Finite Group Databasebeta

← #27272550

Subgroups — #27272550

Order 13568.

568 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283054C211yesmaximal, characteristic
2C2×#16810378C211yesmaximal, characteristic
3#20272540C212yesmaximal, complemented
4#20272537C212yesmaximal, complemented
5C53×#1728C211yesmaximal, characteristic
6C23×#88322—21no—
7C2×#13133196C411yescharacteristic
8#16810529C411yescharacteristic
9#16810529C2211yescharacteristic
10#16810521C411yescharacteristic
11#16810521C2211yescharacteristic
12#16810507—21no—
13#16810507—21no—
14#16810496—21no—
15#16810486—21no—
16#16810483C411yescharacteristic
17#16810378—21no—
18#16810378C2212yes—
19C22×M16×C53—21no—
20C22×C42×C53C2211yescharacteristic
21C53×#336—21no—
22C53×#336C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C22×#88322—24no—
25C22×#88322—22no—
26C22×#88322—42no—
27C22×#88322—42no—
28#13133196—42no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C2×M16×C53—24no—
48C2×M16×C53—42no—
49C2×M16×C53—42no—
50C22×C8×C53—22no—
51C2×C42×C53—42no—
52C2×C42×C53C2×C411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C2×C42×C53C2311yescharacteristic
56C24×C53C22⋊C411yescharacteristic
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—22no—
65C22×C4×C53—41no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53SD1612yes—
70C22×C4×C53Q1612yes—
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C2×Dic106—22no—
79C2×Dic106—41no—
80C2×Dic106—41no—
81M16×C53—48no—
82C2×C8×C53—24no—
83C2×C8×C53—42no—
84C2×C8×C53—42no—
85C42×C53—41no—
86C42×C53—41no—
87C42×C53—41no—
88C42×C53—41no—
89C22⋊Dic53—42no—
90C22⋊Dic53—42no—
91C4⋊Dic53—22no—
92C4⋊Dic53—41no—
93C4⋊Dic53—41no—
94#11719919—42no—
95#11719919—42no—
96#11719919—41no—
97#11719919—41no—
98C2×#88322—24no—
99C2×#88322—48no—
100C2×#88322—42no—
101C2×#88322—42no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53#9912yes—
107C23×C53#10312yes—
108C23×C53#10412yes—
109C23×C53C2×C22⋊C411yescharacteristic
110C2×C4×C53—24no—
111C2×C4×C53—24no—
112C2×C4×C53—24no—
113C2×C4×C53—24no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—44no—
128C2×C4×C53#10312yes—
129C2×C4×C53#10412yes—
130C2×C4×C53#13112yes—
131C2×C4×C53C2×SD1611yescharacteristic
132C2×C4×C53C2×Q1611yescharacteristic
133C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C53#13711yescharacteristic
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147Dic106—82no—
148C8×C53—44no—
149#88322—44no—
150#7681—531nomaximal, Sylow
151C22×C53—22no—
152C22×C53—22no—
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—22no—
157C22×C53#32814yes—
158C22×C53#40811yescharacteristic
159C22×C53C2×#9911yescharacteristic
160C22×C53#41311yescharacteristic
161C22×C53C2×#10311yescharacteristic
162C22×C53#41711yescharacteristic
163C22×C53C2×#10411yescharacteristic
164C22×C53C2×#10411yescharacteristic
165C4×C53—22no—
166C4×C53—21no—
167C4×C53—22no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—22no—
171C4×C53—21no—
172C4×C53—41no—
173Dic53—81no—
174Dic53—81no—
175Dic53—81no—
176Dic53—81no—
177#2855—531no—
178#1728D5311yescharacteristic
179C2×#336—531no—
180#1272—532no—
181#1271—532no—
182C2×C53—24no—
183C2×C53#146811yescharacteristic
184C2×C53#148311yescharacteristic
185C2×C53#149311yescharacteristic
186C2×C53C2×#32811yescharacteristic
187C2×C53#150411yescharacteristic
188C2×C53#152411yescharacteristic
189C2×C53#155011yescharacteristic
190C22×M16—21no—
191C23×C8—1061no—
192C2×C22⋊Q8—531no—
193C22×C42C2×D5311yescharacteristic
194#391—531no—
195#391—531no—
196#389—531no—
197#389—531no—
198#387—1061no—
199#387—1061no—
200#385—1061no—
201#381—1061no—
202#380—531no—
203#336—21no—
204#336—532no—
205#336—1061no—
206#336C2×D5312yes—
207C53#768111yescharacteristic, complemented, Sylow
208C22×Q8—531no—
209C23×C4C4×D5311yescharacteristic
210C23×C4D21211yescharacteristic
211C23×C4C22⋊D5311yescharacteristic
212C2×M16—24no—
213C2×M16—42no—
214C2×M16—42no—
215C22×C8—22no—
216C22×C8—1064no—
217C22×C8—1062no—
218C22×C8—2122no—
219C22×C8—2122no—
220C22⋊Q8—2122no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1061no—
227C2×C22⋊C4—1061no—
228C2×C22⋊C4—1061no—
229C2×C42—42no—
230C2×C42C4×D5311yescharacteristic
231C2×C42D21211yescharacteristic
232C2×C42C22⋊D5311yescharacteristic
233C2×C42C22×D5311yescharacteristic
234#95—1061no—
235#95—1061no—
236#95—1061no—
237#95—1061no—
238#95—1061no—
239#95—1061no—
240#95—1061no—
241C24#1171992111yescharacteristic
242C2×Q8—1062no—
243C2×Q8—2121no—
244C2×Q8—2121no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4#1171992111yescharacteristic
259C22×C4#1171992111yescharacteristic
260C22×C4#1171992111yescharacteristic
261C22×C4#1171992312yes—
262C22×C4#1171992512yes—
263C22×C4C2×C4×D5311yescharacteristic
264C22×C4C2×D21211yescharacteristic
265C22×C4C2×C22⋊D5311yescharacteristic
266M16—48no—
267C2×C8—24no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—1064no—
271C2×C8—2128no—
272C2×C8—2122no—
273C2×C8—2122no—
274C4⋊C4—1062no—
275C4⋊C4—2121no—
276C4⋊C4—2121no—
277C4⋊C4—2122no—
278C4⋊C4—2122no—
279C4⋊C4—2121no—
280C4⋊C4—2121no—
281C22⋊C4—2122no—
282C22⋊C4—2122no—
283C42—41no—
284C42—41no—
285C42—41no—
286C42—41no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23#1313313112yes—
292C23#1313314012yes—
293C23#1313314512yes—
294C23C2×#1171992111yescharacteristic
295Q8—4242no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—44no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4#1313314012yes—
327C2×C4#1313314312yes—
328C2×C4#1313322512yes—
329C2×C4C2×#1171992311yescharacteristic
330C2×C4C2×#1171992511yescharacteristic
331C2×C4#1313324511yescharacteristic
332C2×C4#1313325011yescharacteristic
333C8—44no—
334C8—2124no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22#1681035914yes—
342C22#1681059111yescharacteristic
343C22C2×#1313313111yescharacteristic
344C22#1681060511yescharacteristic
345C22C2×#1313314011yescharacteristic
346C22#1681062511yescharacteristic
347C22C2×#1313314311yescharacteristic
348C22C2×#1313314511yescharacteristic
349C4—21no—
350C4—22no—
351C4—21no—
352C4—21no—
353C4—22no—
354C4—21no—
355C4—22no—
356C4—41no—
357C4—4241no—
358C4—4241no—
359C4—4241no—
360C4—4241no—
361C2—24no—
362C2#2027306411yescharacteristic
363C2#2027312911yescharacteristic
364C2#2027316911yescharacteristic
365C2C2×#1681035911yescharacteristic
366C2#2027322011yescharacteristic
367C2#2027330411yescharacteristic
368C2#2027340611yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1728C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1728D53
hypocenterC53#7681
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1728

Complements

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

NH ≅ G/Ntype
3. #20272540C2semidirect
4. #20272537C2semidirect
207. C53#7681semidirect

Maximal subgroups

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

subgroupindex
#202830542
C2×#168103782
#202725402
#202725372
C53×#17282
#768153