Finite Group Databasebeta

← #27280891

Subgroups — #27280891

Order 13568.

518 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283756C211yesmaximal, characteristic, complemented
2#20274725C211yesmaximal, characteristic, complemented
3#20273275C211yesmaximal, characteristic, complemented
4#20273114C211yesmaximal, characteristic, complemented
5#20272610C211yesmaximal, characteristic, complemented
6#20272575C211yesmaximal, characteristic, complemented
7C53×#3036C211yesmaximal, characteristic
8#16810701—22no—
9#16810701—22no—
10#16810676C2211yescharacteristic
11#16810676C2211yescharacteristic
12C2×C4⋊#88322C2211yescharacteristic
13C2×#13133138—21no—
14#16810587—22no—
15#16810489C2211yescharacteristic
16#16810385—21no—
17#16810357—22no—
18D4⋊#88322—22no—
19C2×AGL(1,ℤ/8)×C53—21no—
20C53×#532C2211yescharacteristic
21C2×C4×D4×C53C2211yescharacteristic
22C53×#463—22no—
23C53×#461C412yescomplemented
24C53×#459C412yescomplemented
25C53×#423C2211yescharacteristic
26C53×#417—21no—
27C2×#11719917—24no—
28C22×#88322—21no—
29C2×C4⋊Dic53—21no—
30C4⋊#88322—21no—
31C4⋊#88322—21no—
32C4⋊#88322—21no—
33C4⋊#88322—21no—
34#13133138—42no—
35C22⋊#88322—24no—
36C22⋊#88322—21no—
37C22⋊#88322D412yes—
38C4×#88322—21no—
39C4×#88322D412yes—
40#13133124—21no—
41#13133124—21no—
42C2×#41×C53—21no—
43C22×D4×C53D411yescharacteristic
44C23×C4×C53—21no—
45AGL(1,ℤ/8)×C53—42no—
46AGL(1,ℤ/8)×C53—42no—
47C2×SD16×C53—22no—
48C2×D8×C53—22no—
49C2×M16×C53—21no—
50C4⋊Q8×C53C2×C411yescharacteristic
51#127×C53C2×C411yescharacteristic
52#124×C53—21no—
53#121×C53—21no—
54#121×C53—21no—
55C4×D4×C53—21no—
56C4×D4×C53—21no—
57C4×D4×C53—21no—
58C4×D4×C53—21no—
59C4×D4×C53C2×C412yes—
60C4×D4×C53D412yes—
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C22⋊C4×C53—21no—
63C2×C42×C53C2311yescharacteristic
64C4⋊C8×C53—21no—
65C4⋊C8×C53C2×C412yes—
66#103×C53—22no—
67#102×C53—22no—
68C24×C53—21no—
69#41×C53—41no—
70#41×C53—41no—
71#41×C53—41no—
72#41×C53—41no—
73C2×Q8×C53—21no—
74C2×D4×C53—21no—
75C2×D4×C53—21no—
76C2×D4×C53—21no—
77C2×D4×C53—21no—
78C2×D4×C53—21no—
79C2×D4×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—41no—
83C2×D4×C53C22⋊C412yes—
84C22×C4×C53—21no—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53—41no—
88C22×C4×C53—82no—
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53#4111yescharacteristic
92C22×Dic53—41no—
93SD16×C53—44no—
94D8×C53—44no—
95M16×C53—41no—
96M16×C53—41no—
97C2×C8×C53—22no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53C22⋊C412yes—
102C22⋊C4×C53—22no—
103C22⋊C4×C53—22no—
104C22⋊C4×C53—41no—
105C22⋊C4×C53—41no—
106C42×C53C22×C411yescharacteristic
107C42×C53C2×D411yescharacteristic
108C42×C53C2×D411yescharacteristic
109C42×C53#4111yescharacteristic
110C4⋊Dic53—41no—
111C4⋊Dic53—41no—
112#11719917—84no—
113C2×#88322—22no—
114C2×#88322—22no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322—41no—
118C23×C53—22no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—81no—
123C23×C53—81no—
124C23×C53#12111yescharacteristic
125Q8×C53—41no—
126Q8×C53—41no—
127D4×C53—22no—
128D4×C53—22no—
129D4×C53—42no—
130D4×C53—41no—
131D4×C53—41no—
132D4×C53—41no—
133D4×C53—41no—
134D4×C53—81no—
135D4×C53—81no—
136D4×C53—81no—
137C2×C4×C53—22no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—44no—
141C2×C4×C53—41no—
142C2×C4×C53—82no—
143C2×C4×C53—81no—
144C2×C4×C53—81no—
145C2×C4×C53—81no—
146C2×C4×C53C2×C22⋊C411yescharacteristic
147C2×C4×C53C4×D411yescharacteristic
148C2×C4×C53C4×D411yescharacteristic
149C2×C4×C53#12011yescharacteristic
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12311yescharacteristic
152C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C53#13711yescharacteristic
156C2×Dic53—81no—
157C2×Dic53—81no—
158C2×Dic53—81no—
159C8×C53—42no—
160#88322—42no—
161#88322—42no—
162#88322—81no—
163#9309—531nomaximal, Sylow
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—42no—
167C22×C53—42no—
168C22×C53—42no—
169C22×C53—82no—
170C22×C53—82no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53—81no—
174C22×C53#38611yescharacteristic
175C22×C53#46811yescharacteristic
176C22×C53#46911yescharacteristic
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—42no—
182C4×C53—81no—
183C4×C53#41711yescharacteristic
184C4×C53#41811yescharacteristic
185C4×C53#46811yescharacteristic
186C4×C53#46911yescharacteristic
187Dic53—161no—
188#3036D5311yescharacteristic
189#2975—531no—
190#1804—531no—
191#1517—531no—
192#1480—531no—
193#1301—531no—
194#1286—531no—
195C2×C53—21no—
196C2×C53—21no—
197C2×C53—82no—
198C2×C53—161no—
199C2×C53#188111yescharacteristic
200C2×C53#189511yescharacteristic
201C2×C53#190411yescharacteristic
202C2×AGL(1,ℤ/8)—21no—
203#532C2×D5311yescharacteristic
204C2×C4×D4C2×D5311yescharacteristic
205#463—22no—
206#461Dic5312yescomplemented
207#459Dic5312yescomplemented
208#436—1062no—
209#436—1062no—
210#431—531no—
211#431—531no—
212#423C2×D5311yescharacteristic
213C2×C4⋊C8—531no—
214#417—21no—
215C2×#102—1061no—
216#407—1062no—
217#382—531no—
218#340—1061no—
219#327—1062no—
220D4⋊C8—1062no—
221C53#930911yescharacteristic, complemented, Sylow
222C2×#41—21no—
223C22×D4C22⋊D5311yescharacteristic
224C23×C4—21no—
225AGL(1,ℤ/8)—42no—
226AGL(1,ℤ/8)—42no—
227C2×SD16—22no—
228C2×D8—22no—
229C2×M16—21no—
230C2×M16—1064no—
231C22×C8—1061no—
232C4⋊Q8C2×Dic5311yescharacteristic
233#127C2×Dic5311yescharacteristic
234#124—21no—
235#121—21no—
236#121—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4C2×Dic5312yes—
242C4×D4C22⋊D5312yes—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4C22⋊D5311yescharacteristic
245C2×C22⋊C4—21no—
246C2×C42C22×D5311yescharacteristic
247C4⋊C8—21no—
248C4⋊C8—1061no—
249C4⋊C8—1061no—
250C4⋊C8—1061no—
251C4⋊C8—1061no—
252C4⋊C8C2×Dic5312yes—
253#103—22no—
254#102—22no—
255#102—2122no—
256C22⋊C8—532no—
257C22⋊C8—1064no—
258C22⋊C8—1061no—
259C4×C8—532no—
260C4×C8—1061no—
261#95—1061no—
262#95—1061no—
263C24—21no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—41no—
278C2×D4C22⋊Dic5312yes—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—82no—
284C22×C4—2121no—
285C22×C4D4×D5311yescharacteristic
286C22×C4#1171994711yescharacteristic
287C22×C4C2×C22⋊D5311yescharacteristic
288SD16—44no—
289D8—44no—
290M16—41no—
291M16—41no—
292M16—4244no—
293C2×C8—22no—
294C2×C8—1062no—
295C2×C8—1062no—
296C2×C8—2121no—
297C2×C8—2121no—
298C2×C8—2121no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—2121no—
303C4⋊C4—2121no—
304C4⋊C4C22⋊Dic5312yes—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C42D4×D5311yescharacteristic
310C42#1171994711yescharacteristic
311C42C22×Dic5311yescharacteristic
312C42C2×C22⋊D5311yescharacteristic
313C23—22no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—81no—
318C23—81no—
319C23#1313318911yescharacteristic
320Q8—41no—
321Q8—41no—
322D4—22no—
323D4—22no—
324D4—42no—
325D4—41no—
326D4—41no—
327D4—41no—
328D4—41no—
329D4—81no—
330D4—81no—
331D4—81no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—41no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—4241no—
342C2×C4—4241no—
343C2×C4—4241no—
344C2×C4C2×C22⋊Dic5311yescharacteristic
345C2×C4D4×Dic5311yescharacteristic
346C2×C4D4×Dic5311yescharacteristic
347C2×C4#1313318511yescharacteristic
348C2×C4#1313318911yescharacteristic
349C2×C4#1313319811yescharacteristic
350C2×C4#1313324211yescharacteristic
351C2×C4#1313324311yescharacteristic
352C2×C4#1313324311yescharacteristic
353C2×C4#1313324711yescharacteristic
354C8—42no—
355C8—4241no—
356C8—2122no—
357C8—2122no—
358C22—21no—
359C22—21no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—82no—
364C22—82no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22#1681050311yescharacteristic
369C22#1681086711yescharacteristic
370C22#1681087411yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—8481no—
378C4#1681062411yescharacteristic
379C4#1681062811yescharacteristic
380C4#1681086711yescharacteristic
381C4#1681087411yescharacteristic
382C2—21no—
383C2—21no—
384C2—82no—
385C2—161no—
386C2#2027507911yescharacteristic
387C2#2027515411yescharacteristic
388C2#2027520411yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3036C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3036D53
hypocenterC53#9309
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #20283756C2semidirect
2. #20274725C2semidirect
3. #20273275C2semidirect
4. #20273114C2semidirect
5. #20272610C2semidirect
6. #20272575C2semidirect
23. C53×#461C4semidirect
24. C53×#459C4semidirect
206. #461Dic53semidirect
207. #459Dic53semidirect
221. C53#9309semidirect

Maximal subgroups

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

subgroupindex
#202837562
#202747252
#202732752
#202731142
#202726102
#202725752
C53×#30362
#930953