Finite Group Databasebeta

← #27272238

Subgroups — #27272238

Order 13568.

612 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#389×D53C211yesmaximal, characteristic
2#20274782C211yesmaximal, characteristic
3#20274407C211yesmaximal, characteristic
4#20272539C211yesmaximal, characteristic, complemented
5#20272538C211yesmaximal, characteristic, complemented
6#20272537C211yesmaximal, characteristic, complemented
7C53×#1271C211yesmaximal, characteristic, complemented
8C22×#11719912—21no—
9C2×C4⋊C4×D53—21no—
10C2×C4⋊C4×D53C411yescharacteristic
11C2×C4⋊C4×D53C411yescharacteristic
12C2×C42×D53C2211yescharacteristic
13C2×#13133129—21no—
14#16810522C411yescharacteristic
15#16810521C2211yescharacteristic
16#16810520—21no—
17#16810519C411yescharacteristic
18#16810518—21no—
19#95×D53—21no—
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22#16810377—21no—
23#16810377—21no—
24C53×#389C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×C4×D53—21no—
28C22×C4×D53—21no—
29C22×C4×D53C2×C411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C22×C4×D53D411yescharacteristic
32C2×#11719912—24no—
33C2×#11719912—42no—
34C2×#11719912—42no—
35C22×#88322—21no—
36C22×#88322—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919—21no—
43C2×#11719919—21no—
44C4⋊C4×D53—22no—
45C4⋊C4×D53—22no—
46C4⋊C4×D53—42no—
47C4⋊C4×D53—42no—
48C4⋊C4×D53—42no—
49C2×C4×Dic53C2×C411yescharacteristic
50C2×C4×Dic53D411yescharacteristic
51C2×C4×Dic53D411yescharacteristic
52C42×D53—42no—
53#13133129—42no—
54#13133124—21no—
55#13133124—21no—
56#13133124—21no—
57C22×C8×C53—21no—
58C22×C8×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53C2×C411yescharacteristic
61C2×C4⋊C4×C53C2×C411yescharacteristic
62C2×C42×C53C2311yescharacteristic
63#95×C53—21no—
64C23×D53C22⋊C411yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53C22⋊C411yescharacteristic
73C22×Dic53C22⋊C411yescharacteristic
74C22×Dic53C22⋊C411yescharacteristic
75C2×C4×D53—22no—
76C2×C4×D53—22no—
77C2×C4×D53—22no—
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—22no—
81C2×C4×D53—24no—
82C2×C4×D53—22no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53—41no—
86C2×C4×D53—42no—
87C2×C4×D53—42no—
88C2×C4×D53—42no—
89C2×C4×D53—42no—
90C2×C4×D53—42no—
91C2×C4×D53D812yes—
92C2×C4×D53SD1612yes—
93C2×C8×C53—22no—
94C2×C8×C53—41no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C2×C8×C53—41no—
98C2×C8×C53—41no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53M1612yescomplemented
104C42×C53—41no—
105C4⋊Dic53—21no—
106C4⋊Dic53—42no—
107C4⋊Dic53—41no—
108C4⋊Dic53M1612yescomplemented
109#11719919—24no—
110#11719919—42no—
111#11719919—41no—
112#11719919—41no—
113#11719919—42no—
114C4×Dic53—41no—
115C4×Dic53—41no—
116C4×Dic53—41no—
117C2×#88322—22no—
118C2×#88322—41no—
119C2×#88322—41no—
120C2×#88322—41no—
121C2×#88322—41no—
122C2×#88322—41no—
123#11719912—48no—
124C23×C53C2×C22⋊C411yescharacteristic
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53#9912yes—
130C22×D53#10212yes—
131C22×D53#10412yes—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—22no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53C2×M1611yescharacteristic
145C2×C4×C53C2×M1611yescharacteristic
146C2×C4×C53C2×D811yescharacteristic
147C2×C4×C53C2×SD1611yescharacteristic
148C2×C4×C53#13711yescharacteristic
149C2×C4×C53#13711yescharacteristic
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—22no—
155C2×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53—21no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53#10112yes—
164C2×Dic53#10212yes—
165C4×D53—24no—
166C4×D53—24no—
167C4×D53—24no—
168C4×D53—24no—
169C4×D53—42no—
170C4×D53—42no—
171C4×D53—48no—
172C4×D53—84no—
173C8×C53—42no—
174C8×C53—81no—
175#88322—42no—
176#88322—81no—
177#7610—531nomaximal, Sylow
178C22×C53#40711yescharacteristic
179C22×C53C2×#9911yescharacteristic
180C22×C53C2×#10111yescharacteristic
181C22×C53C2×#10211yescharacteristic
182C22×C53#41911yescharacteristic
183C22×C53C2×#10411yescharacteristic
184C22×C53#42111yescharacteristic
185C2×D53—22no—
186C2×D53—22no—
187C2×D53—22no—
188C2×D53—22no—
189C2×D53—22no—
190C2×D53—22no—
191C2×D53#32714yes—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—42no—
199C4×C53—81no—
200Dic53—22no—
201Dic53—22no—
202Dic53—42no—
203Dic53—41no—
204Dic53—81no—
205C2×#389—531no—
206#1817—531no—
207#1728—531no—
208#1271—531no—
209#1271—531no—
210#1271—531no—
211#1271D5311yescharacteristic, complemented
212C2×C53#146311yescharacteristic
213C2×C53#148611yescharacteristic
214C2×C53#149111yescharacteristic
215C2×C53C2×#32711yescharacteristic
216C2×C53#151111yescharacteristic
217C2×C53#152811yescharacteristic
218C2×C53#154611yescharacteristic
219D53—24no—
220C22×M16—1061no—
221C22×C4⋊C4—531no—
222C22×C4⋊C4—531no—
223C22×C4⋊C4—1061no—
224C22×C42—531no—
225C2×C22⋊C8—1061no—
226#389—531no—
227#389—531no—
228#389—531no—
229#389—1061no—
230#389—1061no—
231#389C2×D5311yescharacteristic
232C2×#95—1061no—
233#336—531no—
234#336—531no—
235#336—1061no—
236#336—1061no—
237#336C2×D5311yescharacteristic
238#336C2×D5311yescharacteristic
239C53#761011yescharacteristic, complemented, Sylow
240C23×C4—531no—
241C23×C4—531no—
242C23×C4—531no—
243C23×C4—1061no—
244C23×C4—1061no—
245C2×M16—1064no—
246C2×M16—2122no—
247C2×M16—2122no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—1061no—
251C22×C8—1061no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—531no—
254C2×C4⋊C4—531no—
255C2×C4⋊C4—1062no—
256C2×C4⋊C4—1061no—
257C2×C4⋊C4—1062no—
258C2×C4⋊C4—1061no—
259C2×C4⋊C4—1061no—
260C2×C4⋊C4—1061no—
261C2×C4⋊C4—1061no—
262C2×C4⋊C4—2122no—
263C2×C4⋊C4—2122no—
264C2×C4⋊C4—2122no—
265C2×C4⋊C4C4×D5311yescharacteristic
266C2×C4⋊C4C4×D5311yescharacteristic
267C2×C42—531no—
268C2×C42—531no—
269C2×C42—531no—
270C2×C42—2122no—
271C2×C42C22×D5311yescharacteristic
272C22⋊C8—2122no—
273#95—21no—
274#95—1061no—
275#95—1061no—
276#95—1061no—
277C24—531no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—531no—
281C22×C4—531no—
282C22×C4—532no—
283C22×C4—532no—
284C22×C4—531no—
285C22×C4—1062no—
286C22×C4—1062no—
287C22×C4—1062no—
288C22×C4—1062no—
289C22×C4—1062no—
290C22×C4—1064no—
291C22×C4—1061no—
292C22×C4—1061no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4—1061no—
297C22×C4—2121no—
298C22×C4—2122no—
299C22×C4—2122no—
300C22×C4—2122no—
301C22×C4—2122no—
302C22×C4—2122no—
303C22×C4C2×C4×D5311yescharacteristic
304C22×C4D4×D5311yescharacteristic
305C22×C4D4×D5311yescharacteristic
306M16—2128no—
307C2×C8—22no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—1062no—
314C2×C8—2121no—
315C2×C8—2121no—
316C2×C8—2121no—
317C2×C8—2121no—
318C2×C8—2121no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—532no—
324C4⋊C4—1064no—
325C4⋊C4—1061no—
326C4⋊C4—2121no—
327C4⋊C4—2122no—
328C4⋊C4—2122no—
329C4⋊C4—2122no—
330C4⋊C4—2121no—
331C4⋊C4—2121no—
332C4⋊C4#1171991212yescomplemented
333C42—41no—
334C42—2121no—
335C42—2121no—
336C42—2121no—
337C23—532no—
338C23—532no—
339C23—532no—
340C23—1061no—
341C23—1061no—
342C23—1061no—
343C23—1061no—
344C23C22⋊C4×D5311yescharacteristic
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—532no—
358C2×C4—532no—
359C2×C4—1064no—
360C2×C4—1064no—
361C2×C4—1064no—
362C2×C4—1064no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—1062no—
368C2×C4—1061no—
369C2×C4—1061no—
370C2×C4—1061no—
371C2×C4—4244no—
372C2×C4—2122no—
373C2×C4—2122no—
374C2×C4—2128no—
375C2×C4—2121no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4—2121no—
379C2×C4—2121no—
380C2×C4M16×D5311yescharacteristic
381C2×C4C2×#1171991211yescharacteristic
382C2×C4D8×D5311yescharacteristic
383C2×C4SD16×D5311yescharacteristic
384C2×C4#1313324611yescharacteristic
385C2×C4#1313324911yescharacteristic
386C8—42no—
387C8—81no—
388C8—4241no—
389C8—2122no—
390C22—534no—
391C22—1062no—
392C22—1062no—
393C22—1062no—
394C22—1062no—
395C22—1062no—
396C22—1062no—
397C22#1681058511yescharacteristic
398C22#99×D5311yescharacteristic
399C22#101×D5311yescharacteristic
400C22#102×D5311yescharacteristic
401C22#1681063011yescharacteristic
402C22#104×D5311yescharacteristic
403C22#1681063811yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—42no—
411C4—81no—
412C4—1062no—
413C4—1062no—
414C4—4241no—
415C4—2121no—
416C4—2122no—
417C2—1064no—
418C2#2027304311yescharacteristic
419C2#2027313811yescharacteristic
420C2#2027316011yescharacteristic
421C2#327×D5311yescharacteristic
422C2#2027324611yescharacteristic
423C2#2027331511yescharacteristic
424C2#2027338511yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1271C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1271D53
hypocenterC53#7610
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #20272539C2semidirect
5. #20272538C2semidirect
6. #20272537C2semidirect
7. C53×#1271C2semidirect
103. C4⋊C4×C53M16semidirect
108. C4⋊Dic53M16semidirect
211. #1271D53semidirect
239. C53#7610semidirect
332. C4⋊C4#11719912semidirect

Maximal subgroups

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

subgroupindex
#389×D532
#202747822
#202744072
#202725392
#202725382
#202725372
C53×#12712
#761053