𝔽16)11yescharacteristic
56C2×D79—81no—
57C2×D79—81no—
58C2×D79—81no—
59C4×C79—21no—
60C4×C79—41no—
61C4×C79—81no—
62C4×C79—81no—
63Dic79—81no—
64#7525—791nomaximal, Sylow
65C2×C79—21no—
66C2×C79#140411yescharacteristic
67D79—161no—
68#1662—791no—
69#1413—791no—
70#1356D7911yescharacteristic, complemented
71C79#752511yescharacteristic, complemented, Sylow
72#562—791no—
73#491C2×D7911yescharacteristic
74#489—1581no—
75#356—1581no—
76#347—21no—
77#330—791no—
78#329—1581no—
79C4⋊Q8D31611yescharacteristic
80#125C4×D7911yescharacteristic
81#124—1581no—
82#123—3161no—
83C22⋊Q8—3161no—
84#121—3161no—
85M32—41no—
86#103—41no—
87#103—3161no—
88#102—3161no—
89#101—3161no—
90#97—1581no—
91#97C22⋊D7911yescharacteristic
92C2×Q8—21no—
93C2×Q8—1581no—
94C2×D4—3161no—
95C2×D4—1581no—
96C22×C4—3161no—
97M16—6321no—
98C2×C8—21no—
99C2×C8—3161no—
100C4⋊C4—21no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—3161no—
104C22⋊C4—3161no—
105C22⋊C4—3161no—
106C22⋊C4—3161no—
107C42(C2×D79)⋊C411yescharacteristic
108C16—42no—
109C23—3161no—
110Q8—81no—
111Q8—6321no—
112D4—6321no—
113D4—12641no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—3161no—
118C2×C4—12641no—
119C2×C4#1457907511yescharacteristic
120C8—41no—
121C8—6321no—
122C8—12641no—
123C22—6321no—
124C22—6321no—
125C22—6321no—
126C22#1909835111yescharacteristic
127C4—21no—
128C4—41no—
129C4—81no—
130C4—81no—
131C4—6321no—
132C2—21no—
133C2—12641no—
134C2#2376155211yescharacteristic
subgroupG/H
centerC2#23761552
commutator#125×C79C2×C4
Frattini#491C2×D79
FittingC79×#1356C2
socleC2×C79#1404
prime radical#19099311C4
cosocle#491×C79C22
hypercenter#1356D79
hypocenterC79#7525
LCSG ▷ #125×C79 ▷ C42×C79 ▷ C2×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #491 ◁ #1356

Complements

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

NH ≅ G/Ntype
2. #23761573C2semidirect
3. C79×#1356C2semidirect
70. #1356D79semidirect
71. C79#7525semidirect

Maximal subgroups

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

subgroupindex
#237627212
#237615732
C79×#13562
#752579