Finite Group Databasebeta

← #28309131

Subgroups — #28309131

Order 13568.

464 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C4×#13133277C211yesmaximal, characteristic
2#20289970C211yesmaximal, characteristic
3#20289699C214yesmaximal, complemented
4#20279017C211yesmaximal, characteristic
5C2×#13133277—21no—
6#16811594—21no—
7C4×#11719940—21no—
8#16811503C2212yes—
9#16811469C2212yes—
10#16811466—21no—
11C2×#13133200—21no—
12C2×C4×C22⋊D53C2211yescharacteristic
13#16810554C2212yes—
14#16810549C412yes—
15#16810519—21no—
16#16810486—21no—
17#16810483C411yescharacteristic
18#16810480—21no—
19C22×C4⋊C4×C53C411yescharacteristic
20C2×#11719936—41no—
21C2×#11719940—22no—
22#13133277—24no—
23#13133274—41no—
24C22⋊#88324—22no—
25#13133272—41no—
26C4×#88324—41no—
27C22×C22⋊D53D411yescharacteristic
28C22×C4×D53D411yescharacteristic
29#13133200—44no—
30C4×C22⋊D53—41no—
31C4×C22⋊D53—41no—
32C2×C4⋊Dic53—21no—
33C2×#11719919—22no—
34C2×#11719919C2311yescharacteristic
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53D411yescharacteristic
37C2×#11719921—22no—
38C2×#11719921C2×C411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40#13133124—22no—
41#13133124—22no—
42#13133124C2×C412yes—
43C23×C4×C53—21no—
44C23×C4×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—41no—
47C2×C4⋊C4×C53—41no—
48C2×C4⋊C4×C53C2×C412yes—
49C24×C53C22⋊C411yescharacteristic
50C23×D53C22⋊C411yescharacteristic
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—22no—
54C22×C4×C53—42no—
55C22×C4×C53—42no—
56C22×C4×C53—42no—
57C22×C4×C53—42no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C2×C22⋊D53—22no—
61C2×C22⋊D53—22no—
62C2×C22⋊D53—41no—
63C22×Dic53—22no—
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C2×D411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C2×C4×D53—21no—
68C2×C4×D53—21no—
69C2×C4×D53—42no—
70#11719940—24no—
71#11719940—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74#11719936—82no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C22⋊Dic53—41no—
79C22⋊Dic53—42no—
80#11719921—41no—
81#11719921—44no—
82C4⋊Dic53—42no—
83#11719919—41no—
84#11719919—44no—
85C4×Dic53—21no—
86C4×Dic53—21no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—41no—
90C23×C53—41no—
91C23×C53#10412yes—
92C23×C53C2×C22⋊C411yescharacteristic
93C22×D53—21no—
94C22×D53—21no—
95C22×D53—21no—
96C22×D53—41no—
97C22×D53—41no—
98C2×C4×C53—22no—
99C2×C4×C53—21no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—41no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—84no—
107C2×C4×C53#9911yescharacteristic
108C2×C4×C53#9911yescharacteristic
109C22⋊D53—44no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53AGL(1,ℤ/8)12yes—
120C4×D53—42no—
121C4×D53—81no—
122#88324—44no—
123#88324—82no—
124#11780—531nomaximal, Sylow
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—22no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53—41no—
131C22×C53—41no—
132C22×C53C2×#9911yescharacteristic
133C22×C53#41811yescharacteristic
134C22×C53C2×#10411yescharacteristic
135C2×D53—42no—
136C2×D53—41no—
137C2×D53—41no—
138C2×D53—41no—
139C2×D53—81no—
140C4×C53—21no—
141C4×C53—41no—
142C4×C53—82no—
143C4×C53#35212yes—
144Dic53—22no—
145Dic53—41no—
146Dic53—41no—
147Dic53—82no—
148#2486—531no—
149C4×#100—531no—
150#1465—531no—
151#1346—534no—
152C2×C53—21no—
153C2×C53—21no—
154C2×C53—42no—
155C2×C53#150711yescharacteristic
156C2×C53C2×#35211yescharacteristic
157C2×C53#213511yescharacteristic
158D53—81no—
159C2×#123—1061no—
160C2×C4×D4—531no—
161C22×C4⋊C4#615711yescharacteristic
162C2×#100—1061no—
163#407—1061no—
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#324—532no—
173#323—1061no—
174C53#1178011yescharacteristic, complemented, Sylow
175C22×D4—531no—
176C23×C4—21no—
177C23×C4—531no—
178C23×C4C2×#615711yescharacteristic
179C2×M16—1062no—
180C2×M16—2121no—
181#123—2124no—
182C4×D4—2121no—
183C4×D4—2121no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—531no—
188C2×C4⋊C4—1062no—
189C2×C4⋊C4—1061no—
190C2×C4⋊C4C2×#615712yes—
191C2×C22⋊C4—531no—
192C2×C22⋊C4—531no—
193C2×C22⋊C4—1062no—
194C2×C22⋊C4—1061no—
195C2×C42—531no—
196#100—1064no—
197C22⋊C8—1062no—
198C22⋊C8—2121no—
199#97—2121no—
200C4×C8—2121no—
201#95—532no—
202#95—1062no—
203#95—1062no—
204C24—531no—
205C24C22⋊#615711yescharacteristic
206C2×D4—1062no—
207C2×D4—1062no—
208C2×D4—2121no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—1061no—
220C22×C4—1061no—
221C22×C4—1062no—
222C22×C4—2122no—
223C22×C4C22⋊#615711yescharacteristic
224C22×C4C22×#615711yescharacteristic
225M16—1064no—
226M16—4242no—
227M16—2122no—
228C2×C8—2122no—
229C2×C8—2122no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—2121no—
234C4⋊C4—2122no—
235C4⋊C4—2124no—
236C22⋊C4—2121no—
237C22⋊C4—2124no—
238C22⋊C4—2121no—
239C22⋊C4—2122no—
240C42—1061no—
241C42—1061no—
242C23—21no—
243C23—21no—
244C23—41no—
245C23—41no—
246C23—1061no—
247C23—1061no—
248C23—1061no—
249C23—2121no—
250C23—2121no—
251C23#1313328112yes—
252C23C2×C22⋊#615711yescharacteristic
253D4—2124no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—84no—
263C2×C4—532no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—4241no—
269C2×C4—2122no—
270C2×C4—2122no—
271C2×C4—2122no—
272C2×C4—2122no—
273C2×C4—2121no—
274C2×C4—2121no—
275C2×C4#1313327511yescharacteristic
276C2×C4#1313327511yescharacteristic
277C8—4242no—
278C8—2124no—
279C22—21no—
280C22—21no—
281C22—22no—
282C22—42no—
283C22—42no—
284C22—41no—
285C22—41no—
286C22—4241no—
287C22—2122no—
288C22—2121no—
289C22—2121no—
290C22—2121no—
291C22C2×#1313327511yescharacteristic
292C22#1681161511yescharacteristic
293C22C2×#1313328111yescharacteristic
294C4—21no—
295C4—41no—
296C4—82no—
297C4—1062no—
298C4—4242no—
299C4—2121no—
300C4—2121no—
301C4#1681152512yes—
302C2—21no—
303C2—21no—
304C2—42no—
305C2—4241no—
306C2#2029011111yescharacteristic
307C2C2×#1681152511yescharacteristic
308C2#2029101711yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53C2×#99
prime radicalC22×C22⋊D53D4
cosocleC2×#11719919C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#11780
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20289699C2semidirect
174. C53#11780semidirect

Maximal subgroups

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

subgroupindex
C4×#131332772
#202899702
#202896992
#202790172
#1178053