Finite Group Databasebeta

← #27272430

Subgroups — #27272430

Order 13568.

518 conjugacy classes of subgroups.

The subgroup lattice has 354 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#20272539C212yesmaximal, complemented
5C53×#1817C211yesmaximal, 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×C42×C53C2211yescharacteristic
20C2×C22⋊C8×C53—21no—
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
47C22×C8×C53—22no—
48C2×C42×C53—42no—
49C2×C42×C53C2×C411yescharacteristic
50C2×C42×C53D411yescharacteristic
51C2×C42×C53D411yescharacteristic
52C2×C42×C53C2311yescharacteristic
53C22⋊C8×C53—42no—
54C24×C53C22⋊C411yescharacteristic
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—41no—
62C22×C4×C53—41no—
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53SD1612yes—
67C22×C4×C53Q1612yes—
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C2×Dic106—22no—
76C2×Dic106—41no—
77C2×Dic106—41no—
78C2×C8×C53—42no—
79C2×C8×C53—42no—
80C2×C8×C53—42no—
81C42×C53—41no—
82C42×C53—41no—
83C42×C53—41no—
84C42×C53—41no—
85C22⋊Dic53—42no—
86C22⋊Dic53—42no—
87C4⋊Dic53—22no—
88C4⋊Dic53—41no—
89C4⋊Dic53—41no—
90#11719919—42no—
91#11719919—42no—
92#11719919—42no—
93C2×#88322—24no—
94C2×#88322—48no—
95C2×#88322—42no—
96C2×#88322—42no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53#9912yes—
102C23×C53#10312yes—
103C23×C53#10412yes—
104C23×C53C2×C22⋊C411yescharacteristic
105C2×C4×C53—24no—
106C2×C4×C53—24no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
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—44no—
122C2×C4×C53—44no—
123C2×C4×C53#10312yes—
124C2×C4×C53#13112yes—
125C2×C4×C53C2×SD1611yescharacteristic
126C2×C4×C53C2×Q1611yescharacteristic
127C2×C4×C53#13512yes—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140Dic106—82no—
141C8×C53—82no—
142#88322—44no—
143#7655—531nomaximal, Sylow
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—22no—
149C22×C53—22no—
150C22×C53#32814yes—
151C22×C53#40811yescharacteristic
152C22×C53C2×#9911yescharacteristic
153C22×C53#41311yescharacteristic
154C22×C53C2×#10311yescharacteristic
155C22×C53#41611yescharacteristic
156C22×C53C2×#10411yescharacteristic
157C22×C53#42111yescharacteristic
158C4×C53—22no—
159C4×C53—21no—
160C4×C53—22no—
161C4×C53—21no—
162C4×C53—22no—
163C4×C53—41no—
164C4×C53—41no—
165Dic53—81no—
166Dic53—81no—
167Dic53—81no—
168Dic53—81no—
169#2855—531no—
170#1817D5311yescharacteristic
171C2×#336—531no—
172#1272—532no—
173#1271—532no—
174C2×C53—24no—
175C2×C53#146811yescharacteristic
176C2×C53#148111yescharacteristic
177C2×C53#149411yescharacteristic
178C2×C53C2×#32811yescharacteristic
179C2×C53#151611yescharacteristic
180C2×C53#153211yescharacteristic
181C2×C53#154211yescharacteristic
182C23×C8—1061no—
183C2×C22⋊Q8—531no—
184C22×C42C2×D5311yescharacteristic
185C2×C22⋊C8—21no—
186#391—531no—
187#391—531no—
188#389—531no—
189#389—531no—
190#387—1061no—
191#387—1061no—
192#385—1061no—
193#381—1061no—
194#380—531no—
195#336—21no—
196#336—532no—
197#336—1061no—
198#336C2×D5312yes—
199C53#765511yescharacteristic, complemented, Sylow
200C22×Q8—531no—
201C23×C4C4×D5311yescharacteristic
202C23×C4D21211yescharacteristic
203C23×C4C22⋊D5311yescharacteristic
204C22×C8—22no—
205C22×C8—1064no—
206C22×C8—1062no—
207C22×C8—2122no—
208C22×C8—2122no—
209C22⋊Q8—2122no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—531no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4—1061no—
216C2×C22⋊C4—1061no—
217C2×C22⋊C4—1061no—
218C2×C42—42no—
219C2×C42C4×D5311yescharacteristic
220C2×C42D21211yescharacteristic
221C2×C42C22⋊D5311yescharacteristic
222C2×C42C22×D5311yescharacteristic
223C22⋊C8—42no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231C24#1171992111yescharacteristic
232C2×Q8—1062no—
233C2×Q8—2121no—
234C2×Q8—2121no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4—1061no—
247C22×C4#1171991312yes—
248C22×C4Dic21212yes—
249C22×C4#1171992111yescharacteristic
250C22×C4#1171992111yescharacteristic
251C22×C4#1171992111yescharacteristic
252C22×C4C2×C4×D5311yescharacteristic
253C22×C4C2×D21211yescharacteristic
254C22×C4C2×C22⋊D5311yescharacteristic
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—1064no—
259C2×C8—2122no—
260C2×C8—2128no—
261C2×C8—2122no—
262C4⋊C4—1062no—
263C4⋊C4—2121no—
264C4⋊C4—2121no—
265C4⋊C4—2122no—
266C4⋊C4—2122no—
267C4⋊C4—2122no—
268C22⋊C4—2122no—
269C22⋊C4—2122no—
270C42—41no—
271C42—41no—
272C42—41no—
273C42—41no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23#1313313112yes—
279C23#1313314212yes—
280C23#1313314312yes—
281C23C2×#1171992111yescharacteristic
282Q8—4242no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4#1313314212yes—
314C2×C4#1313322512yes—
315C2×C4C2×#1171991311yescharacteristic
316C2×C4C2×Dic21211yescharacteristic
317C2×C4#1313323712yes—
318C8—82no—
319C8—2124no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22#1681036114yes—
327C22#1681059111yescharacteristic
328C22C2×#1313313111yescharacteristic
329C22#1681060511yescharacteristic
330C22C2×#1313314211yescharacteristic
331C22#1681061711yescharacteristic
332C22C2×#1313314311yescharacteristic
333C22#1681064111yescharacteristic
334C4—21no—
335C4—22no—
336C4—21no—
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—4241no—
342C4—4241no—
343C4—4241no—
344C4—4241no—
345C2—24no—
346C2#2027306411yescharacteristic
347C2#2027311911yescharacteristic
348C2#2027317311yescharacteristic
349C2C2×#1681036111yescharacteristic
350C2#2027326911yescharacteristic
351C2#2027333611yescharacteristic
352C2#2027337011yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1817C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1817D53
hypocenterC53#7655
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
3. #20272540C2semidirect
4. #20272539C2semidirect
199. C53#7655semidirect

Maximal subgroups

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

subgroupindex
#202830542
C2×#168103782
#202725402
#202725392
C53×#18172
#765553