Finite Group Databasebeta

← #28308171

Subgroups — #28308171

Order 13568.

814 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 88 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#20289643C214yesmaximal, complemented
3C2×#16810526C211yesmaximal, characteristic
4#16811593—22no—
5#16811468—22no—
6#16811468C2214yescomplemented
7C22×C4⋊Dic53C2211yescharacteristic
8C22×C22⋊Dic53—22no—
9C22×C4×Dic53C411yescharacteristic
10#16810526—22no—
11#16810526C412yes—
12#16810526C2212yes—
13C23×D4×C53C411yescharacteristic
14C2×#11719940—42no—
15C22⋊#88324—24no—
16C22⋊#88324—42no—
17C23×Dic53—21no—
18C23×Dic53D412yes—
19C2×C4⋊Dic53—41no—
20C2×C4⋊Dic53—41no—
21C2×C4⋊Dic53C2×C411yescharacteristic
22C2×C4⋊Dic53D412yes—
23C2×C4⋊Dic53C2311yescharacteristic
24C2×C22⋊Dic53—24no—
25C2×C22⋊Dic53—24no—
26C2×C22⋊Dic53—48no—
27C2×C4×Dic53—21no—
28C2×C4×Dic53—42no—
29C2×C4×Dic53C2×C412yes—
30C25×C53—21no—
31C22×D4×C53—24no—
32C22×D4×C53—21no—
33C22×D4×C53—41no—
34C22×D4×C53—41no—
35C22×D4×C53—42no—
36C22×D4×C53C2×C412yes—
37C23×C4×C53C2×C411yescharacteristic
38C24×C53—24no—
39C24×C53—22no—
40C24×C53—42no—
41C24×C53—44no—
42C24×C53—44no—
43C24×C53—41no—
44C24×C53—41no—
45C24×C53C22⋊C411yescharacteristic
46C2×D4×C53—24no—
47C2×D4×C53—22no—
48C2×D4×C53—48no—
49C2×D4×C53—48no—
50C2×D4×C53—42no—
51C2×D4×C53—42no—
52C2×D4×C53—42no—
53C2×D4×C53—42no—
54C2×D4×C53M1614yescomplemented
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×Dic53—24no—
62C22×Dic53—22no—
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—42no—
66C22×Dic53—42no—
67C22×Dic53—44no—
68C22×Dic53C22⋊C412yes—
69C22×Dic53C2×D412yes—
70#11719940—84no—
71C2×#88324—44no—
72C22⋊Dic53—48no—
73C22⋊Dic53—48no—
74C4⋊Dic53—41no—
75C4⋊Dic53—42no—
76C4⋊Dic53—41no—
77C4×Dic53—42no—
78C4×Dic53—42no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—21no—
82C23×C53—21no—
83C23×C53—24no—
84C23×C53—44no—
85C23×C53—44no—
86C23×C53—416no—
87C23×C53—44no—
88C23×C53—44no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53—88no—
92C23×C53—82no—
93C23×C53—82no—
94C23×C53—82no—
95C23×C53—82no—
96C23×C53#9911yescharacteristic
97C23×C53#9911yescharacteristic
98C23×C53C2×C22⋊C411yescharacteristic
99D4×C53—44no—
100D4×C53—48no—
101D4×C53—44no—
102C2×C4×C53—24no—
103C2×C4×C53—24no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53#10012yes—
109C2×C4×C53C2×M1612yes—
110C2×Dic53—24no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—48no—
114C2×Dic53—42no—
115C2×Dic53—44no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—82no—
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—48no—
131C22×C53—44no—
132C22×C53—84no—
133C22×C53—88no—
134C22×C53—88no—
135C22×C53—82no—
136C22×C53—82no—
137C22×C53#35312yes—
138C22×C53#35512yes—
139C22×C53#40711yescharacteristic
140C22×C53C2×#9911yescharacteristic
141C22×C53C2×#10011yescharacteristic
142C4×C53—22no—
143C4×C53—22no—
144Dic53—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—84no—
155C2×C53#146211yescharacteristic
156C2×C53C2×#35311yescharacteristic
157C2×C53C2×#35511yescharacteristic
158C23×D4#615711yescharacteristic
159C22×C4⋊C4—531no—
160C22×C22⋊C4—1062no—
161C22×C42—531no—
162#407—1062no—
163#390—532no—
164#390—532no—
165#390—1062no—
166#324—534no—
167#324—1062no—
168C53#1150511yescharacteristic, complemented, Sylow
169C25—21no—
170C22×D4—24no—
171C22×D4—21no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4—42no—
175C22×D4C2×#615712yes—
176C23×C4—532no—
177C23×C4—1061no—
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—532no—
189C2×C42—1061no—
190C2×C42—2122no—
191C22⋊C8—1064no—
192C22⋊C8—2122no—
193C24—24no—
194C24—22no—
195C24—44no—
196C24—42no—
197C24—44no—
198C24—41no—
199C24—41no—
200C24C22⋊#615711yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—48no—
206C2×D4—48no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4#1171994014yescomplemented
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—532no—
215C22×C4—532no—
216C22×C4—1062no—
217C22×C4—1064no—
218C22×C4—1062no—
219C22×C4—1062no—
220C22×C4—2124no—
221C22×C4—2122no—
222C22×C4—2122no—
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—2122no—
233C42—2122no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—24no—
239C23—44no—
240C23—44no—
241C23—44no—
242C23—416no—
243C23—44no—
244C23—42no—
245C23—42no—
246C23—88no—
247C23—82no—
248C23—82no—
249C23—82no—
250C23—82no—
251C23#1313327511yescharacteristic
252C23#1313327511yescharacteristic
253C23C2×C22⋊#615711yescharacteristic
254D4—44no—
255D4—48no—
256D4—44no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—1064no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—4242no—
267C2×C4—2128no—
268C2×C4—2122no—
269C2×C4—2124no—
270C2×C4—2122no—
271C2×C4—2122no—
272C2×C4—2122no—
273C2×C4#1313327712yes—
274C2×C4C2×#1171994012yes—
275C8—4244no—
276C22—24no—
277C22—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—48no—
285C22—44no—
286C22—84no—
287C22—88no—
288C22—88no—
289C22—82no—
290C22—82no—
291C22#1681152712yes—
292C22#1681153112yes—
293C22#1681159311yescharacteristic
294C22C2×#1313327511yescharacteristic
295C22C2×#1313327711yescharacteristic
296C4—22no—
297C4—22no—
298C4—2124no—
299C4—2122no—
300C4—2122no—
301C2—21no—
302C2—21no—
303C2—22no—
304C2—22no—
305C2—84no—
306C2#2028996011yescharacteristic
307C2C2×#1681152711yescharacteristic
308C2C2×#1681153111yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC23×D4×C53C4
socleC22×C53C2×#99
cosocleC2×C4⋊Dic53C23
hypercenterC23×D4#6157
hypocenterC53#11505
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #20289963C2semidirect
2. #20289643C2semidirect
6. #16811468C22semidirect
54. C2×D4×C53M16semidirect
168. C53#11505semidirect
209. C2×D4#11719940semidirect

Maximal subgroups

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

subgroupindex
#202899632
#202896432
C2×#168105262
#1150553