Finite Group Databasebeta

← #27272381

Subgroups — #27272381

Order 13568.

646 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 163 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#20274695C212yesmaximal
2#20272541C214yesmaximal, complemented
3C2×C53×#391C211yesmaximal, characteristic
4C2×C4⋊#88322—22no—
5C2×C22⋊#88322—22no—
6#16810378C2214yes—
7C23×Q8×C53C411yescharacteristic
8C22×C4⋊C4×C53C411yescharacteristic
9C22×C42×C53C2211yescharacteristic
10C53×#391—22no—
11C53×#391C412yes—
12C53×#391C2212yes—
13C2×#95×C53—22no—
14C22×#88322—24no—
15C4⋊#88322—44no—
16C22⋊#88322—44no—
17C22×Q8×C53—24no—
18C22×Q8×C53—21no—
19C22×Q8×C53—42no—
20C22×Q8×C53—42no—
21C22×Q8×C53C2×C412yes—
22C23×C4×C53—21no—
23C23×C4×C53—21no—
24C23×C4×C53C2×C411yescharacteristic
25C23×C4×C53D412yes—
26C2×C4⋊C4×C53—21no—
27C2×C4⋊C4×C53—42no—
28C2×C4⋊C4×C53C2×C412yes—
29C2×C42×C53—42no—
30C2×C42×C53C2×C411yescharacteristic
31C2×C42×C53D412yes—
32C2×C42×C53C2311yescharacteristic
33#95×C53—24no—
34#95×C53—24no—
35C24×C53C22⋊C411yescharacteristic
36C2×Q8×C53—24no—
37C2×Q8×C53—22no—
38C2×Q8×C53—416no—
39C2×Q8×C53—42no—
40C2×Q8×C53—42no—
41C2×Q8×C53—42no—
42C2×Q8×C53—42no—
43C2×Q8×C53M1614yes—
44C22×C4×C53—22no—
45C22×C4×C53—22no—
46C22×C4×C53—22no—
47C22×C4×C53—24no—
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53—42no—
51C22×C4×C53—42no—
52C22×C4×C53—42no—
53C22×C4×C53—42no—
54C22×C4×C53—44no—
55C22×C4×C53—42no—
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C412yes—
58C22×C4×C53SD1614yes—
59C22×C4×C53Q1614yes—
60C22×C4×C53C22×C411yescharacteristic
61C22×C4×C53C2×D412yes—
62C4⋊C4×C53—42no—
63C4⋊C4×C53—42no—
64C42×C53—24no—
65C42×C53—41no—
66C42×C53—41no—
67C2×#88322—44no—
68C2×#88322—44no—
69C2×#88322—44no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53—22no—
73C23×C53#10012yes—
74C23×C53#10314yes—
75C23×C53C2×C22⋊C411yescharacteristic
76Q8×C53—48no—
77Q8×C53—44no—
78Q8×C53—44no—
79C2×C4×C53—28no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—24no—
89C2×C4×C53—44no—
90C2×C4×C53—44no—
91C2×C4×C53—48no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—44no—
96C2×C4×C53—84no—
97C2×C4×C53#9912yes—
98C2×C4×C53#10314yes—
99C2×C4×C53C2×M1612yes—
100C2×C4×C53C2×SD1612yes—
101C2×C4×C53C2×Q1612yes—
102#88322—84no—
103#7643—531nomaximal, Sylow
104C22×C53—22no—
105C22×C53—24no—
106C22×C53—24no—
107C22×C53—22no—
108C22×C53#33314yes—
109C22×C53#40711yescharacteristic
110C22×C53C2×#9911yescharacteristic
111C22×C53C2×#10011yescharacteristic
112C22×C53C2×#10312yes—
113C22×C53#42112yes—
114C4×C53—22no—
115C4×C53—22no—
116C4×C53—22no—
117C4×C53—24no—
118C4×C53—22no—
119C4×C53—44no—
120C4×C53—82no—
121C2×#391D5311yescharacteristic
122#1797—532no—
123#1272—534no—
124C2×C53—24no—
125C2×C53#146211yescharacteristic
126C2×C53#148712yes—
127C2×C53#152012yes—
128C2×C53#153311yescharacteristic
129C2×C53C2×#33311yescharacteristic
130C23×Q8Dic5311yescharacteristic
131C22×C4⋊C4Dic5311yescharacteristic
132C22×C42C2×D5311yescharacteristic
133C2×C4⋊C8—1062no—
134C2×C22⋊C8—1062no—
135#391—22no—
136#391Dic5312yes—
137#391C2×D5312yes—
138C2×#95—22no—
139#336—534no—
140C53#764311yescharacteristic, complemented, Sylow
141C22×Q8—24no—
142C22×Q8—21no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8C2×Dic5312yes—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4C2×Dic5311yescharacteristic
149C23×C4C22⋊D5312yes—
150C22×C8—1064no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4C2×Dic5312yes—
154C2×C42—42no—
155C2×C42C2×Dic5311yescharacteristic
156C2×C42C22⋊D5312yes—
157C2×C42C22×D5311yescharacteristic
158C4⋊C8—2124no—
159C22⋊C8—2124no—
160#95—24no—
161#95—24no—
162C24C22⋊Dic5311yescharacteristic
163C2×Q8—24no—
164C2×Q8—22no—
165C2×Q8—416no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8#1171991714yes—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—44no—
182C22×C4—42no—
183C22×C4#1171992414yes—
184C22×C4#1171992514yes—
185C22×C4C22⋊Dic5311yescharacteristic
186C22×C4C22⋊Dic5312yes—
187C22×C4C22×Dic5311yescharacteristic
188C22×C4C2×C22⋊D5312yes—
189C2×C8—2124no—
190C2×C8—2124no—
191C2×C8—2124no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C42—24no—
195C42—41no—
196C42—41no—
197C23—21no—
198C23—21no—
199C23—22no—
200C23#1313313412yes—
201C23#1313314114yes—
202C23C2×C22⋊Dic5311yescharacteristic
203Q8—44no—
204Q8—48no—
205Q8—44no—
206C2×C4—28no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—48no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—84no—
224C2×C4#1313313212yes—
225C2×C4#1313314114yes—
226C2×C4C2×#1171991712yes—
227C2×C4C2×#1171992412yes—
228C2×C4C2×#1171992512yes—
229C8—4244no—
230C22—24no—
231C22—22no—
232C22—22no—
233C22—24no—
234C22#1681037214yes—
235C22#1681058711yescharacteristic
236C22C2×#1313313211yescharacteristic
237C22C2×#1313313411yescharacteristic
238C22C2×#1313314112yes—
239C22#1681064012yes—
240C4—22no—
241C4—24no—
242C4—22no—
243C4—22no—
244C4—22no—
245C4—44no—
246C4—82no—
247C2—24no—
248C2#2027304211yescharacteristic
249C2#2027314412yes—
250C2#2027328712yes—
251C2#2027333911yescharacteristic
252C2C2×#1681037211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#391C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenterC2×#391D53
hypocenterC53#7643
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. #20272541C2semidirect
140. C53#7643semidirect

Maximal subgroups

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

subgroupindex
#202746952
#202725412
C2×C53×#3912
#764353