Finite Group Databasebeta

← #27302791

Subgroups — #27302791

Order 13568.

618 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810861C211yesmaximal, characteristic, complemented
2C2×#16810856C211yesmaximal, characteristic
3C2×#16810427C211yesmaximal, characteristic, complemented
4C2×#16810424C211yesmaximal, characteristic, complemented
5C2×#16810402C211yesmaximal, characteristic, complemented
6#20272807C211yesmaximal, characteristic, complemented
7C22×SD32×C53C211yesmaximal, characteristic, complemented
8C22×C8×D53C2211yescharacteristic
9C2×#13133193—21no—
10C2×#13133188—21no—
11C22×#11719908C2211yescharacteristic
12#16810861—22no—
13#16810861—22no—
14#16810856—22no—
15#16810856—22no—
16C2×#13133151C2211yescharacteristic
17C2×#13133151C2211yescharacteristic
18C2×#13133141—21no—
19C2×#13133138—21no—
20#16810427—22no—
21#16810424—22no—
22#16810402—22no—
23C22×Q16×C53C2211yescharacteristic
24C22×D8×C53C2211yescharacteristic
25C2×SD32×C53—22no—
26C2×SD32×C53—22no—
27C2×SD32×C53—22no—
28C22×C16×C53C2211yescharacteristic
29C22×C22⋊D53—41no—
30C22×C4×D53D411yescharacteristic
31C22×#88322D411yescharacteristic
32C2×C8×D53—21no—
33C2×C8×D53—21no—
34C2×C8×D53—21no—
35C2×C8×D53—21no—
36C2×C8×D53—21no—
37C2×C8×D53—21no—
38#13133193—24no—
39#13133193—42no—
40#13133188—24no—
41#13133188—42no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—21no—
44C2×#11719919—41no—
45C2×C22⋊Dic53—41no—
46C2×#11719921—41no—
47C2×#11719908—21no—
48C2×#11719908—21no—
49C2×#11719908—21no—
50#13133151—21no—
51#13133151—21no—
52#13133151D412yes—
53#13133151D412yescomplemented
54#13133141—24no—
55#13133138—24no—
56C22×Q8×C53—21no—
57C22×D4×C53—21no—
58C2×Q16×C53—21no—
59C2×Q16×C53—21no—
60C2×Q16×C53—21no—
61C2×Q16×C53—21no—
62C2×Q16×C53—21no—
63C2×Q16×C53D412yescomplemented
64C2×D8×C53—21no—
65C2×D8×C53—21no—
66C2×D8×C53—21no—
67C2×D8×C53—21no—
68C2×D8×C53—21no—
69C2×D8×C53D412yes—
70C22×C8×C53C2311yescharacteristic
71SD32×C53—28no—
72C2×C16×C53—21no—
73C2×C16×C53—21no—
74C2×C16×C53D412yescomplemented
75C24×C53—41no—
76C23×D53D811yescharacteristic
77C2×Q8×C53—22no—
78C2×Q8×C53—41no—
79C2×Q8×C53—41no—
80C2×Q8×C53—41no—
81C2×Q8×C53—41no—
82C2×Q8×C53—41no—
83C2×D4×C53—22no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53C2×D411yescharacteristic
91C2×C22⋊D53—44no—
92C2×C22⋊D53—82no—
93C2×C22⋊D53—82no—
94C22×Dic53—41no—
95C22×Dic53—41no—
96C22×Dic53D811yescharacteristic
97C2×C4×D53—21no—
98C2×C4×D53—21no—
99C2×C4×D53—21no—
100C2×C4×D53—21no—
101C2×C4×D53—21no—
102C2×C4×D53—21no—
103Q16×C53—22no—
104Q16×C53—22no—
105Q16×C53—41no—
106Q16×C53—41no—
107D8×C53—22no—
108D8×C53—22no—
109D8×C53—41no—
110D8×C53—41no—
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53C2×D411yescharacteristic
113C2×C8×C53C2×D411yescharacteristic
114C2×C8×C53C2×D411yescharacteristic
115C2×C8×C53C2×D411yescharacteristic
116C2×C8×C53#4111yescharacteristic
117C22⋊Dic53—44no—
118#11719921—44no—
119C4⋊Dic53—22no—
120C4⋊Dic53—22no—
121C4⋊Dic53—41no—
122C4⋊Dic53—41no—
123#11719919—44no—
124C2×#88322—21no—
125C2×#88322—21no—
126C2×#88322—21no—
127C8×D53—41no—
128C8×D53—41no—
129C8×D53—41no—
130C8×D53—41no—
131C16×C53—22no—
132#11719908—41no—
133C23×C53—42no—
134C23×C53—81no—
135C23×C53—81no—
136C23×C53—81no—
137C23×C53—81no—
138C23×C53—81no—
139C23×C53—81no—
140C23×C53C2×D811yescharacteristic
141C22×D53—21no—
142C22×D53—21no—
143C22×D53—21no—
144C22×D53—21no—
145C22×D53—21no—
146C22×D53—21no—
147C22×D53SD3212yescomplemented
148Q8×C53—42no—
149Q8×C53—42no—
150Q8×C53—81no—
151Q8×C53—81no—
152D4×C53—42no—
153D4×C53—42no—
154D4×C53—81no—
155D4×C53—81no—
156C2×C4×C53—42no—
157C2×C4×C53—81no—
158C2×C4×C53—81no—
159C2×C4×C53#12011yescharacteristic
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#13511yescharacteristic
163C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
165C22⋊D53—88no—
166C2×Dic53—21no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169C2×Dic53—42no—
170C2×Dic53—42no—
171C2×Dic53—81no—
172C2×Dic53—81no—
173C2×Dic53—81no—
174C2×Dic53—81no—
175C4×D53—41no—
176C4×D53—41no—
177C4×D53—41no—
178C4×D53—41no—
179C8×C53#12011yescharacteristic
180C8×C53#12111yescharacteristic
181C8×C53#12111yescharacteristic
182C8×C53#12711yescharacteristic
183#88322—41no—
184#13305—531nomaximal, Sylow
185C22×C53—82no—
186C22×C53—82no—
187C22×C53—82no—
188C22×C53—161no—
189C22×C53—161no—
190C22×C53—161no—
191C22×C53—161no—
192C22×C53#44711yescharacteristic
193C22×C53#44711yescharacteristic
194C22×C53#46611yescharacteristic
195C22×C53C2×SD3211yescharacteristic
196C22×C53#50811yescharacteristic
197C22×C53#50911yescharacteristic
198C22×C53#51011yescharacteristic
199C2×D53—22no—
200C2×D53—22no—
201C2×D53—22no—
202C2×D53—41no—
203C2×D53—41no—
204C2×D53—41no—
205C2×D53—41no—
206C4×C53—82no—
207C4×C53#39211yescharacteristic
208C4×C53#44911yescharacteristic
209C4×C53#44911yescharacteristic
210C4×C53#46911yescharacteristic
211Dic53—41no—
212Dic53—82no—
213Dic53—82no—
214C22×SD32D5311yescharacteristic, complemented
215C2×#467—531no—
216C2×#466—531no—
217C2×#358—531no—
218C2×#357—531no—
219C2×C22⋊C16—531no—
220#1380—531no—
221C2×C53—162no—
222C2×C53#199911yescharacteristic
223C2×C53#218511yescharacteristic
224C2×C53#218611yescharacteristic
225C2×C53#218711yescharacteristic
226C2×C53#218911yescharacteristic
227C2×C53#221711yescharacteristic
228C2×C53#222011yescharacteristic
229D53—42no—
230C22×Q16C2×D5311yescharacteristic
231C22×D8C2×D5311yescharacteristic
232C23×C8—531no—
233C2×C22⋊Q8—1061no—
234C2×#121—1061no—
235C2×SD32—22no—
236C2×SD32—22no—
237C2×SD32—22no—
238C22×C16—531no—
239C22×C16C2×D5311yescharacteristic
240#467—1062no—
241#467—1062no—
242#466—1062no—
243#466—1062no—
244C2×#107—531no—
245C2×#107—531no—
246C2×#103—1061no—
247C2×#102—1061no—
248#358—1062no—
249#357—1062no—
250C22⋊C16—1062no—
251C53#1330511yescharacteristic, complemented, Sylow
252C22×Q8—21no—
253C22×D4—21no—
254C22×D4—2121no—
255C23×C4—531no—
256C2×Q16—21no—
257C2×Q16—21no—
258C2×Q16—21no—
259C2×Q16—21no—
260C2×Q16—21no—
261C2×Q16C22⋊D5312yescomplemented
262C2×D8—21no—
263C2×D8—21no—
264C2×D8—21no—
265C2×D8—21no—
266C2×D8—21no—
267C2×D8C22⋊D5312yes—
268C22×C8—531no—
269C22×C8—1061no—
270C22×C8—1061no—
271C22×C8—1061no—
272C22×C8—1061no—
273C22×C8—1061no—
274C22×C8—1061no—
275C22×C8C22×D5311yescharacteristic
276C22⋊Q8—1064no—
277C22⋊Q8—2122no—
278#121—1064no—
279#121—2122no—
280C2×C4⋊C4—1061no—
281C2×C4⋊C4—1061no—
282C2×C4⋊C4—2121no—
283C2×C22⋊C4—2121no—
284C2×C22⋊C4—2121no—
285SD32—28no—
286C2×C16—21no—
287C2×C16—21no—
288C2×C16—1061no—
289C2×C16—1061no—
290C2×C16—1061no—
291C2×C16C22⋊D5312yescomplemented
292#107—532no—
293#107—532no—
294#107—1061no—
295#107—1061no—
296#103—1064no—
297#102—1064no—
298C24—41no—
299C24—531no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×D4—22no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—4242no—
313C2×D4—4242no—
314C2×D4—2124no—
315C22×C4—41no—
316C22×C4—531no—
317C22×C4—1061no—
318C22×C4—1061no—
319C22×C4—1061no—
320C22×C4—1061no—
321C22×C4—1061no—
322C22×C4—1061no—
323C22×C4—2121no—
324C22×C4—2121no—
325C22×C4D4×D5311yescharacteristic
326Q16—22no—
327Q16—22no—
328Q16—41no—
329Q16—41no—
330D8—22no—
331D8—22no—
332D8—41no—
333D8—41no—
334C2×C8—1061no—
335C2×C8—1061no—
336C2×C8—1061no—
337C2×C8—2121no—
338C2×C8—2121no—
339C2×C8—2121no—
340C2×C8—2121no—
341C2×C8D4×D5311yescharacteristic
342C2×C8D4×D5311yescharacteristic
343C2×C8#1171994711yescharacteristic
344C2×C8C2×C22⋊D5311yescharacteristic
345C2×C8C2×C22⋊D5311yescharacteristic
346C2×C8C2×C22⋊D5311yescharacteristic
347C4⋊C4—1062no—
348C4⋊C4—1062no—
349C4⋊C4—2124no—
350C4⋊C4—2121no—
351C4⋊C4—2121no—
352C22⋊C4—2124no—
353C22⋊C4—2124no—
354C16—22no—
355C16—2121no—
356C23—42no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—532no—
364C23—1061no—
365C23—1061no—
366C23—1061no—
367C23—1061no—
368C23—1061no—
369C23—1061no—
370C23D8×D5311yescharacteristic
371Q8—42no—
372Q8—42no—
373Q8—81no—
374Q8—81no—
375D4—42no—
376D4—42no—
377D4—81no—
378D4—81no—
379D4—4248no—
380C2×C4—42no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—1061no—
384C2×C4—1061no—
385C2×C4—1061no—
386C2×C4—4241no—
387C2×C4—4241no—
388C2×C4—4241no—
389C2×C4—4241no—
390C2×C4—2122no—
391C2×C4—2122no—
392C2×C4—2121no—
393C2×C4—2121no—
394C2×C4—2121no—
395C2×C4—2121no—
396C2×C4#1313318511yescharacteristic
397C2×C4#1313318511yescharacteristic
398C2×C4#1313318811yescharacteristic
399C2×C4#1313323611yescharacteristic
400C2×C4#1313324111yescharacteristic
401C2×C4#1313324111yescharacteristic
402C8—2121no—
403C8#1313318611yescharacteristic
404C8#1313318911yescharacteristic
405C8#1313318911yescharacteristic
406C8#1313321311yescharacteristic
407C22—82no—
408C22—82no—
409C22—82no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22—161no—
414C22—1062no—
415C22—1062no—
416C22—1062no—
417C22—2121no—
418C22—2121no—
419C22—2121no—
420C22—2121no—
421C22#1681075511yescharacteristic
422C22#1681075511yescharacteristic
423C22#1681085611yescharacteristic
424C22SD32×D5311yescharacteristic
425C22#1681105411yescharacteristic
426C22#1681106011yescharacteristic
427C22#1681106211yescharacteristic
428C4—82no—
429C4—4242no—
430C4—4242no—
431C4—2121no—
432C4#1681053111yescharacteristic
433C4#1681076511yescharacteristic
434C4#1681076511yescharacteristic
435C4#1681087311yescharacteristic
436C2—162no—
437C2—2122no—
438C2#2027571011yescharacteristic
439C2#2027663011yescharacteristic
440C2#2027663711yescharacteristic
441C2#2027664211yescharacteristic
442C2#2027665211yescharacteristic
443C2#2027680811yescharacteristic
444C2#2027682511yescharacteristic
subgroupG/H
centerC23D8×D53
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC22×SD32×C53C2
socleC23×C53C2×D8
prime radicalC2×#16810856C2
hypercenterC22×SD32D53
hypocenterC53#13305
LCSG ▷ C22×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C22×SD32

Complements

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

NH ≅ G/Ntype
1. C2×#16810861C2semidirect
3. C2×#16810427C2semidirect
4. C2×#16810424C2semidirect
5. C2×#16810402C2semidirect
6. #20272807C2semidirect
7. C22×SD32×C53C2semidirect
53. #13133151D4semidirect
63. C2×Q16×C53D4semidirect
74. C2×C16×C53D4semidirect
147. C22×D53SD32semidirect
214. C22×SD32D53semidirect
251. C53#13305semidirect
261. C2×Q16C22⋊D53semidirect
291. C2×C16C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168108612
C2×#168108562
C2×#168104272
C2×#168104242
C2×#168104022
#202728072
C22×SD32×C532
#1330553