Finite Group Databasebeta

← #28301667

Subgroups — #28301667

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊#88324C211yesmaximal, characteristic, complemented
2#20290430C211yesmaximal, characteristic, complemented
3C2×#16811471C212yesmaximal
4#20289551C212yesmaximal, complemented
5#20277173C211yesmaximal, characteristic
6C2×C4⋊#88324C2212yes—
7C2×C4×#88324—21no—
8C2×C4×#88324C2212yes—
9#16811489—21no—
10#16811489—21no—
11#16811489—21no—
12#16811471C418yescomplemented
13C2×#13133179C2211yescharacteristic
14C2×#13133175—21no—
15#16810525C411yescharacteristic
16#16810524C411yescharacteristic
17#16810522C411yescharacteristic
18#16810515—21no—
19#16810513—21no—
20C4⋊C4×Dic53C2211yescharacteristic
21#16810470—21no—
22C42×Dic53C2211yescharacteristic
23C53×#389C411yescharacteristic
24C22×#88324—22no—
25C22×#88324—22no—
26C4⋊#88324—22no—
27C4⋊#88324C2×C414yes—
28C4×#88324—22no—
29C4×#88324—42no—
30C4×#88324C2×C414yes—
31C22×D212C2×C411yescharacteristic
32C22×C4×D53—21no—
33#13133179—22no—
34#13133179C2×C414yes—
35#13133175—42no—
36#13133175—42no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43C2×#11719921—21no—
44C2×C4×Dic53—21no—
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53D411yescharacteristic
49C2×C4×Dic53D411yescharacteristic
50C2×C4×Dic53C2311yescharacteristic
51#13133124—21no—
52#13133124—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C4⋊C4×C53C2×C411yescharacteristic
56C2×C42×C53C2×C411yescharacteristic
57#95×C53—21no—
58C23×D53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—21no—
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C2×D212—22no—
70C2×D212—22no—
71C2×D212—21no—
72C2×D212C4212yes—
73C2×C4×D53—24no—
74C2×C4×D53—41no—
75C2×C4×D53—41no—
76C2×C4×D53—42no—
77C2×#88324—24no—
78C2×#88324—24no—
79C2×#88324—42no—
80C2×#88324—42no—
81C2×#88324—42no—
82C2×#88324—42no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—42no—
86C4⋊C4×C53C4212yes—
87C42×C53—22no—
88#11719921—24no—
89#11719921—42no—
90#11719921—42no—
91#11719921—42no—
92#11719921—42no—
93C4⋊Dic53—22no—
94C4⋊Dic53—42no—
95C4×Dic53—22no—
96C4×Dic53—22no—
97C4×Dic53—41no—
98C4×Dic53—41no—
99C4×Dic53—41no—
100C4×Dic53—41no—
101C4×Dic53D812yes—
102C4×Dic53SD1612yes—
103C4×Dic53C22×C412yes—
104C4×Dic53#4112yes—
105C23×C53C2×C22⋊C411yescharacteristic
106C22×D53—22no—
107C22×D53—41no—
108C22×D53—41no—
109C22×D53—41no—
110C22×D53—41no—
111C22×D53—41no—
112C22×D53—41no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—22no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53#10212yes—
125C2×C4×C53C2×C4211yescharacteristic
126C2×C4×C53#11711yescharacteristic
127C2×C4×C53#13112yes—
128C2×C4×C53#13112yes—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53#10212yes—
140C2×Dic53C4×D412yes—
141C2×Dic53C4×D412yes—
142C2×Dic53C2×D811yescharacteristic
143C2×Dic53C2×SD1611yescharacteristic
144C2×Dic53#13512yes—
145D212—24no—
146D212—42no—
147C4×D53—44no—
148C4×D53—44no—
149#88324—44no—
150#88324—44no—
151#8183—531nomaximal, Sylow
152C22×C53C4×C22⋊C411yescharacteristic
153C22×C53#40811yescharacteristic
154C22×C53#40811yescharacteristic
155C22×C53C2×#10211yescharacteristic
156C22×C53#41611yescharacteristic
157C22×C53#42111yescharacteristic
158C22×C53#42111yescharacteristic
159C2×D53—42no—
160C2×D53—42no—
161C2×D53—42no—
162C2×D53—81no—
163C2×D53—81no—
164C2×D53—81no—
165C2×D53—81no—
166C4×C53—22no—
167C4×C53—41no—
168C4×C53—42no—
169C4×C53—81no—
170C4×C53#40512yes—
171C4×C53#43312yes—
172Dic53—22no—
173Dic53—22no—
174Dic53—41no—
175Dic53—81no—
176Dic53C4×D812yes—
177Dic53C4×SD1612yes—
178#2293—531no—
179C4×C4⋊C8—531no—
180#1744—531no—
181C2×D4⋊C8—532no—
182#1279—532no—
183C2×C53#148011yescharacteristic
184C2×C53#148011yescharacteristic
185C2×C53#148211yescharacteristic
186C2×C53#148211yescharacteristic
187C2×C53#175011yescharacteristic
188C2×C53#175911yescharacteristic
189C2×C53C4×#10211yescharacteristic
190D53—82no—
191C2×C4×D4—531no—
192C2×#117—1061no—
193C2×C4⋊C8—532no—
194C2×C4×C8—532no—
195C2×C4×C8—1061no—
196#390—531no—
197#390—531no—
198#389—531no—
199#389#615711yescharacteristic
200#388—1061no—
201#388—1061no—
202C4×C4⋊C4—531no—
203C4×C22⋊C4—1061no—
204C43—531no—
205#336—1061no—
206#336—1061no—
207#336—1061no—
208D4⋊C8—538no—
209C53#818311yescharacteristic, complemented, Sylow
210C22×D4—531no—
211C23×C4—1061no—
212C22×C8—1062no—
213C22×C8—1062no—
214C4×D4—534no—
215C4×D4—1062no—
216#117—2122no—
217#117—2122no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4C2×#615711yescharacteristic
222C2×C4⋊C4C2×#615711yescharacteristic
223C2×C22⋊C4—1061no—
224C2×C22⋊C4—1061no—
225C2×C22⋊C4—1061no—
226C2×C22⋊C4—1061no—
227C2×C22⋊C4—1061no—
228C2×C42—531no—
229C2×C42—531no—
230C2×C42—531no—
231C2×C42—531no—
232C2×C42—531no—
233C2×C42—531no—
234C2×C42—1061no—
235C2×C42C2×#615711yescharacteristic
236C4⋊C8—534no—
237C4⋊C8—1062no—
238C4×C8—534no—
239C4×C8—1062no—
240C4×C8—2122no—
241#95—21no—
242#95—1061no—
243#95—1061no—
244C24—1061no—
245C2×D4—532no—
246C2×D4—1061no—
247C2×D4—1062no—
248C2×D4—1062no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—531no—
254C22×C4—531no—
255C22×C4—1064no—
256C22×C4—1061no—
257C22×C4—2121no—
258C22×C4—2121no—
259C22×C4—2122no—
260C22×C4C22⋊#615711yescharacteristic
261C22×C4C22⋊#615711yescharacteristic
262C22×C4C22×#615711yescharacteristic
263C2×C8—1064no—
264C2×C8—1064no—
265C2×C8—2122no—
266C2×C8—2122no—
267C2×C8—2122no—
268C2×C8—2122no—
269C4⋊C4—21no—
270C4⋊C4—22no—
271C4⋊C4—42no—
272C4⋊C4—1062no—
273C4⋊C4—2122no—
274C4⋊C4C4×#615712yes—
275C22⋊C4—1064no—
276C22⋊C4—2122no—
277C22⋊C4—2122no—
278C22⋊C4—2122no—
279C22⋊C4—2122no—
280C42—22no—
281C42—532no—
282C42—532no—
283C42—532no—
284C42—532no—
285C42—1062no—
286C42—1062no—
287C42—2121no—
288C42—2121no—
289C42—2121no—
290C42—2121no—
291C23—1062no—
292C23—2121no—
293C23—2121no—
294C23—2121no—
295C23—2121no—
296C23—2121no—
297C23—2121no—
298C23C2×C22⋊#615711yescharacteristic
299D4—1064no—
300D4—2122no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—532no—
313C2×C4—531no—
314C2×C4—531no—
315C2×C4—532no—
316C2×C4—532no—
317C2×C4—532no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—2124no—
326C2×C4—2124no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4#1313327912yes—
331C2×C4C2×C4×#615711yescharacteristic
332C2×C4#1313329611yescharacteristic
333C2×C4#1313330412yes—
334C2×C4#1313330512yes—
335C8—2124no—
336C8—2124no—
337C22—4241no—
338C22—4241no—
339C22—4241no—
340C22—4241no—
341C22—2122no—
342C22—2122no—
343C22—2122no—
344C22C4×C22⋊#615711yescharacteristic
345C22#1681159611yescharacteristic
346C22#1681159811yescharacteristic
347C22C2×#1313327911yescharacteristic
348C22#1681161311yescharacteristic
349C22#1681162011yescharacteristic
350C22#1681162211yescharacteristic
351C4—22no—
352C4—42no—
353C4—41no—
354C4—81no—
355C4—532no—
356C4—532no—
357C4—1062no—
358C4—1062no—
359C4—4241no—
360C4—2121no—
361C4#1681158812yes—
362C4#1681164812yes—
363C2—4242no—
364C2#2029002011yescharacteristic
365C2#2029002211yescharacteristic
366C2#2029002811yescharacteristic
367C2#2029002911yescharacteristic
368C2#2029044411yescharacteristic
369C2#2029047911yescharacteristic
370C2C4×#1313327911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#8183
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊#88324C2semidirect
2. #20290430C2semidirect
4. #20289551C2semidirect
12. #16811471C4semidirect
209. C53#8183semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#883242
#202904302
C2×#168114712
#202895512
#202771732
#818353