Finite Group Databasebeta

← #27278843

Subgroups — #27278843

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810910C211yesmaximal, characteristic
2#20277666C211yesmaximal, characteristic
3#20275664C211yesmaximal, characteristic, complemented
4C2×#16810352C211yesmaximal, characteristic
5#20272598C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C53×#1956C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9C2×#13133175—21no—
10#16810910—22no—
11#16810910C414yescomplemented
12C2×#13133148—21no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133137—21no—
16#16810524—21no—
17#16810513—22no—
18#16810480—21no—
19C4⋊C4×Dic53—21no—
20C4×C4⋊Dic53C2211yescharacteristic
21#16810385—21no—
22#16810385—21no—
23#16810352—22no—
24C2×C4⋊Q8×C53C2211yescharacteristic
25C2×#103×C53—22no—
26C2×C4×C8×C53C2211yescharacteristic
27C53×#384C2211yescharacteristic
28C22×D212D411yescharacteristic
29C22×C4×D53—21no—
30C22×#88322—21no—
31C22×#88322—21no—
32C4×D212—21no—
33C4×D212—21no—
34C4×D212C2×C412yes—
35C4×D212D412yes—
36#13133175—42no—
37#13133175—42no—
38C2×C4⋊Dic53D411yescharacteristic
39C2×C4⋊Dic53D411yescharacteristic
40C2×C4⋊Dic53D411yescharacteristic
41C2×#11719921—22no—
42C2×#11719921—22no—
43C2×#11719921—21no—
44C2×C4×Dic53—22no—
45C2×C4×Dic53—21no—
46#13133148—24no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50C4⋊#88322C2×C412yes—
51#13133137—24no—
52#13133124—22no—
53#13133124—21no—
54C22×Q8×C53—21no—
55C22×C8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53C2×C412yes—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53D412yes—
65C2×C42×C53C2311yescharacteristic
66#103×C53—44no—
67C4×C8×C53—21no—
68C4×C8×C53—21no—
69C23×D53—21no—
70C2×Q8×C53—22no—
71C2×Q8×C53—42no—
72C2×Q8×C53—42no—
73C2×Q8×C53—41no—
74C22×C4×C53—22no—
75C22×C4×C53—41no—
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212C22⋊C412yes—
88C2×C4×D53—22no—
89C2×C4×D53—22no—
90C2×C4×D53—41no—
91C2×C4×D53—41no—
92C2×C4×D53—41no—
93C2×C4×D53—41no—
94C2×C8×C53—22no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—42no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C42×C53C22×C411yescharacteristic
107C42×C53C2×D411yescharacteristic
108C42×C53#4111yescharacteristic
109C42×C53#4111yescharacteristic
110#11719921—22no—
111#11719921—22no—
112#11719921—44no—
113#11719921—82no—
114C4⋊Dic53—21no—
115C4⋊Dic53—21no—
116C4⋊Dic53—21no—
117C4⋊Dic53—21no—
118C4⋊Dic53C22⋊C412yes—
119C4⋊Dic53SD1612yes—
120C4×Dic53—44no—
121C4×Dic53—41no—
122C4×Dic53—41no—
123C2×#88322—22no—
124C2×#88322—41no—
125C2×#88322—41no—
126C2×#88322—41no—
127C2×#88322—41no—
128C2×#88322—41no—
129C23×C53#12011yescharacteristic
130C22×D53—22no—
131C22×D53—41no—
132C22×D53—41no—
133C22×D53—41no—
134C22×D53—41no—
135C22×D53—41no—
136C22×D53—41no—
137Q8×C53—44no—
138Q8×C53—82no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—81no—
146C2×C4×C53—81no—
147C2×C4×C53C2×C22⋊C411yescharacteristic
148C2×C4×C53C4×D411yescharacteristic
149C2×C4×C53C4×D411yescharacteristic
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#12311yescharacteristic
153C2×C4×C53C2×SD1611yescharacteristic
154C2×C4×C53#13511yescharacteristic
155C2×C4×C53#13511yescharacteristic
156C2×C4×C53#13511yescharacteristic
157C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C53#13711yescharacteristic
159C2×C4×C53#13711yescharacteristic
160C2×C4×C53#13711yescharacteristic
161C2×Dic53—22no—
162C2×Dic53—22no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170D212—22no—
171D212—22no—
172D212—41no—
173D212—41no—
174C4×D53—42no—
175C4×D53—42no—
176C4×D53—42no—
177C4×D53—42no—
178C8×C53—42no—
179#88322—42no—
180#88322—81no—
181#8976—531nomaximal, Sylow
182C22×C53#38611yescharacteristic
183C22×C53#45011yescharacteristic
184C22×C53#45211yescharacteristic
185C22×C53#45211yescharacteristic
186C22×C53#45211yescharacteristic
187C22×C53#45311yescharacteristic
188C22×C53#45411yescharacteristic
189C2×D53—42no—
190C2×D53—42no—
191C2×D53—42no—
192C2×D53—81no—
193C2×D53—81no—
194C2×D53—81no—
195C2×D53—81no—
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—82no—
199C4×C53—82no—
200C4×C53#41611yescharacteristic
201C4×C53#41911yescharacteristic
202C4×C53C4×SD1611yescharacteristic
203C4×C53#44011yescharacteristic
204C4×C53#46111yescharacteristic
205C4×C53#46311yescharacteristic
206C4×C53#48511yescharacteristic
207C4×C53#48511yescharacteristic
208Dic53—42no—
209Dic53—42no—
210Dic53—81no—
211C2×#476—531no—
212#2354—531no—
213#1987—531no—
214#1956D5311yescharacteristic, complemented
215C2×D4⋊C8—531no—
216#1296—531no—
217#1279—531no—
218C2×C53#163111yescharacteristic
219C2×C53#163411yescharacteristic
220C2×C53#163511yescharacteristic
221C2×C53#163811yescharacteristic
222C2×C53#187311yescharacteristic
223C2×C53#187711yescharacteristic
224C2×C53#188311yescharacteristic
225D53—82no—
226C2×C4⋊Q8C2×D5311yescharacteristic
227C2×C4×D4—531no—
228C2×#117—1061no—
229#476—534no—
230#476—1062no—
231C2×#106—1061no—
232C2×C4⋊C8—531no—
233C2×C4⋊C8—531no—
234C2×#103—22no—
235C2×#102—1061no—
236C2×C4×C8C2×D5311yescharacteristic
237#390—1061no—
238#388—1062no—
239#384C2×D5311yescharacteristic
240#380—1061no—
241C4×C4⋊C4—531no—
242C4×C4⋊C4—1061no—
243#340—1061no—
244#340—1061no—
245D4⋊C8—1062no—
246C53#897611yescharacteristic, complemented, Sylow
247C22×Q8—21no—
248C22×D4—531no—
249C23×C4—1061no—
250C22×C8—21no—
251C22×C8—1061no—
252C22×C8—1061no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8C4×D5312yes—
257C4×D4—532no—
258C4×D4—532no—
259C4×D4—1061no—
260C4×D4—1061no—
261#117—2122no—
262#117—2122no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—531no—
268C2×C4⋊C4—531no—
269C2×C4⋊C4—531no—
270C2×C4⋊C4C22⋊D5312yes—
271C2×C22⋊C4—1061no—
272C2×C22⋊C4—1062no—
273C2×C22⋊C4—1062no—
274C2×C42—1062no—
275C2×C42—1061no—
276C2×C42C22×D5311yescharacteristic
277#106—1064no—
278C4⋊C8—532no—
279C4⋊C8—1061no—
280C4⋊C8—1061no—
281C4⋊C8—1061no—
282#103—44no—
283#102—1064no—
284C4×C8—21no—
285C4×C8—21no—
286#95—1062no—
287#95—1061no—
288C24—1061no—
289C2×Q8—22no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—41no—
293C2×D4—532no—
294C2×D4—1061no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—1061no—
298C2×D4—1061no—
299C22×C4—22no—
300C22×C4—41no—
301C22×C4—1061no—
302C22×C4—1061no—
303C22×C4—1062no—
304C22×C4—1062no—
305C22×C4—1061no—
306C22×C4—2121no—
307C22×C4—2121no—
308C22×C4—2121no—
309C22×C4—2121no—
310C22×C4D4×D5311yescharacteristic
311C22×C4D4×D5311yescharacteristic
312C22×C4C2×C22⋊D5311yescharacteristic
313C2×C8—22no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—1062no—
317C2×C8—2121no—
318C2×C8—2121no—
319C2×C8—2121no—
320C2×C8—2121no—
321C2×C8—2121no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—532no—
332C4⋊C4—532no—
333C4⋊C4—1061no—
334C4⋊C4—1061no—
335C4⋊C4—1061no—
336C4⋊C4—1061no—
337C22⋊C4—1062no—
338C22⋊C4—1062no—
339C22⋊C4—4242no—
340C22⋊C4—2124no—
341C42—2124no—
342C42—2121no—
343C42—2121no—
344C42C2×C4×D5311yescharacteristic
345C42#1171994511yescharacteristic
346C42D4×D5311yescharacteristic
347C42#1171994711yescharacteristic
348C23—1062no—
349C23—2121no—
350C23—2121no—
351C23—2121no—
352C23—2121no—
353C23—2121no—
354C23—2121no—
355C23#1313318511yescharacteristic
356Q8—44no—
357Q8—82no—
358D4—1062no—
359D4—1062no—
360D4—2121no—
361D4—2121no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—1062no—
371C2×C4—1062no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4—2121no—
375C2×C4—2121no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4—2121no—
379C2×C4—2122no—
380C2×C4—2122no—
381C2×C4—2122no—
382C2×C4—2122no—
383C2×C4C22⋊C4×D5311yescharacteristic
384C2×C4#1313317711yescharacteristic
385C2×C4C4×C22⋊D5311yescharacteristic
386C2×C4#1313318811yescharacteristic
387C2×C4#1313319011yescharacteristic
388C2×C4#1313319911yescharacteristic
389C2×C4SD16×D5311yescharacteristic
390C2×C4#1313323811yescharacteristic
391C2×C4#1313324011yescharacteristic
392C2×C4#1313324011yescharacteristic
393C2×C4#1313324411yescharacteristic
394C2×C4#1313324611yescharacteristic
395C2×C4#1313324911yescharacteristic
396C2×C4#1313324911yescharacteristic
397C8—42no—
398C8—4241no—
399C8—2122no—
400C22—4241no—
401C22—4241no—
402C22—4241no—
403C22—4241no—
404C22—2122no—
405C22—2122no—
406C22—2122no—
407C22#1681049911yescharacteristic
408C22#1681077211yescharacteristic
409C22#1681078211yescharacteristic
410C22#1681078411yescharacteristic
411C22#1681078411yescharacteristic
412C22#1681079011yescharacteristic
413C22#1681079511yescharacteristic
414C4—21no—
415C4—21no—
416C4—82no—
417C4—82no—
418C4—4241no—
419C4—2122no—
420C4—2122no—
421C4#1681061811yescharacteristic
422C4#1681063011yescharacteristic
423C4#1681070811yescharacteristic
424C4#1681072611yescharacteristic
425C4#1681082711yescharacteristic
426C4#1681084111yescharacteristic
427C4#1681096211yescharacteristic
428C4#1681096311yescharacteristic
429C2—4242no—
430C2#2027383911yescharacteristic
431C2#2027385811yescharacteristic
432C2#2027386311yescharacteristic
433C2#2027388411yescharacteristic
434C2#2027502411yescharacteristic
435C2#2027504611yescharacteristic
436C2#2027509411yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1956C2
socleC23×C53#120
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1956D53
hypocenterC53#8976
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
3. #20275664C2semidirect
5. #20272598C2semidirect
6. #20272555C2semidirect
7. C53×#1956C2semidirect
11. #16810910C4semidirect
214. #1956D53semidirect
246. C53#8976semidirect

Maximal subgroups

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

subgroupindex
C2×#168109102
#202776662
#202756642
C2×#168103522
#202725982
#202725552
C53×#19562
#897653