Finite Group Databasebeta

← #27294680

Subgroups — #27294680

Order 13568.

464 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 87 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#20279023C211yesmaximal, characteristic
2C4×#13133134C211yesmaximal, characteristic
3#20272722C214yesmaximal, complemented
4C53×#1465C211yesmaximal, characteristic
5C2×#13133198—21no—
6C22×C4⋊Dic53C411yescharacteristic
7C2×#13133134—21no—
8C4×#11719917—21no—
9#16810553C2212yes—
10#16810548C412yes—
11#16810521—21no—
12#16810484—21no—
13#16810483—21no—
14#16810483C411yescharacteristic
15#16810392C2212yes—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#407—21no—
18C53×#324C2212yes—
19C53×#323—21no—
20C23×Dic53—21no—
21C2×#11719917—22no—
22#13133198—44no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—41no—
26C2×C4⋊Dic53—41no—
27C2×C4⋊Dic53C2×C412yes—
28C2×#11719919—22no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53—21no—
31#13133134—24no—
32#13133128—41no—
33C4×#88322—41no—
34#13133124—22no—
35#13133124—22no—
36#13133124C2×C412yes—
37C22×D4×C53D411yescharacteristic
38C23×C4×C53C2×C411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C2×M16×C53—41no—
41C4×D4×C53—41no—
42C4×D4×C53—41no—
43C2×C4⋊C4×C53C2311yescharacteristic
44C2×C22⋊C4×C53C2×C411yescharacteristic
45C2×C22⋊C4×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47C22⋊C8×C53—22no—
48C22⋊C8×C53—41no—
49C24×C53C22⋊C411yescharacteristic
50C24×C53C22⋊C411yescharacteristic
51C2×D4×C53—22no—
52C2×D4×C53—22no—
53C2×D4×C53—41no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—42no—
59C22×C4×C53—42no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×Dic53—22no—
66C22×Dic53—22no—
67C22×Dic53—42no—
68C22×Dic53—42no—
69C22×Dic53—42no—
70M16×C53—82no—
71C2×C8×C53—42no—
72C4⋊C4×C53—41no—
73C22⋊C4×C53—41no—
74C22⋊C4×C53—41no—
75C42×C53—21no—
76C42×C53—21no—
77C22⋊Dic53—44no—
78C22⋊Dic53—42no—
79C4⋊Dic53—42no—
80C4⋊Dic53—41no—
81C4⋊Dic53—41no—
82C4⋊Dic53—42no—
83#11719919—44no—
84#11719917—24no—
85#11719917—42no—
86C2×#88322—42no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53#10412yes—
97C23×C53C2×C22⋊C411yescharacteristic
98D4×C53—44no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—81no—
112C2×C4×C53#9911yescharacteristic
113C2×C4×C53#9911yescharacteristic
114C2×C4×C53AGL(1,ℤ/8)12yes—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—84no—
122C8×C53—82no—
123#88322—44no—
124#11780—531nomaximal, Sylow
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—22no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53—42no—
131C22×C53—41no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—81no—
137C22×C53C2×#9911yescharacteristic
138C22×C53#41811yescharacteristic
139C22×C53C2×#10411yescharacteristic
140C4×C53—21no—
141C4×C53—22no—
142C4×C53—41no—
143C4×C53—41no—
144C4×C53—41no—
145C4×C53#35212yes—
146Dic53—82no—
147Dic53—82no—
148#2486—531no—
149C4×#100—531no—
150#1465D5311yescharacteristic
151#1346—534no—
152C2×C53—21no—
153C2×C53—21no—
154C2×C53—42no—
155C2×C53—81no—
156C2×C53#150711yescharacteristic
157C2×C53C2×#35211yescharacteristic
158C2×C53#213511yescharacteristic
159C2×#123—1061no—
160C2×C4×D4C2×D5311yescharacteristic
161C22×C4⋊C4—531no—
162C2×#100—1061no—
163#407—21no—
164C4×M16—1061no—
165#398—532no—
166#397—532no—
167#389—1061no—
168#381—1061no—
169#380—531no—
170#380—1061no—
171#343—532no—
172#324C2×D5312yes—
173#323—21no—
174C53#1178011yescharacteristic, complemented, Sylow
175C22×D4D21211yescharacteristic
176C23×C4—1061no—
177C23×C4C4×D5311yescharacteristic
178C23×C4C22⋊D5311yescharacteristic
179C2×M16—41no—
180C2×M16—1062no—
181#123—2124no—
182C4×D4—41no—
183C4×D4—41no—
184C2×C4⋊C4—532no—
185C2×C4⋊C4—1062no—
186C2×C4⋊C4—1061no—
187C2×C4⋊C4—1061no—
188C2×C4⋊C4—2121no—
189C2×C4⋊C4—2121no—
190C2×C4⋊C4C22×D5311yescharacteristic
191C2×C22⋊C4—1061no—
192C2×C22⋊C4—1062no—
193C2×C22⋊C4C4×D5311yescharacteristic
194C2×C22⋊C4C22⋊D5311yescharacteristic
195C2×C42D21211yescharacteristic
196#100—1064no—
197C22⋊C8—22no—
198C22⋊C8—41no—
199#97—2121no—
200C4×C8—2121no—
201#95—532no—
202#95—1062no—
203#95—1062no—
204C24#1171992111yescharacteristic
205C24#1171992111yescharacteristic
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—1062no—
216C22×C4—1062no—
217C22×C4—2122no—
218C22×C4—2122no—
219C22×C4—2122no—
220C22×C4#1171992111yescharacteristic
221C22×C4#1171992111yescharacteristic
222C22×C4C2×C4×D5311yescharacteristic
223C22×C4C2×D21211yescharacteristic
224C22×C4C2×C22⋊D5311yescharacteristic
225M16—82no—
226M16—1064no—
227M16—2122no—
228C2×C8—42no—
229C2×C8—2122no—
230C4⋊C4—41no—
231C4⋊C4—2122no—
232C4⋊C4—2122no—
233C4⋊C4—2121no—
234C4⋊C4—2121no—
235C4⋊C4—2124no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—2122no—
239C22⋊C4—2124no—
240C42—21no—
241C42—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23#1313314312yes—
252C23C2×#1171992111yescharacteristic
253D4—44no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—81no—
267C2×C4—4244no—
268C2×C4—2122no—
269C2×C4—2122no—
270C2×C4—2122no—
271C2×C4—2122no—
272C2×C4—2122no—
273C2×C4—2122no—
274C2×C4#1313313111yescharacteristic
275C2×C4#1313313111yescharacteristic
276C2×C4#1313324512yes—
277C8—82no—
278C8—2124no—
279C22—21no—
280C22—21no—
281C22—22no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—81no—
291C22C2×#1313313111yescharacteristic
292C22#1681062911yescharacteristic
293C22C2×#1313314311yescharacteristic
294C4—22no—
295C4—21no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—4242no—
300C4—4242no—
301C4#1681041212yes—
302C2—21no—
303C2—21no—
304C2—42no—
305C2—81no—
306C2#2027323211yescharacteristic
307C2C2×#1681041211yescharacteristic
308C2#2027640911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1465C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C4⋊C4×C53C23
hypercenter#1465D53
hypocenterC53#11780
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #20272722C2semidirect
174. C53#11780semidirect

Maximal subgroups

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

subgroupindex
#202790232
C4×#131331342
#202727222
C53×#14652
#1178053