Finite Group Databasebeta

← #27272369

Subgroups — #27272369

Order 13568.

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

normal subgroups only

This group has 862 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, complemented
2#20272536C214yesmaximal, complemented
3C2×C53×#390C211yesmaximal, characteristic
4C2×C4⋊#88322—22no—
5C2×C22⋊#88322—22no—
6#16810378C2214yescomplemented
7C23×D4×C53C411yescharacteristic
8C22×C4⋊C4×C53C411yescharacteristic
9C22×C22⋊C4×C53—22no—
10C22×C42×C53C2211yescharacteristic
11C53×#390—22no—
12C53×#390C412yes—
13C53×#390C2212yes—
14C22×#88322—24no—
15C4⋊#88322—44no—
16C22⋊#88322—44no—
17C25×C53—21no—
18C22×D4×C53—24no—
19C22×D4×C53—21no—
20C22×D4×C53—42no—
21C22×D4×C53—42no—
22C22×D4×C53C2×C412yes—
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×C22⋊C4×C53—24no—
30C2×C22⋊C4×C53—24no—
31C2×C22⋊C4×C53—44no—
32C2×C22⋊C4×C53—44no—
33C2×C42×C53—42no—
34C2×C42×C53C2×C411yescharacteristic
35C2×C42×C53D412yes—
36C2×C42×C53C2311yescharacteristic
37C24×C53—24no—
38C24×C53—22no—
39C24×C53—42no—
40C24×C53—44no—
41C24×C53—42no—
42C24×C53—44no—
43C24×C53C22⋊C411yescharacteristic
44C2×D4×C53—24no—
45C2×D4×C53—22no—
46C2×D4×C53—416no—
47C2×D4×C53—42no—
48C2×D4×C53—42no—
49C2×D4×C53—42no—
50C2×D4×C53—42no—
51C2×D4×C53M1614yes—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—42no—
57C22×C4×C53—42no—
58C22×C4×C53—42no—
59C22×C4×C53—44no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C412yes—
62C22×C4×C53D814yes—
63C22×C4×C53SD1614yes—
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D412yes—
66C4⋊C4×C53—42no—
67C4⋊C4×C53—42no—
68C22⋊C4×C53—48no—
69C22⋊C4×C53—44no—
70C22⋊C4×C53—44no—
71C42×C53—24no—
72C42×C53—41no—
73C42×C53—41no—
74C2×#88322—44no—
75C2×#88322—44no—
76C2×#88322—44no—
77C23×C53—21no—
78C23×C53—21no—
79C23×C53—22no—
80C23×C53—24no—
81C23×C53—48no—
82C23×C53—416no—
83C23×C53—42no—
84C23×C53—42no—
85C23×C53—44no—
86C23×C53—44no—
87C23×C53—88no—
88C23×C53—84no—
89C23×C53—84no—
90C23×C53#10012yes—
91C23×C53#10214yes—
92C23×C53C2×C22⋊C411yescharacteristic
93D4×C53—44no—
94D4×C53—48no—
95D4×C53—44no—
96C2×C4×C53—28no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—44no—
106C2×C4×C53—44no—
107C2×C4×C53—42no—
108C2×C4×C53—44no—
109C2×C4×C53—84no—
110C2×C4×C53#9912yes—
111C2×C4×C53#10214yes—
112C2×C4×C53C2×M1612yes—
113C2×C4×C53C2×D812yes—
114C2×C4×C53C2×SD1612yes—
115#88322—84no—
116#7640—531nomaximal, Sylow
117C22×C53—22no—
118C22×C53—24no—
119C22×C53—24no—
120C22×C53—22no—
121C22×C53—44no—
122C22×C53—48no—
123C22×C53—816no—
124C22×C53—82no—
125C22×C53—84no—
126C22×C53—82no—
127C22×C53#33114yes—
128C22×C53#40711yescharacteristic
129C22×C53C2×#9911yescharacteristic
130C22×C53C2×#10011yescharacteristic
131C22×C53C2×#10212yes—
132C22×C53#42112yes—
133C4×C53—22no—
134C4×C53—22no—
135C4×C53—22no—
136C4×C53—24no—
137C4×C53—22no—
138C4×C53—82no—
139C2×#390D5311yescharacteristic
140#1797—532no—
141#1270—534no—
142C2×C53—24no—
143C2×C53—84no—
144C2×C53#146211yescharacteristic
145C2×C53#148612yes—
146C2×C53#151912yes—
147C2×C53#153311yescharacteristic
148C2×C53C2×#33111yescharacteristic
149C23×D4Dic5311yescharacteristic
150C22×C4⋊C4Dic5311yescharacteristic
151C22×C22⋊C4—22no—
152C22×C42C2×D5311yescharacteristic
153C2×C4⋊C8—1062no—
154C2×C22⋊C8—1062no—
155#390—22no—
156#390Dic5312yes—
157#390C2×D5312yes—
158#336—534no—
159C53#764011yescharacteristic, complemented, Sylow
160C25—21no—
161C22×D4—24no—
162C22×D4—21no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4C2×Dic5312yes—
166C23×C4—21no—
167C23×C4C2×Dic5311yescharacteristic
168C23×C4C22⋊D5312yes—
169C22×C8—1064no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4C2×Dic5312yes—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C42—42no—
178C2×C42C2×Dic5311yescharacteristic
179C2×C42C22⋊D5312yes—
180C2×C42C22×D5311yescharacteristic
181C4⋊C8—2124no—
182C22⋊C8—2124no—
183C24—24no—
184C24—22no—
185C24—42no—
186C24—44no—
187C24—42no—
188C24—44no—
189C24C22⋊Dic5311yescharacteristic
190C2×D4—24no—
191C2×D4—22no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—416no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4#1171991714yes—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—44no—
206C22×C4#1171992214yes—
207C22×C4#1171992314yes—
208C22×C4C22⋊Dic5311yescharacteristic
209C22×C4C22⋊Dic5312yes—
210C22×C4C22×Dic5311yescharacteristic
211C22×C4C2×C22⋊D5312yes—
212C2×C8—2124no—
213C2×C8—2124no—
214C2×C8—2124no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C22⋊C4—48no—
218C22⋊C4—44no—
219C22⋊C4—44no—
220C42—24no—
221C42—41no—
222C42—41no—
223C23—21no—
224C23—21no—
225C23—22no—
226C23—24no—
227C23—48no—
228C23—416no—
229C23—42no—
230C23—42no—
231C23—44no—
232C23—44no—
233C23—88no—
234C23—84no—
235C23—84no—
236C23#1313313412yes—
237C23#1313313814yes—
238C23C2×C22⋊Dic5311yescharacteristic
239D4—44no—
240D4—48no—
241D4—44no—
242C2×C4—28no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—84no—
256C2×C4#1313313212yes—
257C2×C4#1313313814yes—
258C2×C4C2×#1171991712yes—
259C2×C4C2×#1171992212yes—
260C2×C4C2×#1171992312yes—
261C8—4244no—
262C22—24no—
263C22—22no—
264C22—22no—
265C22—24no—
266C22—44no—
267C22—48no—
268C22—816no—
269C22—84no—
270C22—82no—
271C22—82no—
272C22#1681036714yes—
273C22#1681058711yescharacteristic
274C22C2×#1313313211yescharacteristic
275C22C2×#1313313411yescharacteristic
276C22C2×#1313313812yes—
277C22#1681064012yes—
278C4—22no—
279C4—24no—
280C4—22no—
281C4—22no—
282C4—22no—
283C4—82no—
284C2—24no—
285C2—84no—
286C2#2027304211yescharacteristic
287C2#2027314012yes—
288C2#2027328312yes—
289C2#2027333911yescharacteristic
290C2C2×#1681036711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#390C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D4×C53C4
cosocleC2×C42×C53C23
hypercenterC2×#390D53
hypocenterC53#7640
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. #20274695C2semidirect
2. #20272536C2semidirect
6. #16810378C22semidirect
159. C53#7640semidirect

Maximal subgroups

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

subgroupindex
#202746952
#202725362
C2×C53×#3902
#764053