Finite Group Databasebeta

← #27465295

Subgroups — #27465295

Order 13568.

888 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 187 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#20277307C212yesmaximal
2#20276148C214yesmaximal, complemented
3#20274910C214yesmaximal, complemented
4#20274811C214yesmaximal, complemented
5C53×#1780C211yesmaximal, characteristic
6C2×#13133176C2212yes—
7C22×C4⋊Dic53C2212yes—
8C2×#13133149C2214yes—
9#16810530—22no—
10#16810526—22no—
11#16810503—22no—
12#16810503—22no—
13#16810483C2214yes—
14#16810483C2214yes—
15#16810386C2214yes—
16#16810386C2218yes—
17C22×#41×C53C2211yescharacteristic
18C2×C22⋊C8×C53C2214yes—
19C2×C22⋊C8×C53C2212yes—
20C23×Dic53—22no—
21#13133176—28no—
22C2×C4⋊Dic53—22no—
23C2×C4⋊Dic53—44no—
24C2×C4⋊Dic53C2314yes—
25C2×C4⋊Dic53C2314yes—
26C2×C22⋊Dic53—22no—
27C2×C22⋊Dic53—24no—
28C2×C22⋊Dic53—24no—
29C2×C4×Dic53—22no—
30#13133149—44no—
31#13133124—24no—
32#13133124—24no—
33C2×#41×C53—28no—
34C2×#41×C53—42no—
35C22×Q8×C53D411yescharacteristic
36C22×D4×C53D411yescharacteristic
37C22×D4×C53D412yes—
38C23×C4×C53C2311yescharacteristic
39C23×C4×C53C2312yes—
40C22×C8×C53C2314yes—
41C22⋊C8×C53—28no—
42C22⋊C8×C53—42no—
43C24×C53C2×D411yescharacteristic
44C24×C53C2×D412yes—
45#41×C53—416no—
46C2×Q8×C53—24no—
47C2×Q8×C53—41no—
48C2×D4×C53—24no—
49C2×D4×C53—24no—
50C2×D4×C53—24no—
51C2×D4×C53—42no—
52C2×D4×C53—41no—
53C22×C4×C53—24no—
54C22×C4×C53—24no—
55C22×C4×C53—24no—
56C22×C4×C53—22no—
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53C2×D412yes—
59C22×C4×C53#4118yes—
60C22×C4×C53C2411yescharacteristic
61C22×Dic53—24no—
62C22×Dic53—24no—
63C22×Dic53—44no—
64C22×Dic53—84no—
65C2×C8×C53—28no—
66C2×C8×C53—24no—
67C22⋊Dic53—44no—
68C22⋊Dic53—84no—
69C22⋊Dic53—84no—
70C4⋊Dic53—28no—
71C4⋊Dic53—28no—
72C4⋊Dic53—44no—
73C4×Dic53—44no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—24no—
77C23×C53—24no—
78C23×C53—22no—
79C23×C53—21no—
80C23×C53—24no—
81C23×C53#12314yes—
82C23×C53C22×D411yescharacteristic
83Q8×C53—44no—
84D4×C53—48no—
85D4×C53—44no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—216no—
89C2×C4×C53—22no—
90C2×C4×C53—21no—
91C2×C4×C53—44no—
92C2×C4×C53#12314yes—
93C2×C4×C53C2×#4114yes—
94C2×C4×C53H(5,𝔽2)12yes—
95C2×Dic53—44no—
96C2×Dic53—44no—
97C2×Dic53—44no—
98C2×Dic53—44no—
99C2×Dic53—44no—
100C2×Dic53—44no—
101C2×Dic53—88no—
102C8×C53—44no—
103#27609—531nomaximal, Sylow
104C22×C53—28no—
105C22×C53—24no—
106C22×C53—44no—
107C22×C53—42no—
108C22×C53—42no—
109C22×C53—44no—
110C22×C53—42no—
111C22×C53—41no—
112C22×C53C2×#12312yes—
113C22×C53#53411yescharacteristic
114C22×C53#57714yes—
115C4×C53—24no—
116C4×C53—41no—
117C4×C53#55114yes—
118Dic53—84no—
119Dic53—84no—
120#2311—532no—
121#2077—534no—
122#1853—534no—
123#1825—534no—
124#1780D5311yescharacteristic
125C2×C53—44no—
126C2×C53—81no—
127C2×C53#248311yescharacteristic
128C2×C53#309914yes—
129C2×C53#320012yes—
130C22×#41C2×D5311yescharacteristic
131C2×#117—532no—
132C22×C4⋊C4—532no—
133C2×#106—534no—
134C2×C22⋊C8C2×D5314yes—
135C2×C22⋊C8C2×D5312yes—
136#391—1062no—
137#390—1062no—
138#386—1062no—
139#386—1062no—
140#380—534no—
141#380—534no—
142#340—534no—
143#340—538no—
144C53#2760911yescharacteristic, complemented, Sylow
145C2×#41—28no—
146C2×#41—42no—
147C22×Q8D21211yescharacteristic
148C22×D4D21211yescharacteristic
149C22×D4D21212yes—
150C23×C4—1062no—
151C23×C4C22×D5311yescharacteristic
152C23×C4C22×D5312yes—
153C22×C8C22×D5314yes—
154#117—1068no—
155C2×C4⋊C4—534no—
156C2×C4⋊C4—534no—
157C2×C4⋊C4—1062no—
158C2×C4⋊C4—2124no—
159C2×C22⋊C4—1064no—
160C2×C22⋊C4—1064no—
161C2×C22⋊C4—1062no—
162C2×C42—1062no—
163#106—2124no—
164C22⋊C8—28no—
165C22⋊C8—42no—
166#95—1064no—
167#95—1064no—
168C24C2×D21211yescharacteristic
169C24C2×D21212yes—
170#41—416no—
171C2×Q8—24no—
172C2×Q8—41no—
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—24no—
176C2×D4—42no—
177C2×D4—41no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—1064no—
183C22×C4—1064no—
184C22×C4—4244no—
185C22×C4—2124no—
186C22×C4C2×D21211yescharacteristic
187C22×C4C2×D21212yes—
188C22×C4#1171994718yes—
189C22×C4C23×D5311yescharacteristic
190C2×C8—28no—
191C2×C8—24no—
192C4⋊C4—1068no—
193C4⋊C4—1068no—
194C4⋊C4—2124no—
195C22⋊C4—4244no—
196C22⋊C4—4244no—
197C22⋊C4—2124no—
198C42—2124no—
199C23—22no—
200C23—24no—
201C23—24no—
202C23—22no—
203C23—22no—
204C23—24no—
205C23—21no—
206C23#1313320114yes—
207C23C22×D21211yescharacteristic
208Q8—44no—
209D4—48no—
210D4—44no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—216no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—44no—
217C2×C4—4248no—
218C2×C4—2124no—
219C2×C4—2124no—
220C2×C4—2124no—
221C2×C4—2124no—
222C2×C4—2124no—
223C2×C4—2124no—
224C2×C4#1313320114yes—
225C2×C4C2×#1171994714yes—
226C2×C4#1313326212yes—
227C8—44no—
228C22—28no—
229C22—24no—
230C22—44no—
231C22—42no—
232C22—42no—
233C22—44no—
234C22—42no—
235C22—41no—
236C22C2×#1313320112yes—
237C22#1681118511yescharacteristic
238C22#1681143514yes—
239C4—24no—
240C4—41no—
241C4—4244no—
242C4—4244no—
243C4#1681130414yes—
244C2—44no—
245C2—81no—
246C2#2027899611yescharacteristic
247C2#2028470614yes—
248C2#2028558812yes—
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1780C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
hypercenter#1780D53
hypocenterC53#27609
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×#41 ◁ #1780

Complements

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

NH ≅ G/Ntype
2. #20276148C2semidirect
3. #20274910C2semidirect
4. #20274811C2semidirect
144. C53#27609semidirect

Maximal subgroups

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

subgroupindex
#202773072
#202761482
#202749102
#202748112
C53×#17802
#2760953