Finite Group Databasebeta

← #27301486

Subgroups — #27301486

Order 13568.

484 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M32×D53C211yesmaximal, characteristic
2C2×#16810861C211yesmaximal, characteristic
3C2×#16810439C211yesmaximal, characteristic, complemented
4#20274750C211yesmaximal, characteristic
5#20272825C211yesmaximal, characteristic, complemented
6#20272824C211yesmaximal, characteristic, complemented
7C2×C53×#362C211yesmaximal, characteristic, complemented
8C22×C8×D53C2211yescharacteristic
9C2×#13133193—21no—
10C2×#13133157C2211yescharacteristic
11C2×#13133156—21no—
12M32×D53—22no—
13M32×D53—22no—
14C2×C16×D53—21no—
15#16810861—21no—
16#16810861—21no—
17#16810861C412yes—
18#16810861C412yes—
19C2×#13133153C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133141—21no—
22#16810586—21no—
23#16810439—22no—
24#16810387—21no—
25C22×Q16×C53C2211yescharacteristic
26C2×M32×C53C2211yescharacteristic
27C2×#108×C53C2211yescharacteristic
28C53×#362—22no—
29C22×C4×D53D411yescharacteristic
30C2×#11719917—21no—
31C22×#88322D411yescharacteristic
32C2×C8×D53—21no—
33C2×C8×D53—21no—
34C2×C8×D53—21no—
35C2×C8×D53—21no—
36C2×C8×D53C2×C411yescharacteristic
37C2×C8×D53C2×C411yescharacteristic
38C2×C8×D53D411yescharacteristic
39C2×C8×D53D411yescharacteristic
40#13133193—22no—
41#13133193—22no—
42#13133193—41no—
43#13133193—41no—
44C2×C4⋊Dic53—21no—
45C2×#11719919—41no—
46C2×#11719921—41no—
47#13133157—21no—
48#13133157D412yes—
49#13133156—24no—
50C2×#11719908—21no—
51C16×D53—24no—
52#13133153—21no—
53#13133153—21no—
54#13133151—21no—
55#13133151C2×C412yes—
56#13133141—22no—
57#13133141—22no—
58#13133129—41no—
59#13133129—41no—
60C22×Q8×C53—21no—
61C2×Q16×C53—21no—
62C2×Q16×C53—21no—
63C2×Q16×C53—21no—
64C2×Q16×C53—21no—
65C2×Q16×C53—21no—
66C2×Q16×C53C2×C412yes—
67C2×M16×C53—21no—
68C22×C8×C53C2311yescharacteristic
69M32×C53—21no—
70M32×C53D412yes—
71C2×C16×C53—21no—
72#108×C53—21no—
73#108×C53—21no—
74C23×D53SD1611yescharacteristic
75C2×Q8×C53—22no—
76C2×Q8×C53—41no—
77C2×Q8×C53—41no—
78C2×Q8×C53—41no—
79C2×Q8×C53—41no—
80C2×Q8×C53—41no—
81C22×C4×C53—41no—
82C22×C4×C53C2×D411yescharacteristic
83C22×Dic53—41no—
84C22×Dic53SD1611yescharacteristic
85C2×C4×D53—21no—
86C2×C4×D53—21no—
87C2×C4×D53—21no—
88C2×C4×D53—21no—
89C2×C4×D53C22⋊C411yescharacteristic
90C2×C4×D53C22⋊C411yescharacteristic
91C2×C4×D53D811yescharacteristic
92C2×C4×D53D811yescharacteristic
93Q16×C53—22no—
94Q16×C53—22no—
95Q16×C53—41no—
96Q16×C53—41no—
97M16×C53—41no—
98M16×C53—41no—
99C2×C8×C53—41no—
100C2×C8×C53C22×C411yescharacteristic
101C2×C8×C53C2×D411yescharacteristic
102C2×C8×C53C2×D411yescharacteristic
103C2×C8×C53C2×D411yescharacteristic
104C2×C8×C53#4111yescharacteristic
105C2×C8×C53#4111yescharacteristic
106#11719921—42no—
107#11719921—42no—
108C4⋊Dic53—22no—
109C4⋊Dic53—41no—
110#11719919—42no—
111#11719919—42no—
112#11719917—41no—
113#11719917—41no—
114C2×#88322—21no—
115C2×#88322—21no—
116C2×#88322—41no—
117C2×#88322C22⋊C411yescharacteristic
118C2×#88322C22⋊C411yescharacteristic
119C8×D53—21no—
120C8×D53—21no—
121C8×D53—21no—
122C8×D53—21no—
123C8×D53—21no—
124C8×D53—21no—
125C8×D53—21no—
126C8×D53—21no—
127C16×C53—22no—
128#11719908—22no—
129C23×C53C2×SD1611yescharacteristic
130C22×D53—21no—
131C22×D53—21no—
132C22×D53—22no—
133C22×D53—41no—
134C22×D53#10211yescharacteristic
135C22×D53#10211yescharacteristic
136Q8×C53—42no—
137Q8×C53—42no—
138Q8×C53—81no—
139Q8×C53—81no—
140C2×C4×C53—42no—
141C2×C4×C53—81no—
142C2×C4×C53—81no—
143C2×C4×C53C2×C22⋊C411yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53C2×D811yescharacteristic
147C2×C4×C53#13511yescharacteristic
148C2×C4×C53#13511yescharacteristic
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—42no—
152C2×Dic53—82no—
153C2×Dic53#10211yescharacteristic
154C2×Dic53#10211yescharacteristic
155C4×D53—21no—
156C4×D53—21no—
157C4×D53—21no—
158C4×D53—21no—
159C4×D53—21no—
160C4×D53—21no—
161C4×D53—21no—
162C4×D53—21no—
163C8×C53—81no—
164C8×C53C4×D411yescharacteristic
165C8×C53C4×D411yescharacteristic
166C8×C53#12011yescharacteristic
167C8×C53#12311yescharacteristic
168#88322—21no—
169#88322—21no—
170#88322—81no—
171#13066—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53C2×#10211yescharacteristic
175C22×C53#46511yescharacteristic
176C22×C53#46511yescharacteristic
177C2×D53—21no—
178C2×D53—21no—
179C2×D53—21no—
180C2×D53—21no—
181C2×D53—21no—
182C2×D53—42no—
183C2×D53—42no—
184C2×D53#36212yescomplemented
185C4×C53—82no—
186C4×C53#38611yescharacteristic
187C4×C53#41611yescharacteristic
188C4×C53#46611yescharacteristic
189C4×C53#46611yescharacteristic
190Dic53—21no—
191Dic53—21no—
192Dic53—82no—
193C22×M32—531no—
194C2×#467—531no—
195C2×#362—531no—
196C2×#362D5311yescharacteristic, complemented
197#1810—531no—
198#1387—531no—
199#1387—531no—
200C2×C53—21no—
201C2×C53—21no—
202C2×C53#189311yescharacteristic
203C2×C53C2×#36211yescharacteristic
204C2×C53#214811yescharacteristic
205D53—22no—
206D53—41no—
207C22×Q16C2×D5311yescharacteristic
208C23×C8—531no—
209C2×C22⋊Q8—1061no—
210C2×M32—531no—
211C2×M32—1061no—
212C2×M32—1062no—
213C2×M32—1062no—
214C2×M32C2×D5311yescharacteristic
215C22×C16—1061no—
216#467—532no—
217#467—532no—
218#467—1061no—
219#467—1061no—
220C2×#108—531no—
221C2×#108C2×D5311yescharacteristic
222C2×#107—531no—
223C2×#103—1061no—
224#407—1061no—
225#362—22no—
226#362—1062no—
227#341—1061no—
228C53#1306611yescharacteristic, complemented, Sylow
229C22×Q8—21no—
230C23×C4—531no—
231C2×Q16—21no—
232C2×Q16—21no—
233C2×Q16—21no—
234C2×Q16—21no—
235C2×Q16—21no—
236C2×Q16C4×D5312yes—
237C2×M16—21no—
238C2×M16—1061no—
239C22×C8—531no—
240C22×C8—531no—
241C22×C8—531no—
242C22×C8—531no—
243C22×C8—531no—
244C22×C8—1061no—
245C22×C8—1061no—
246C22×C8—1061no—
247C22×C8—1061no—
248C22×C8C22×D5311yescharacteristic
249C22⋊Q8—1062no—
250C22⋊Q8—1062no—
251C22⋊Q8—2121no—
252C22⋊Q8—2121no—
253C2×C4⋊C4—1061no—
254C2×C4⋊C4—2121no—
255C2×C22⋊C4—2121no—
256M32—21no—
257M32—532no—
258M32—1064no—
259M32—1061no—
260M32C22⋊D5312yes—
261C2×C16—21no—
262C2×C16—1064no—
263C2×C16—1061no—
264#108—21no—
265#108—21no—
266#108—1061no—
267#108—1061no—
268#107—532no—
269#107—1061no—
270#103—1062no—
271#103—1062no—
272C22⋊C8—2121no—
273C22⋊C8—2121no—
274C24—531no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C22×C4—41no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—531no—
285C22×C4—531no—
286C22×C4—531no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4—2121no—
292C22×C4D4×D5311yescharacteristic
293Q16—22no—
294Q16—22no—
295Q16—41no—
296Q16—41no—
297M16—41no—
298M16—41no—
299M16—2121no—
300M16—2121no—
301C2×C8—41no—
302C2×C8—531no—
303C2×C8—531no—
304C2×C8—1061no—
305C2×C8—1061no—
306C2×C8—1061no—
307C2×C8—1061no—
308C2×C8—1061no—
309C2×C8—1061no—
310C2×C8—1061no—
311C2×C8—1061no—
312C2×C8—1061no—
313C2×C8—1061no—
314C2×C8—2121no—
315C2×C8C2×C4×D5311yescharacteristic
316C2×C8#1171994511yescharacteristic
317C2×C8D4×D5311yescharacteristic
318C2×C8D4×D5311yescharacteristic
319C2×C8#1171994711yescharacteristic
320C2×C8C2×C22⋊D5311yescharacteristic
321C4⋊C4—1062no—
322C4⋊C4—2122no—
323C4⋊C4—2122no—
324C4⋊C4—2121no—
325C22⋊C4—2122no—
326C22⋊C4—2122no—
327C16—22no—
328C16—1062no—
329C23—531no—
330C23—531no—
331C23—1062no—
332C23—1061no—
333C23—1061no—
334C23—2121no—
335C23SD16×D5311yescharacteristic
336Q8—42no—
337Q8—42no—
338Q8—81no—
339Q8—81no—
340C2×C4—42no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—531no—
344C2×C4—531no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—4242no—
356C2×C4—2122no—
357C2×C4C22⋊C4×D5311yescharacteristic
358C2×C4#1313318811yescharacteristic
359C2×C4#1313319011yescharacteristic
360C2×C4D8×D5311yescharacteristic
361C2×C4#1313323611yescharacteristic
362C2×C4#1313323811yescharacteristic
363C8—81no—
364C8—1061no—
365C8—1061no—
366C8—4241no—
367C8#1313317711yescharacteristic
368C8C4×C22⋊D5311yescharacteristic
369C8#1313318511yescharacteristic
370C8#1313319911yescharacteristic
371C22—21no—
372C22—21no—
373C22—532no—
374C22—1061no—
375C22—1061no—
376C22—1061no—
377C22—1061no—
378C22—1061no—
379C22—2122no—
380C22—2122no—
381C22#102×D5311yescharacteristic
382C22#1681084911yescharacteristic
383C22#1681085111yescharacteristic
384C4—82no—
385C4—1061no—
386C4—1061no—
387C4—4242no—
388C4#1681049911yescharacteristic
389C4#1681061611yescharacteristic
390C4#1681085611yescharacteristic
391C4#1681085811yescharacteristic
392C2—21no—
393C2—21no—
394C2—1062no—
395C2—2121no—
396C2#2027514011yescharacteristic
397C2#362×D5311yescharacteristic
398C2#2027646311yescharacteristic
subgroupG/H
centerC22#102×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#362C2
socleC22×C53C2×#102
prime radicalC23×D53SD16
cosocleC22×C8×C53C23
hypercenterC2×#362D53
hypocenterC53#13066
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#362

Complements

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

NH ≅ G/Ntype
3. C2×#16810439C2semidirect
5. #20272825C2semidirect
6. #20272824C2semidirect
7. C2×C53×#362C2semidirect
184. C2×D53#362semidirect
196. C2×#362D53semidirect
228. C53#13066semidirect

Maximal subgroups

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

subgroupindex
C2×M32×D532
C2×#168108612
C2×#168104392
#202747502
#202728252
#202728242
C2×C53×#3622
#1306653