Finite Group Databasebeta

← #27283226

Subgroups — #27283226

Order 13568.

624 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283067C211yesmaximal, characteristic
2C2×#16810361C214yesmaximal
3#20272942C211yesmaximal, characteristic
4C53×#1797C211yesmaximal, characteristic
5C2×#13133217C411yescharacteristic, complemented
6C2×#13133196C2212yes—
7#16810529C412yescomplemented
8#16810521—21no—
9#16810521—21no—
10#16810493C411yescharacteristic, complemented
11#16810483—21no—
12#16810483—21no—
13C2×#13133124—21no—
14C2×#13133124—21no—
15C2×#13133124C2212yes—
16#16810361—28no—
17C22×C42×C53C2211yescharacteristic
18C2×C4⋊C8×C53—21no—
19C2×C22⋊C8×C53C2212yes—
20C53×#336—21no—
21C22×Dic106C2×C412yes—
22C23×Dic53—22no—
23#13133217—42no—
24#13133196—24no—
25#13133196—24no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53C2×C412yes—
31C2×#11719919—22no—
32C2×C22⋊Dic53—22no—
33#13133124—22no—
34#13133124—22no—
35#13133124—22no—
36#13133124—42no—
37#13133124—42no—
38#13133124D414yes—
39C23×C4×C53D411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C23×C4×C53C2311yescharacteristic
42C22×C8×C53—22no—
43C2×C42×C53—42no—
44C2×C42×C53C2×C411yescharacteristic
45C2×C42×C53C2×C411yescharacteristic
46C2×C42×C53D411yescharacteristic
47C2×C42×C53D411yescharacteristic
48C4⋊C8×C53—42no—
49C22⋊C8×C53—24no—
50C24×C53C2×D411yescharacteristic
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—41no—
54C22×C4×C53—41no—
55C22×C4×C53—41no—
56C22×C4×C53—41no—
57C22×C4×C53—41no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53SD1612yes—
63C22×C4×C53Q1612yes—
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D412yes—
67C22×C4×C53#4112yes—
68C22×Dic53—22no—
69C22×Dic53—22no—
70C22×Dic53—24no—
71C22×Dic53—42no—
72C22×Dic53—42no—
73C22×Dic53—42no—
74C22×Dic53—42no—
75C22×Dic53—42no—
76C22×Dic53—42no—
77C2×Dic106—24no—
78C2×Dic106—42no—
79C2×Dic106—42no—
80C2×C8×C53—42no—
81C2×C8×C53—42no—
82C2×C8×C53—42no—
83C42×C53—22no—
84C42×C53—22no—
85C42×C53—41no—
86C42×C53—41no—
87C22⋊Dic53—44no—
88C4⋊Dic53—24no—
89C4⋊Dic53—42no—
90C4⋊Dic53—42no—
91C4⋊Dic53—42no—
92C4⋊Dic53—42no—
93#11719919—42no—
94#11719919—42no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53C2×C22⋊C411yescharacteristic
100C23×C53#12112yes—
101C23×C53C2×SD1611yescharacteristic
102C23×C53C2×Q1611yescharacteristic
103C23×C53#13712yes—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
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—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—44no—
122C2×C4×C53#9912yes—
123C2×C4×C53#10312yes—
124C2×C4×C53#10312yes—
125C2×C4×C53C4×D412yes—
126C2×C4×C53#12411yescharacteristic
127C2×C4×C53#12711yescharacteristic
128C2×Dic53—44no—
129C2×Dic53—44no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—44no—
137C2×Dic53—82no—
138C2×Dic53—82no—
139C2×Dic53—82no—
140C2×Dic53—82no—
141Dic106—84no—
142C8×C53—82no—
143#9714—531nomaximal, Sylow
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53#39011yescharacteristic
153C22×C53C2×#9911yescharacteristic
154C22×C53#41011yescharacteristic
155C22×C53C2×#10311yescharacteristic
156C22×C53#41911yescharacteristic
157C22×C53#42111yescharacteristic
158C22×C53#42111yescharacteristic
159C22×C53#46112yes—
160C22×C53#46212yes—
161C4×C53—21no—
162C4×C53—22no—
163C4×C53—21no—
164C4×C53—22no—
165C4×C53—22no—
166C4×C53—22no—
167C4×C53—41no—
168Dic53—84no—
169Dic53—82no—
170Dic53—82no—
171#2857—531no—
172#1797D5311yescharacteristic
173C2×#328—534no—
174#1434—531no—
175C2×C53—22no—
176C2×C53—22no—
177C2×C53#151111yescharacteristic
178C2×C53#151511yescharacteristic
179C2×C53#151811yescharacteristic
180C2×C53#152011yescharacteristic
181C2×C53#196311yescharacteristic
182C2×C53#196411yescharacteristic
183C2×C53#198111yescharacteristic
184C2×C4⋊Q8—531no—
185C2×C22⋊Q8—532no—
186C22×C42C2×D5311yescharacteristic
187C2×C4⋊C8—21no—
188C2×C22⋊C8C2×D5312yes—
189#391—532no—
190#389—1061no—
191#389—1061no—
192#384—531no—
193#380—1061no—
194#380—1061no—
195C2×#95—532no—
196C2×#95—1061no—
197C2×#95—1061no—
198#336—21no—
199#328—1068no—
200C53#971411yescharacteristic, complemented, Sylow
201C22×Q8—532no—
202C23×C4—1062no—
203C23×C4D21211yescharacteristic
204C23×C4D21211yescharacteristic
205C23×C4C22×D5311yescharacteristic
206C22×C8—22no—
207C4⋊Q8—2122no—
208C22⋊Q8—1064no—
209C22⋊Q8—1064no—
210C2×C4⋊C4—532no—
211C2×C4⋊C4—1062no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4—1061no—
216C2×C22⋊C4—1062no—
217C2×C42—42no—
218C2×C42C4×D5311yescharacteristic
219C2×C42C4×D5311yescharacteristic
220C2×C42C22⋊D5311yescharacteristic
221C2×C42C22⋊D5311yescharacteristic
222C4⋊C8—42no—
223C22⋊C8—24no—
224#95—534no—
225#95—1062no—
226#95—1062no—
227#95—1062no—
228#95—2122no—
229#95—2122no—
230C24C2×D21211yescharacteristic
231C2×Q8—1064no—
232C2×Q8—2122no—
233C2×Q8—2122no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—1062no—
242C22×C4—1062no—
243C22×C4—1064no—
244C22×C4—2122no—
245C22×C4—2122no—
246C22×C4—2122no—
247C22×C4—2122no—
248C22×C4—2122no—
249C22×C4—2122no—
250C22×C4#1171991312yes—
251C22×C4Dic21212yes—
252C22×C4#1171992111yescharacteristic
253C22×C4#1171992111yescharacteristic
254C22×C4#1171992111yescharacteristic
255C22×C4#1171992111yescharacteristic
256C22×C4C2×C4×D5311yescharacteristic
257C22×C4D4×D5312yes—
258C22×C4#1171994912yes—
259C22×C4C2×C22⋊D5311yescharacteristic
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C4⋊C4—1064no—
264C4⋊C4—2122no—
265C4⋊C4—2122no—
266C4⋊C4—2122no—
267C4⋊C4—2122no—
268C4⋊C4—2122no—
269C4⋊C4—2122no—
270C22⋊C4—2124no—
271C42—22no—
272C42—22no—
273C42—41no—
274C42—41no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23C2×#1171992111yescharacteristic
280C23#1313319212yes—
281C23C2×#1171991311yescharacteristic
282C23C2×Dic21211yescharacteristic
283C23#1313325012yes—
284Q8—4244no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
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—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—4242no—
304C2×C4—4242no—
305C2×C4—4242no—
306C2×C4—4242no—
307C2×C4—2124no—
308C2×C4—2124no—
309C2×C4—2122no—
310C2×C4—2122no—
311C2×C4—2124no—
312C2×C4—2122no—
313C2×C4—2122no—
314C2×C4—2122no—
315C2×C4—2122no—
316C2×C4#1313313112yes—
317C2×C4#1313314212yes—
318C2×C4#1313314212yes—
319C2×C4#1313317912yes—
320C2×C4#1313320611yescharacteristic
321C2×C4#1313321311yescharacteristic
322C8—82no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22#1681052411yescharacteristic
332C22C2×#1313313111yescharacteristic
333C22#1681059611yescharacteristic
334C22C2×#1313314211yescharacteristic
335C22#1681063311yescharacteristic
336C22#1681063911yescharacteristic
337C22#1681063911yescharacteristic
338C22#1681082912yes—
339C22#1681083412yes—
340C4—21no—
341C4—22no—
342C4—21no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—41no—
347C4—4244no—
348C4—4242no—
349C4—4242no—
350C2—22no—
351C2—22no—
352C2#2027324911yescharacteristic
353C2#2027326711yescharacteristic
354C2#2027328011yescharacteristic
355C2#2027328811yescharacteristic
356C2#2027554611yescharacteristic
357C2#2027555011yescharacteristic
358C2#2027563711yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1797C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1797D53
hypocenterC53#9714
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
5. C2×#13133217C4semidirect
7. #16810529C4semidirect
10. #16810493C4semidirect
200. C53#9714semidirect

Maximal subgroups

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

subgroupindex
#202830672
C2×#168103612
#202729422
C53×#17972
#971453