Finite Group Databasebeta

← #27285316

Subgroups — #27285316

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810969C211yesmaximal, characteristic, complemented
2C2×#16810903C211yesmaximal, characteristic, complemented
3#20275638C211yesmaximal, characteristic, complemented
4C2×#16810367C211yesmaximal, characteristic, complemented
5C2×D4⋊#88322C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#2541C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9#16810969—22no—
10#16810969—22no—
11#16810903—22no—
12#16810903—22no—
13C2×#13133150—21no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133140—22no—
17C2×#13133138—21no—
18C2×#13133138—22no—
19C2×C4×#88322C2211yescharacteristic
20#16810367—22no—
21D4⋊#88322—22no—
22C2×#127×C53—21no—
23C2×#127×C53C2211yescharacteristic
24C2×#121×C53—22no—
25C2×C4×D4×C53C2211yescharacteristic
26C53×#392—22no—
27C53×#390—21no—
28C4×C4⋊C4×C53C2211yescharacteristic
29C22×Dic106—21no—
30C22×#88322—21no—
31C22×#88322—21no—
32C22×#88322—21no—
33#13133217—21no—
34#13133217—21no—
35#13133217—21no—
36#13133217D412yescomplemented
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39#13133150—24no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322D412yescomplemented
44#13133140—44no—
45#13133138—24no—
46#13133138—28no—
47C4×#88322—21no—
48C4×#88322D412yescomplemented
49C22×D4×C53—24no—
50C22×D4×C53—21no—
51C22×D4×C53—21no—
52C22×D4×C53D411yescharacteristic
53C22×D4×C53D412yes—
54C23×C4×C53—21no—
55#127×C53—21no—
56#127×C53—21no—
57#127×C53—21no—
58#127×C53—21no—
59#127×C53—42no—
60#127×C53—42no—
61#121×C53—44no—
62#121×C53—44no—
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C4⋊C4×C53D412yes—
69C2×C22⋊C4×C53—24no—
70C2×C22⋊C4×C53—21no—
71C2×C42×C53—21no—
72C2×C42×C53C2311yescharacteristic
73#95×C53—21no—
74C24×C53—21no—
75C24×C53—22no—
76C2×D4×C53—21no—
77C2×D4×C53—22no—
78C2×D4×C53—21no—
79C2×D4×C53—22no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—44no—
84C2×D4×C53—48no—
85C2×D4×C53—48no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—42no—
91C2×D4×C53—41no—
92C2×D4×C53—41no—
93C2×D4×C53—42no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53—84no—
97C2×D4×C53SD1612yes—
98C22×C4×C53—22no—
99C22×C4×C53—21no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53—42no—
103C22×C4×C53—42no—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×Dic53—41no—
108C2×Dic106—22no—
109C2×Dic106—42no—
110C2×Dic106—42no—
111C2×Dic106—41no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53D814yes—
116C22⋊C4×C53—48no—
117C22⋊C4×C53—41no—
118C22⋊C4×C53—41no—
119C42×C53—41no—
120C42×C53—41no—
121C42×C53C2×D411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C42×C53C2×D411yescharacteristic
124C42×C53#4111yescharacteristic
125C4⋊Dic53—22no—
126C4⋊Dic53—22no—
127C4⋊Dic53—41no—
128C4⋊Dic53—41no—
129C2×#88322—22no—
130C2×#88322—22no—
131C2×#88322—42no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322—42no—
136C23×C53—22no—
137C23×C53—41no—
138C23×C53—42no—
139C23×C53—41no—
140C23×C53—42no—
141C23×C53—42no—
142C23×C53—44no—
143C23×C53—42no—
144C23×C53—42no—
145C23×C53—44no—
146C23×C53#12011yescharacteristic
147D4×C53—22no—
148D4×C53—22no—
149D4×C53—44no—
150D4×C53—41no—
151D4×C53—41no—
152D4×C53—82no—
153D4×C53—816no—
154D4×C53—81no—
155D4×C53—81no—
156D4×C53—82no—
157D4×C53—81no—
158D4×C53—81no—
159D4×C53—81no—
160D4×C53—81no—
161C2×C4×C53—22no—
162C2×C4×C53—24no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53—41no—
166C2×C4×C53—42no—
167C2×C4×C53—82no—
168C2×C4×C53—81no—
169C2×C4×C53—81no—
170C2×C4×C53#12011yescharacteristic
171C2×C4×C53#12011yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12111yescharacteristic
175C2×C4×C53#12711yescharacteristic
176C2×C4×C53C2×D812yes—
177C2×C4×C53C2×SD1611yescharacteristic
178C2×C4×C53#13511yescharacteristic
179C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C53AGL(1,ℤ/8)12yes—
181C2×C4×C53#13711yescharacteristic
182C2×Dic53—42no—
183C2×Dic53—82no—
184Dic106—44no—
185Dic106—81no—
186Dic106—81no—
187#88322—42no—
188#88322—42no—
189#88322—81no—
190#10070—531nomaximal, Sylow
191C22×C53—44no—
192C22×C53—42no—
193C22×C53—82no—
194C22×C53—81no—
195C22×C53—81no—
196C22×C53—84no—
197C22×C53—84no—
198C22×C53—82no—
199C22×C53—82no—
200C22×C53—82no—
201C22×C53#39211yescharacteristic
202C22×C53#44712yes—
203C22×C53#45011yescharacteristic
204C22×C53#45211yescharacteristic
205C22×C53#45512yes—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—44no—
209C4×C53—81no—
210C4×C53#44811yescharacteristic
211C4×C53#44911yescharacteristic
212C4×C53#45912yes—
213C4×C53#46511yescharacteristic
214C4×C53#46811yescharacteristic
215C4×C53#49311yescharacteristic
216C4×C53#49611yescharacteristic
217Dic53—82no—
218C2×#486—531no—
219C2×#474—531no—
220#2541D5311yescharacteristic
221#1981—531no—
222C2×#331—531no—
223C2×D4⋊C8—531no—
224#1279—531no—
225C2×C53—82no—
226C2×C53—162no—
227C2×C53#163912yes—
228C2×C53#164112yes—
229C2×C53#200011yescharacteristic
230C2×C53#200411yescharacteristic
231C2×C53#201111yescharacteristic
232C2×C4⋊Q8—531no—
233C2×#127—21no—
234C2×#127C2×D5311yescharacteristic
235C2×#121—22no—
236C2×C4×D4C2×D5311yescharacteristic
237#486—1062no—
238#486—1062no—
239#474—1062no—
240#474—1062no—
241C2×#107—1061no—
242C2×C4⋊C8—531no—
243C2×C4⋊C8—531no—
244C2×#103—1062no—
245C2×#102—1062no—
246C2×#102—1061no—
247C2×C4×C8—531no—
248#392—22no—
249#390—21no—
250C4×C4⋊C4C2×D5311yescharacteristic
251#331—1062no—
252D4⋊C8—1062no—
253C53#1007011yescharacteristic, complemented, Sylow
254C22×Q8—1061no—
255C22×D4—24no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4C22⋊D5311yescharacteristic
259C22×D4C22⋊D5312yes—
260C23×C4—21no—
261C22×C8—1061no—
262C22×C8—1061no—
263C22×C8—1061no—
264C4⋊Q8—532no—
265C4⋊Q8—1061no—
266C4⋊Q8—1061no—
267C4⋊Q8—1061no—
268#127—21no—
269#127—21no—
270#127—21no—
271#127—21no—
272#127—42no—
273#127—42no—
274#121—44no—
275#121—44no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C2×C4⋊C4—1061no—
281C2×C4⋊C4—1061no—
282C2×C4⋊C4C22⋊D5311yescharacteristic
283C2×C4⋊C4C22⋊D5312yes—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—21no—
286C2×C42—21no—
287C2×C42C22×D5311yescharacteristic
288#107—1064no—
289C4⋊C8—532no—
290C4⋊C8—1061no—
291C4⋊C8—1061no—
292C4⋊C8—1061no—
293#103—2124no—
294#102—1064no—
295#102—1068no—
296C4×C8—532no—
297C4×C8—1061no—
298#95—21no—
299C24—21no—
300C24—22no—
301C2×Q8—1062no—
302C2×Q8—2122no—
303C2×Q8—2122no—
304C2×Q8—2121no—
305C2×D4—21no—
306C2×D4—22no—
307C2×D4—21no—
308C2×D4—22no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—44no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—42no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—42no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—84no—
326C2×D4#1171992312yes—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—2121no—
334C22×C4C2×C22⋊D5311yescharacteristic
335C22×C4C2×C22⋊D5311yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337C2×C8—1062no—
338C2×C8—1062no—
339C2×C8—2121no—
340C2×C8—2122no—
341C2×C8—2122no—
342C2×C8—2121no—
343C2×C8—2121no—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—1062no—
348C4⋊C4—1062no—
349C4⋊C4—2121no—
350C4⋊C4—2121no—
351C4⋊C4#1171992214yes—
352C22⋊C4—48no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C42—41no—
356C42—41no—
357C42D4×D5311yescharacteristic
358C42D4×D5311yescharacteristic
359C42D4×D5311yescharacteristic
360C42#1171994711yescharacteristic
361C23—22no—
362C23—42no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—44no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—44no—
371C23#1313318611yescharacteristic
372Q8—4241no—
373Q8—4241no—
374Q8—2124no—
375D4—22no—
376D4—22no—
377D4—44no—
378D4—41no—
379D4—41no—
380D4—82no—
381D4—816no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4—82no—
386D4—81no—
387D4—81no—
388D4—81no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4—81no—
398C2×C4—4242no—
399C2×C4—2122no—
400C2×C4#1313318511yescharacteristic
401C2×C4#1313318511yescharacteristic
402C2×C4#1313318811yescharacteristic
403C2×C4#1313318911yescharacteristic
404C2×C4#1313318911yescharacteristic
405C2×C4#1313321311yescharacteristic
406C2×C4C2×#1171992212yes—
407C2×C4C2×#1171992311yescharacteristic
408C2×C4#1313323711yescharacteristic
409C2×C4#1313324312yes—
410C2×C4#1313324311yescharacteristic
411C2×C4#1313324711yescharacteristic
412C8—4241no—
413C8—2122no—
414C8—2122no—
415C22—42no—
416C22—44no—
417C22—81no—
418C22—81no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—84no—
424C22—84no—
425C22#1681053111yescharacteristic
426C22#1681075712yes—
427C22#1681077411yescharacteristic
428C22#1681078511yescharacteristic
429C22#1681080112yes—
430C4—21no—
431C4—21no—
432C4—44no—
433C4—81no—
434C4—4242no—
435C4#1681076111yescharacteristic
436C4#1681076611yescharacteristic
437C4#1681081512yes—
438C4#1681085011yescharacteristic
439C4#1681086711yescharacteristic
440C4#1681099911yescharacteristic
441C4#1681101011yescharacteristic
442C2—82no—
443C2—162no—
444C2#2027389412yes—
445C2#2027390812yes—
446C2#2027571811yescharacteristic
447C2#2027574211yescharacteristic
448C2#2027578611yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#2541C2
socleC23×C53#120
hypercenter#2541D53
hypocenterC53#10070
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. C2×#16810969C2semidirect
2. C2×#16810903C2semidirect
3. #20275638C2semidirect
4. C2×#16810367C2semidirect
5. C2×D4⋊#88322C2semidirect
6. #20272556C2semidirect
36. #13133217D4semidirect
43. C4⋊#88322D4semidirect
48. C4×#88322D4semidirect
253. C53#10070semidirect

Maximal subgroups

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

subgroupindex
C2×#168109692
C2×#168109032
#202756382
C2×#168103672
C2×D4⋊#883222
#202725562
C53×#25412
#1007053