Finite Group Databasebeta

← #27279148

Subgroups — #27279148

Order 13568.

660 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810554C211yesmaximal, characteristic
2#20277332C211yesmaximal, characteristic
3#20274801C211yesmaximal, characteristic
4C2×#16810348C211yesmaximal, characteristic
5#20272595C211yesmaximal, characteristic
6#20272593C211yesmaximal, characteristic
7C53×#1823C211yesmaximal, characteristic
8C2×#13133173—21no—
9C22×#11719919—21no—
10C22×#11719919C2211yescharacteristic
11C2×C22⋊#88322C2211yescharacteristic
12#16810554—22no—
13#16810554C414yescomplemented
14#16810520—22no—
15#16810518—22no—
16#16810483C2211yescharacteristic
17#16810481—21no—
18C2×#13133124—21no—
19C2×#13133124C2211yescharacteristic
20#16810385—22no—
21#16810348—22no—
22C22×C4⋊C4×C53C2211yescharacteristic
23C2×C22⋊C8×C53C2211yescharacteristic
24C53×#380C2211yescharacteristic
25C53×#340—22no—
26C23×Dic53—21no—
27C23×Dic53D411yescharacteristic
28C23×Dic53D411yescharacteristic
29C22×#88322—21no—
30#13133173—42no—
31#13133173—42no—
32C2×C4⋊Dic53D412yes—
33C2×#11719919—22no—
34C2×#11719919—22no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—24no—
40C2×#11719919—42no—
41C2×#11719919—42no—
42C2×#11719919C2×C412yes—
43C2×#11719919D412yes—
44C2×C22⋊Dic53—21no—
45C2×C4×Dic53—22no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48#13133124—22no—
49#13133124—21no—
50#13133124—21no—
51#13133124—24no—
52#13133124C2×C412yes—
53C23×C4×C53—21no—
54C23×C4×C53C2311yescharacteristic
55C22×C8×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53C2×C412yes—
62C2×C4⋊C4×C53D412yes—
63C2×C22⋊C4×C53—21no—
64C22⋊C8×C53—21no—
65C22⋊C8×C53—21no—
66#95×C53—21no—
67C24×C53C2×D411yescharacteristic
68C22×C4×C53—22no—
69C22×C4×C53—22no—
70C22×C4×C53—42no—
71C22×C4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53—41no—
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53#4111yescharacteristic
82C22×C4×C53#4111yescharacteristic
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—24no—
88C22×Dic53—22no—
89C22×Dic53—21no—
90C22×Dic53—21no—
91C22×Dic53—22no—
92C22×Dic53—42no—
93C22×Dic53—41no—
94C22×Dic53—41no—
95C22×Dic53—42no—
96C22×Dic53C22⋊C412yes—
97C22×Dic53C22⋊C412yes—
98C22×Dic53D812yes—
99C22×Dic53SD1612yes—
100C2×C8×C53—42no—
101C2×C8×C53—41no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—42no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C22⋊C4×C53—41no—
108C22⋊C4×C53—41no—
109C22⋊Dic53—41no—
110C22⋊Dic53—41no—
111C4⋊Dic53—42no—
112#11719919—22no—
113#11719919—22no—
114#11719919—22no—
115#11719919—22no—
116#11719919—44no—
117#11719919—48no—
118#11719919—82no—
119C4×Dic53—44no—
120C2×#88322—42no—
121C2×#88322—41no—
122C23×C53—22no—
123C23×C53—22no—
124C23×C53C2×C22⋊C411yescharacteristic
125C23×C53#12011yescharacteristic
126C23×C53#12311yescharacteristic
127C23×C53C2×D811yescharacteristic
128C23×C53C2×SD1611yescharacteristic
129C23×C53AGL(1,ℤ/8)11yescharacteristic
130C23×C53#13711yescharacteristic
131C2×C4×C53—22no—
132C2×C4×C53—22no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53—81no—
144C2×C4×C53—81no—
145C2×C4×C53—81no—
146C2×C4×C53—81no—
147C2×C4×C53C4×D411yescharacteristic
148C2×C4×C53C4×D411yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53#12111yescharacteristic
151C2×Dic53—22no—
152C2×Dic53—22no—
153C2×Dic53—22no—
154C2×Dic53—22no—
155C2×Dic53—22no—
156C2×Dic53—22no—
157C2×Dic53—24no—
158C2×Dic53—22no—
159C2×Dic53—44no—
160C2×Dic53—44no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166C2×Dic53—42no—
167C2×Dic53—42no—
168C2×Dic53—81no—
169C2×Dic53#10214yes—
170C8×C53—81no—
171#88322—81no—
172#9027—531nomaximal, Sylow
173C22×C53—22no—
174C22×C53—22no—
175C22×C53—22no—
176C22×C53—22no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53#38611yescharacteristic
182C22×C53C2×#10211yescharacteristic
183C22×C53#41711yescharacteristic
184C22×C53#44711yescharacteristic
185C22×C53#44811yescharacteristic
186C22×C53#45612yes—
187C22×C53#45812yes—
188C22×C53#48011yescharacteristic
189C22×C53#48111yescharacteristic
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—82no—
195C4×C53—82no—
196Dic53—24no—
197Dic53—44no—
198Dic53—42no—
199Dic53—82no—
200C2×#398—531no—
201#2316—531no—
202#1823—531no—
203#1823D5311yescharacteristic
204C2×#324—531no—
205#1294—531no—
206#1294—531no—
207C2×C53—21no—
208C2×C53—21no—
209C2×C53—21no—
210C2×C53—21no—
211C2×C53#161012yes—
212C2×C53#161112yes—
213C2×C53#187411yescharacteristic
214C2×C53#188611yescharacteristic
215C2×C53#189011yescharacteristic
216C2×#117—1061no—
217C22×C4⋊C4—531no—
218C22×C4⋊C4—1061no—
219C22×C4⋊C4C2×D5311yescharacteristic
220C2×C22⋊C8—531no—
221C2×C22⋊C8C2×D5311yescharacteristic
222#398—534no—
223#398—1062no—
224#389—1062no—
225#389—1062no—
226#380—531no—
227#380—1061no—
228#380C2×D5311yescharacteristic
229C2×#95—531no—
230C2×#95—1061no—
231#340—22no—
232#340—1062no—
233#324—1062no—
234C53#902711yescharacteristic, complemented, Sylow
235C23×C4—21no—
236C23×C4—531no—
237C23×C4—531no—
238C23×C4—1061no—
239C23×C4C22×D5311yescharacteristic
240C22×C8—21no—
241C22×C8—1061no—
242#117—2122no—
243#117—2122no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—532no—
250C2×C4⋊C4—532no—
251C2×C4⋊C4—532no—
252C2×C4⋊C4—1061no—
253C2×C4⋊C4—1061no—
254C2×C4⋊C4—1061no—
255C2×C4⋊C4—1061no—
256C2×C4⋊C4—1062no—
257C2×C4⋊C4—1064no—
258C2×C4⋊C4—1062no—
259C2×C4⋊C4—2122no—
260C2×C4⋊C4—2122no—
261C2×C4⋊C4C4×D5312yes—
262C2×C4⋊C4C22⋊D5312yes—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—1061no—
265C2×C42—1062no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—1061no—
269C22⋊C8—1061no—
270#95—21no—
271#95—532no—
272#95—1064no—
273#95—1062no—
274#95—1061no—
275#95—1061no—
276C24D4×D5311yescharacteristic
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—532no—
287C22×C4—532no—
288C22×C4—532no—
289C22×C4—532no—
290C22×C4—1061no—
291C22×C4—1061no—
292C22×C4—1061no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—1064no—
296C22×C4—1062no—
297C22×C4—1061no—
298C22×C4—1062no—
299C22×C4—2122no—
300C22×C4—2122no—
301C22×C4—2121no—
302C22×C4—2121no—
303C22×C4C2×C4×D5311yescharacteristic
304C22×C4#1171994511yescharacteristic
305C22×C4D4×D5311yescharacteristic
306C22×C4D4×D5311yescharacteristic
307C22×C4#1171994711yescharacteristic
308C22×C4C2×C22⋊D5311yescharacteristic
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—2122no—
312C2×C8—2121no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—1062no—
319C4⋊C4—1062no—
320C4⋊C4—1062no—
321C4⋊C4—1062no—
322C4⋊C4—4242no—
323C4⋊C4—2122no—
324C4⋊C4—2128no—
325C4⋊C4—2124no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—2121no—
329C22⋊C4—2121no—
330C42—2124no—
331C23—22no—
332C23—22no—
333C23C22⋊C4×D5311yescharacteristic
334C23#1313318511yescharacteristic
335C23#1313319911yescharacteristic
336C23D8×D5311yescharacteristic
337C23SD16×D5311yescharacteristic
338C23#1313324111yescharacteristic
339C23#1313324611yescharacteristic
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—534no—
357C2×C4—1062no—
358C2×C4—1062no—
359C2×C4—1062no—
360C2×C4—1062no—
361C2×C4—1062no—
362C2×C4—1062no—
363C2×C4—1064no—
364C2×C4—1062no—
365C2×C4—4241no—
366C2×C4—2122no—
367C2×C4—2122no—
368C2×C4—2122no—
369C2×C4—2124no—
370C2×C4—2124no—
371C2×C4—2122no—
372C2×C4—2122no—
373C2×C4—2122no—
374C2×C4—2122no—
375C2×C4#1313317711yescharacteristic
376C2×C4C4×C22⋊D5311yescharacteristic
377C2×C4#1313318811yescharacteristic
378C2×C4#1313319011yescharacteristic
379C8—81no—
380C8—4241no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22#1681049911yescharacteristic
390C22#102×D5311yescharacteristic
391C22#1681062111yescharacteristic
392C22#1681075511yescharacteristic
393C22#1681076011yescharacteristic
394C22#1681080412yes—
395C22#1681081212yes—
396C22#1681093511yescharacteristic
397C22#1681094011yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—82no—
403C4—82no—
404C4—1064no—
405C4—4242no—
406C4—2124no—
407C4—2122no—
408C2—21no—
409C2—21no—
410C2—21no—
411C2—21no—
412C2#2027371612yes—
413C2#2027372112yes—
414C2#2027502911yescharacteristic
415C2#2027510511yescharacteristic
416C2#2027512511yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1823C2
socleC23×C53#120
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1823D53
hypocenterC53#9027
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
13. #16810554C4semidirect
234. C53#9027semidirect

Maximal subgroups

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

subgroupindex
C2×#168105542
#202773322
#202748012
C2×#168103482
#202725952
#202725932
C53×#18232
#902753