Finite Group Databasebeta

← #27284554

Subgroups — #27284554

Order 13568.

692 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 109 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#20285463C211yesmaximal, characteristic
2C2×#16810936C211yesmaximal, characteristic, complemented
3#20275551C211yesmaximal, characteristic, complemented
4#20274801C211yesmaximal, characteristic, complemented
5#20273283C211yesmaximal, characteristic, complemented
6#20272563C211yesmaximal, characteristic, complemented
7C53×#2501C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9#16810936—22no—
10#16810936—21no—
11#16810936—22no—
12#16810934—21no—
13#16810857—22no—
14#16810757—21no—
15#16810756—21no—
16C2×#13133150—21no—
17C2×#13133137—21no—
18C2×#13133131—22no—
19C2×C22⋊#88322C2211yescharacteristic
20C2×C22⋊#88322C2211yescharacteristic
21#16810483C2211yescharacteristic
22#16810385—22no—
23#16810357—22no—
24C2×#123×C53—22no—
25C2×#121×C53C2211yescharacteristic
26C22×C4⋊C4×C53C2211yescharacteristic
27C53×#399—22no—
28C53×#390—21no—
29C53×#384—21no—
30C4×C22⋊C4×C53C2211yescharacteristic
31C22×D212D411yescharacteristic
32C22×C22⋊D53—21no—
33C2×#11719922—22no—
34C22×#88322—21no—
35C22×#88322D412yes—
36#13133191—21no—
37#13133191—21no—
38#13133191—21no—
39#13133191—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53D411yescharacteristic
42C2×C22⋊Dic53—21no—
43C2×#11719921—21no—
44#13133150—22no—
45#13133150—24no—
46#13133138—22no—
47#13133137—22no—
48#13133137—24no—
49#13133131—44no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322—21no—
53C22⋊#88322D412yes—
54#13133124—21no—
55C22×D4×C53—21no—
56C22×D4×C53D411yescharacteristic
57C23×C4×C53—21no—
58C23×C4×C53C2311yescharacteristic
59#123×C53—44no—
60#123×C53—44no—
61#121×C53—21no—
62#121×C53—21no—
63#121×C53—21no—
64#121×C53—21no—
65C2×C4⋊C4×C53—24no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53D412yes—
72C2×C4⋊C4×C53D411yescharacteristic
73C2×C22⋊C4×C53—24no—
74C2×C22⋊C4×C53—21no—
75C2×C22⋊C4×C53D412yes—
76C2×C42×C53—21no—
77#95×C53—21no—
78C24×C53—21no—
79C24×C53C2×D411yescharacteristic
80C23×D53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—42no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—42no—
87C2×D4×C53—42no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53—21no—
91C22×C4×C53—42no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—42no—
95C22×C4×C53—42no—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53#4111yescharacteristic
102C2×C22⋊D53—41no—
103C2×C22⋊D53—41no—
104C2×C22⋊D53—42no—
105C2×C22⋊D53—42no—
106C22×Dic53—21no—
107C22×Dic53—41no—
108C2×D212—21no—
109C2×D212—21no—
110C2×D212—42no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—22no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—48no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C22⋊C4×C53—22no—
119C22⋊C4×C53—48no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C22⋊C4×C53D814yescomplemented
123C42×C53—42no—
124C22⋊Dic53—42no—
125#11719922—82no—
126#11719921—41no—
127#11719921—41no—
128C4⋊Dic53—21no—
129C4⋊Dic53—21no—
130C4⋊Dic53—42no—
131C2×#88322—22no—
132C2×#88322—22no—
133C2×#88322—42no—
134C2×#88322—42no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53—42no—
138C23×C53—42no—
139C23×C53—41no—
140C23×C53—42no—
141C23×C53#12011yescharacteristic
142C23×C53#12111yescharacteristic
143C23×C53#12711yescharacteristic
144C23×C53C2×D812yes—
145C23×C53AGL(1,ℤ/8)12yes—
146C22×D53—41no—
147C22×D53—42no—
148C22×D53—81no—
149C22×D53—81no—
150D4×C53—42no—
151D4×C53—81no—
152D4×C53—82no—
153D4×C53—81no—
154D4×C53—81no—
155C2×C4×C53—22no—
156C2×C4×C53—22no—
157C2×C4×C53—22no—
158C2×C4×C53—24no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—42no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53—41no—
166C2×C4×C53—44no—
167C2×C4×C53—42no—
168C2×C4×C53—82no—
169C2×C4×C53—81no—
170C2×C4×C53—81no—
171C2×C4×C53#12011yescharacteristic
172C2×C4×C53#12011yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12111yescharacteristic
175C22⋊D53—82no—
176C22⋊D53—81no—
177C22⋊D53—81no—
178C2×Dic53—41no—
179C2×Dic53—42no—
180C2×Dic53—81no—
181C2×Dic53—81no—
182C2×Dic53—81no—
183D212—42no—
184D212—42no—
185#88322—42no—
186#88322—42no—
187#9952—531nomaximal, Sylow
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—42no—
195C22×C53—42no—
196C22×C53—84no—
197C22×C53—81no—
198C22×C53—81no—
199C22×C53—81no—
200C22×C53#39211yescharacteristic
201C22×C53#44712yes—
202C22×C53#45912yes—
203C22×C53#49311yescharacteristic
204C22×C53#49611yescharacteristic
205C2×D53—81no—
206C2×D53—81no—
207C2×D53—81no—
208C2×D53—162no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—44no—
213C4×C53—82no—
214C4×C53—81no—
215C4×C53#45812yes—
216Dic53—81no—
217Dic53—161no—
218#3186—531no—
219C2×#480—531no—
220#2501D5311yescharacteristic, complemented
221#1964—531no—
222#1823—531no—
223#1519—531no—
224#1282—531no—
225C2×C53—21no—
226C2×C53—22no—
227C2×C53—22no—
228C2×C53—21no—
229C2×C53—161no—
230C2×C53#200911yescharacteristic
231C2×C53#201111yescharacteristic
232C2×C53#202211yescharacteristic
233D53—161no—
234C2×#123—22no—
235C2×#121—531no—
236C2×#121C2×D5311yescharacteristic
237C22×C4⋊C4C2×D5311yescharacteristic
238#480—1062no—
239#480—1061no—
240#480—1062no—
241#480—1061no—
242#466—1062no—
243#447—1061no—
244#447—1061no—
245C2×#107—1061no—
246C2×#102—1061no—
247C2×#99—1062no—
248C2×C22⋊C8—531no—
249C2×C22⋊C8—531no—
250#399—22no—
251#390—21no—
252#384—21no—
253#380—531no—
254C4×C22⋊C4C2×D5311yescharacteristic
255#340—1062no—
256#327—1062no—
257C53#995211yescharacteristic, complemented, Sylow
258C22×D4—21no—
259C22×D4—531no—
260C22×D4—1061no—
261C22×D4C22⋊D5311yescharacteristic
262C23×C4—21no—
263C23×C4C22×D5311yescharacteristic
264C2×D8—1062no—
265C22×C8—532no—
266C22×C8—1061no—
267#123—44no—
268#123—44no—
269#121—21no—
270#121—21no—
271#121—21no—
272#121—21no—
273#121—1061no—
274#121—1061no—
275#121—1061no—
276#121—1061no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—531no—
284C2×C4⋊C4—1061no—
285C2×C4⋊C4C22⋊D5312yes—
286C2×C4⋊C4C22⋊D5311yescharacteristic
287C2×C22⋊C4—24no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—1061no—
290C2×C22⋊C4—1061no—
291C2×C22⋊C4C22⋊D5312yes—
292C2×C42—21no—
293#107—1064no—
294#107—1062no—
295#102—1064no—
296#102—1062no—
297#102—1062no—
298#99—2124no—
299C22⋊C8—532no—
300C22⋊C8—1061no—
301C22⋊C8—1061no—
302C22⋊C8—1061no—
303#95—21no—
304#95—1061no—
305C24—21no—
306C24—1061no—
307C24C2×C22⋊D5311yescharacteristic
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—42no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—1061no—
316C2×D4—1061no—
317C2×D4—2122no—
318C2×D4—2121no—
319C2×D4—2121no—
320C2×D4—2122no—
321C2×D4—2122no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—1061no—
331C22×C4—2121no—
332C22×C4D4×D5311yescharacteristic
333C22×C4D4×D5311yescharacteristic
334C22×C4D4×D5311yescharacteristic
335C22×C4#1171994711yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337C22×C4C2×C22⋊D5311yescharacteristic
338D8—4242no—
339C2×C8—1062no—
340C2×C8—1062no—
341C2×C8—2122no—
342C2×C8—2122no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—21no—
347C4⋊C4—48no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—1061no—
351C4⋊C4—1061no—
352C4⋊C4—2122no—
353C22⋊C4—22no—
354C22⋊C4—48no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—2122no—
358C22⋊C4—2121no—
359C22⋊C4—2121no—
360C22⋊C4#1171992214yescomplemented
361C42—42no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—41no—
368C23—4241no—
369C23—4241no—
370C23—2122no—
371C23—2121no—
372C23#1313318611yescharacteristic
373C23#1313318811yescharacteristic
374C23#1313321311yescharacteristic
375C23C2×#1171992212yes—
376C23#1313324312yes—
377D4—42no—
378D4—81no—
379D4—81no—
380D4—82no—
381D4—81no—
382D4—4241no—
383D4—4242no—
384D4—4241no—
385D4—2122no—
386D4—2122no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4—81no—
403C2×C4—4241no—
404C2×C4—4241no—
405C2×C4—4241no—
406C2×C4—2121no—
407C2×C4—2122no—
408C2×C4#1313318511yescharacteristic
409C2×C4#1313318511yescharacteristic
410C2×C4#1313318911yescharacteristic
411C2×C4#1313318911yescharacteristic
412C8—2122no—
413C8—2122no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—42no—
421C22—42no—
422C22—84no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—8482no—
427C22—4241no—
428C22—4241no—
429C22—4241no—
430C22#1681053111yescharacteristic
431C22#1681075712yes—
432C22#1681081512yes—
433C22#1681099911yescharacteristic
434C22#1681101011yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—44no—
439C4—82no—
440C4—81no—
441C4—8481no—
442C4—4241no—
443C4#1681081212yes—
444C2—22no—
445C2—21no—
446C2—22no—
447C2—21no—
448C2—161no—
449C2—8481no—
450C2#2027577511yescharacteristic
451C2#2027578611yescharacteristic
452C2#2027584311yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#2501C2
socleC22×C53#392
prime radical#20285463C2
hypercenter#2501D53
hypocenterC53#9952
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2501

Complements

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

NH ≅ G/Ntype
2. C2×#16810936C2semidirect
3. #20275551C2semidirect
4. #20274801C2semidirect
5. #20273283C2semidirect
6. #20272563C2semidirect
7. C53×#2501C2semidirect
122. C22⋊C4×C53D8semidirect
220. #2501D53semidirect
257. C53#9952semidirect
360. C22⋊C4#11719922semidirect

Maximal subgroups

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

subgroupindex
#202854632
C2×#168109362
#202755512
#202748012
#202732832
#202725632
C53×#25012
#995253