Finite Group Databasebeta

← #27282153

Subgroups — #27282153

Order 13568.

538 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#20284651C211yesmaximal, characteristic
2#20274815C211yesmaximal, characteristic, complemented
3#20274790C211yesmaximal, characteristic, complemented
4#20273244C211yesmaximal, characteristic
5#20272623C211yesmaximal, characteristic, complemented
6#20272568C211yesmaximal, characteristic, complemented
7C53×#2333C211yesmaximal, characteristic
8C2×#13133176C2211yescharacteristic
9#16810944—22no—
10#16810939C412yescomplemented
11#16810934C412yescomplemented
12#16810663—21no—
13#16810628—21no—
14#16810587C2211yescharacteristic
15#16810587C2211yescharacteristic
16#16810486C2211yescharacteristic
17#16810389—21no—
18#16810389—21no—
19#16810383—21no—
20#16810383—21no—
21#16810357—22no—
22C2×#121×C53C2211yescharacteristic
23C53×#390—21no—
24C53×#388—21no—
25C53×#387—21no—
26C53×#386—21no—
27C53×#386—21no—
28C53×#381C2211yescharacteristic
29C4×C22⋊C4×C53—21no—
30C4×C22⋊C4×C53C2211yescharacteristic
31C2×#11719917—21no—
32C2×#11719917—21no—
33#13133176—21no—
34#13133176—21no—
35#13133176—21no—
36#13133176—21no—
37C2×C4⋊Dic53C2×C411yescharacteristic, complemented
38C2×C4⋊Dic53C2×C411yescharacteristic, complemented
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C4×Dic53—21no—
42#13133150—22no—
43#13133148—22no—
44#13133138—22no—
45#13133138—22no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322C2×C412yescomplemented
50#13133124—21no—
51#13133124—21no—
52C22×D4×C53—21no—
53C22×D4×C53D411yescharacteristic
54C23×C4×C53—21no—
55C23×C4×C53C2311yescharacteristic
56#121×C53—21no—
57#121×C53—21no—
58#121×C53C2×C412yes—
59#121×C53D412yes—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53D411yescharacteristic
68C2×C22⋊C4×C53D411yescharacteristic
69C2×C42×C53—21no—
70C2×C42×C53D411yescharacteristic
71C2×C42×C53D411yescharacteristic
72#95×C53—21no—
73#95×C53—21no—
74#95×C53—21no—
75#95×C53—21no—
76#95×C53—21no—
77C24×C53—21no—
78C24×C53C2×D411yescharacteristic
79C2×D4×C53—22no—
80C2×D4×C53—22no—
81C2×D4×C53—21no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—81no—
85C2×D4×C53—81no—
86C2×D4×C53C22⋊C412yes—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—42no—
95C22×C4×C53—81no—
96C22×C4×C53—81no—
97C22×C4×C53C22×C411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53#4111yescharacteristic
102C22×C4×C53#4111yescharacteristic
103C22×Dic53—21no—
104C22×Dic53—21no—
105C22×Dic53—41no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—81no—
108C4⋊C4×C53C22⋊C412yes—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—22no—
111C22⋊C4×C53—42no—
112C22⋊C4×C53—42no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C22⋊C4×C53—81no—
116C22⋊C4×C53—81no—
117C42×C53—41no—
118C42×C53—41no—
119C42×C53—81no—
120C22⋊Dic53—41no—
121C22⋊Dic53—41no—
122C22⋊Dic53—41no—
123C22⋊Dic53—41no—
124C4⋊Dic53—21no—
125C4⋊Dic53—21no—
126C4⋊Dic53—21no—
127C4⋊Dic53—21no—
128C4×Dic53—41no—
129C4×Dic53—41no—
130#11719917—41no—
131#11719917—41no—
132#11719917—41no—
133#11719917—41no—
134C2×#88322—22no—
135C2×#88322—41no—
136C23×C53—21no—
137C23×C53—21no—
138C23×C53—22no—
139C23×C53—41no—
140C23×C53—41no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—41no—
146C23×C53—41no—
147C23×C53C4×D411yescharacteristic
148C23×C53#12011yescharacteristic
149C23×C53#12111yescharacteristic
150D4×C53—42no—
151D4×C53—81no—
152D4×C53—82no—
153D4×C53—82no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—22no—
159C2×C4×C53—22no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—84no—
178C2×C4×C53C2×C22⋊C411yescharacteristic
179C2×C4×C53C4×D411yescharacteristic
180C2×C4×C53#12111yescharacteristic
181C2×C4×C53#12311yescharacteristic
182C2×Dic53—41no—
183C2×Dic53—41no—
184C2×Dic53—81no—
185C2×Dic53—81no—
186C2×Dic53—81no—
187C2×Dic53—81no—
188C2×Dic53—81no—
189#88322—42no—
190#88322—81no—
191#9525—531nomaximal, Sylow
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—42no—
201C22×C53—42no—
202C22×C53—42no—
203C22×C53—81no—
204C22×C53—81no—
205C22×C53—81no—
206C22×C53—81no—
207C22×C53—81no—
208C22×C53—81no—
209C22×C53#38611yescharacteristic
210C22×C53#45811yescharacteristic
211C22×C53#45811yescharacteristic
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—42no—
217C4×C53—42no—
218C4×C53—81no—
219C4×C53—81no—
220C4×C53—81no—
221Dic53—81no—
222Dic53—81no—
223Dic53—161no—
224#3094—531no—
225#2333D5311yescharacteristic
226#1826—531no—
227#1820—531no—
228#1510—531no—
229#1306—531no—
230#1284—531no—
231C2×C53—21no—
232C2×C53—21no—
233C2×C53—41no—
234C2×C53—41no—
235C2×C53—82no—
236C2×C53#189011yescharacteristic
237C2×C53#190411yescharacteristic
238C2×C53#190611yescharacteristic
239C2×#121C2×D5311yescharacteristic
240C2×#117—531no—
241#482—1062no—
242#481—532no—
243#480—532no—
244#428—1061no—
245#418—1061no—
246#407—531no—
247#407—531no—
248#390—21no—
249#388—21no—
250#387—21no—
251#386—21no—
252#386—21no—
253#381—531no—
254#381C2×D5311yescharacteristic
255C4×C22⋊C4—21no—
256C4×C22⋊C4C2×D5311yescharacteristic
257#342—1061no—
258#342—1061no—
259#339—1061no—
260#339—1061no—
261#327—1062no—
262C53#952511yescharacteristic, complemented, Sylow
263C22×D4—21no—
264C22×D4C22⋊D5311yescharacteristic
265C23×C4—21no—
266C23×C4C22×D5311yescharacteristic
267C2×M16—1061no—
268C2×M16—1061no—
269#121—21no—
270#121—21no—
271#121C4×D5312yes—
272#121D21212yes—
273#117—1061no—
274#117—1061no—
275#117—1061no—
276#117—1061no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—531no—
279C2×C4⋊C4—531no—
280C2×C4⋊C4C22⋊D5311yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—1061no—
287C2×C22⋊C4—1061no—
288C2×C22⋊C4C22⋊D5311yescharacteristic
289C2×C22⋊C4C22⋊D5311yescharacteristic
290C2×C42—21no—
291C2×C42—1061no—
292C2×C42C22⋊D5311yescharacteristic
293C2×C42C22⋊D5311yescharacteristic
294#107—1062no—
295#106—1062no—
296#102—1062no—
297#102—1062no—
298C22⋊C8—532no—
299C22⋊C8—1061no—
300C22⋊C8—1061no—
301C22⋊C8—1061no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—1061no—
308#95—1061no—
309C24—21no—
310C24C2×C22⋊D5311yescharacteristic
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—21no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—81no—
317C2×D4—81no—
318C2×D4#1171992112yes—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4—1061no—
330C22×C4—1061no—
331C22×C4—2121no—
332C22×C4C2×C4×D5311yescharacteristic
333C22×C4C2×D21211yescharacteristic
334C22×C4#1171994511yescharacteristic
335C22×C4#1171994511yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337C22×C4C2×C22⋊D5311yescharacteristic
338M16—2121no—
339M16—2121no—
340M16—2121no—
341M16—2121no—
342C2×C8—1062no—
343C2×C8—2121no—
344C4⋊C4—21no—
345C4⋊C4—81no—
346C4⋊C4—1061no—
347C4⋊C4—1061no—
348C4⋊C4—1061no—
349C4⋊C4—1061no—
350C4⋊C4#1171992112yes—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—81no—
358C22⋊C4—81no—
359C22⋊C4—2121no—
360C22⋊C4—2121no—
361C22⋊C4—2121no—
362C22⋊C4—2121no—
363C42—41no—
364C42—41no—
365C42—81no—
366C42—2121no—
367C42—2121no—
368C23—21no—
369C23—21no—
370C23—22no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23C4×C22⋊D5311yescharacteristic
380C23#1313318611yescharacteristic
381C23#1313319111yescharacteristic
382D4—42no—
383D4—81no—
384D4—82no—
385D4—82no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—84no—
410C2×C4—4241no—
411C2×C4—4241no—
412C2×C4—4241no—
413C2×C4—4241no—
414C2×C4—4241no—
415C2×C4—2121no—
416C2×C4—2121no—
417C2×C4C2×#1171992111yescharacteristic
418C2×C4C4×C22⋊D5311yescharacteristic
419C2×C4#1313319111yescharacteristic
420C2×C4#1313320011yescharacteristic
421C8—4241no—
422C8—2122no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22#1681050211yescharacteristic
441C22#1681081311yescharacteristic
442C22#1681081311yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—42no—
448C4—42no—
449C4—81no—
450C4—81no—
451C4—81no—
452C4—8481no—
453C4—4241no—
454C4—4241no—
455C2—21no—
456C2—21no—
457C2—41no—
458C2—41no—
459C2—82no—
460C2#2027512811yescharacteristic
461C2#2027520211yescharacteristic
462C2#2027521611yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2333C2
socleC22×C53#386
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#2333D53
hypocenterC53#9525
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2333

Complements

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

NH ≅ G/Ntype
2. #20274815C2semidirect
3. #20274790C2semidirect
5. #20272623C2semidirect
6. #20272568C2semidirect
10. #16810939C4semidirect
11. #16810934C4semidirect
37. C2×C4⋊Dic53C2×C4semidirect
38. C2×C4⋊Dic53C2×C4semidirect
49. C22⋊#88322C2×C4semidirect
262. C53#9525semidirect

Maximal subgroups

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

subgroupindex
#202846512
#202748152
#202747902
#202732442
#202726232
#202725682
C53×#23332
#952553