Finite Group Databasebeta

← #27302796

Subgroups — #27302796

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
1C22×#13133162C211yesmaximal, characteristic, complemented
2C2×#16810864C211yesmaximal, characteristic, complemented
3C2×#16810859C211yesmaximal, characteristic
4C2×#16810428C211yesmaximal, characteristic, complemented
5C2×#16810425C211yesmaximal, characteristic, complemented
6#20272807C211yesmaximal, characteristic, complemented
7C2×C53×C22⋊C16C211yesmaximal, characteristic, complemented
8C22×Dic212C2211yescharacteristic
9C22×D424C2211yescharacteristic
10C2×#13133196—21no—
11C2×#13133191—21no—
12C2×#13133162—22no—
13C2×#13133162—22no—
14C2×#13133162—22no—
15#16810864—22no—
16#16810864—22no—
17#16810859—22no—
18#16810859—22no—
19C2×#13133151C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133142—21no—
22C2×#13133139—21no—
23#16810428—22no—
24#16810425—22no—
25C23×C8×C53C2211yescharacteristic
26C22×C16×C53C2211yescharacteristic
27C22×C16×C53C2211yescharacteristic
28C53×C22⋊C16—22no—
29C22×Dic106—21no—
30C22×D212—21no—
31C22×C22⋊D53—41no—
32C2×Dic212—21no—
33C2×Dic212—21no—
34C2×Dic212—21no—
35C2×Dic212—21no—
36C2×Dic212—21no—
37C2×Dic212D412yescomplemented
38C2×D424—21no—
39C2×D424—21no—
40C2×D424—21no—
41C2×D424—21no—
42C2×D424—21no—
43C2×D424D412yes—
44#13133196—24no—
45#13133196—42no—
46#13133191—24no—
47#13133191—42no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53—21no—
50C2×#11719919—41no—
51C2×C22⋊Dic53—41no—
52C2×#11719921—41no—
53#13133162—28no—
54#13133151—21no—
55#13133151—21no—
56#13133151D412yes—
57#13133151D412yescomplemented
58#13133142—24no—
59#13133139—24no—
60C23×C4×C53D411yescharacteristic
61C22×C8×C53—21no—
62C22×C8×C53—21no—
63C22×C8×C53—21no—
64C22×C8×C53—21no—
65C22×C8×C53—21no—
66C22×C8×C53—21no—
67C22×C8×C53D411yescharacteristic
68C22×C8×C53C2311yescharacteristic
69C2×C16×C53—21no—
70C2×C16×C53—21no—
71C2×C16×C53—21no—
72C2×C16×C53—21no—
73C2×C16×C53—21no—
74C2×C16×C53D412yescomplemented
75C24×C53D811yescharacteristic
76C23×D53—41no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53D811yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C2×C22⋊D53—44no—
86C2×C22⋊D53—82no—
87C2×C22⋊D53—82no—
88C22×Dic53—41no—
89C22×Dic53—41no—
90C22×Dic53—41no—
91C2×D212—22no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×Dic106—22no—
98C2×Dic106—41no—
99C2×Dic106—41no—
100C2×Dic106—41no—
101C2×Dic106—41no—
102C2×Dic106—41no—
103C2×C8×C53—21no—
104C2×C8×C53—21no—
105C2×C8×C53—21no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C2×C8×C53—41no—
109C2×C8×C53—41no—
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53C2×D411yescharacteristic
113C2×C8×C53C2×D411yescharacteristic
114C2×C8×C53C2×D411yescharacteristic
115C2×C8×C53#4111yescharacteristic
116C22⋊Dic53—44no—
117#11719921—44no—
118C4⋊Dic53—22no—
119C4⋊Dic53—22no—
120C4⋊Dic53—41no—
121C4⋊Dic53—41no—
122#11719919—44no—
123Dic212—22no—
124Dic212—22no—
125Dic212—41no—
126Dic212—41no—
127D424—22no—
128D424—22no—
129D424—41no—
130D424—41no—
131C16×C53—22no—
132C16×C53—41no—
133C23×C53—21no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—21no—
137C23×C53—21no—
138C23×C53—21no—
139C23×C53SD3212yescomplemented
140C23×C53C2×D811yescharacteristic
141C22×D53—42no—
142C22×D53—81no—
143C22×D53—81no—
144C22×D53—81no—
145C22×D53—81no—
146C22×D53—81no—
147C22×D53—81no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53#12011yescharacteristic
156C2×C4×C53#12011yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#13511yescharacteristic
159C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
161C22⋊D53—88no—
162C2×Dic53—42no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165C2×Dic53—81no—
166C2×Dic53—81no—
167C2×Dic53—81no—
168C2×Dic53—81no—
169C2×Dic53—81no—
170C2×Dic53—81no—
171D212—42no—
172D212—42no—
173D212—81no—
174D212—81no—
175Dic106—42no—
176Dic106—42no—
177Dic106—81no—
178Dic106—81no—
179C8×C53—41no—
180C8×C53#12011yescharacteristic
181C8×C53#12111yescharacteristic
182C8×C53#12111yescharacteristic
183C8×C53#12711yescharacteristic
184#13305—531nomaximal, Sylow
185C22×C53—22no—
186C22×C53—22no—
187C22×C53—22no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53#44711yescharacteristic
193C22×C53#44711yescharacteristic
194C22×C53#46611yescharacteristic
195C22×C53C2×SD3211yescharacteristic
196C22×C53#50811yescharacteristic
197C22×C53#50911yescharacteristic
198C22×C53#51011yescharacteristic
199C2×D53—82no—
200C2×D53—82no—
201C2×D53—82no—
202C2×D53—161no—
203C2×D53—161no—
204C2×D53—161no—
205C2×D53—161no—
206C4×C53—41no—
207C4×C53#39211yescharacteristic
208C4×C53#44911yescharacteristic
209C4×C53#44911yescharacteristic
210C4×C53#46911yescharacteristic
211Dic53—82no—
212Dic53—82no—
213Dic53—82no—
214C22×SD32—531no—
215C2×#467—531no—
216C2×#466—531no—
217C2×#358—531no—
218C2×#357—531no—
219C2×C22⋊C16D5311yescharacteristic, complemented
220#1380—531no—
221C2×C53—42no—
222C2×C53#199911yescharacteristic
223C2×C53#218511yescharacteristic
224C2×C53#218611yescharacteristic
225C2×C53#218711yescharacteristic
226C2×C53#218911yescharacteristic
227C2×C53#221711yescharacteristic
228C2×C53#222011yescharacteristic
229D53—162no—
230C22×Q16—531no—
231C22×D8—531no—
232C23×C8C2×D5311yescharacteristic
233C2×C22⋊Q8—1061no—
234C2×#121—1061no—
235C2×SD32—1062no—
236C2×SD32—1062no—
237C2×SD32—1062no—
238C22×C16C2×D5311yescharacteristic
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—22no—
251C53#1330511yescharacteristic, complemented, Sylow
252C22×Q8—1061no—
253C22×D4—1061no—
254C22×D4—2121no—
255C23×C4D21211yescharacteristic
256C2×Q16—532no—
257C2×Q16—1061no—
258C2×Q16—1061no—
259C2×Q16—1061no—
260C2×Q16—1061no—
261C2×Q16—1061no—
262C2×D8—532no—
263C2×D8—1061no—
264C2×D8—1061no—
265C2×D8—1061no—
266C2×D8—1061no—
267C2×D8—1061no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—21no—
273C22×C8—21no—
274C22×C8D21211yescharacteristic
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—1068no—
286C2×C16—21no—
287C2×C16—21no—
288C2×C16—21no—
289C2×C16—21no—
290C2×C16—21no—
291C2×C16C22⋊D5312yescomplemented
292#107—532no—
293#107—532no—
294#107—1061no—
295#107—1061no—
296#103—1064no—
297#102—1064no—
298C24—2121no—
299C24D42411yescharacteristic
300C2×Q8—1062no—
301C2×Q8—2121no—
302C2×Q8—2121no—
303C2×Q8—2121no—
304C2×Q8—2121no—
305C2×Q8—2121no—
306C2×D4—1062no—
307C2×D4—4242no—
308C2×D4—4242no—
309C2×D4—2124no—
310C2×D4—2121no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2121no—
314C2×D4—2121no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—2121no—
322C22×C4—2121no—
323C22×C4—2121no—
324C22×C4D42411yescharacteristic
325C22×C4C2×D21211yescharacteristic
326Q16—1062no—
327Q16—1062no—
328Q16—2121no—
329Q16—2121no—
330D8—1062no—
331D8—1062no—
332D8—2121no—
333D8—2121no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8#1171994511yescharacteristic
342C2×C8D4×D5311yescharacteristic
343C2×C8D4×D5311yescharacteristic
344C2×C8D4×D5311yescharacteristic
345C2×C8D4×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—41no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—4241no—
363C23—4241no—
364C23—4241no—
365C23—4241no—
366C23—4241no—
367C23—4241no—
368C23—2122no—
369C23#1313316212yescomplemented
370C23C2×D42411yescharacteristic
371Q8—4241no—
372Q8—4241no—
373Q8—2122no—
374Q8—2122no—
375D4—4248no—
376D4—4241no—
377D4—4241no—
378D4—2122no—
379D4—2122no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—4241no—
388C2×C4—4241no—
389C2×C4—4241no—
390C2×C4—4241no—
391C2×C4—4241no—
392C2×C4—4241no—
393C2×C4—2122no—
394C2×C4—2122no—
395C2×C4—2122no—
396C2×C4#1313318711yescharacteristic
397C2×C4#1313318711yescharacteristic
398C2×C4#1313319111yescharacteristic
399C2×C4#1313323911yescharacteristic
400C2×C4#1313324511yescharacteristic
401C2×C4#1313324511yescharacteristic
402C8—41no—
403C8#1313318511yescharacteristic
404C8#1313319011yescharacteristic
405C8#1313319011yescharacteristic
406C8#1313321311yescharacteristic
407C22—22no—
408C22—22no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—8481no—
415C22—8481no—
416C22—8481no—
417C22—8481no—
418C22—4242no—
419C22—4242no—
420C22—4242no—
421C22#1681075911yescharacteristic
422C22#1681075911yescharacteristic
423C22#1681085911yescharacteristic
424C22C2×#1313316211yescharacteristic
425C22#1681105511yescharacteristic
426C22#1681106111yescharacteristic
427C22#1681106611yescharacteristic
428C4—41no—
429C4—4242no—
430C4—4242no—
431C4—4242no—
432C4#1681053211yescharacteristic
433C4#1681077011yescharacteristic
434C4#1681077011yescharacteristic
435C4#1681087611yescharacteristic
436C2—42no—
437C2—8482no—
438C2#2027571311yescharacteristic
439C2#2027663411yescharacteristic
440C2#2027664011yescharacteristic
441C2#2027664611yescharacteristic
442C2#2027665711yescharacteristic
443C2#2027681111yescharacteristic
444C2#2027682811yescharacteristic
subgroupG/H
centerC23C2×D424
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC2×C53×C22⋊C16C2
socleC23×C53C2×D8
prime radicalC2×#16810859C2
hypercenterC2×C22⋊C16D53
hypocenterC53#13305
LCSG ▷ C22×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
1. C22×#13133162C2semidirect
2. C2×#16810864C2semidirect
4. C2×#16810428C2semidirect
5. C2×#16810425C2semidirect
6. #20272807C2semidirect
7. C2×C53×C22⋊C16C2semidirect
37. C2×Dic212D4semidirect
57. #13133151D4semidirect
74. C2×C16×C53D4semidirect
139. C23×C53SD32semidirect
219. C2×C22⋊C16D53semidirect
251. C53#13305semidirect
291. C2×C16C22⋊D53semidirect
369. C23#13133162semidirect

Maximal subgroups

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

subgroupindex
C22×#131331622
C2×#168108642
C2×#168108592
C2×#168104282
C2×#168104252
#202728072
C2×C53×C22⋊C162
#1330553