Finite Group Databasebeta

← #27284664

Subgroups — #27284664

Order 13568.

658 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810958C211yesmaximal, characteristic, complemented
2C2×#16810941C211yesmaximal, characteristic
3#20280643C211yesmaximal, characteristic
4C2×#16810359C211yesmaximal, characteristic, complemented
5C2×#16810356C211yesmaximal, characteristic
6#20272558C211yesmaximal, characteristic, complemented
7C53×#1445C211yesmaximal, characteristic, complemented
8C2×#13133196C2211yescharacteristic
9C2×#13133191C2211yescharacteristic
10#16810958—22no—
11#16810958—22no—
12#16810941—22no—
13#16810941—22no—
14C2×#13133150—21no—
15C2×#13133148—21no—
16C2×#13133140—21no—
17C2×#13133137—21no—
18C2×C22⋊#88322C2211yescharacteristic
19C2×C22⋊#88322C2211yescharacteristic
20#16810563—21no—
21#16810550—21no—
22#16810541—21no—
23#16810536—21no—
24#16810502—22no—
25#16810359—22no—
26#16810356—22no—
27C22×C4⋊C4×C53C2211yescharacteristic
28C2×#95×C53—22no—
29C2×#95×C53C2211yescharacteristic
30C2×#95×C53C2211yescharacteristic
31C22×Dic106D411yescharacteristic
32C22×D212D411yescharacteristic
33C22×C22⋊D53—21no—
34C22×#88322—21no—
35C22×#88322—21no—
36#13133196—21no—
37#13133196—21no—
38#13133196—21no—
39#13133196—21no—
40#13133191—21no—
41#13133191—21no—
42#13133191—21no—
43#13133191—21no—
44C2×C4⋊Dic53D411yescharacteristic
45C2×C4⋊Dic53D411yescharacteristic
46C2×#11719919—21no—
47C2×C22⋊Dic53—21no—
48C2×#11719921—22no—
49C2×#11719921—22no—
50C2×#11719921—21no—
51#13133150—24no—
52#13133148—24no—
53#13133140—24no—
54#13133137—24no—
55C22⋊#88322—21no—
56C22⋊#88322—21no—
57C22⋊#88322D412yes—
58C22⋊#88322D412yes—
59#13133124—22no—
60#13133124—22no—
61C23×C4×C53—21no—
62C23×C4×C53—21no—
63C23×C4×C53D412yes—
64C23×C4×C53C2311yescharacteristic
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53D412yes—
71#95×C53—21no—
72#95×C53—21no—
73#95×C53—21no—
74#95×C53—21no—
75#95×C53—44no—
76C24×C53C2×D411yescharacteristic
77C23×D53—21no—
78C22×C4×C53—22no—
79C22×C4×C53—22no—
80C22×C4×C53—22no—
81C22×C4×C53—22no—
82C22×C4×C53—22no—
83C22×C4×C53—42no—
84C22×C4×C53—42no—
85C22×C4×C53—42no—
86C22×C4×C53—42no—
87C22×C4×C53—41no—
88C22×C4×C53—42no—
89C22×C4×C53—42no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—42no—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53#4111yescharacteristic
100C2×C22⋊D53—41no—
101C2×C22⋊D53—41no—
102C2×C22⋊D53—42no—
103C2×C22⋊D53—42no—
104C22×Dic53—21no—
105C22×Dic53—21no—
106C22×Dic53—21no—
107C2×D212—21no—
108C2×D212—21no—
109C2×D212—42no—
110C2×Dic106—21no—
111C2×Dic106—21no—
112C2×Dic106—42no—
113C4⋊C4×C53—22no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—41no—
117C22⋊Dic53—41no—
118C22⋊Dic53—41no—
119#11719921—41no—
120#11719921—41no—
121#11719921—82no—
122#11719921—82no—
123C4⋊Dic53—21no—
124C4⋊Dic53—21no—
125C4⋊Dic53—21no—
126C4⋊Dic53—21no—
127#11719919—41no—
128#11719919—41no—
129C2×#88322—22no—
130C2×#88322—22no—
131C2×#88322—42no—
132C2×#88322—42no—
133C23×C53—21no—
134C23×C53—21no—
135C23×C53—22no—
136C23×C53#12011yescharacteristic
137C23×C53#12111yescharacteristic
138C23×C53#12711yescharacteristic
139C23×C53#13512yes—
140C23×C53#13712yes—
141C22×D53—41no—
142C22×D53—42no—
143C22×D53—41no—
144C22×D53—41no—
145C22×D53—42no—
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—44no—
153C2×C4×C53—42no—
154C2×C4×C53—44no—
155C2×C4×C53—44no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53—42no—
159C2×C4×C53—42no—
160C2×C4×C53—82no—
161C2×C4×C53—81no—
162C2×C4×C53—82no—
163C2×C4×C53—81no—
164C2×C4×C53—81no—
165C2×C4×C53—81no—
166C2×C4×C53—82no—
167C2×C4×C53#12011yescharacteristic
168C2×C4×C53#12011yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C22⋊D53—82no—
172C22⋊D53—81no—
173C22⋊D53—81no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—42no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180C2×Dic53—42no—
181D212—42no—
182D212—81no—
183Dic106—42no—
184Dic106—81no—
185#88322—42no—
186#88322—42no—
187#9969—531nomaximal, Sylow
188C22×C53—22no—
189C22×C53—22no—
190C22×C53—22no—
191C22×C53—22no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53#39211yescharacteristic
197C22×C53#45212yes—
198C22×C53#45311yescharacteristic
199C22×C53#45411yescharacteristic
200C22×C53#45812yes—
201C22×C53#46412yes—
202C22×C53#49511yescharacteristic
203C22×C53#49711yescharacteristic
204C2×D53—82no—
205C2×D53—82no—
206C2×D53—81no—
207C2×D53—81no—
208C2×D53—81no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—82no—
214C4×C53—82no—
215C4×C53—82no—
216Dic53—81no—
217Dic53—81no—
218Dic53—81no—
219C2×#484—531no—
220C2×#481—531no—
221#2639—531no—
222C2×#328—531no—
223C2×#327—531no—
224#1445D5311yescharacteristic, complemented
225#1280—531no—
226C2×C53—21no—
227C2×C53—22no—
228C2×C53—21no—
229C2×C53#161512yes—
230C2×C53#161812yes—
231C2×C53#200611yescharacteristic
232C2×C53#201511yescharacteristic
233C2×C53#202211yescharacteristic
234D53—161no—
235C2×C22⋊Q8—531no—
236C2×#121—531no—
237C22×C4⋊C4C2×D5311yescharacteristic
238#484—1062no—
239#484—1062no—
240#481—1062no—
241#481—1062no—
242C2×#107—1061no—
243C2×#106—1061no—
244C2×#103—1061no—
245C2×#102—1061no—
246C2×C22⋊C8—531no—
247C2×C22⋊C8—531no—
248#400—1061no—
249#397—1061no—
250#394—1061no—
251#393—1061no—
252#386—1062no—
253C2×#95—22no—
254C2×#95C2×D5311yescharacteristic
255C2×#95C2×D5311yescharacteristic
256#328—1062no—
257#327—1062no—
258C53#996911yescharacteristic, complemented, Sylow
259C22×Q8—531no—
260C22×D4—531no—
261C22×D4—1061no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4C22⋊D5312yes—
265C23×C4C22×D5311yescharacteristic
266C22×C8—1061no—
267C22×C8—1061no—
268C22⋊Q8—1061no—
269C22⋊Q8—1061no—
270C22⋊Q8—1061no—
271C22⋊Q8—1061no—
272#121—1061no—
273#121—1061no—
274#121—1061no—
275#121—1061no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—531no—
282C2×C4⋊C4—531no—
283C2×C4⋊C4—1061no—
284C2×C4⋊C4D21212yes—
285C2×C22⋊C4—1062no—
286C2×C22⋊C4—1062no—
287C2×C22⋊C4—1061no—
288C2×C22⋊C4—1061no—
289#107—1064no—
290#106—1064no—
291#103—1064no—
292#102—1064no—
293C22⋊C8—532no—
294C22⋊C8—532no—
295C22⋊C8—1061no—
296C22⋊C8—1061no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—44no—
302#95—1062no—
303#95—1062no—
304C24—1061no—
305C24C2×C22⋊D5311yescharacteristic
306C2×Q8—1061no—
307C2×Q8—1061no—
308C2×Q8—2122no—
309C2×D4—1061no—
310C2×D4—1061no—
311C2×D4—2122no—
312C2×D4—2121no—
313C2×D4—2121no—
314C2×D4—2122no—
315C2×D4—2122no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—1061no—
333C22×C4—1061no—
334C22×C4—1061no—
335C22×C4C2×D21211yescharacteristic
336C22×C4#1171994511yescharacteristic
337C22×C4D4×D5311yescharacteristic
338C22×C4D4×D5311yescharacteristic
339C22×C4D4×D5311yescharacteristic
340C22×C4D4×D5311yescharacteristic
341C2×C8—1062no—
342C2×C8—1062no—
343C2×C8—2122no—
344C2×C8—2122no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—1061no—
350C4⋊C4—1061no—
351C4⋊C4—1061no—
352C4⋊C4—1061no—
353C4⋊C4—2121no—
354C4⋊C4—2121no—
355C22⋊C4—4242no—
356C22⋊C4—4242no—
357C22⋊C4—2121no—
358C22⋊C4—2121no—
359C22⋊C4—2121no—
360C22⋊C4—2121no—
361C23—21no—
362C23—21no—
363C23—22no—
364C23—2122no—
365C23—2121no—
366C23—2121no—
367C23—2121no—
368C23—2122no—
369C23#1313318511yescharacteristic
370C23#1313319111yescharacteristic
371C23#1313321311yescharacteristic
372C23#1313323712yes—
373C23#1313324812yes—
374Q8—4241no—
375Q8—2122no—
376D4—4241no—
377D4—4241no—
378D4—4242no—
379D4—4241no—
380D4—2122no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4—2122no—
403C2×C4—2121no—
404C2×C4—2121no—
405C2×C4—2121no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4—2122no—
409C2×C4#1313318711yescharacteristic
410C2×C4#1313318711yescharacteristic
411C2×C4#1313319011yescharacteristic
412C2×C4#1313319011yescharacteristic
413C8—2122no—
414C8—2122no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—21no—
422C22—21no—
423C22—4241no—
424C22—4242no—
425C22—4241no—
426C22—4242no—
427C22—4241no—
428C22#1681053211yescharacteristic
429C22#1681078312yes—
430C22#1681079011yescharacteristic
431C22#1681079511yescharacteristic
432C22#1681081212yes—
433C22#1681084512yes—
434C22#1681100511yescharacteristic
435C22#1681101511yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—82no—
441C4—82no—
442C4—82no—
443C4—4241no—
444C4—4241no—
445C4—4241no—
446C2—21no—
447C2—21no—
448C2—22no—
449C2—8481no—
450C2#2027374412yes—
451C2#2027376112yes—
452C2#2027575611yescharacteristic
453C2#2027580811yescharacteristic
454C2#2027584611yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1445C2
socleC23×C53#120
prime radicalC2×#13133191C22
hypercenter#1445D53
hypocenterC53#9969
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1445

Complements

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

NH ≅ G/Ntype
1. C2×#16810958C2semidirect
4. C2×#16810359C2semidirect
6. #20272558C2semidirect
7. C53×#1445C2semidirect
224. #1445D53semidirect
258. C53#9969semidirect

Maximal subgroups

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

subgroupindex
C2×#168109582
C2×#168109412
#202806432
C2×#168103592
C2×#168103562
#202725582
C53×#14452
#996953