Finite Group Databasebeta

← #28299030

Subgroups — #28299030

Order 13568.

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

normal subgroups only

This group has 252 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#20289812C211yesmaximal, characteristic
2#20289646C211yesmaximal, characteristic
3#20275817C211yesmaximal, characteristic
4C2×#13133278—21no—
5#16811468—21no—
6C22×Dic212—21no—
7C2×#13133217C411yescharacteristic
8C2×#13133142—21no—
9#16810529C2211yescharacteristic
10C53×#336C411yescharacteristic
11C2×#11719940—41no—
12#13133278—22no—
13#13133278—41no—
14C22⋊#88324—41no—
15C22×Dic106—21no—
16C22×Dic106D411yescharacteristic
17C2×Dic212—42no—
18C2×Dic212—42no—
19C2×Dic212—41no—
20C2×Dic212—41no—
21#13133217—42no—
22C2×C4⋊Dic53—41no—
23C2×C4⋊Dic53D411yescharacteristic
24#13133142—42no—
25#13133124—41no—
26C22×C8×C53—21no—
27C2×C42×C53C2×C411yescharacteristic
28C22×C4×C53—21no—
29C22×C4×C53C22⋊C411yescharacteristic
30C22×Dic53—21no—
31C22×Dic53—21no—
32C22×Dic53—41no—
33C2×Dic106—22no—
34C2×Dic106—21no—
35C2×Dic106—41no—
36C2×Dic106—41no—
37C2×Dic106—41no—
38C2×Dic106—41no—
39C2×Dic106—41no—
40C2×Dic106—41no—
41C2×Dic106—41no—
42C2×Dic106D811yescharacteristic
43C2×Dic106SD1611yescharacteristic
44#11719940—42no—
45#11719940—81no—
46C2×#88324—81no—
47C2×#88324—81no—
48C2×C8×C53—22no—
49C2×C8×C53—41no—
50C2×C8×C53—41no—
51C42×C53—41no—
52C4⋊Dic53—42no—
53C4⋊Dic53—41no—
54C4⋊Dic53—42no—
55Dic212—44no—
56Dic212—44no—
57C23×C53#9911yescharacteristic
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—22no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53#10211yescharacteristic
64C2×C4×C53#10411yescharacteristic
65C2×Dic53—21no—
66C2×Dic53—21no—
67C2×Dic53—41no—
68C2×Dic53—41no—
69C2×Dic53—42no—
70C2×Dic53—41no—
71C2×Dic53—81no—
72C2×Dic53—81no—
73C2×Dic53—81no—
74Dic106—41no—
75Dic106—41no—
76Dic106—42no—
77Dic106—42no—
78Dic106—81no—
79Dic106—81no—
80Dic106—81no—
81#88324—82no—
82#88324—161no—
83C8×C53—42no—
84#7405—531nomaximal, Sylow
85C22×C53—21no—
86C22×C53—21no—
87C22×C53#32711yescharacteristic
88C22×C53AΣL(1,𝔽16)11yescharacteristic
89C22×C53#35311yescharacteristic
90C4×C53—21no—
91C4×C53—21no—
92C4×C53—42no—
93Dic53—41no—
94Dic53—41no—
95Dic53—82no—
96Dic53—81no—
97#2016—531no—
98#1394—531no—
99#1319—531no—
100C2×C53—21no—
101C2×C53—21no—
102C2×C53#134311yescharacteristic
103C2×C53#140711yescharacteristic
104C2×C53#141111yescharacteristic
105C22×Q16—1061no—
106C2×C4⋊Q8—531no—
107C2×#103—1061no—
108C2×#101—1061no—
109#391—531no—
110#336#615711yescharacteristic
111#324—1061no—
112C53#740511yescharacteristic, complemented, Sylow
113C22×Q8—531no—
114C22×Q8—1061no—
115C2×Q16—2122no—
116C2×Q16—2121no—
117C2×Q16—2121no—
118C2×Q16—2122no—
119C2×M16—2121no—
120C22×C8—21no—
121C4⋊Q8—2122no—
122C2×C4⋊C4—531no—
123C2×C4⋊C4—2121no—
124C2×C42C2×#615711yescharacteristic
125#103—2122no—
126#101—1062no—
127#101—2121no—
128C22⋊C8—2121no—
129#95—2121no—
130C2×Q8—531no—
131C2×Q8—531no—
132C2×Q8—1062no—
133C2×Q8—1061no—
134C2×Q8—2121no—
135C2×Q8—2121no—
136C2×Q8—2121no—
137C2×Q8—2121no—
138C2×Q8—2121no—
139C2×Q8—2121no—
140C2×Q8—2121no—
141C22×C4—21no—
142C22×C4—1061no—
143C22×C4—1061no—
144C22×C4—2121no—
145C22×C4C22⋊#615711yescharacteristic
146Q16—2124no—
147Q16—2124no—
148M16—4241no—
149M16—2122no—
150C2×C8—22no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—4241no—
154C2×C8—4241no—
155C4⋊C4—2121no—
156C4⋊C4—2122no—
157C4⋊C4—2122no—
158C42—41no—
159C23#1313327611yescharacteristic
160Q8—4241no—
161Q8—4241no—
162Q8—4241no—
163Q8—2122no—
164Q8—2122no—
165Q8—2121no—
166Q8—2121no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—1061no—
173C2×C4—1061no—
174C2×C4—4241no—
175C2×C4—4241no—
176C2×C4—4241no—
177C2×C4—2122no—
178C2×C4—2121no—
179C2×C4—2121no—
180C2×C4—2121no—
181C2×C4#1313327911yescharacteristic
182C2×C4#1313328211yescharacteristic
183C8—42no—
184C8—8481no—
185C8—4242no—
186C22—21no—
187C22—21no—
188C22#1681147511yescharacteristic
189C22#1681152411yescharacteristic
190C22#1681152811yescharacteristic
191C4—21no—
192C4—21no—
193C4—42no—
194C4—4241no—
195C4—4242no—
196C4—2121no—
197C4—2121no—
198C2—21no—
199C2—21no—
200C2#2028969411yescharacteristic
201C2#2028984511yescharacteristic
202C2#2028985311yescharacteristic
subgroupG/H
centerC22#16811528
commutatorC2×C42×C53C2×C4
FrattiniC2×C42C2×#6157
FittingC53×#336C4
socleC22×C53#353
prime radicalC23×C53#99
cosocle#16810529C22
hypercenter#336#6157
hypocenterC53#7405
LCSG ▷ C2×C42×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #336

Complements

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

NH ≅ G/Ntype
112. C53#7405semidirect

Maximal subgroups

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

subgroupindex
#202898122
#202896462
#202758172
#740553