Finite Group Databasebeta

← #27272350

Subgroups — #27272350

Order 13568.

520 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 107 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#20277262C211yesmaximal, characteristic
2#20274567C212yesmaximal, complemented
3#20272540C212yesmaximal, complemented
4#20272536C211yesmaximal, characteristic
5C53×#1270C211yesmaximal, characteristic
6C2×#13133204C411yescharacteristic
7C2×#13133182C411yescharacteristic
8C2×D4×Dic53C411yescharacteristic
9C2×#13133146—22no—
10C2×#13133127—22no—
11#16810529C2211yescharacteristic
12#16810511—22no—
13#16810488C411yescharacteristic
14C22⋊C4×Dic53—22no—
15C42×Dic53C2211yescharacteristic
16#16810378C2212yes—
17C53×#390C2211yescharacteristic
18C53×#336C2212yes—
19C22×Dic106C2×C411yescharacteristic
20C23×Dic53—21no—
21C22×#88322—22no—
22#13133204—22no—
23#13133204—22no—
24#13133182—42no—
25D4×Dic53—42no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—21no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—21no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53D412yes—
34C2×C4×Dic53D412yes—
35#13133146—28no—
36#13133127—44no—
37#13133124—22no—
38#13133124—22no—
39C22×D4×C53C2×C411yescharacteristic
40C22×C8×C53—22no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53—22no—
43C2×C42×C53C2311yescharacteristic
44C24×C53—21no—
45C2×D4×C53—21no—
46C2×D4×C53—41no—
47C2×D4×C53—41no—
48C2×D4×C53M1612yes—
49C22×C4×C53—21no—
50C22×C4×C53C22×C411yescharacteristic
51C22×C4×C53C2×D412yes—
52C22×Dic53—21no—
53C22×Dic53—21no—
54C22×Dic53—42no—
55C22×Dic53—44no—
56C22×Dic53C22⋊C412yes—
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C22⋊C411yescharacteristic
59C2×Dic106—21no—
60C2×Dic106—41no—
61C2×Dic106—41no—
62C2×Dic106M1612yes—
63C2×C8×C53—24no—
64C2×C8×C53—42no—
65C2×C8×C53—42no—
66C4⋊C4×C53—41no—
67C22⋊C4×C53—42no—
68C22⋊C4×C53—42no—
69C42×C53—41no—
70C22⋊Dic53—28no—
71C22⋊Dic53—42no—
72C4⋊Dic53—41no—
73#11719919—42no—
74C4×Dic53—21no—
75C4×Dic53—21no—
76C4×Dic53—22no—
77C4×Dic53—22no—
78C4×Dic53—22no—
79C4×Dic53—24no—
80C4×Dic53—42no—
81C2×#88322—24no—
82C2×#88322—42no—
83C2×#88322—42no—
84C23×C53—22no—
85C23×C53—41no—
86C23×C53—41no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53C2×C22⋊C411yescharacteristic
90D4×C53—42no—
91D4×C53—42no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—41no—
97C2×C4×C53—42no—
98C2×C4×C53C2×M1611yescharacteristic
99C2×C4×C53C2×M1611yescharacteristic
100C2×C4×C53AGL(1,ℤ/8)12yes—
101C2×C4×C53#13712yes—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—21no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111C2×Dic53—42no—
112C2×Dic53—84no—
113C2×Dic53#10012yes—
114C2×Dic53#10414yes—
115Dic106—42no—
116Dic106—42no—
117C8×C53—44no—
118#88322—44no—
119#7635—531nomaximal, Sylow
120C22×C53—42no—
121C22×C53—44no—
122C22×C53—81no—
123C22×C53—82no—
124C22×C53—81no—
125C22×C53#40711yescharacteristic
126C22×C53#41011yescharacteristic
127C22×C53C2×#10011yescharacteristic
128C22×C53#41712yes—
129C22×C53C2×#10412yes—
130C4×C53—21no—
131C4×C53—21no—
132C4×C53—22no—
133C4×C53—22no—
134C4×C53—81no—
135Dic53—41no—
136Dic53—42no—
137Dic53—42no—
138Dic53—81no—
139Dic53#32914yes—
140C4×#124—531no—
141#1767—532no—
142#1272—532no—
143#1270—531no—
144#1270D5311yescharacteristic
145C2×C53—82no—
146C2×C53#146611yescharacteristic
147C2×C53#148912yes—
148C2×C53#150912yes—
149C2×C53C2×#32911yescharacteristic
150C2×C53#156011yescharacteristic
151C2×#124—531no—
152C2×C4×Q8—531no—
153C2×C4×D4—531no—
154C2×C4⋊C8—1062no—
155C2×#97—1062no—
156#391—531no—
157#390C2×D5311yescharacteristic
158#388—1062no—
159#382—531no—
160C4×C22⋊C4—1062no—
161C43—531no—
162#336—532no—
163#336C2×D5312yes—
164C53#763511yescharacteristic, complemented, Sylow
165C22×Q8—531no—
166C22×D4C4×D5311yescharacteristic
167C23×C4—1061no—
168C22×C8—22no—
169C22×C8—1062no—
170#124—1062no—
171#124—1062no—
172C4×Q8—2122no—
173C4×D4—2122no—
174C2×C4⋊C4—531no—
175C2×C4⋊C4—1061no—
176C2×C4⋊C4C4×D5311yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—1062no—
179C2×C22⋊C4—1061no—
180C2×C42—531no—
181C2×C42—531no—
182C2×C42—532no—
183C2×C42—532no—
184C2×C42—1061no—
185C2×C42C22×D5311yescharacteristic
186C4⋊C8—1068no—
187#97—2124no—
188#95—1062no—
189#95—1062no—
190C24—21no—
191C2×Q8—532no—
192C2×Q8—1061no—
193C2×Q8—2121no—
194C2×Q8—2121no—
195C2×D4—21no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4#1171991212yes—
199C22×C4—21no—
200C22×C4—531no—
201C22×C4—532no—
202C22×C4—531no—
203C22×C4—1061no—
204C22×C4—1061no—
205C22×C4—2122no—
206C22×C4—2124no—
207C22×C4C2×C4×D5311yescharacteristic
208C22×C4D4×D5312yes—
209C2×C8—24no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—1064no—
213C2×C8—2122no—
214C2×C8—2122no—
215C4⋊C4—41no—
216C4⋊C4—2121no—
217C4⋊C4—2122no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—1068no—
221C22⋊C4—2122no—
222C42—41no—
223C42—1064no—
224C42—1062no—
225C42—1061no—
226C42—1061no—
227C42—1062no—
228C42—1062no—
229C42—2122no—
230C23—22no—
231C23—41no—
232C23—41no—
233C23—42no—
234C23—42no—
235C23C22⋊C4×D5311yescharacteristic
236Q8—2122no—
237Q8—2122no—
238D4—42no—
239D4—42no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—532no—
247C2×C4—534no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—1061no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—4244no—
255C2×C4—2121no—
256C2×C4—2122no—
257C2×C4—2121no—
258C2×C4—2121no—
259C2×C4M16×D5311yescharacteristic
260C2×C4C2×#1171991211yescharacteristic
261C2×C4#1313324112yes—
262C2×C4#1313324612yes—
263C8—44no—
264C8—2124no—
265C22—42no—
266C22—44no—
267C22—81no—
268C22—82no—
269C22—81no—
270C22#1681058511yescharacteristic
271C22#1681059511yescharacteristic
272C22#100×D5311yescharacteristic
273C22#1681062112yes—
274C22#104×D5312yes—
275C4—22no—
276C4—22no—
277C4—21no—
278C4—21no—
279C4—81no—
280C4—534no—
281C4—4241no—
282C4—2121no—
283C4—2122no—
284C4—2122no—
285C2—82no—
286C2#2027305411yescharacteristic
287C2#2027315112yes—
288C2#2027323712yes—
289C2#329×D5311yescharacteristic
290C2#2027344711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1270C2
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#1270D53
hypocenterC53#7635
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #20274567C2semidirect
3. #20272540C2semidirect
164. C53#7635semidirect

Maximal subgroups

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

subgroupindex
#202772622
#202745672
#202725402
#202725362
C53×#12702
#763553