Finite Group Databasebeta

← #28299116

Subgroups — #28299116

Order 13568.

194 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 36 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#20289656C211yesmaximal, characteristic
2#20289591C211yesmaximal, characteristic
3#20276062C211yesmaximal, characteristic
4C2×#13133272—21no—
5#16811468—21no—
6#16810666—21no—
7#16810488C2211yescharacteristic
8#16810392C411yescharacteristic
9#16810382—21no—
10C53×#544C411yescharacteristic
11C22×#88324—41no—
12C22⋊#88324—41no—
13#13133272—22no—
14#13133272—41no—
15C2×#11719917—21no—
16C2×C4⋊Dic53D411yescharacteristic
17C2×C4×Dic53D411yescharacteristic
18#13133150—41no—
19#13133148—41no—
20#13133124—41no—
21C4⋊Q8×C53—41no—
22#125×C53—41no—
23C22⋊Q8×C53—41no—
24#117×C53—21no—
25C2×C4⋊C4×C53C2×C411yescharacteristic
26C2×Q8×C53—41no—
27C22×C4×C53—21no—
28C22×C4×C53C22⋊C411yescharacteristic
29C22×Dic53—21no—
30C22×Dic53—21no—
31C2×#88324—42no—
32C2×#88324—42no—
33C2×#88324—42no—
34C2×#88324—81no—
35C4⋊C4×C53—21no—
36C4⋊C4×C53—21no—
37C4⋊C4×C53—41no—
38C4⋊C4×C53—42no—
39C4⋊C4×C53—41no—
40C22⋊C4×C53—41no—
41C42×C53—41no—
42C4⋊Dic53—41no—
43C4×Dic53—21no—
44C4×Dic53SD1611yescharacteristic
45C4×Dic53Q1611yescharacteristic
46#11719917—22no—
47#11719917—41no—
48C2×#88322—41no—
49C23×C53#9911yescharacteristic
50Q8×C53—161no—
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—41no—
54C2×C4×C53—41no—
55C2×C4×C53—41no—
56C2×C4×C53—81no—
57C2×C4×C53#10311yescharacteristic
58C2×C4×C53#10411yescharacteristic
59C2×Dic53—21no—
60C2×Dic53—21no—
61C2×Dic53—41no—
62C2×Dic53—41no—
63C2×Dic53—41no—
64C2×Dic53—81no—
65#88324—44no—
66#88324—161no—
67#88322—42no—
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—81no—
77C4×C53—81no—
78C4×C53—81no—
79Dic53—41no—
80Dic53—41no—
81Dic53—81no—
82#2061—531no—
83#1324—531no—
84#1295—531no—
85C2×C53—21no—
86C2×C53—21no—
87C2×C53#135411yescharacteristic
88C2×C53#140311yescharacteristic
89C2×C53#140411yescharacteristic
90#544#615711yescharacteristic
91#428—1061no—
92C2×#97—1061no—
93#382—531no—
94#343—531no—
95#339—1061no—
96#324—1061no—
97C53#744211yescharacteristic, complemented, Sylow
98C2×M16—1061no—
99C22×C8—2121no—
100C4⋊Q8—41no—
101#125—41no—
102C22⋊Q8—41no—
103#117—21no—
104C2×C4⋊C4—531no—
105C2×C4⋊C4C2×#615711yescharacteristic
106C2×C42—531no—
107#107—2121no—
108#106—2121no—
109C22⋊C8—2121no—
110#97—1062no—
111#97—2121no—
112#95—2121no—
113C2×Q8—41no—
114C22×C4—21no—
115C22×C4—1061no—
116C22×C4—1061no—
117C22×C4C22⋊#615711yescharacteristic
118M16—1062no—
119M16—2121no—
120C2×C8—4241no—
121C2×C8—2121no—
122C2×C8—2122no—
123C2×C8—2122no—
124C2×C8—2122no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C4⋊C4—42no—
130C4⋊C4—2121no—
131C22⋊C4—41no—
132C42—41no—
133C42—531no—
134C42—531no—
135C42—1061no—
136C23#1313327511yescharacteristic
137Q8—161no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—81no—
144C2×C4—1061no—
145C2×C4—1061no—
146C2×C4—4241no—
147C2×C4—2121no—
148C2×C4—2121no—
149C2×C4—2121no—
150C2×C4#1313328011yescharacteristic
151C2×C4#1313328211yescharacteristic
152C8—8481no—
153C8—2122no—
154C8—2124no—
155C22—21no—
156C22—21no—
157C22#1681147711yescharacteristic
158C22#1681152311yescharacteristic
159C22#1681153311yescharacteristic
160C4—21no—
161C4—21no—
162C4—81no—
163C4—81no—
164C4—81no—
165C4—4241no—
166C4—2121no—
167C4—2121no—
168C2—21no—
169C2—21no—
170C2#2028971611yescharacteristic
171C2#2028983611yescharacteristic
172C2#2028983811yescharacteristic
subgroupG/H
centerC22#16811523
commutatorC2×C4⋊C4×C53C2×C4
FrattiniC2×C4⋊C4C2×#6157
FittingC53×#544C4
socleC22×C53AΣL(1,𝔽16)
prime radicalC23×C53#99
cosocle#16810488C22
hypercenter#544#6157
hypocenterC53#7442
LCSG ▷ C2×C4⋊C4×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4 ◁ #544

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
#202896562
#202895912
#202760622
#744253