Finite Group Databasebeta

← #27287886

Subgroups — #27287886

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275522C211yesmaximal, characteristic, complemented
2#20275510C211yesmaximal, characteristic, complemented
3#20274715C211yesmaximal, characteristic, complemented
4C2×#16810373C211yesmaximal, characteristic, complemented
5C2×#16810367C211yesmaximal, characteristic, complemented
6#20272599C211yesmaximal, characteristic, complemented
7C2×C53×#459C211yesmaximal, characteristic
8C2×C4⋊#88322C2211yescharacteristic
9C2×C4⋊#88322C2211yescharacteristic
10C2×#13133138—21no—
11C2×#13133138—21no—
12C2×#13133138—21no—
13C2×C22⋊#88322—21no—
14C2×C22⋊#88322—21no—
15C2×C4×#88322C2211yescharacteristic
16#16810493C2211yescharacteristic
17#16810385—21no—
18#16810373—22no—
19#16810367—22no—
20C22×D8×C53—21no—
21C2×#127×C53C2211yescharacteristic
22C2×C4×D4×C53C2211yescharacteristic
23C53×#459—22no—
24C53×#459—22no—
25C2×C4⋊C8×C53C2211yescharacteristic
26C2×#102×C53—21no—
27C22×#88322—21no—
28C22×#88322—21no—
29C22×#88322D411yescharacteristic
30C22×#88322D411yescharacteristic
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C4⋊#88322—21no—
35C4⋊#88322—21no—
36C4⋊#88322—21no—
37C4⋊#88322—21no—
38#13133138—42no—
39#13133138—42no—
40#13133138—42no—
41C22⋊#88322—42no—
42C22⋊#88322—42no—
43C4×#88322—21no—
44C4×#88322—21no—
45C22×D4×C53—21no—
46C22×D4×C53—21no—
47C22×D4×C53—21no—
48C22×D4×C53D411yescharacteristic
49C23×C4×C53—21no—
50C2×D8×C53—24no—
51C2×D8×C53—42no—
52C2×D8×C53—42no—
53C22×C8×C53—21no—
54#127×C53—21no—
55#127×C53—21no—
56#127×C53—21no—
57#127×C53D412yes—
58C4×D4×C53—21no—
59C4×D4×C53—21no—
60C4×D4×C53—21no—
61C4×D4×C53—21no—
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C22⋊C4×C53—21no—
64C2×C42×C53C2311yescharacteristic
65C4⋊C8×C53—21no—
66C4⋊C8×C53D412yes—
67#102×C53—24no—
68C24×C53—21no—
69C24×C53—41no—
70C2×D4×C53—21no—
71C2×D4×C53—22no—
72C2×D4×C53—22no—
73C2×D4×C53—21no—
74C2×D4×C53—22no—
75C2×D4×C53—21no—
76C2×D4×C53—21no—
77C2×D4×C53—21no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—81no—
90C2×D4×C53—81no—
91C2×D4×C53SD1612yes—
92C22×C4×C53—21no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53#4111yescharacteristic
102C22×Dic53—41no—
103D8×C53—48no—
104C2×C8×C53—22no—
105C2×C8×C53—41no—
106C2×C8×C53—41no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C22⋊C4×C53—41no—
110C22⋊C4×C53—41no—
111C42×C53C2×D411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C42×C53#4111yescharacteristic
114C42×C53#4111yescharacteristic
115C4⋊Dic53—41no—
116C4⋊Dic53—41no—
117C4⋊Dic53—41no—
118C4⋊Dic53—41no—
119C4⋊Dic53—81no—
120C2×#88322—21no—
121C2×#88322—21no—
122C2×#88322—21no—
123C2×#88322—21no—
124C2×#88322—21no—
125C2×#88322—41no—
126C2×#88322—41no—
127C2×#88322—41no—
128C2×#88322—41no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322D812yescomplemented
132C23×C53—22no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53—42no—
140C23×C53—81no—
141C23×C53—81no—
142C23×C53—81no—
143C23×C53—81no—
144C23×C53—81no—
145C23×C53—81no—
146C23×C53#12111yescharacteristic
147D4×C53—22no—
148D4×C53—22no—
149D4×C53—41no—
150D4×C53—42no—
151D4×C53—42no—
152D4×C53—42no—
153D4×C53—42no—
154D4×C53—41no—
155D4×C53—82no—
156D4×C53—82no—
157D4×C53—81no—
158D4×C53—81no—
159D4×C53—81no—
160D4×C53—81no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—81no—
167C2×C4×C53—81no—
168C2×C4×C53—81no—
169C2×C4×C53—81no—
170C2×C4×C53#12011yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12311yescharacteristic
174C2×C4×C53#12311yescharacteristic
175C2×C4×C53#12411yescharacteristic
176C2×C4×C53C2×D811yescharacteristic
177C2×C4×C53C2×SD1611yescharacteristic
178C2×C4×C53#13511yescharacteristic
179C2×C4×C53#13511yescharacteristic
180C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
182C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
183C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
184C2×Dic53—81no—
185C2×Dic53—81no—
186C2×Dic53—81no—
187C8×C53—42no—
188#88322—22no—
189#88322—41no—
190#88322—81no—
191#88322—81no—
192#10473—531nomaximal, Sylow
193C22×C53—42no—
194C22×C53—42no—
195C22×C53—42no—
196C22×C53—81no—
197C22×C53—81no—
198C22×C53—81no—
199C22×C53—81no—
200C22×C53—82no—
201C22×C53—82no—
202C22×C53—82no—
203C22×C53—161no—
204C22×C53—161no—
205C22×C53—161no—
206C22×C53—161no—
207C22×C53#39411yescharacteristic
208C22×C53#45911yescharacteristic
209C22×C53#46311yescharacteristic
210C22×C53#46511yescharacteristic
211C22×C53#46911yescharacteristic
212C22×C53#47111yescharacteristic
213C22×C53#47211yescharacteristic
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—81no—
217C4×C53#44911yescharacteristic
218C4×C53#45011yescharacteristic
219C4×C53#45911yescharacteristic
220C4×C53#46311yescharacteristic
221C4×C53#46611yescharacteristic
222C4×C53#46911yescharacteristic
223C4×C53#48111yescharacteristic
224C4×C53#48211yescharacteristic
225Dic53—161no—
226C2×#459D5311yescharacteristic
227#1957—531no—
228#1954—531no—
229#1802—531no—
230C2×#334—531no—
231C2×#331—531no—
232#1296—531no—
233C2×C53—82no—
234C2×C53—162no—
235C2×C53#166511yescharacteristic
236C2×C53#167711yescharacteristic
237C2×C53#168711yescharacteristic
238C2×C53#170111yescharacteristic
239C2×C53#203411yescharacteristic
240C2×C53#205111yescharacteristic
241C2×C53#205411yescharacteristic
242C22×D8—21no—
243C2×#127C2×D5311yescharacteristic
244C2×C4×D4C2×D5311yescharacteristic
245#459—22no—
246#459—22no—
247C2×C4⋊C8—531no—
248C2×C4⋊C8—531no—
249C2×C4⋊C8C2×D5311yescharacteristic
250C2×#102—21no—
251C2×#102—1061no—
252C2×#102—1061no—
253C2×#102—1061no—
254C2×C22⋊C8—1061no—
255C2×C22⋊C8—1061no—
256C2×C4×C8—531no—
257#384—531no—
258#340—1061no—
259#334—1062no—
260#331—1062no—
261C53#1047311yescharacteristic, complemented, Sylow
262C22×D4—21no—
263C22×D4—21no—
264C22×D4—21no—
265C22×D4C22⋊D5311yescharacteristic
266C23×C4—21no—
267C2×D8—24no—
268C2×D8—42no—
269C2×D8—42no—
270C22×C8—21no—
271C22×C8—531no—
272C22×C8—531no—
273C22×C8—1061no—
274C22×C8—1061no—
275#127—21no—
276#127—21no—
277#127—21no—
278#127C22⋊D5312yes—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4—21no—
283C2×C4⋊C4—1061no—
284C2×C4⋊C4—1061no—
285C2×C4⋊C4—1061no—
286C2×C4⋊C4C22⋊D5311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C42C22×D5311yescharacteristic
289C4⋊C8—21no—
290C4⋊C8—1061no—
291C4⋊C8—1061no—
292C4⋊C8—1061no—
293C4⋊C8—1061no—
294C4⋊C8C22⋊D5312yes—
295#102—24no—
296#102—2122no—
297#102—2122no—
298#102—2122no—
299C22⋊C8—2122no—
300C22⋊C8—2122no—
301C4×C8—1061no—
302C4×C8—1061no—
303C24—21no—
304C24—41no—
305C2×D4—21no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—21no—
309C2×D4—22no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—81no—
325C2×D4—81no—
326C2×D4#1171992312yes—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—2121no—
335C22×C4D4×D5311yescharacteristic
336C22×C4#1171994711yescharacteristic
337C22×C4C2×C22⋊D5311yescharacteristic
338D8—48no—
339C2×C8—22no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—532no—
343C2×C8—1061no—
344C2×C8—1061no—
345C2×C8—1061no—
346C2×C8—1061no—
347C2×C8—1061no—
348C2×C8—2121no—
349C2×C8—2121no—
350C2×C8—2121no—
351C2×C8—2121no—
352C2×C8—2121no—
353C2×C8—2121no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—4241no—
357C4⋊C4—2121no—
358C4⋊C4—2121no—
359C4⋊C4—2121no—
360C4⋊C4—2121no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C42#1171994711yescharacteristic
364C42#1171994711yescharacteristic
365C42C2×C22⋊D5311yescharacteristic
366C42C2×C22⋊D5311yescharacteristic
367C23—22no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—42no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23#1313318911yescharacteristic
382D4—22no—
383D4—22no—
384D4—41no—
385D4—42no—
386D4—42no—
387D4—42no—
388D4—42no—
389D4—41no—
390D4—82no—
391D4—82no—
392D4—81no—
393D4—81no—
394D4—81no—
395D4—81no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—4241no—
406C2×C4—4241no—
407C2×C4—4241no—
408C2×C4#1313318611yescharacteristic
409C2×C4#1313318811yescharacteristic
410C2×C4#1313318911yescharacteristic
411C2×C4#1313319811yescharacteristic
412C2×C4#1313319811yescharacteristic
413C2×C4#1313320411yescharacteristic
414C2×C4D8×D5311yescharacteristic
415C2×C4C2×#1171992311yescharacteristic
416C2×C4#1313323611yescharacteristic
417C2×C4#1313323711yescharacteristic
418C2×C4#1313324111yescharacteristic
419C2×C4#1313324111yescharacteristic
420C2×C4#1313324211yescharacteristic
421C2×C4#1313324311yescharacteristic
422C8—42no—
423C8—1062no—
424C8—4241no—
425C8—4241no—
426C8—2121no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—82no—
435C22—82no—
436C22—82no—
437C22—161no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22#1681053711yescharacteristic
442C22#1681081611yescharacteristic
443C22#1681083611yescharacteristic
444C22#1681085011yescharacteristic
445C22#1681087411yescharacteristic
446C22#1681088411yescharacteristic
447C22#1681089111yescharacteristic
448C4—21no—
449C4—21no—
450C4—81no—
451C4—8481no—
452C4#1681076811yescharacteristic
453C4#1681077411yescharacteristic
454C4#1681081611yescharacteristic
455C4#1681083611yescharacteristic
456C4#1681085611yescharacteristic
457C4#1681087311yescharacteristic
458C4#1681093911yescharacteristic
459C4#1681094411yescharacteristic
460C2—82no—
461C2—162no—
462C2#2027405411yescharacteristic
463C2#2027412211yescharacteristic
464C2#2027419211yescharacteristic
465C2#2027427411yescharacteristic
466C2#2027590111yescharacteristic
467C2#2027600411yescharacteristic
468C2#2027602311yescharacteristic
subgroupG/H
centerC23#13133189
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC2×C53×#459C2
socleC23×C53#121
hypercenterC2×#459D53
hypocenterC53#10473
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. #20275522C2semidirect
2. #20275510C2semidirect
3. #20274715C2semidirect
4. C2×#16810373C2semidirect
5. C2×#16810367C2semidirect
6. #20272599C2semidirect
131. C2×#88322D8semidirect
261. C53#10473semidirect

Maximal subgroups

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

subgroupindex
#202755222
#202755102
#202747152
C2×#168103732
C2×#168103672
#202725992
C2×C53×#4592
#1047353