Finite Group Databasebeta

← #28298522

Subgroups — #28298522

Order 13568.

276 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 52 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#20290616C211yesmaximal, characteristic
2#20289968C211yesmaximal, characteristic
3#20274482C211yesmaximal, characteristic
4#16811643—21no—
5#16811595—21no—
6#16811503—21no—
7#16811503C412yes—
8#16811469—21no—
9#16811467C412yes—
10C2×#13133174C2211yescharacteristic
11C2×#13133129—21no—
12C2×#13133128C411yescharacteristic
13#16810377—21no—
14C2×C4×C8×C53C411yescharacteristic
15C2×#11719940—22no—
16#13133284—41no—
17#13133274—41no—
18#13133274—41no—
19C22⋊#88324—22no—
20#13133272—41no—
21C22×C4×D53D411yescharacteristic
22C22×#88322—21no—
23#13133174—41no—
24#13133174—41no—
25C2×#11719919C2×C411yescharacteristic
26C2×#11719919D411yescharacteristic
27C2×#11719921C2×C411yescharacteristic
28C2×#11719921Q811yescharacteristic
29C2×C4×Dic53D411yescharacteristic
30#13133129—24no—
31#13133128—41no—
32C22×C8×C53—21no—
33C2×C42×C53C2×C411yescharacteristic
34C4×C8×C53—41no—
35C23×D53C4⋊C411yescharacteristic
36C22×C4×C53C4211yescharacteristic
37C22×C4×C53C22⋊C411yescharacteristic
38C22×C4×C53C4⋊C411yescharacteristic
39C22×Dic53C22⋊C411yescharacteristic
40C22×Dic53C22⋊C411yescharacteristic
41C22×Dic53C4⋊C411yescharacteristic
42C2×C4×D53—21no—
43C2×C4×D53—21no—
44C2×C4×D53—22no—
45C2×C4×D53—41no—
46#11719940—42no—
47#11719940—42no—
48C2×#88324—42no—
49C2×#88324—42no—
50C2×C8×C53—22no—
51C2×C8×C53—41no—
52C2×C8×C53—41no—
53C42×C53—21no—
54C42×C53—21no—
55#11719921—41no—
56#11719921—41no—
57#11719919—41no—
58#11719919—41no—
59C4×Dic53—21no—
60C4×Dic53—21no—
61C2×#88322—22no—
62C2×#88322—41no—
63C2×#88322—41no—
64C23×C53#9511yescharacteristic
65C22×D53—21no—
66C22×D53—21no—
67C22×D53—21no—
68C22×D53—41no—
69C22×D53—41no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—41no—
75C2×C4×C53—41no—
76C2×C4×C53#9911yescharacteristic
77C2×C4×C53#9911yescharacteristic
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—41no—
83C2×Dic53—41no—
84C2×Dic53#10412yes—
85C4×D53—42no—
86C4×D53—42no—
87#88324—82no—
88#88324—82no—
89C8×C53—42no—
90#88322—42no—
91#7216—531nomaximal, Sylow
92C22×C53—21no—
93C22×C53—21no—
94C22×C53H(3,ℤ/4)11yescharacteristic
95C22×C53#33911yescharacteristic
96C22×C53#34211yescharacteristic
97C2×D53—42no—
98C2×D53—41no—
99C2×D53—41no—
100C2×D53—41no—
101C2×D53—81no—
102C4×C53—21no—
103C4×C53—21no—
104C4×C53—41no—
105C4×C53—41no—
106Dic53—22no—
107Dic53—41no—
108Dic53—41no—
109#1822—531no—
110#1750—531no—
111#1464—531no—
112C2×C53—21no—
113C2×C53—21no—
114C2×C53#128411yescharacteristic
115C2×C53#139211yescharacteristic
116C2×C53#139511yescharacteristic
117D53—81no—
118C2×#117—531no—
119#432—1061no—
120#407—1061no—
121C2×C22⋊C8—1061no—
122C2×#97—531no—
123C2×C4×C8#615711yescharacteristic
124#343—532no—
125#343—1061no—
126#336—1061no—
127#324—1061no—
128#323—532no—
129C53#721611yescharacteristic, complemented, Sylow
130C23×C4—531no—
131C2×M16—1062no—
132C22×C8—21no—
133C22×C8—1061no—
134#117—2121no—
135#117—2121no—
136C2×C4⋊C4—531no—
137C2×C4⋊C4—531no—
138C2×C22⋊C4—531no—
139C2×C22⋊C4—531no—
140C2×C42—531no—
141C2×C42C2×#615711yescharacteristic
142C4⋊C8—2121no—
143C22⋊C8—1064no—
144C22⋊C8—1062no—
145C22⋊C8—2121no—
146C22⋊C8—2121no—
147#97—2121no—
148#97—2121no—
149C4×C8—41no—
150C24—531no—
151C22×C4—531no—
152C22×C4—531no—
153C22×C4—531no—
154C22×C4—1062no—
155C22×C4—1061no—
156C22×C4—1061no—
157C22×C4—2121no—
158C22×C4C4×#615711yescharacteristic
159C22×C4C4⋊#615711yescharacteristic
160C22×C4C22⋊#615711yescharacteristic
161M16—2122no—
162M16—2122no—
163C2×C8—22no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—1062no—
167C2×C8—2122no—
168C2×C8—2122no—
169C2×C8—2121no—
170C2×C8—2121no—
171C4⋊C4—2121no—
172C4⋊C4—2121no—
173C22⋊C4—2121no—
174C22⋊C4—2121no—
175C42—21no—
176C42—21no—
177C42—1061no—
178C42—1061no—
179C23—1061no—
180C23—1061no—
181C23—1061no—
182C23—2121no—
183C23—2121no—
184C23(C2×#6157)⋊C411yescharacteristic
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—532no—
192C2×C4—1061no—
193C2×C4—1061no—
194C2×C4—1061no—
195C2×C4—1061no—
196C2×C4—2122no—
197C2×C4—2122no—
198C2×C4—2121no—
199C2×C4—2121no—
200C2×C4#1313327511yescharacteristic
201C2×C4#1313327611yescharacteristic
202C8—42no—
203C8—4242no—
204C8—4242no—
205C8—2122no—
206C22—21no—
207C22—21no—
208C22—4241no—
209C22—2122no—
210C22—2121no—
211C22—2121no—
212C22—2121no—
213C22#1681149011yescharacteristic
214C22#1681149411yescharacteristic
215C22#1681150111yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—1062no—
221C4—2121no—
222C4—2121no—
223C2—21no—
224C2—21no—
225C2—4241no—
226C2#2028956311yescharacteristic
227C2#2028980611yescharacteristic
228C2#2028981511yescharacteristic
subgroupG/H
centerC22#16811501
commutatorC22×C4×C53C42
FrattiniC2×C42C2×#6157
FittingC2×C4×C8×C53C4
socleC22×C53#342
prime radicalC23×D53C4⋊C4
cosocleC2×#13133174C22
hypercenterC2×C4×C8#6157
hypocenterC53#7216
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×C8

Complements

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

NH ≅ G/Ntype
129. C53#7216semidirect

Maximal subgroups

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

subgroupindex
#202906162
#202899682
#202744822
#721653