Finite Group Databasebeta

← #28303349

Subgroups — #28303349

Order 13568.

554 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 112 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#20290441C211yesmaximal, characteristic
2#20290441C211yesmaximal, characteristic
3#20289591C214yesmaximal, complemented
4C2×#16810488C211yesmaximal, characteristic
5C2×C22⋊#88324—21no—
6C2×C22⋊#88324—21no—
7C2×#13133272C2212yes—
8C2×#13133272C2212yes—
9#16811489—21no—
10#16811489—21no—
11C22×C4⋊Dic53C411yescharacteristic
12C22×C4×Dic53C2211yescharacteristic
13#16810488—21no—
14#16810488—21no—
15#16810488C412yes—
16#16810488C2212yes—
17C2×#13133124—21no—
18C2×#13133124—21no—
19C22×C4⋊C4×C53C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22C22⋊#88324—24no—
23C22⋊#88324—24no—
24#13133272—24no—
25#13133272—24no—
26C23×Dic53—21no—
27C23×Dic53D411yescharacteristic
28C23×Dic53D411yescharacteristic
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—42no—
31C2×C4⋊Dic53C2×C412yes—
32C2×C4×Dic53—24no—
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53D411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53C2311yescharacteristic
37#13133124—22no—
38#13133124—22no—
39#13133124—22no—
40#13133124—22no—
41C23×C4×C53—21no—
42C23×C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—42no—
45C2×C4⋊C4×C53C2×C412yes—
46C24×C53C22⋊C411yescharacteristic
47C22×C4×C53—22no—
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—42no—
52C22×C4×C53—42no—
53C22×C4×C53—42no—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22×C411yescharacteristic
56C22×Dic53—22no—
57C22×Dic53—22no—
58C22×Dic53—22no—
59C22×Dic53—22no—
60C22×Dic53—22no—
61C22×Dic53—41no—
62C22×Dic53—41no—
63C22×Dic53—42no—
64C22×Dic53—42no—
65C22×Dic53—42no—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C22×Dic53C2×D411yescharacteristic
70C2×#88324—24no—
71C2×#88324—24no—
72C2×#88324—42no—
73C2×#88324—42no—
74C2×#88324—42no—
75C2×#88324—42no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—42no—
78C4⋊Dic53—42no—
79C4⋊Dic53—42no—
80C4×Dic53—22no—
81C4×Dic53—22no—
82C4×Dic53—22no—
83C4×Dic53#4114yes—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53#10112yes—
89C23×C53#10412yes—
90C23×C53#10412yes—
91C23×C53C2×C22⋊C411yescharacteristic
92C2×C4×C53—28no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—84no—
101C2×C4×C53#10012yes—
102C2×C4×C53#11712yes—
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—44no—
114C2×Dic53—44no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—84no—
119C2×Dic53#12312yes—
120C2×Dic53#12312yes—
121#88324—44no—
122#88324—44no—
123#8719—531nomaximal, Sylow
124C22×C53—22no—
125C22×C53—22no—
126C22×C53—22no—
127C22×C53—22no—
128C22×C53—22no—
129C22×C53—22no—
130C22×C53#33014yes—
131C22×C53#38111yescharacteristic
132C22×C53C2×#10011yescharacteristic
133C22×C53C2×#10111yescharacteristic
134C22×C53C2×#10411yescharacteristic
135C22×C53C2×#10411yescharacteristic
136C22×C53#42111yescharacteristic
137C22×C53#42111yescharacteristic
138C4×C53—22no—
139C4×C53—22no—
140C4×C53—82no—
141Dic53—22no—
142Dic53—22no—
143Dic53—41no—
144Dic53—41no—
145Dic53—82no—
146C2×#382—531no—
147#1749—531no—
148#1749—531no—
149#1295—534no—
150C2×C53—24no—
151C2×C53C2×#33011yescharacteristic
152C2×C53#152711yescharacteristic
153C2×C53#152711yescharacteristic
154C2×C53#153411yescharacteristic
155C2×C53#182111yescharacteristic
156C2×C53#182611yescharacteristic
157C2×C53#182611yescharacteristic
158C22×C4⋊C4—531no—
159C22×C4⋊C4#615711yescharacteristic
160C22×C42—531no—
161C2×C22⋊C8—1061no—
162C2×C22⋊C8—1061no—
163C2×#97—532no—
164C2×#97—532no—
165#382—532no—
166#382—532no—
167#382—1061no—
168#382—1061no—
169C2×#95—1061no—
170C2×#95—1061no—
171#336—1061no—
172#336—1061no—
173C53#871911yescharacteristic, complemented, Sylow
174C23×C4—21no—
175C23×C4—531no—
176C23×C4—531no—
177C23×C4—1061no—
178C23×C4C2×#615711yescharacteristic
179C22×C8—1062no—
180C22×C8—1062no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—532no—
184C2×C4⋊C4—1061no—
185C2×C4⋊C4—2122no—
186C2×C4⋊C4C2×#615712yes—
187C2×C42—531no—
188C2×C42—531no—
189C2×C42—531no—
190C2×C42—531no—
191C2×C42—1064no—
192C22⋊C8—1064no—
193C22⋊C8—1064no—
194#97—1064no—
195#97—1064no—
196#95—1062no—
197#95—1062no—
198#95—1062no—
199#95—1062no—
200C24C22⋊#615711yescharacteristic
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—531no—
209C22×C4—531no—
210C22×C4—531no—
211C22×C4—531no—
212C22×C4—1062no—
213C22×C4—1062no—
214C22×C4—1062no—
215C22×C4—1062no—
216C22×C4—1062no—
217C22×C4—2121no—
218C22×C4—2121no—
219C22×C4—2122no—
220C22×C4—2122no—
221C22×C4—2122no—
222C22×C4C22⋊#615711yescharacteristic
223C22×C4C22×#615711yescharacteristic
224C2×C8—1064no—
225C2×C8—1064no—
226C2×C8—2122no—
227C2×C8—2122no—
228C2×C8—2122no—
229C2×C8—2122no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—2122no—
233C4⋊C4—2122no—
234C42—534no—
235C42—1062no—
236C42—1062no—
237C42—1062no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23#1313327812yes—
243C23#1313328112yes—
244C23#1313328212yes—
245C23C2×C22⋊#615711yescharacteristic
246C2×C4—28no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—84no—
255C2×C4—532no—
256C2×C4—532no—
257C2×C4—1061no—
258C2×C4—1061no—
259C2×C4—1061no—
260C2×C4—1061no—
261C2×C4—1061no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—4244no—
268C2×C4—2124no—
269C2×C4—2124no—
270C2×C4—2122no—
271C2×C4—2122no—
272C2×C4—2122no—
273C2×C4#1313327712yes—
274C2×C4#1313329612yes—
275C8—2124no—
276C8—2124no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22#1681148014yes—
284C22#1681156611yescharacteristic
285C22C2×#1313327711yescharacteristic
286C22C2×#1313327811yescharacteristic
287C22C2×#1313328111yescharacteristic
288C22C2×#1313328211yescharacteristic
289C22#1681162111yescharacteristic
290C22#1681162211yescharacteristic
291C4—22no—
292C4—22no—
293C4—82no—
294C4—1062no—
295C4—1062no—
296C4—4242no—
297C4—2121no—
298C4—2121no—
299C2—24no—
300C2C2×#1681148011yescharacteristic
301C2#2029017411yescharacteristic
302C2#2029017611yescharacteristic
303C2#2029019711yescharacteristic
304C2#2029061411yescharacteristic
305C2#2029062111yescharacteristic
306C2#2029062211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#8719
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20289591C2semidirect
173. C53#8719semidirect

Maximal subgroups

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

subgroupindex
#202904412
#202895912
C2×#168104882
#871953