Finite Group Databasebeta

← #28308173

Subgroups — #28308173

Order 13568.

814 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 84 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#20289963C212yesmaximal, complemented
2#20289644C214yesmaximal, complemented
3C2×#16810525C211yesmaximal, characteristic
4#16811593—22no—
5#16811468—22no—
6#16811468C2214yescomplemented
7C23×D212C411yescharacteristic
8C22×C4⋊Dic53C2211yescharacteristic
9C22×#11719921—22no—
10#16810525—22no—
11#16810525C412yes—
12#16810525C2212yes—
13C22×C42×C53C411yescharacteristic
14C2×#11719940—42no—
15C22⋊#88324—24no—
16C22⋊#88324—42no—
17C24×D53—21no—
18C22×D212—21no—
19C22×D212—24no—
20C22×D212—41no—
21C22×D212—41no—
22C22×D212—42no—
23C22×D212C2×C412yes—
24C23×Dic53D412yes—
25C2×C4⋊Dic53—41no—
26C2×C4⋊Dic53—41no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53D412yes—
29C2×C4⋊Dic53C2311yescharacteristic
30C2×#11719921—24no—
31C2×#11719921—24no—
32C2×#11719921—48no—
33C23×C4×C53—21no—
34C23×C4×C53C2×C411yescharacteristic
35C2×C42×C53—21no—
36C2×C42×C53—42no—
37C2×C42×C53C2×C412yes—
38C24×C53C22⋊C411yescharacteristic
39C23×D53—24no—
40C23×D53—22no—
41C23×D53—44no—
42C23×D53—42no—
43C23×D53—44no—
44C23×D53—41no—
45C23×D53—41no—
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53—24no—
51C22×C4×C53—22no—
52C22×C4×C53—42no—
53C22×C4×C53—42no—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22×C411yescharacteristic
56C22×Dic53—22no—
57C22×Dic53—22no—
58C22×Dic53—44no—
59C22×Dic53C22⋊C412yes—
60C22×Dic53C2×D412yes—
61C2×D212—24no—
62C2×D212—22no—
63C2×D212—48no—
64C2×D212—48no—
65C2×D212—42no—
66C2×D212—42no—
67C2×D212—42no—
68C2×D212—42no—
69C2×D212M1614yescomplemented
70#11719940—84no—
71C2×#88324—44no—
72C42×C53—42no—
73C42×C53—42no—
74#11719921—48no—
75#11719921—48no—
76C4⋊Dic53—41no—
77C4⋊Dic53—42no—
78C4⋊Dic53—41no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—21no—
82C23×C53—21no—
83C23×C53#9911yescharacteristic
84C23×C53#9911yescharacteristic
85C23×C53C2×C22⋊C411yescharacteristic
86C22×D53—24no—
87C22×D53—44no—
88C22×D53—44no—
89C22×D53—416no—
90C22×D53—44no—
91C22×D53—44no—
92C22×D53—42no—
93C22×D53—42no—
94C22×D53—88no—
95C22×D53—82no—
96C22×D53—82no—
97C22×D53—82no—
98C22×D53—82no—
99C2×C4×C53—24no—
100C2×C4×C53—24no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—24no—
106C2×C4×C53—48no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53#10012yes—
110C2×C4×C53C2×M1612yes—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—44no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53—82no—
117D212—44no—
118D212—48no—
119D212—44no—
120#88324—84no—
121#11505—531nomaximal, Sylow
122C22×C53—24no—
123C22×C53—24no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53#35312yes—
131C22×C53#35512yes—
132C22×C53#40711yescharacteristic
133C22×C53C2×#9911yescharacteristic
134C22×C53C2×#10011yescharacteristic
135C2×D53—48no—
136C2×D53—44no—
137C2×D53—84no—
138C2×D53—88no—
139C2×D53—88no—
140C2×D53—82no—
141C2×D53—82no—
142C4×C53—22no—
143C4×C53—22no—
144C4×C53—44no—
145Dic53—42no—
146Dic53—42no—
147C2×#390—531no—
148#1463—532no—
149#1318—534no—
150C2×C53—21no—
151C2×C53—21no—
152C2×C53—22no—
153C2×C53—22no—
154C2×C53#146211yescharacteristic
155C2×C53C2×#35311yescharacteristic
156C2×C53C2×#35511yescharacteristic
157D53—84no—
158C23×D4—531no—
159C22×C4⋊C4—531no—
160C22×C22⋊C4—1062no—
161C22×C42#615711yescharacteristic
162#407—1062no—
163#390—532no—
164#390—532no—
165#390—1062no—
166#324—534no—
167#324—1062no—
168C53#1150511yescharacteristic, complemented, Sylow
169C25—1061no—
170C22×D4—532no—
171C22×D4—1064no—
172C22×D4—1061no—
173C22×D4—2121no—
174C22×D4—2121no—
175C22×D4—2122no—
176C23×C4—21no—
177C23×C4—532no—
178C23×C4C2×#615711yescharacteristic
179C2×M16—2122no—
180C2×C4⋊C4—532no—
181C2×C4⋊C4—531no—
182C2×C4⋊C4—531no—
183C2×C4⋊C4—2121no—
184C2×C4⋊C4—2121no—
185C2×C22⋊C4—1064no—
186C2×C22⋊C4—1064no—
187C2×C22⋊C4—2128no—
188C2×C42—21no—
189C2×C42—42no—
190C2×C42C2×#615712yes—
191C22⋊C8—1064no—
192C22⋊C8—2122no—
193C24—1064no—
194C24—1062no—
195C24—2124no—
196C24—2122no—
197C24—2124no—
198C24—2121no—
199C24—2121no—
200C24C22⋊#615711yescharacteristic
201C2×D4—534no—
202C2×D4—1064no—
203C2×D4—1062no—
204C2×D4—2122no—
205C2×D4—2122no—
206C2×D4—2128no—
207C2×D4—2128no—
208C2×D4—2122no—
209C2×D4—2122no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—532no—
219C22×C4—532no—
220C22×C4—1062no—
221C22×C4—1062no—
222C22×C4—2124no—
223C22×C4C22⋊#615711yescharacteristic
224C22×C4C22×#615711yescharacteristic
225M16—4244no—
226C2×C8—2124no—
227C4⋊C4—2122no—
228C4⋊C4—2121no—
229C4⋊C4—2121no—
230C22⋊C4—2128no—
231C22⋊C4—2128no—
232C42—42no—
233C42—42no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—1064no—
239C23—4248no—
240C23—4242no—
241C23—4242no—
242C23—4242no—
243C23—4242no—
244C23—2124no—
245C23—2124no—
246C23—2124no—
247C23—21216no—
248C23—2124no—
249C23—2122no—
250C23—2122no—
251C23#1313327611yescharacteristic
252C23#1313327611yescharacteristic
253C23C2×C22⋊#615711yescharacteristic
254D4—2124no—
255D4—2128no—
256D4—2124no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—24no—
264C2×C4—48no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—4242no—
270C2×C4—2124no—
271C2×C4—2122no—
272C2×C4—2122no—
273C2×C4#1313327712yes—
274C2×C4D53⋊M1612yes—
275C8—4244no—
276C22—24no—
277C22—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—4244no—
285C22—4248no—
286C22—4248no—
287C22—4242no—
288C22—4242no—
289C22—2128no—
290C22—2124no—
291C22#1681152812yes—
292C22#1681153212yes—
293C22#1681159511yescharacteristic
294C22C2×#1313327611yescharacteristic
295C22C2×#1313327711yescharacteristic
296C4—22no—
297C4—22no—
298C4—44no—
299C4—2122no—
300C4—2122no—
301C2—21no—
302C2—22no—
303C2—22no—
304C2—21no—
305C2—4244no—
306C2#2028996211yescharacteristic
307C2C2×#1681152811yescharacteristic
308C2C2×#1681153211yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×C42×C53C4
socleC22×C53C2×#99
prime radicalC23×D212C4
cosocleC2×C4⋊Dic53C23
hypercenterC22×C42#6157
hypocenterC53#11505
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
1. #20289963C2semidirect
2. #20289644C2semidirect
6. #16811468C22semidirect
69. C2×D212M16semidirect
168. C53#11505semidirect

Maximal subgroups

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

subgroupindex
#202899632
#202896442
C2×#168105252
#1150553