Finite Group Databasebeta

← #28301539

Subgroups — #28301539

Order 13568.

698 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20290441C212yesmaximal
2C2×#16811469C212yesmaximal
3#20289553C212yesmaximal
4C2×C4×#11719919C211yesmaximal, characteristic
5C2×C22⋊#88324C2214yes—
6C2×#13133272—22no—
7#16811489—22no—
8#16811469C418yescomplemented
9C22×#11719919C411yescharacteristic
10C22×#11719919C2211yescharacteristic
11C22×C4×Dic53C411yescharacteristic
12C22×C4×Dic53C2211yescharacteristic
13C4×#11719919—22no—
14C4×#11719919—22no—
15C2×#13133124C411yescharacteristic
16C2×#13133124C2211yescharacteristic
17C22×C42×C53C411yescharacteristic
18C22×#88324—24no—
19C22⋊#88324—24no—
20C22⋊#88324C2×C418yes—
21#13133272—44no—
22C23×Dic53C2×C411yescharacteristic
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2311yescharacteristic
26C2×#11719919—22no—
27C2×#11719919—22no—
28C2×#11719919—22no—
29C2×#11719919—22no—
30C2×#11719919—22no—
31C2×#11719919C2×C414yes—
32C2×C4×Dic53—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53—22no—
35C2×C4×Dic53—22no—
36C2×C4×Dic53—42no—
37C2×C4×Dic53D412yes—
38C2×C4×Dic53D412yes—
39#13133124—22no—
40#13133124—22no—
41C23×C4×C53C2×C411yescharacteristic
42C23×C4×C53C2×C411yescharacteristic
43C23×C4×C53C2×C411yescharacteristic
44C2×C42×C53—21no—
45C2×C42×C53—21no—
46C2×C42×C53—42no—
47C24×C53C22×C411yescharacteristic
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—41no—
57C22×C4×C53—41no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53C4212yes—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53M1612yes—
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—41no—
73C22×Dic53—41no—
74C22×Dic53—41no—
75C22×Dic53—41no—
76C22×Dic53—41no—
77C22×Dic53—41no—
78C22×Dic53C4212yes—
79C22×Dic53C22⋊C412yes—
80C22×Dic53M1612yes—
81C22×Dic53M1612yes—
82C22×Dic53M1612yes—
83C22×Dic53C22×C412yes—
84C22×Dic53C2×D412yes—
85C22×Dic53#4112yes—
86C2×#88324—28no—
87C2×#88324—44no—
88C2×#88324—44no—
89C42×C53—42no—
90C42×C53—42no—
91#11719919—28no—
92#11719919—28no—
93C4×Dic53—24no—
94C4×Dic53—24no—
95C4×Dic53—24no—
96C4×Dic53—42no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53C2×C4211yescharacteristic
102C23×C53C2×C22⋊C411yescharacteristic
103C23×C53#11711yescharacteristic
104C23×C53C2×M1611yescharacteristic
105C23×C53C2×M1611yescharacteristic
106C23×C53C2×M1611yescharacteristic
107C23×C53C2×M1611yescharacteristic
108C2×C4×C53—24no—
109C2×C4×C53—24no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53#9912yes—
125C2×C4×C53#10112yes—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53—22no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—22no—
132C2×Dic53—22no—
133C2×Dic53—22no—
134C2×Dic53—22no—
135C2×Dic53—22no—
136C2×Dic53—22no—
137C2×Dic53—22no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—42no—
146C2×Dic53#9714yescomplemented
147C2×Dic53C4×D414yes—
148#88324—48no—
149#8153—531nomaximal, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53C4×C22⋊C411yescharacteristic
163C22×C53C2×#9711yescharacteristic
164C22×C53C4×M1611yescharacteristic
165C22×C53#40711yescharacteristic
166C22×C53#40711yescharacteristic
167C22×C53C2×#9911yescharacteristic
168C22×C53#41011yescharacteristic
169C22×C53C2×#10111yescharacteristic
170C22×C53#41311yescharacteristic
171C22×C53#43211yescharacteristic
172C22×C53#43211yescharacteristic
173C4×C53—22no—
174C4×C53—22no—
175C4×C53—42no—
176C4×C53—42no—
177Dic53—24no—
178Dic53—24no—
179Dic53—42no—
180Dic53—42no—
181C2×C4×C4⋊C4—531no—
182#1749—532no—
183C2×#324—532no—
184#1280—532no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53#146311yescharacteristic
190C2×C53#146411yescharacteristic
191C2×C53#146511yescharacteristic
192C2×C53#146711yescharacteristic
193C2×C53#174911yescharacteristic
194C2×C53C4×#9911yescharacteristic
195C2×C53C4×#10111yescharacteristic
196C22×C4⋊C4—531no—
197C22×C4⋊C4—531no—
198C22×C42—531no—
199C22×C42—531no—
200C22×C42#615711yescharacteristic
201C2×C22⋊C8—534no—
202C2×#97—1062no—
203C4×C4⋊C4—1062no—
204C4×C4⋊C4—1062no—
205C2×#95—531no—
206C2×#95—531no—
207#336—1062no—
208#324—538no—
209C53#815311yescharacteristic, complemented, Sylow
210C23×C4—531no—
211C23×C4—531no—
212C23×C4—531no—
213C23×C4—531no—
214C23×C4C2×#615711yescharacteristic
215C23×C4C2×#615711yescharacteristic
216C23×C4C2×#615711yescharacteristic
217C22×C8—1064no—
218C2×C4⋊C4—534no—
219C2×C4⋊C4—1062no—
220C2×C4⋊C4—1062no—
221C2×C4⋊C4—1062no—
222C2×C4⋊C4—1062no—
223C2×C4⋊C4—1062no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—42no—
227C2×C42—532no—
228C2×C42—532no—
229C2×C42—1062no—
230C2×C42—1062no—
231C2×C42—1061no—
232C2×C42—1061no—
233C2×C42—2122no—
234C22⋊C8—538no—
235C22⋊C8—1064no—
236#97—2124no—
237#95—1062no—
238#95—1062no—
239C24C22×#615711yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—532no—
253C22×C4—532no—
254C22×C4—532no—
255C22×C4—532no—
256C22×C4—532no—
257C22×C4—532no—
258C22×C4—532no—
259C22×C4—532no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4—2121no—
269C22×C4—2121no—
270C22×C4—2121no—
271C22×C4—2121no—
272C22×C4—2121no—
273C22×C4—2121no—
274C22×C4#1171993612yes—
275C22×C4C4×#615712yes—
276C22×C4C22⋊#615711yescharacteristic
277C22×C4C22⋊#615711yescharacteristic
278C2×C8—1068no—
279C2×C8—2124no—
280C2×C8—2124no—
281C4⋊C4—1068no—
282C4⋊C4—1068no—
283C42—42no—
284C42—42no—
285C42—1064no—
286C42—1064no—
287C42—1064no—
288C42—2122no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23C2×C4×#615711yescharacteristic
294C23C2×C22⋊#615711yescharacteristic
295C23#1313329611yescharacteristic
296C23D53⋊M1611yescharacteristic
297C23D53⋊M1611yescharacteristic
298C23D53⋊M1611yescharacteristic
299C23C2×#1171993611yescharacteristic
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—534no—
317C2×C4—534no—
318C2×C4—1062no—
319C2×C4—1062no—
320C2×C4—1062no—
321C2×C4—1062no—
322C2×C4—1062no—
323C2×C4—1062no—
324C2×C4—1062no—
325C2×C4—1062no—
326C2×C4—1062no—
327C2×C4—1062no—
328C2×C4—1062no—
329C2×C4—1062no—
330C2×C4—2122no—
331C2×C4—2122no—
332C2×C4—2122no—
333C2×C4—2122no—
334C2×C4—2122no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2122no—
338C2×C4#1313327512yes—
339C2×C4#1313327812yes—
340C8—2128no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22C4×C22⋊#615711yescharacteristic
354C22#1681158211yescharacteristic
355C22C4×#1171993611yescharacteristic
356C22#1681159411yescharacteristic
357C22#1681159511yescharacteristic
358C22C2×#1313327511yescharacteristic
359C22#1681160211yescharacteristic
360C22C2×#1313327811yescharacteristic
361C22#1681160911yescharacteristic
362C22#1681164511yescharacteristic
363C22#1681164611yescharacteristic
364C4—22no—
365C4—22no—
366C4—42no—
367C4—42no—
368C4—1064no—
369C4—1064no—
370C4—2122no—
371C4—2122no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2#2028996511yescharacteristic
377C2#2028996811yescharacteristic
378C2#2028997011yescharacteristic
379C2#2028997711yescharacteristic
380C2#2029044211yescharacteristic
381C2C4×#1313327511yescharacteristic
382C2C4×#1313327811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C2×C42
FrattiniC24C22×#6157
FittingC22×C42×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22×C4
cosocleC23×Dic53C23
hypercenterC22×C42#6157
hypocenterC53#8153
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
8. #16811469C4semidirect
146. C2×Dic53#97semidirect
209. C53#8153semidirect

Maximal subgroups

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

subgroupindex
#202904412
C2×#168114692
#202895532
C2×C4×#117199192
#815353