Finite Group Databasebeta

← #27271718

Subgroups — #27271718

Order 13568.

194 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 41 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#20276065C211yesmaximal, characteristic
2#20272659C211yesmaximal, characteristic
3C53×#1295C211yesmaximal, characteristic
4#16811254C411yescharacteristic
5#16810663—21no—
6#16810392C411yescharacteristic
7#16810383—21no—
8#16810349—21no—
9C2×#97×C53—21no—
10C53×#382C2211yescharacteristic
11C2×#11719917—21no—
12#13133216—41no—
13#13133208—41no—
14#13133196—41no—
15#13133176—21no—
16#13133150—41no—
17#13133148—41no—
18C22⋊#88322—41no—
19C22×C8×C53—41no—
20C2×C4⋊C4×C53C2×C411yescharacteristic
21C2×C4⋊C4×C53D411yescharacteristic
22C2×C42×C53D411yescharacteristic
23#97×C53—22no—
24#97×C53—41no—
25#95×C53—41no—
26C22×C4×C53—21no—
27C22×C4×C53—21no—
28C22×C4×C53—21no—
29C22×C4×C53C22⋊C411yescharacteristic
30C2×Dic106—41no—
31C2×C8×C53—42no—
32C2×C8×C53—42no—
33C2×C8×C53—42no—
34C4⋊C4×C53—41no—
35C4⋊C4×C53—41no—
36C42×C53—21no—
37C42×C53SD1611yescharacteristic
38C42×C53Q1611yescharacteristic
39C22⋊Dic53—41no—
40C4⋊Dic53—21no—
41C4⋊Dic53—21no—
42C4⋊Dic53—41no—
43#11719919—42no—
44C4×Dic53—41no—
45#11719917—22no—
46#11719917—41no—
47C2×#88322—41no—
48C2×#88322—81no—
49C23×C53#9911yescharacteristic
50C2×C4×C53—21no—
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—21no—
54C2×C4×C53—41no—
55C2×C4×C53—41no—
56C2×C4×C53—41no—
57C2×C4×C53—41no—
58C2×C4×C53—81no—
59C2×C4×C53—81no—
60C2×C4×C53#10311yescharacteristic
61C2×C4×C53#10411yescharacteristic
62C2×Dic53—41no—
63C2×Dic53—41no—
64Dic106—161no—
65C8×C53—44no—
66#88322—42no—
67#88322—161no—
68#7442—531nomaximal, Sylow
69C22×C53—21no—
70C22×C53—21no—
71C22×C53#32811yescharacteristic
72C22×C53AΣL(1,𝔽16)11yescharacteristic
73C22×C53#35611yescharacteristic
74C4×C53—21no—
75C4×C53—21no—
76C4×C53—41no—
77C4×C53—41no—
78C4×C53—81no—
79C4×C53—81no—
80Dic53—81no—
81Dic53—81no—
82#2061—531no—
83#1324—531no—
84#1295D5311yescharacteristic
85C2×C53—21no—
86C2×C53—21no—
87C2×C53#135411yescharacteristic
88C2×C53#140311yescharacteristic
89C2×C53#140411yescharacteristic
90#544—531no—
91#428—1061no—
92C2×#97—21no—
93#382C2×D5311yescharacteristic
94#343—531no—
95#339—1061no—
96#324—1061no—
97C53#744211yescharacteristic, complemented, Sylow
98C2×M16—1061no—
99C22×C8—41no—
100C4⋊Q8—2121no—
101#125—2121no—
102C22⋊Q8—2121no—
103#117—1061no—
104C2×C4⋊C4C4×D5311yescharacteristic
105C2×C4⋊C4D21211yescharacteristic
106C2×C42C22⋊D5311yescharacteristic
107#107—2121no—
108#106—2121no—
109C22⋊C8—2121no—
110#97—22no—
111#97—41no—
112#95—41no—
113C2×Q8—2121no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4#1171992111yescharacteristic
118M16—1062no—
119M16—2121no—
120C2×C8—42no—
121C2×C8—42no—
122C2×C8—42no—
123C2×C8—4241no—
124C2×C8—2121no—
125C4⋊C4—41no—
126C4⋊C4—41no—
127C4⋊C4—1061no—
128C4⋊C4—1061no—
129C4⋊C4—2121no—
130C4⋊C4—2122no—
131C22⋊C4—2121no—
132C42—21no—
133C42—2121no—
134C42#1171992311yescharacteristic
135C42#1171992511yescharacteristic
136C23#1313313111yescharacteristic
137Q8—8481no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—81no—
147C2×C4—81no—
148C2×C4—2121no—
149C2×C4—2121no—
150C2×C4#1313314011yescharacteristic
151C2×C4#1313314511yescharacteristic
152C8—44no—
153C8—8481no—
154C8—2122no—
155C22—21no—
156C22—21no—
157C22#1681035911yescharacteristic
158C22#1681041111yescharacteristic
159C22#1681042111yescharacteristic
160C4—21no—
161C4—21no—
162C4—41no—
163C4—41no—
164C4—81no—
165C4—81no—
166C4—4241no—
167C4—4241no—
168C2—21no—
169C2—21no—
170C2#2027274411yescharacteristic
171C2#2027286411yescharacteristic
172C2#2027286711yescharacteristic
subgroupG/H
centerC22#16810411
commutatorC2×C4⋊C4×C53C2×C4
Frattini#382C2×D53
FittingC53×#1295C2
socleC22×C53AΣL(1,𝔽16)
prime radicalC23×C53#99
cosocleC53×#382C22
hypercenter#1295D53
hypocenterC53#7442
LCSG ▷ C2×C4⋊C4×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #382 ◁ #1295

Complements

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

NH ≅ G/Ntype
97. C53#7442semidirect

Maximal subgroups

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

subgroupindex
#202760652
#202726592
C53×#12952
#744253