Finite Group Databasebeta

← #27294779

Subgroups — #27294779

Order 13568.

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

normal subgroups only

This group has 478 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
1C2×#16810553C211yesmaximal, characteristic
2#20274795C211yesmaximal, characteristic
3#20272722C214yesmaximal, complemented
4C53×#1463C211yesmaximal, characteristic
5C22×C4⋊Dic53C411yescharacteristic
6C22×#11719919—21no—
7#16810587—21no—
8#16810553—21no—
9#16810553—21no—
10#16810553C412yes—
11#16810553C2212yes—
12C2×#13133124—21no—
13C2×#13133124C411yescharacteristic
14#16810392—21no—
15#16810392C2212yes—
16C22×C4⋊C4×C53C2211yescharacteristic
17C53×#407—21no—
18C53×#324—21no—
19C53×#324C2212yes—
20C23×Dic53—21no—
21C23×Dic53—21no—
22C2×#11719917—22no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—41no—
25C2×C4⋊Dic53—41no—
26C2×C4⋊Dic53C2×C412yes—
27C2×#11719919—22no—
28C2×#11719919—22no—
29C2×#11719919—44no—
30C22⋊#88322—41no—
31C22⋊#88322—41no—
32#13133124—21no—
33#13133124—22no—
34#13133124—22no—
35#13133124C2×C412yes—
36C23×C4×C53C2×C411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C2×M16×C53—41no—
40C2×C4⋊C4×C53—41no—
41C2×C4⋊C4×C53—41no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C4⋊C4×C53C2311yescharacteristic
46C22⋊C8×C53—22no—
47C22⋊C8×C53—41no—
48C24×C53C22⋊C411yescharacteristic
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—22no—
56C22×C4×C53—41no—
57C22×C4×C53—42no—
58C22×C4×C53—42no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×Dic53—24no—
66C22×Dic53—22no—
67C22×Dic53—22no—
68C22×Dic53—42no—
69C22×Dic53—42no—
70C22×Dic53—42no—
71C22×Dic53—42no—
72C22×Dic53—41no—
73C22×Dic53—41no—
74M16×C53—82no—
75C2×C8×C53—42no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊Dic53—42no—
81C4⋊Dic53—42no—
82#11719919—44no—
83#11719919—44no—
84#11719917—42no—
85#11719917—42no—
86C2×#88322—42no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53#9911yescharacteristic
92C23×C53#9911yescharacteristic
93C23×C53C2×C22⋊C411yescharacteristic
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
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—41no—
112C2×C4×C53—81no—
113C2×C4×C53AGL(1,ℤ/8)12yes—
114C2×Dic53—24no—
115C2×Dic53—48no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—82no—
122C2×Dic53—82no—
123C8×C53—82no—
124#88322—82no—
125#11803—531nomaximal, Sylow
126C22×C53—24no—
127C22×C53—24no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—21no—
133C22×C53—21no—
134C22×C53#35212yes—
135C22×C53#35612yes—
136C22×C53C2×#9911yescharacteristic
137C22×C53#41811yescharacteristic
138C22×C53#42111yescharacteristic
139C4×C53—22no—
140C4×C53—41no—
141C4×C53—41no—
142C4×C53—41no—
143C4×C53—41no—
144C4×C53—41no—
145Dic53—44no—
146Dic53—82no—
147C2×#398—531no—
148#1821—531no—
149#1463D5311yescharacteristic
150#1346—534no—
151C2×C53—21no—
152C2×C53—22no—
153C2×C53—22no—
154C2×C53—21no—
155C2×C53#151011yescharacteristic
156C2×C53C2×#35211yescharacteristic
157C2×C53C2×#35611yescharacteristic
158C22×C4⋊C4—531no—
159C22×C4⋊C4—1061no—
160C22×C4⋊C4C2×D5311yescharacteristic
161#407—21no—
162#407—1061no—
163#398—532no—
164#398—532no—
165#398—1061no—
166#398—1061no—
167C2×#95—531no—
168C2×#95—1061no—
169#343—532no—
170#343—1061no—
171#324—21no—
172#324C2×D5312yes—
173C53#1180311yescharacteristic, complemented, Sylow
174C23×C4—1061no—
175C23×C4—1061no—
176C23×C4C4×D5311yescharacteristic
177C23×C4D21211yescharacteristic
178C23×C4C22⋊D5311yescharacteristic
179C2×M16—41no—
180C2×M16—1062no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—532no—
184C2×C4⋊C4—1062no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4—1062no—
187C2×C4⋊C4—2121no—
188C2×C4⋊C4—2121no—
189C2×C4⋊C4—2124no—
190C2×C4⋊C4C4×D5311yescharacteristic
191C2×C4⋊C4D21211yescharacteristic
192C2×C4⋊C4C22⋊D5311yescharacteristic
193C2×C4⋊C4C22×D5311yescharacteristic
194C22⋊C8—22no—
195C22⋊C8—41no—
196C22⋊C8—2121no—
197C22⋊C8—2121no—
198#95—532no—
199#95—1062no—
200#95—1061no—
201#95—1062no—
202C24#1171992111yescharacteristic
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—1064no—
214C22×C4—1062no—
215C22×C4—1062no—
216C22×C4—2122no—
217C22×C4—2122no—
218C22×C4—2121no—
219C22×C4—2121no—
220C22×C4—2122no—
221C22×C4—2122no—
222C22×C4#1171992111yescharacteristic
223C22×C4#1171992111yescharacteristic
224C22×C4#1171992111yescharacteristic
225C22×C4C2×C4×D5311yescharacteristic
226C22×C4C2×D21211yescharacteristic
227C22×C4C2×C22⋊D5311yescharacteristic
228M16—82no—
229M16—2122no—
230M16—2122no—
231C2×C8—42no—
232C2×C8—2122no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—2122no—
238C4⋊C4—2124no—
239C4⋊C4—2122no—
240C4⋊C4—2124no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23#1313313111yescharacteristic
246C23#1313313111yescharacteristic
247C23C2×#1171992111yescharacteristic
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—81no—
267C2×C4—1064no—
268C2×C4—4242no—
269C2×C4—4242no—
270C2×C4—2122no—
271C2×C4—2122no—
272C2×C4—2122no—
273C2×C4—2128no—
274C2×C4—2122no—
275C2×C4—2122no—
276C2×C4#1313324512yes—
277C8—82no—
278C8—4242no—
279C22—24no—
280C22—24no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22#1681041212yes—
288C22#1681042112yes—
289C22C2×#1313313111yescharacteristic
290C22#1681062911yescharacteristic
291C22#1681063911yescharacteristic
292C4—22no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—4242no—
299C4—2124no—
300C2—21no—
301C2—21no—
302C2—22no—
303C2—22no—
304C2#2027324511yescharacteristic
305C2C2×#1681041211yescharacteristic
306C2C2×#1681042111yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1463C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1463D53
hypocenterC53#11803
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #20272722C2semidirect
173. C53#11803semidirect

Maximal subgroups

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

subgroupindex
C2×#168105532
#202747952
#202727222
C53×#14632
#1180353