Finite Group Databasebeta

← #27284663

Subgroups — #27284663

Order 13568.

658 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 113 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
2#20280648C211yesmaximal, characteristic
3C2×#16810361C211yesmaximal, characteristic, complemented
4C2×#16810357C211yesmaximal, characteristic
5#20272983C211yesmaximal, characteristic, complemented
6#20272557C211yesmaximal, characteristic, complemented
7C2×C53×#481C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810958—22no—
10#16810958—22no—
11C2×#13133150—21no—
12C2×#13133140—21no—
13C2×C22⋊#88322C2211yescharacteristic
14#16810561—21no—
15#16810548—21no—
16#16810537—21no—
17#16810535—21no—
18#16810503—22no—
19C2×#13133124—22no—
20C2×#13133124C2211yescharacteristic
21C2×#13133124C2211yescharacteristic
22#16810361—22no—
23#16810357—22no—
24C2×#121×C53C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C53×#481—22no—
27C53×#481—22no—
28C2×#106×C53—21no—
29C2×#102×C53—21no—
30C2×C22⋊C8×C53C2211yescharacteristic
31C22×Dic106D411yescharacteristic
32C23×Dic53—21no—
33C23×Dic53D412yes—
34C22×#88322—21no—
35#13133196—21no—
36#13133196—21no—
37#13133196—21no—
38#13133196—21no—
39C2×C4⋊Dic53D411yescharacteristic
40C2×#11719919—21no—
41C2×C22⋊Dic53—22no—
42C2×C22⋊Dic53—22no—
43C2×C22⋊Dic53—21no—
44#13133150—24no—
45#13133140—24no—
46C22⋊#88322—21no—
47C22⋊#88322D412yes—
48#13133124—22no—
49#13133124—22no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54#13133124—44no—
55C22×D4×C53—21no—
56C22×D4×C53D411yescharacteristic
57C23×C4×C53—21no—
58C23×C4×C53C2311yescharacteristic
59C22×C8×C53—21no—
60#121×C53—21no—
61#121×C53—21no—
62#121×C53—21no—
63#121×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53D412yes—
70C2×C4⋊C4×C53D411yescharacteristic
71C2×C22⋊C4×C53—21no—
72#106×C53—24no—
73#102×C53—24no—
74C22⋊C8×C53—21no—
75C22⋊C8×C53D412yes—
76C24×C53—21no—
77C24×C53C2×D411yescharacteristic
78C2×D4×C53—21no—
79C2×D4×C53—21no—
80C2×D4×C53—42no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—42no—
84C2×D4×C53—42no—
85C22×C4×C53—21no—
86C22×C4×C53—22no—
87C22×C4×C53—42no—
88C22×C4×C53—42no—
89C22×C4×C53—42no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53#4111yescharacteristic
98C22×Dic53—22no—
99C22×Dic53—22no—
100C22×Dic53—21no—
101C22×Dic53—21no—
102C22×Dic53—22no—
103C22×Dic53—22no—
104C22×Dic53—42no—
105C22×Dic53—42no—
106C22×Dic53—41no—
107C22×Dic53—42no—
108C22×Dic53—41no—
109C22×Dic53—42no—
110C2×Dic106—21no—
111C2×Dic106—21no—
112C2×Dic106—42no—
113C2×C8×C53—22no—
114C2×C8×C53—42no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—41no—
120C4⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C22⋊C4×C53—41no—
123C22⋊Dic53—41no—
124C22⋊Dic53—41no—
125C22⋊Dic53—82no—
126C22⋊Dic53—82no—
127C4⋊Dic53—21no—
128C4⋊Dic53—21no—
129#11719919—41no—
130#11719919—41no—
131C2×#88322—22no—
132C2×#88322—42no—
133C23×C53—21no—
134C23×C53—21no—
135C23×C53—22no—
136C23×C53—41no—
137C23×C53—42no—
138C23×C53—41no—
139C23×C53—41no—
140C23×C53—42no—
141C23×C53#12011yescharacteristic
142C23×C53#12111yescharacteristic
143C23×C53#12711yescharacteristic
144C23×C53#13512yes—
145C23×C53#13712yes—
146D4×C53—42no—
147D4×C53—81no—
148D4×C53—82no—
149D4×C53—81no—
150D4×C53—81no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—44no—
158C2×C4×C53—42no—
159C2×C4×C53—41no—
160C2×C4×C53—42no—
161C2×C4×C53—82no—
162C2×C4×C53—81no—
163C2×C4×C53—81no—
164C2×C4×C53#12011yescharacteristic
165C2×C4×C53#12011yescharacteristic
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×Dic53—44no—
169C2×Dic53—42no—
170C2×Dic53—44no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—42no—
174C2×Dic53—42no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—42no—
178C2×Dic53—42no—
179C2×Dic53—82no—
180C2×Dic53—81no—
181C2×Dic53—81no—
182C2×Dic53—82no—
183Dic106—42no—
184Dic106—81no—
185C8×C53—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—82no—
197C22×C53—82no—
198C22×C53—81no—
199C22×C53—81no—
200C22×C53—81no—
201C22×C53#39211yescharacteristic
202C22×C53#45212yes—
203C22×C53#45311yescharacteristic
204C22×C53#45411yescharacteristic
205C22×C53#45812yes—
206C22×C53#46412yes—
207C22×C53#49511yescharacteristic
208C22×C53#49711yescharacteristic
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—82no—
214C4×C53—81no—
215Dic53—82no—
216Dic53—82no—
217Dic53—81no—
218Dic53—81no—
219C2×#484—531no—
220C2×#481D5311yescharacteristic
221#2639—531no—
222C2×#328—531no—
223C2×#327—531no—
224#1445—531no—
225#1280—531no—
226C2×C53—21no—
227C2×C53—22no—
228C2×C53—21no—
229C2×C53—161no—
230C2×C53#161512yes—
231C2×C53#161812yes—
232C2×C53#200611yescharacteristic
233C2×C53#201511yescharacteristic
234C2×C53#202211yescharacteristic
235C2×C22⋊Q8—531no—
236C2×#121C2×D5311yescharacteristic
237C22×C4⋊C4C2×D5311yescharacteristic
238#484—1062no—
239#484—1062no—
240#481—22no—
241#481—22no—
242C2×#107—1061no—
243C2×#106—21no—
244C2×#103—1061no—
245C2×#102—21no—
246C2×C22⋊C8—531no—
247C2×C22⋊C8C2×D5311yescharacteristic
248#400—1061no—
249#397—1061no—
250#394—1061no—
251#393—1061no—
252#386—1062no—
253C2×#95—531no—
254C2×#95—531no—
255C2×#95—1062no—
256#328—1062no—
257#327—1062no—
258C53#996911yescharacteristic, complemented, Sylow
259C22×Q8—531no—
260C22×D4—21no—
261C22×D4C22⋊D5311yescharacteristic
262C23×C4—21no—
263C23×C4—532no—
264C23×C4—1061no—
265C23×C4C22×D5311yescharacteristic
266C22×C8—21no—
267C22×C8—1061no—
268C22⋊Q8—1061no—
269C22⋊Q8—1061no—
270C22⋊Q8—1061no—
271C22⋊Q8—1061no—
272#121—21no—
273#121—21no—
274#121—21no—
275#121—21no—
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—1061no—
283C2×C4⋊C4C22⋊D5312yes—
284C2×C4⋊C4C22⋊D5311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—1062no—
287C2×C22⋊C4—1062no—
288C2×C22⋊C4—1061no—
289#107—1064no—
290#106—24no—
291#103—1064no—
292#102—24no—
293C22⋊C8—21no—
294C22⋊C8—532no—
295C22⋊C8—1061no—
296C22⋊C8C22⋊D5312yes—
297#95—1062no—
298#95—1062no—
299#95—1061no—
300#95—1061no—
301#95—1061no—
302#95—1061no—
303#95—2124no—
304C24—21no—
305C24D4×D5311yescharacteristic
306C2×Q8—1061no—
307C2×Q8—1061no—
308C2×Q8—2122no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—42no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—42no—
315C2×D4—42no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—1062no—
324C22×C4—1062no—
325C22×C4—1062no—
326C22×C4—1061no—
327C22×C4—1061no—
328C22×C4—1062no—
329C22×C4—2121no—
330C22×C4—2122no—
331C22×C4—2121no—
332C22×C4—2122no—
333C22×C4—2122no—
334C22×C4—2122no—
335C22×C4D4×D5311yescharacteristic
336C22×C4D4×D5311yescharacteristic
337C22×C4#1171994711yescharacteristic
338C22×C4C2×C22⋊D5311yescharacteristic
339C22×C4C2×C22⋊D5311yescharacteristic
340C22×C4C2×C22⋊D5311yescharacteristic
341C2×C8—22no—
342C2×C8—42no—
343C2×C8—1062no—
344C2×C8—2122no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—1061no—
352C4⋊C4—1061no—
353C4⋊C4—2121no—
354C4⋊C4—2121no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—4242no—
358C22⋊C4—4242no—
359C22⋊C4—2121no—
360C22⋊C4—2121no—
361C23—21no—
362C23—21no—
363C23—22no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23#1313318511yescharacteristic
370C23#1313318911yescharacteristic
371C23#1313321311yescharacteristic
372C23#1313323812yes—
373C23#1313324612yes—
374Q8—4241no—
375Q8—2122no—
376D4—42no—
377D4—81no—
378D4—81no—
379D4—82no—
380D4—81no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—4242no—
395C2×C4—4241no—
396C2×C4—4241no—
397C2×C4—4242no—
398C2×C4—2122no—
399C2×C4—2124no—
400C2×C4—2124no—
401C2×C4—2122no—
402C2×C4—2121no—
403C2×C4—2121no—
404C2×C4—2121no—
405C2×C4—2122no—
406C2×C4—2121no—
407C2×C4—2122no—
408C2×C4—2122no—
409C2×C4#1313318511yescharacteristic
410C2×C4#1313318611yescharacteristic
411C2×C4#1313318811yescharacteristic
412C2×C4#1313318911yescharacteristic
413C8—42no—
414C8—2122no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—21no—
422C22—21no—
423C22—81no—
424C22—82no—
425C22—81no—
426C22—82no—
427C22—81no—
428C22#1681053111yescharacteristic
429C22#1681078212yes—
430C22#1681078911yescharacteristic
431C22#1681079411yescharacteristic
432C22#1681081212yes—
433C22#1681084312yes—
434C22#1681100411yescharacteristic
435C22#1681101411yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—82no—
441C4—81no—
442C4—4242no—
443C4—4241no—
444C4—4242no—
445C4—4241no—
446C2—21no—
447C2—21no—
448C2—22no—
449C2—161no—
450C2#2027374312yes—
451C2#2027376012yes—
452C2#2027575511yescharacteristic
453C2#2027580711yescharacteristic
454C2#2027584311yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC2×C53×#481C2
socleC23×C53#120
prime radicalC2×#121×C53C22
hypercenterC2×#481D53
hypocenterC53#9969
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
1. C2×#16810958C2semidirect
3. C2×#16810361C2semidirect
5. #20272983C2semidirect
6. #20272557C2semidirect
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
#202806482
C2×#168103612
C2×#168103572
#202729832
#202725572
C2×C53×#4812
#996953