Finite Group Databasebeta

← #28298524

Subgroups — #28298524

Order 13568.

276 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 50 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#20290617C211yesmaximal, characteristic
2#20290445C211yesmaximal, characteristic
3#20273046C211yesmaximal, characteristic
4#16811593—21no—
5C2×C22⋊#88324—21no—
6C2×#13133272C411yescharacteristic
7C2×C4×#88324C411yescharacteristic
8#16811503—21no—
9#16811503C412yes—
10#16811489—21no—
11C2×#13133173C2211yescharacteristic
12#16810677—21no—
13#16810348—21no—
14#16810347C411yescharacteristic
15C53×#323C411yescharacteristic
16C2×#11719940—22no—
17C22×#88324—21no—
18C22×#88324—21no—
19C22⋊#88324—24no—
20C22⋊#88324—41no—
21C22⋊#88324—41no—
22#13133272—41no—
23C4×#88324—41no—
24C23×Dic53D411yescharacteristic
25#13133173—41no—
26#13133173—41no—
27C2×#11719919C2×C411yescharacteristic
28C2×#11719919D411yescharacteristic
29C2×C22⋊Dic53Q811yescharacteristic
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53D411yescharacteristic
32#13133146—41no—
33C22⋊#88322—21no—
34#13133127—41no—
35C2×C22⋊C4×C53C2×C411yescharacteristic
36C22⋊C8×C53—21no—
37C24×C53C4⋊C411yescharacteristic
38C22×C4×C53C4211yescharacteristic
39C22×C4×C53C22⋊C411yescharacteristic
40C22×Dic53—21no—
41C22×Dic53—21no—
42C22×Dic53—22no—
43C22×Dic53—41no—
44C22×Dic53C22⋊C411yescharacteristic
45C22×Dic53C22⋊C411yescharacteristic
46C22×Dic53C4⋊C411yescharacteristic
47C22×Dic53C4⋊C411yescharacteristic
48#11719940—42no—
49#11719940—42no—
50C2×#88324—22no—
51C2×#88324—22no—
52C2×#88324—41no—
53C2×#88324—41no—
54C2×#88324—41no—
55C2×#88324—41no—
56C2×#88324—42no—
57C2×C8×C53—41no—
58C22⋊C4×C53—41no—
59C22⋊Dic53—41no—
60#11719919—41no—
61#11719919—41no—
62C4×Dic53—21no—
63C4×Dic53—21no—
64C4×Dic53—21no—
65C4×Dic53—21no—
66C2×#88322—41no—
67C23×C53—21no—
68C23×C53—21no—
69C23×C53—21no—
70C23×C53—41no—
71C23×C53—41no—
72C23×C53#9511yescharacteristic
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—41no—
76C2×C4×C53—41no—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—21no—
83C2×Dic53—42no—
84C2×Dic53—42no—
85C2×Dic53—41no—
86C2×Dic53—41no—
87C2×Dic53#9911yescharacteristic
88C2×Dic53#9911yescharacteristic
89C2×Dic53#10412yes—
90#88324—42no—
91#88324—42no—
92#88324—82no—
93C8×C53—81no—
94#88322—81no—
95#7216—531nomaximal, Sylow
96C22×C53—21no—
97C22×C53—21no—
98C22×C53—42no—
99C22×C53—41no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—81no—
103C22×C53H(3,ℤ/4)11yescharacteristic
104C22×C53#33911yescharacteristic
105C22×C53#34211yescharacteristic
106C4×C53—41no—
107C4×C53—41no—
108Dic53—21no—
109Dic53—21no—
110Dic53—22no—
111Dic53—41no—
112Dic53—41no—
113#1822—531no—
114#1750—531no—
115#1464—531no—
116C2×C53—21no—
117C2×C53—21no—
118C2×C53—81no—
119C2×C53#128411yescharacteristic
120C2×C53#139211yescharacteristic
121C2×C53#139511yescharacteristic
122C2×#117—531no—
123#432—1061no—
124#407—1061no—
125C2×C22⋊C8—1061no—
126C2×#97—531no—
127C2×C4×C8—531no—
128#343—532no—
129#343—1061no—
130#336—1061no—
131#324—1061no—
132#323—531no—
133#323#615711yescharacteristic
134C53#721611yescharacteristic, complemented, Sylow
135C23×C4—531no—
136C2×M16—1062no—
137C22×C8—1061no—
138C22×C8—1061no—
139#117—2121no—
140#117—2121no—
141C2×C4⋊C4—531no—
142C2×C4⋊C4—531no—
143C2×C22⋊C4—531no—
144C2×C22⋊C4C2×#615711yescharacteristic
145C2×C42—531no—
146C2×C42—531no—
147C4⋊C8—2121no—
148C22⋊C8—21no—
149C22⋊C8—1064no—
150C22⋊C8—1061no—
151C22⋊C8—2121no—
152C22⋊C8—2121no—
153#97—2121no—
154#97—2121no—
155C4×C8—2121no—
156C24C4⋊#615711yescharacteristic
157C22×C4—531no—
158C22×C4—531no—
159C22×C4—531no—
160C22×C4—531no—
161C22×C4—1062no—
162C22×C4—1061no—
163C22×C4—1061no—
164C22×C4—2121no—
165C22×C4C4×#615711yescharacteristic
166C22×C4C22⋊#615711yescharacteristic
167M16—2122no—
168M16—2122no—
169C2×C8—41no—
170C2×C8—1062no—
171C2×C8—1062no—
172C2×C8—2121no—
173C2×C8—2122no—
174C2×C8—2121no—
175C2×C8—2121no—
176C2×C8—2121no—
177C2×C8—2121no—
178C4⋊C4—2121no—
179C4⋊C4—2121no—
180C22⋊C4—41no—
181C22⋊C4—2121no—
182C42—1061no—
183C42—1061no—
184C42—1061no—
185C42—1061no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—41no—
190C23—41no—
191C23(C2×#6157)⋊C411yescharacteristic
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—532no—
197C2×C4—531no—
198C2×C4—531no—
199C2×C4—1061no—
200C2×C4—1061no—
201C2×C4—1061no—
202C2×C4—1061no—
203C2×C4—1061no—
204C2×C4—1061no—
205C2×C4—2122no—
206C2×C4—2122no—
207C2×C4—2121no—
208C2×C4—2121no—
209C8—81no—
210C8—4241no—
211C8—4242no—
212C8—2122no—
213C8—2122no—
214C22—21no—
215C22—21no—
216C22—42no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22#1681149111yescharacteristic
222C22#1681149511yescharacteristic
223C22#1681150011yescharacteristic
224C4—41no—
225C4—41no—
226C4—1062no—
227C4—1061no—
228C4—1061no—
229C4—2121no—
230C4—2121no—
231C2—21no—
232C2—21no—
233C2—81no—
234C2#2028956411yescharacteristic
235C2#2028980511yescharacteristic
236C2#2028981411yescharacteristic
subgroupG/H
centerC22#16811500
commutatorC22×C4×C53C42
FrattiniC2×C22⋊C4C2×#6157
FittingC53×#323C4
socleC22×C53#342
prime radicalC24×C53C4⋊C4
cosocleC2×#13133173C22
hypercenter#323#6157
hypocenterC53#7216
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #323

Complements

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

NH ≅ G/Ntype
134. C53#7216semidirect

Maximal subgroups

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

subgroupindex
#202906172
#202904452
#202730462
#721653