Finite Group Databasebeta

← #27271924

Subgroups — #27271924

Order 13568.

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

normal subgroups only

This group has 156 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#20273791C211yesmaximal, characteristic
2#20272880C211yesmaximal, characteristic
3C53×#1359C211yesmaximal, characteristic
4#16811353C411yescharacteristic, complemented
5#16810820—21no—
6#16810416—21no—
7#16810372C411yescharacteristic, complemented
8#16810355—21no—
9C53×#492C2211yescharacteristic
10C53×#364—21no—
11C2×#11719923—41no—
12#13133216—41no—
13#13133215—41no—
14C4×Dic106—21no—
15#13133182—41no—
16C4⋊#88322—21no—
17#13133138—41no—
18#13133132—41no—
19C2×SD16×C53—41no—
20C4⋊Q8×C53C2×C411yescharacteristic
21#127×C53D411yescharacteristic
22M32×C53—41no—
23C4×C8×C53D411yescharacteristic
24C2×Q8×C53—21no—
25C2×D4×C53—21no—
26C2×D4×C53—41no—
27C2×Dic106—21no—
28C2×Dic106—41no—
29SD16×C53—81no—
30SD16×C53—81no—
31C2×C8×C53—21no—
32C4⋊C4×C53—41no—
33C42×C53C22⋊C411yescharacteristic
34C22⋊Dic53—81no—
35#11719923—82no—
36C4⋊Dic53—21no—
37#11719919—42no—
38#11719919—41no—
39C4×Dic53—41no—
40C4×Dic53—41no—
41C2×#88322—41no—
42C16×C53—81no—
43C23×C53—41no—
44Q8×C53—81no—
45D4×C53—81no—
46D4×C53—81no—
47D4×C53—81no—
48C2×C4×C53—21no—
49C2×C4×C53—41no—
50C2×C4×C53#9911yescharacteristic
51C2×Dic53—41no—
52C2×Dic53—41no—
53C2×Dic53—161no—
54Dic106—22no—
55Dic106—41no—
56Dic106—161no—
57C8×C53—21no—
58C8×C53—21no—
59#88322—81no—
60#7520—531nomaximal, Sylow
61C22×C53—81no—
62C22×C53—161no—
63C22×C53AΣL(1,𝔽16)11yescharacteristic
64C4×C53—21no—
65C4×C53—21no—
66C4×C53—21no—
67C4×C53—81no—
68Dic53—42no—
69Dic53—81no—
70Dic53—321no—
71#1623—531no—
72#1409—531no—
73#1359D5311yescharacteristic
74C2×C53—21no—
75C2×C53—161no—
76C2×C53#140211yescharacteristic
77Q82—531no—
78#492C2×D5311yescharacteristic
79#460—1061no—
80#364—21no—
81#353—1061no—
82#333—531no—
83Q8⋊C8—1061no—
84C53#752011yescharacteristic, complemented, Sylow
85C2×SD16—41no—
86C2×SD16—2121no—
87C4⋊Q8—2121no—
88C4⋊Q8—2121no—
89C4⋊Q8C4×D5311yescharacteristic
90#127D21211yescharacteristic
91C4×Q8—1061no—
92C4×Q8—2121no—
93M32—41no—
94C4⋊C8—1061no—
95#102—2121no—
96#99—2121no—
97C4×C8C22⋊D5311yescharacteristic
98C2×Q8—21no—
99C2×Q8—1061no—
100C2×Q8—2121no—
101C2×D4—21no—
102C2×D4—41no—
103SD16—81no—
104SD16—81no—
105SD16—4242no—
106C2×C8—21no—
107C2×C8—2121no—
108C4⋊C4—41no—
109C4⋊C4—1061no—
110C4⋊C4—2122no—
111C4⋊C4—2121no—
112C22⋊C4—4241no—
113C42—2121no—
114C42—2121no—
115C42#1171992111yescharacteristic
116C16—81no—
117C23—41no—
118Q8—81no—
119Q8—8481no—
120Q8—1062no—
121Q8—2121no—
122D4—81no—
123D4—81no—
124D4—81no—
125C2×C4—21no—
126C2×C4—41no—
127C2×C4—8481no—
128C2×C4—2121no—
129C2×C4—2121no—
130C2×C4#1313313111yescharacteristic
131C8—21no—
132C8—21no—
133C8—4241no—
134C22—81no—
135C22—161no—
136C22#1681041111yescharacteristic
137C4—21no—
138C4—21no—
139C4—21no—
140C4—81no—
141C4—16961no—
142C4—4241no—
143C4—2122no—
144C2—21no—
145C2—161no—
146C2#2027286111yescharacteristic
subgroupG/H
centerC2#20272861
commutatorC4⋊Q8×C53C2×C4
Frattini#492C2×D53
FittingC53×#1359C2
socleC2×C53#1402
prime radical#127×C53D4
cosocleC53×#492C22
hypercenter#1359D53
hypocenterC53#7520
LCSG ▷ C4⋊Q8×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ #1359

Complements

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

NH ≅ G/Ntype
4. #16811353C4semidirect
7. #16810372C4semidirect
84. C53#7520semidirect

Maximal subgroups

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

subgroupindex
#202737912
#202728802
C53×#13592
#752053