Finite Group Databasebeta

← #34176108

Subgroups — #34176108

Order 20224.

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

normal subgroups only

This group has 158 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#23762692C211yesmaximal, characteristic
2#23761433C211yesmaximal, characteristic, complemented
3C79×#1412C211yesmaximal, characteristic, complemented
4#19099311C411yescharacteristic, complemented
5#19098950—21no—
6#19098341—21no—
7#19098305C411yescharacteristic, complemented
8#19098302—21no—
9#490×C79C2211yescharacteristic
10#353×C79—21no—
11C2×#11929612—41no—
12C2×#11929609—41no—
13#14579149—21no—
14#14579146—41no—
15#14579139—41no—
16#14579136—41no—
17#14579101—41no—
18#14579072—21no—
19#127×C79D411yescharacteristic
20#125×C79C2×C411yescharacteristic
21#102×C79—41no—
22#99×C79—41no—
23#97×C79D411yescharacteristic
24C2×Dic158—21no—
25C2×D316—21no—
26C2×D316—41no—
27C2×C4×D79—41no—
28#11929612—82no—
29#11929609—82no—
30C2×C79⋊C8—41no—
31Dic79⋊C4—41no—
32(C2×D79)⋊C4—41no—
33(C2×D79)⋊C4—41no—
34(C2×D79)⋊C4—41no—
35C79⋊C16—42no—
36C2×D4×C79—21no—
37C2×D4×C79—41no—
38C2×C8×C79—21no—
39C4⋊C4×C79—21no—
40C4⋊C4×C79—41no—
41C22⋊C4×C79—81no—
42C42×C79C22⋊C411yescharacteristic
43C23×C79—41no—
44C22×D79—41no—
45D4×C79—81no—
46D4×C79—81no—
47D4×C79—81no—
48C2×C4×C79—21no—
49C2×C4×C79—41no—
50C2×C4×C79—161no—
51C2×C4×C79#9911yescharacteristic
52C2×Dic79—41no—
53D316—81no—
54D316—161no—
55C4×D79—161no—
56Dic158—81no—
57C8×C79—41no—
58C79⋊C8—42no—
59C22×C79—81no—
60C22×C79—161no—
61C22×C79AΣL(1,𝔽16)11yescharacteristic
62C2×D79—81no—
63C2×D79—81no—
64C2×D79—81no—
65C4×C79—21no—
66C4×C79—41no—
67C4×C79—81no—
68C4×C79—321no—
69Dic79—81no—
70#7523—791nomaximal, Sylow
71C2×C79—21no—
72C2×C79—161no—
73C2×C79#140411yescharacteristic
74D79—161no—
75#1656—791no—
76#1412D7911yescharacteristic, complemented
77#1355—791no—
78C79#752311yescharacteristic, complemented, Sylow
79#562—791no—
80#496—1581no—
81#490C2×D7911yescharacteristic
82#353—21no—
83#347—1581no—
84#330—791no—
85#329—1581no—
86C2×SD16—3161no—
87C2×D8—3161no—
88#127C22⋊D7911yescharacteristic
89#125C4×D7911yescharacteristic
90#124—1581no—
91#123—3161no—
92C22⋊Q8—3161no—
93#121—3161no—
94M32—3161no—
95#102—41no—
96#99—41no—
97#97—1581no—
98#97D31611yescharacteristic
99C2×Q8—1581no—
100C2×D4—21no—
101C2×D4—41no—
102C2×D4—3161no—
103C2×D4—1581no—
104C22×C4—3161no—
105SD16—6322no—
106D8—6322no—
107C2×C8—21no—
108C2×C8—3161no—
109C4⋊C4—21no—
110C4⋊C4—41no—
111C4⋊C4—3161no—
112C22⋊C4—81no—
113C22⋊C4—3161no—
114C22⋊C4—3161no—
115C22⋊C4—3161no—
116C42(C2×D79)⋊C411yescharacteristic
117C16—3162no—
118C23—41no—
119C23—3161no—
120Q8—6321no—
121D4—81no—
122D4—81no—
123D4—81no—
124D4—6321no—
125D4—12641no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—161no—
129C2×C4—3161no—
130C2×C4—12641no—
131C2×C4#1457907511yescharacteristic
132C8—41no—
133C8—3162no—
134C22—81no—
135C22—161no—
136C22—6321no—
137C22—6321no—
138C22—6321no—
139C22#1909834911yescharacteristic
140C4—21no—
141C4—41no—
142C4—81no—
143C4—321no—
144C4—6321no—
145C2—21no—
146C2—161no—
147C2—12641no—
148C2#2376155011yescharacteristic
subgroupG/H
centerC2#23761550
commutator#125×C79C2×C4
Frattini#490C2×D79
FittingC79×#1412C2
socleC2×C79#1404
prime radical#23762692C2
cosocle#490×C79C22
hypercenter#1412D79
hypocenterC79#7523
LCSG ▷ #125×C79 ▷ C42×C79 ▷ C2×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #490 ◁ #1412

Complements

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

NH ≅ G/Ntype
2. #23761433C2semidirect
3. C79×#1412C2semidirect
4. #19099311C4semidirect
7. #19098305C4semidirect
76. #1412D79semidirect
78. C79#7523semidirect

Maximal subgroups

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

subgroupindex
#237626922
#237614332
C79×#14122
#752379