Finite Group Databasebeta

← #27280260

Subgroups — #27280260

Order 13568.

594 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
1C2×#16810904C211yesmaximal, characteristic
2C2×#16810688C211yesmaximal, characteristic
3#20275516C211yesmaximal, characteristic, complemented
4#20274712C211yesmaximal, characteristic
5C2×#16810370C211yesmaximal, characteristic, complemented
6#20272549C211yesmaximal, characteristic, complemented
7C53×#1296C211yesmaximal, characteristic, complemented
8C22×C8×D53—21no—
9C2×#13133217C2211yescharacteristic
10C2×C4×D212C2211yescharacteristic
11#16810904—22no—
12#16810904C414yescomplemented
13#16810688—22no—
14#16810688—22no—
15C2×#13133151—21no—
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133140—21no—
18C2×#13133140—21no—
19C2×#13133139—21no—
20C2×#13133137—21no—
21C2×#13133129—21no—
22C2×C4×#88322C2211yescharacteristic
23#16810385—21no—
24#16810370—22no—
25C2×C4⋊C8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C53×#384C2211yescharacteristic
28C22×Dic106—21no—
29C22×D212D411yescharacteristic
30C22×C4×D53—21no—
31C22×#88322—21no—
32C22×#88322D411yescharacteristic
33C22×#88322D411yescharacteristic
34C2×C8×D53—24no—
35C2×C8×D53—42no—
36C2×C8×D53—42no—
37#13133217—21no—
38#13133217—21no—
39#13133217—21no—
40#13133217C2×C412yes—
41C4×D212—21no—
42C4×D212—21no—
43C4×D212C2×C412yes—
44C4×D212D412yes—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53D411yescharacteristic
48C2×#11719921—21no—
49#13133151—24no—
50C4⋊#88322—21no—
51C4⋊#88322—21no—
52#13133140—42no—
53#13133140—42no—
54#13133139—24no—
55#13133137—42no—
56#13133129—24no—
57C4×#88322—21no—
58C4×#88322—21no—
59C22×C8×C53—21no—
60C22×C8×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C42×C53C2311yescharacteristic
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67C4⋊C8×C53C2×C412yes—
68C4⋊C8×C53D412yes—
69C23×D53—21no—
70C22×C4×C53—41no—
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53#4111yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—41no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212C22⋊C412yes—
80C2×D212SD1612yes—
81C2×D212Q1612yes—
82C2×C4×D53—22no—
83C2×C4×D53—22no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×Dic106—22no—
89C2×Dic106—41no—
90C2×Dic106—41no—
91C2×Dic106—41no—
92C2×Dic106—41no—
93C2×Dic106—41no—
94C2×C8×C53—22no—
95C2×C8×C53—22no—
96C2×C8×C53—41no—
97C2×C8×C53—41no—
98C2×C8×C53—41no—
99C2×C8×C53—41no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—81no—
105C42×C53C22×C411yescharacteristic
106C42×C53C2×D411yescharacteristic
107C42×C53C2×D411yescharacteristic
108C42×C53#4111yescharacteristic
109#11719921—22no—
110#11719921—22no—
111C4⋊Dic53—21no—
112C4⋊Dic53—22no—
113C4⋊Dic53—22no—
114C4⋊Dic53—41no—
115C4⋊Dic53—41no—
116C4⋊Dic53C22⋊C412yes—
117C2×#88322—21no—
118C2×#88322—21no—
119C2×#88322—21no—
120C2×#88322—21no—
121C2×#88322—21no—
122C2×#88322—41no—
123C2×#88322—41no—
124C2×#88322—41no—
125C2×#88322D812yes—
126C8×D53—48no—
127C23×C53#12111yescharacteristic
128C22×D53—22no—
129C22×D53—22no—
130C22×D53—22no—
131C22×D53—41no—
132C22×D53—41no—
133C22×D53—41no—
134C22×D53—41no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—81no—
138C2×C4×C53—81no—
139C2×C4×C53—81no—
140C2×C4×C53C2×C22⋊C411yescharacteristic
141C2×C4×C53C4×D411yescharacteristic
142C2×C4×C53C4×D411yescharacteristic
143C2×C4×C53#12011yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12311yescharacteristic
146C2×C4×C53C2×D811yescharacteristic
147C2×C4×C53C2×SD1611yescharacteristic
148C2×C4×C53C2×Q1611yescharacteristic
149C2×C4×C53#13511yescharacteristic
150C2×C4×C53#13511yescharacteristic
151C2×C4×C53#13511yescharacteristic
152C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
154C2×Dic53—22no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—42no—
158C2×Dic53—81no—
159C2×Dic53—81no—
160D212—22no—
161D212—22no—
162D212—22no—
163D212#10314yescomplemented
164C4×D53—42no—
165C4×D53—42no—
166C4×D53—42no—
167C4×D53—42no—
168Dic106—42no—
169Dic106—42no—
170Dic106—81no—
171Dic106—81no—
172C8×C53—42no—
173C8×C53—42no—
174#88322—22no—
175#88322—41no—
176#88322—81no—
177#9219—531nomaximal, Sylow
178C22×C53#38611yescharacteristic
179C22×C53#45911yescharacteristic
180C22×C53#46311yescharacteristic
181C22×C53#46511yescharacteristic
182C22×C53#46711yescharacteristic
183C22×C53#47111yescharacteristic
184C22×C53#47311yescharacteristic
185C2×D53—24no—
186C2×D53—42no—
187C2×D53—42no—
188C2×D53—42no—
189C2×D53—42no—
190C2×D53—42no—
191C2×D53—42no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—161no—
195C4×C53C2×#10311yescharacteristic
196C4×C53#41611yescharacteristic
197C4×C53C4×D811yescharacteristic
198C4×C53#44211yescharacteristic
199C4×C53#44711yescharacteristic
200C4×C53#44911yescharacteristic
201C4×C53#46511yescharacteristic
202C4×C53#46711yescharacteristic
203Dic53—42no—
204Dic53—82no—
205C2×#474—531no—
206C2×C8×D4—531no—
207#1956—531no—
208#1802—531no—
209C2×#332—531no—
210#1296D5311yescharacteristic, complemented
211#1277—531no—
212C2×C53#166511yescharacteristic
213C2×C53#166911yescharacteristic
214C2×C53#167911yescharacteristic
215C2×C53#168311yescharacteristic
216C2×C53#187911yescharacteristic
217C2×C53#189211yescharacteristic
218C2×C53#190811yescharacteristic
219D53—44no—
220C23×C8—1061no—
221C2×C4⋊Q8—531no—
222C2×C4×D4—531no—
223#474—534no—
224#474—1062no—
225C8×D4—1062no—
226C8×D4—1062no—
227C2×#107—1061no—
228C2×C4⋊C8—531no—
229C2×C4⋊C8C2×D5311yescharacteristic
230C2×C4⋊C8C2×D5311yescharacteristic
231C2×#103—1061no—
232C2×#103—1061no—
233C2×#102—1061no—
234C2×#102—1061no—
235C2×C22⋊C8—1061no—
236C2×C4×C8—531no—
237#384C2×D5311yescharacteristic
238#340—1061no—
239#332—1062no—
240C53#921911yescharacteristic, complemented, Sylow
241C22×Q8—1061no—
242C22×D4—531no—
243C23×C4—1061no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—531no—
247C22×C8—531no—
248C22×C8—1064no—
249C22×C8—1061no—
250C22×C8—2122no—
251C22×C8—2122no—
252C4⋊Q8—532no—
253C4⋊Q8—1061no—
254C4⋊Q8—1061no—
255C4⋊Q8—1061no—
256C4×D4—532no—
257C4×D4—532no—
258C4×D4—1061no—
259C4×D4—1061no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—531no—
264C2×C4⋊C4—1061no—
265C2×C4⋊C4—1061no—
266C2×C22⋊C4—1061no—
267C2×C42C22×D5311yescharacteristic
268#107—1064no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—1061no—
272C4⋊C8—1061no—
273C4⋊C8C4×D5312yes—
274C4⋊C8D21212yes—
275#103—2122no—
276#103—2122no—
277#102—1064no—
278#102—2122no—
279C22⋊C8—1064no—
280C4×C8—1061no—
281C4×C8—1061no—
282C24—1061no—
283C2×Q8—1062no—
284C2×Q8—2121no—
285C2×Q8—2121no—
286C2×Q8—2121no—
287C2×Q8—2121no—
288C2×Q8—2121no—
289C2×D4—532no—
290C2×D4—532no—
291C2×D4—532no—
292C2×D4—1061no—
293C2×D4—1061no—
294C2×D4—1061no—
295C22×C4—41no—
296C22×C4—1061no—
297C22×C4—1062no—
298C22×C4—1062no—
299C22×C4—2121no—
300C22×C4—2121no—
301C22×C4—2121no—
302C22×C4—2121no—
303C22×C4—2121no—
304C22×C4#1171994511yescharacteristic
305C22×C4D4×D5311yescharacteristic
306C22×C4D4×D5311yescharacteristic
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—532no—
314C2×C8—1061no—
315C2×C8—1061no—
316C2×C8—1061no—
317C2×C8—1061no—
318C2×C8—1061no—
319C2×C8—2128no—
320C2×C8—2121no—
321C2×C8—2121no—
322C2×C8—2121no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—81no—
328C4⋊C4—532no—
329C4⋊C4—1061no—
330C4⋊C4—1062no—
331C4⋊C4—1062no—
332C4⋊C4—2121no—
333C4⋊C4—2121no—
334C22⋊C4—1062no—
335C22⋊C4—1062no—
336C42C2×C4×D5311yescharacteristic
337C42C2×D21211yescharacteristic
338C42D4×D5311yescharacteristic
339C42#1171994911yescharacteristic
340C23—1062no—
341C23—1062no—
342C23—1062no—
343C23—2121no—
344C23—2121no—
345C23—2121no—
346C23—2121no—
347C23#1313319011yescharacteristic
348Q8—4241no—
349Q8—4241no—
350Q8—2122no—
351Q8—2122no—
352D4—534no—
353D4—1062no—
354D4—1062no—
355D4—1062no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—1062no—
362C2×C4—4241no—
363C2×C4—4241no—
364C2×C4—2122no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4—2122no—
368C2×C4—2122no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4C22⋊C4×D5311yescharacteristic
372C2×C4#1313317911yescharacteristic
373C2×C4C4×D21211yescharacteristic
374C2×C4#1313318711yescharacteristic
375C2×C4#1313319211yescharacteristic
376C2×C4#1313320211yescharacteristic
377C2×C4D8×D5311yescharacteristic
378C2×C4SD16×D5311yescharacteristic
379C2×C4Q16×D5311yescharacteristic
380C2×C4#1313323611yescharacteristic
381C2×C4#1313323811yescharacteristic
382C2×C4#1313324011yescharacteristic
383C2×C4#1313324411yescharacteristic
384C2×C4#1313324411yescharacteristic
385C8—42no—
386C8—42no—
387C8—1062no—
388C8—4241no—
389C8—2121no—
390C22—1064no—
391C22—2122no—
392C22—2122no—
393C22—2122no—
394C22—2122no—
395C22—2122no—
396C22—2122no—
397C22#1681050111yescharacteristic
398C22#1681081711yescharacteristic
399C22#1681084111yescharacteristic
400C22#1681085111yescharacteristic
401C22#1681086311yescharacteristic
402C22#1681088511yescharacteristic
403C22#1681089711yescharacteristic
404C4—21no—
405C4—21no—
406C4—161no—
407C4—4242no—
408C4—2122no—
409C4#103×D5311yescharacteristic
410C4#1681061611yescharacteristic
411C4#1681070511yescharacteristic
412C4#1681073411yescharacteristic
413C4#1681075811yescharacteristic
414C4#1681077111yescharacteristic
415C4#1681085311yescharacteristic
416C4#1681086511yescharacteristic
417C2—2124no—
418C2#2027405511yescharacteristic
419C2#2027407911yescharacteristic
420C2#2027414111yescharacteristic
421C2#2027416911yescharacteristic
422C2#2027506411yescharacteristic
423C2#2027513711yescharacteristic
424C2#2027523111yescharacteristic
subgroupG/H
centerC23#13133190
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1296C2
socleC23×C53#121
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1296D53
hypocenterC53#9219
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. #20275516C2semidirect
5. C2×#16810370C2semidirect
6. #20272549C2semidirect
7. C53×#1296C2semidirect
12. #16810904C4semidirect
163. D212#103semidirect
210. #1296D53semidirect
240. C53#9219semidirect

Maximal subgroups

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

subgroupindex
C2×#168109042
C2×#168106882
#202755162
#202747122
C2×#168103702
#202725492
C53×#12962
#921953