Finite Group Databasebeta

← #27281856

Subgroups — #27281856

Order 13568.

934 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 135 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#20278071C211yesmaximal, characteristic
2#20274703C211yesmaximal, characteristic, complemented
3#20274647C212yesmaximal, complemented
4#20272583C212yesmaximal, complemented
5C53×#2972C211yesmaximal, characteristic, complemented
6C23×D212C2211yescharacteristic
7C2×#13133191—21no—
8C2×#13133185—21no—
9C2×#13133139—24no—
10C2×#13133132—21no—
11#16810540C412yescomplemented
12#16810532C412yescomplemented
13#16810525—21no—
14#16810483C2212yes—
15#16810483C2211yescharacteristic
16#16810478—21no—
17#16810390—21no—
18C2×C4×D4×C53C2211yescharacteristic
19C8×D4×C53—24no—
20C53×#431—21no—
21C2×C22⋊C8×C53—21no—
22C2×C22⋊C8×C53C2212yes—
23C24×D53—21no—
24C22×D212—22no—
25C22×D212—21no—
26C22×D212—22no—
27C22×D212—44no—
28C22×D212C2×C412yes—
29C22×C22⋊D53—22no—
30#13133191—44no—
31#13133185—42no—
32#13133185—42no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—22no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—22no—
37C2×#11719921—22no—
38C2×#11719921—22no—
39C2×#11719921—22no—
40#13133139—48no—
41#13133132—42no—
42#13133124—22no—
43#13133124C2×C412yes—
44C22×D4×C53D411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53C2311yescharacteristic
47C22×C8×C53—22no—
48C22×C8×C53D414yescomplemented
49C4×D4×C53—22no—
50C4×D4×C53—22no—
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C4⋊C8×C53—22no—
56C22⋊C8×C53—22no—
57C22⋊C8×C53—22no—
58C22⋊C8×C53—22no—
59C4×C8×C53—22no—
60C24×C53C22⋊C411yescharacteristic
61C24×C53C2×D411yescharacteristic
62C23×D53—22no—
63C23×D53—44no—
64C23×D53—42no—
65C23×D53—41no—
66C23×D53—41no—
67C23×D53—42no—
68C23×D53—82no—
69C2×D4×C53—22no—
70C2×D4×C53—22no—
71C2×D4×C53D812yes—
72C2×D4×C53SD1612yes—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—42no—
76C22×C4×C53—42no—
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22×C411yescharacteristic
81C22×C4×C53C2×D412yes—
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53#4112yes—
84C2×C22⋊D53—44no—
85C2×C22⋊D53—44no—
86C2×C22⋊D53—82no—
87C22×Dic53—22no—
88C22×Dic53—42no—
89C2×D212—28no—
90C2×D212—24no—
91C2×D212—24no—
92C2×D212—44no—
93C2×D212—48no—
94C2×D212—42no—
95C2×D212—44no—
96C2×D212—41no—
97C2×D212—41no—
98C2×C8×C53—24no—
99C2×C8×C53—44no—
100C2×C8×C53—82no—
101C4⋊C4×C53—22no—
102C22⋊C4×C53—22no—
103C22⋊C4×C53—22no—
104C42×C53—21no—
105C42×C53—21no—
106C22⋊Dic53—41no—
107C22⋊Dic53—41no—
108C22⋊Dic53—42no—
109C22⋊Dic53—42no—
110#11719921—44no—
111#11719921—44no—
112#11719921—82no—
113C4⋊Dic53—42no—
114C4⋊Dic53—44no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53#10212yescomplemented
122C23×C53C2×C22⋊C411yescharacteristic
123C23×C53#12112yes—
124C23×C53C2×D811yescharacteristic
125C23×C53C2×SD1611yescharacteristic
126C23×C53#13512yes—
127C22×D53—24no—
128C22×D53—48no—
129C22×D53—42no—
130C22×D53—42no—
131C22×D53—42no—
132C22×D53—42no—
133C22×D53—42no—
134C22×D53—42no—
135C22×D53—84no—
136C22×D53—84no—
137C22×D53—88no—
138C22×D53—84no—
139C22×D53—82no—
140C22×D53—82no—
141D4×C53—24no—
142D4×C53—24no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—22no—
146C2×C4×C53—24no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—81no—
155C2×C4×C53#10212yescomplemented
156C2×C4×C53C4×D412yes—
157C2×C4×C53#12011yescharacteristic
158C2×C4×C53#12311yescharacteristic
159C22⋊D53—88no—
160C2×Dic53—42no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163C2×Dic53—82no—
164C2×Dic53—82no—
165C2×Dic53—82no—
166D212—28no—
167D212—44no—
168D212—44no—
169D212—44no—
170C8×C53—44no—
171#9468—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—22no—
174C22×C53—21no—
175C22×C53—22no—
176C22×C53—22no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—42no—
180C22×C53—42no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—81no—
184C22×C53#38611yescharacteristic
185C22×C53C2×#10211yescharacteristic
186C22×C53#41611yescharacteristic
187C22×C53#46512yes—
188C22×C53#46612yes—
189C2×D53—44no—
190C2×D53—44no—
191C2×D53—44no—
192C2×D53—82no—
193C2×D53—82no—
194C2×D53—88no—
195C2×D53—88no—
196C2×D53—82no—
197C2×D53—82no—
198C4×C53—22no—
199C4×C53—21no—
200C4×C53—22no—
201C4×C53—41no—
202C4×C53—41no—
203C4×C53B(4,𝔽2)12yes—
204Dic53—82no—
205Dic53—162no—
206#2972D5311yescharacteristic, complemented
207#2408—531no—
208#1799—531no—
209#1786—532no—
210#1289—532no—
211C2×C53—21no—
212C2×C53—22no—
213C2×C53—21no—
214C2×C53—22no—
215C2×C53—42no—
216C2×C53#188911yescharacteristic
217C2×C53#189311yescharacteristic
218C2×C53#191011yescharacteristic
219D53—84no—
220C23×D4—531no—
221C2×#121—1061no—
222C2×#120—1061no—
223C2×C4×D4C2×D5311yescharacteristic
224C8×D4—24no—
225#431—21no—
226C2×#102—1064no—
227C2×#99—1061no—
228C2×C22⋊C8—21no—
229C2×C22⋊C8C2×D5312yes—
230#394—532no—
231#392—532no—
232#390—1061no—
233#380—531no—
234#380—532no—
235#379—1061no—
236#342—1061no—
237C53#946811yescharacteristic, complemented, Sylow
238C25—1061no—
239C22×D4—532no—
240C22×D4—1062no—
241C22×D4—1062no—
242C22×D4—1062no—
243C22×D4—1061no—
244C22×D4—2124no—
245C22×D4D21211yescharacteristic
246C23×C4C22⋊D5311yescharacteristic
247C23×C4C22×D5311yescharacteristic
248C22×C8—22no—
249C22×C8C22⋊D5314yescomplemented
250#121—2124no—
251#120—2122no—
252#120—2122no—
253C4×D4—22no—
254C4×D4—22no—
255C2×C4⋊C4—1061no—
256C2×C4⋊C4—1062no—
257C2×C4⋊C4C4×D5311yescharacteristic
258C2×C22⋊C4—1062no—
259C2×C22⋊C4—1061no—
260C2×C22⋊C4—1062no—
261C2×C22⋊C4—1062no—
262C2×C22⋊C4—1062no—
263C2×C22⋊C4C4×D5311yescharacteristic
264C2×C22⋊C4D21211yescharacteristic
265C2×C42C22⋊D5311yescharacteristic
266C4⋊C8—22no—
267#102—2128no—
268#99—2122no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C4×C8—22no—
273#95—532no—
274#95—1062no—
275C24—1062no—
276C24—4242no—
277C24—2124no—
278C24—2121no—
279C24—2121no—
280C24—2122no—
281C24—2122no—
282C24#1171992111yescharacteristic
283C24C2×D21211yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—1068no—
287C2×D4—1064no—
288C2×D4—1064no—
289C2×D4—4242no—
290C2×D4—2124no—
291C2×D4—2122no—
292C2×D4—2124no—
293C2×D4—2128no—
294C2×D4—2124no—
295C2×D4—2124no—
296C2×D4—2121no—
297C2×D4—2121no—
298C2×D4#1171991312yes—
299C2×D4D42412yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—1062no—
305C22×C4—2122no—
306C22×C4#1171992111yescharacteristic
307C22×C4#1171992111yescharacteristic
308C22×C4#1171992111yescharacteristic
309C22×C4C2×C4×D5311yescharacteristic
310C22×C4#1171994512yes—
311C22×C4C2×C22⋊D5312yes—
312C22×C4C2×C22⋊D5311yescharacteristic
313C2×C8—24no—
314C2×C8—44no—
315C2×C8—82no—
316C4⋊C4—22no—
317C4⋊C4—2124no—
318C4⋊C4—2122no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—4242no—
322C22⋊C4—2124no—
323C22⋊C4—2122no—
324C22⋊C4—2122no—
325C22⋊C4—2121no—
326C22⋊C4—2121no—
327C22⋊C4—2124no—
328C42—21no—
329C42—21no—
330C23—21no—
331C23—21no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—1064no—
337C23—4244no—
338C23—4244no—
339C23—4244no—
340C23—4248no—
341C23—4242no—
342C23—4242no—
343C23—2122no—
344C23—2128no—
345C23—2122no—
346C23—2122no—
347C23—2122no—
348C23—2122no—
349C23—2122no—
350C23#1313313912yescomplemented
351C23C2×#1171992111yescharacteristic
352C23#1313319112yes—
353C23C2×D42411yescharacteristic
354C23C2×#1171991311yescharacteristic
355C23#1313323912yes—
356D4—24no—
357D4—24no—
358D4—1068no—
359D4—4248no—
360D4—2124no—
361D4—2124no—
362D4—2124no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—4242no—
376C2×C4—4242no—
377C2×C4—4242no—
378C2×C4—2122no—
379C2×C4—2122no—
380C2×C4—2122no—
381C2×C4#1313313912yescomplemented
382C2×C4C4×C22⋊D5312yes—
383C2×C4#1313318611yescharacteristic
384C2×C4#1313320011yescharacteristic
385C8—44no—
386C22—21no—
387C22—22no—
388C22—21no—
389C22—22no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—42no—
394C22—42no—
395C22—41no—
396C22—41no—
397C22—81no—
398C22—4248no—
399C22—4248no—
400C22—4242no—
401C22—4242no—
402C22—4242no—
403C22—4242no—
404C22—2124no—
405C22—2124no—
406C22—2124no—
407C22#1681050211yescharacteristic
408C22C2×#1313313911yescharacteristic
409C22#1681062011yescharacteristic
410C22#1681085212yes—
411C22#1681085912yes—
412C4—21no—
413C4—22no—
414C4—22no—
415C4—41no—
416C4—41no—
417C4—8482no—
418C4—4242no—
419C4#1681081012yes—
420C2—22no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—42no—
425C2—4244no—
426C2#2027512311yescharacteristic
427C2#2027514311yescharacteristic
428C2#2027524211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2972C2
socleC22×C53#386
prime radical#20278071C2
cosocleC23×C4×C53C23
hypercenter#2972D53
hypocenterC53#9468
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #20274703C2semidirect
3. #20274647C2semidirect
4. #20272583C2semidirect
5. C53×#2972C2semidirect
11. #16810540C4semidirect
12. #16810532C4semidirect
48. C22×C8×C53D4semidirect
121. C23×C53#102semidirect
155. C2×C4×C53#102semidirect
206. #2972D53semidirect
237. C53#9468semidirect
249. C22×C8C22⋊D53semidirect
350. C23#13133139semidirect
381. C2×C4#13133139semidirect

Maximal subgroups

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

subgroupindex
#202780712
#202747032
#202746472
#202725832
C53×#29722
#946853