Finite Group Databasebeta

← #27280242

Subgroups — #27280242

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810825C211yesmaximal, characteristic
2C2×#16810688C211yesmaximal, characteristic
3#20275677C211yesmaximal, characteristic, complemented
4#20274716C211yesmaximal, characteristic
5#20272599C211yesmaximal, characteristic, complemented
6#20272549C211yesmaximal, characteristic, complemented
7C2×C53×#332C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C22×C8×D53—21no—
10C2×C4×D212C2211yescharacteristic
11#16810825—22no—
12#16810825C414yescomplemented
13#16810688—22no—
14#16810688—22no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133141—21no—
17C2×#13133139—21no—
18C2×#13133129—21no—
19C2×C4×#88322C2211yescharacteristic
20#16810493C2211yescharacteristic
21#16810386—21no—
22#16810385—21no—
23#16810385—21no—
24C2×C4⋊Q8×C53C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C53×#332—22no—
28C22×D212D411yescharacteristic
29C22×C4×D53—21no—
30C2×#11719924—24no—
31C2×#11719924—42no—
32C2×#11719924—42no—
33C22×#88322—21no—
34C22×#88322D411yescharacteristic
35C22×#88322D411yescharacteristic
36C2×C8×D53—24no—
37C2×C8×D53—42no—
38C2×C8×D53—42no—
39C4×D212—21no—
40C4×D212—21no—
41C4×D212C2×C412yes—
42C4×D212D412yes—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53D411yescharacteristic
47C2×#11719921—21no—
48C4⋊#88322—21no—
49C4⋊#88322C2×C412yes—
50#13133141—24no—
51#13133139—42no—
52#13133129—24no—
53C4×#88322—21no—
54C4×#88322—21no—
55C22×Q8×C53—21no—
56C22×C8×C53—21no—
57C22×C8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53C2×C412yes—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C42×C53C2311yescharacteristic
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67C4⋊C8×C53—21no—
68C4⋊C8×C53D412yes—
69C23×D53—21no—
70C2×Q8×C53—22no—
71C2×Q8×C53—41no—
72C2×Q8×C53—41no—
73C2×Q8×C53—41no—
74C2×Q8×C53—41no—
75C2×Q8×C53—41no—
76C22×C4×C53—41no—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×Dic53—21no—
81C22×Dic53—41no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212C22⋊C412yes—
86C2×D212D812yes—
87C2×D212SD1612yes—
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—
97C2×C8×C53—41no—
98C2×C8×C53—41no—
99C2×C8×C53—41no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104C42×C53C22×C411yescharacteristic
105C42×C53C2×D411yescharacteristic
106C42×C53C2×D411yescharacteristic
107C42×C53#4111yescharacteristic
108#11719924—48no—
109#11719921—22no—
110#11719921—22no—
111C4⋊Dic53—21no—
112C4⋊Dic53—41no—
113C4⋊Dic53—41no—
114C4⋊Dic53—41no—
115C4⋊Dic53—41no—
116C4⋊Dic53—81no—
117C4⋊Dic53C22⋊C412yes—
118C2×#88322—21no—
119C2×#88322—21no—
120C2×#88322—22no—
121C2×#88322—21no—
122C2×#88322—21no—
123C2×#88322—21no—
124C2×#88322—41no—
125C2×#88322—41no—
126C2×#88322SD1612yescomplemented
127C8×D53—48no—
128C23×C53#12111yescharacteristic
129C22×D53—22no—
130C22×D53—22no—
131C22×D53—22no—
132C22×D53—41no—
133C22×D53—41no—
134C22×D53—41no—
135C22×D53—41no—
136Q8×C53—42no—
137Q8×C53—42no—
138Q8×C53—81no—
139Q8×C53—81no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—42no—
143C2×C4×C53—81no—
144C2×C4×C53—81no—
145C2×C4×C53C2×C22⋊C411yescharacteristic
146C2×C4×C53C4×D411yescharacteristic
147C2×C4×C53C4×D411yescharacteristic
148C2×C4×C53#12011yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53#12311yescharacteristic
151C2×C4×C53C2×D811yescharacteristic
152C2×C4×C53C2×SD1611yescharacteristic
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×Dic53—22no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—81no—
163C2×Dic53—81no—
164C2×Dic53—81no—
165D212—22no—
166D212—22no—
167D212—22no—
168D212#10214yes—
169C4×D53—42no—
170C4×D53—42no—
171C4×D53—42no—
172C4×D53—42no—
173C8×C53—42no—
174C8×C53—81no—
175#88322—22no—
176#88322—42no—
177#88322—41no—
178#9217—531nomaximal, Sylow
179C22×C53#38611yescharacteristic
180C22×C53#46111yescharacteristic
181C22×C53#46311yescharacteristic
182C22×C53#46511yescharacteristic
183C22×C53#46611yescharacteristic
184C22×C53#47111yescharacteristic
185C22×C53#47311yescharacteristic
186C2×D53—24no—
187C2×D53—42no—
188C2×D53—42no—
189C2×D53—42no—
190C2×D53—42no—
191C2×D53—42no—
192C2×D53—42no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—82no—
196C4×C53C2×#10211yescharacteristic
197C4×C53#41611yescharacteristic
198C4×C53C4×SD1611yescharacteristic
199C4×C53#44011yescharacteristic
200C4×C53#45011yescharacteristic
201C4×C53#45211yescharacteristic
202C4×C53#46511yescharacteristic
203C4×C53#46611yescharacteristic
204Dic53—42no—
205Dic53—161no—
206C2×#461—531no—
207C2×C8×D4—531no—
208#1990—531no—
209#1802—531no—
210C2×#332D5311yescharacteristic, complemented
211#1296—531no—
212#1277—531no—
213C2×C53#167111yescharacteristic
214C2×C53#167311yescharacteristic
215C2×C53#167911yescharacteristic
216C2×C53#168111yescharacteristic
217C2×C53#187711yescharacteristic
218C2×C53#189311yescharacteristic
219C2×C53#190811yescharacteristic
220D53—44no—
221C22×SD16—1061no—
222C23×C8—1061no—
223C2×C4⋊Q8C2×D5311yescharacteristic
224C2×C4×D4—531no—
225#461—534no—
226#461—1062no—
227C8×D4—1062no—
228C8×D4—1062no—
229C2×C4⋊C8—531no—
230C2×C4⋊C8C2×D5311yescharacteristic
231C2×C4⋊C8C2×D5311yescharacteristic
232C2×#103—1061no—
233C2×#102—1061no—
234C2×C22⋊C8—1061no—
235C2×C4×C8—531no—
236#384—531no—
237#340—1061no—
238#340—1061no—
239#340—1061no—
240#332—22no—
241C53#921711yescharacteristic, complemented, Sylow
242C22×Q8—21no—
243C22×D4—531no—
244C23×C4—1061no—
245C2×SD16—1064no—
246C2×SD16—2122no—
247C2×SD16—2122no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—531no—
251C22×C8—531no—
252C22×C8—1064no—
253C22×C8—1061no—
254C22×C8—2122no—
255C22×C8—2122no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8C4×D5312yes—
260C4×D4—532no—
261C4×D4—532no—
262C4×D4—1061no—
263C4×D4—1061no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—531no—
267C2×C4⋊C4—1061no—
268C2×C4⋊C4—1061no—
269C2×C4⋊C4—1061no—
270C2×C22⋊C4—1061no—
271C2×C42C22×D5311yescharacteristic
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—532no—
276C4⋊C8—1061no—
277C4⋊C8C22⋊D5312yes—
278#103—1064no—
279#102—2122no—
280C22⋊C8—1064no—
281C4×C8—1061no—
282C4×C8—1061no—
283C24—1061no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×D4—532no—
291C2×D4—532no—
292C2×D4—532no—
293C2×D4—1061no—
294C2×D4—1061no—
295C2×D4—1061no—
296C22×C4—41no—
297C22×C4—1061no—
298C22×C4—1062no—
299C22×C4—1062no—
300C22×C4—2121no—
301C22×C4—2121no—
302C22×C4—2121no—
303C22×C4—2121no—
304C22×C4—2121no—
305C22×C4#1171994511yescharacteristic
306C22×C4D4×D5311yescharacteristic
307C22×C4D4×D5311yescharacteristic
308SD16—2128no—
309C2×C8—22no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—532no—
316C2×C8—1061no—
317C2×C8—1061no—
318C2×C8—1062no—
319C2×C8—1061no—
320C2×C8—1061no—
321C2×C8—1061no—
322C2×C8—2128no—
323C2×C8—2121no—
324C2×C8—2121no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—532no—
330C4⋊C4—1061no—
331C4⋊C4—4241no—
332C4⋊C4—2121no—
333C4⋊C4—2121no—
334C4⋊C4—2121no—
335C4⋊C4—2121no—
336C22⋊C4—1062no—
337C22⋊C4—1062no—
338C42C2×C4×D5311yescharacteristic
339C42D4×D5311yescharacteristic
340C42#1171994711yescharacteristic
341C42C2×C22⋊D5311yescharacteristic
342C23—1062no—
343C23—1062no—
344C23—1062no—
345C23—2121no—
346C23—2121no—
347C23—2121no—
348C23—2121no—
349C23#1313319011yescharacteristic
350Q8—42no—
351Q8—42no—
352Q8—81no—
353Q8—81no—
354D4—534no—
355D4—1062no—
356D4—1062no—
357D4—1062no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—1062no—
364C2×C4—4241no—
365C2×C4—4241no—
366C2×C4—4241no—
367C2×C4—2121no—
368C2×C4—2121no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4—2122no—
372C2×C4—2122no—
373C2×C4C22⋊C4×D5311yescharacteristic
374C2×C4#1313317711yescharacteristic
375C2×C4C4×C22⋊D5311yescharacteristic
376C2×C4#1313318511yescharacteristic
377C2×C4#1313318811yescharacteristic
378C2×C4#1313319911yescharacteristic
379C2×C4D8×D5311yescharacteristic
380C2×C4SD16×D5311yescharacteristic
381C2×C4SD16×D5311yescharacteristic
382C2×C4#1313323611yescharacteristic
383C2×C4#1313323811yescharacteristic
384C2×C4#1313324011yescharacteristic
385C2×C4#1313324411yescharacteristic
386C2×C4#1313324911yescharacteristic
387C8—42no—
388C8—81no—
389C8—1062no—
390C8—2121no—
391C8—2122no—
392C22—1064no—
393C22—2122no—
394C22—2122no—
395C22—2122no—
396C22—2122no—
397C22—2122no—
398C22—2122no—
399C22#1681049911yescharacteristic
400C22#1681082711yescharacteristic
401C22#1681084111yescharacteristic
402C22#1681085111yescharacteristic
403C22#1681085811yescharacteristic
404C22#1681088911yescharacteristic
405C22#1681089711yescharacteristic
406C4—21no—
407C4—21no—
408C4—82no—
409C4—8481no—
410C4—2122no—
411C4#102×D5311yescharacteristic
412C4#1681061611yescharacteristic
413C4#1681070811yescharacteristic
414C4#1681072611yescharacteristic
415C4#1681077211yescharacteristic
416C4#1681078411yescharacteristic
417C4#1681084911yescharacteristic
418C4#1681085611yescharacteristic
419C2—2124no—
420C2#2027408911yescharacteristic
421C2#2027410311yescharacteristic
422C2#2027414111yescharacteristic
423C2#2027415511yescharacteristic
424C2#2027504611yescharacteristic
425C2#2027514011yescharacteristic
426C2#2027522711yescharacteristic
subgroupG/H
centerC23#13133190
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#332C2
socleC23×C53#121
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenterC2×#332D53
hypocenterC53#9217
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
3. #20275677C2semidirect
5. #20272599C2semidirect
6. #20272549C2semidirect
7. C2×C53×#332C2semidirect
12. #16810825C4semidirect
126. C2×#88322SD16semidirect
210. C2×#332D53semidirect
241. C53#9217semidirect

Maximal subgroups

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

subgroupindex
C2×#168108252
C2×#168106882
#202756772
#202747162
#202725992
#202725492
C2×C53×#3322
#921753