Finite Group Databasebeta

← #27284607

Subgroups — #27284607

Order 13568.

650 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 105 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#20285468C211yesmaximal, characteristic
2#20279182C211yesmaximal, characteristic
3#20275701C211yesmaximal, characteristic, complemented
4#20274801C211yesmaximal, characteristic, complemented
5#20273284C211yesmaximal, characteristic
6#20272561C211yesmaximal, characteristic, complemented
7C2×C53×#483C211yesmaximal, characteristic, complemented
8C2×#13133200—22no—
9C2×#13133191C2211yescharacteristic
10#16810951—21no—
11#16810941—21no—
12#16810854—22no—
13#16810773—21no—
14#16810762—21no—
15C2×C22⋊#88322C2211yescharacteristic
16#16810558—22no—
17#16810524—21no—
18#16810492—21no—
19#16810483C2211yescharacteristic
20C4×C22⋊Dic53C2211yescharacteristic
21#16810390—22no—
22#16810385—22no—
23#16810358—22no—
24C2×C22⋊Q8×C53C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C53×#483—22no—
27C53×#483—22no—
28C2×#106×C53—21no—
29C2×#103×C53—21no—
30C2×C22⋊C8×C53C2211yescharacteristic
31C22×D212D411yescharacteristic
32C22×C22⋊D53—21no—
33C2×#11719924—22no—
34C22×#88322D412yes—
35#13133200—44no—
36#13133200—44no—
37#13133191—21no—
38#13133191—21no—
39#13133191—21no—
40#13133191—21no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53D411yescharacteristic
43C2×#11719919—24no—
44C2×C22⋊Dic53—21no—
45C2×C22⋊Dic53D412yes—
46C2×#11719921—24no—
47C2×#11719921—21no—
48C2×C4×Dic53—21no—
49#13133148—22no—
50#13133141—22no—
51#13133137—22no—
52C22⋊#88322—21no—
53C22⋊#88322—21no—
54#13133124—21no—
55#13133124—21no—
56C22×Q8×C53D411yescharacteristic
57C23×C4×C53—21no—
58C23×C4×C53C2311yescharacteristic
59C22×C8×C53—21no—
60C22⋊Q8×C53—21no—
61C22⋊Q8×C53—21no—
62C22⋊Q8×C53—21no—
63C22⋊Q8×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×C53—21no—
70C2×C4⋊C4×C53D412yes—
71C2×C4⋊C4×C53D411yescharacteristic
72C2×C22⋊C4×C53—21no—
73#106×C53—24no—
74#103×C53—24no—
75C22⋊C8×C53—21no—
76C22⋊C8×C53D412yes—
77C24×C53C2×D411yescharacteristic
78C23×D53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—21no—
81C2×Q8×C53—42no—
82C22×C4×C53—22no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—42no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—42no—
89C22×C4×C53—42no—
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53#4111yescharacteristic
96C2×C22⋊D53—41no—
97C2×C22⋊D53—41no—
98C2×C22⋊D53—42no—
99C2×C22⋊D53—42no—
100C22×Dic53—22no—
101C22×Dic53—21no—
102C22×Dic53—41no—
103C2×D212—21no—
104C2×D212—21no—
105C2×D212—42no—
106C2×C8×C53—22no—
107C2×C8×C53—42no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53—41no—
116C22⋊C4×C53—41no—
117C22⋊C4×C53—41no—
118C22⋊Dic53—22no—
119C22⋊Dic53—42no—
120C22⋊Dic53SD1614yescomplemented
121#11719924—82no—
122#11719921—48no—
123#11719921—41no—
124#11719921—41no—
125C4⋊Dic53—21no—
126C4⋊Dic53—21no—
127C4⋊Dic53—42no—
128#11719919—48no—
129C4×Dic53—42no—
130C2×#88322—22no—
131C2×#88322—42no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53#12011yescharacteristic
135C23×C53#12111yescharacteristic
136C23×C53#12711yescharacteristic
137C23×C53C2×SD1612yes—
138C23×C53AGL(1,ℤ/8)12yes—
139C22×D53—42no—
140C22×D53—42no—
141C22×D53—41no—
142C22×D53—42no—
143Q8×C53—42no—
144Q8×C53—42no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—42no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—44no—
155C2×C4×C53—82no—
156C2×C4×C53—81no—
157C2×C4×C53—81no—
158C2×C4×C53—81no—
159C2×C4×C53—81no—
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12011yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C22⋊D53—82no—
165C22⋊D53—81no—
166C22⋊D53—81no—
167C2×Dic53—24no—
168C2×Dic53—42no—
169C2×Dic53—42no—
170C2×Dic53—41no—
171C2×Dic53—42no—
172C2×Dic53—81no—
173C2×Dic53—81no—
174C2×Dic53—81no—
175D212—42no—
176D212—81no—
177C8×C53—42no—
178#88322—42no—
179#9960—531nomaximal, Sylow
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—42no—
187C22×C53—42no—
188C22×C53#39211yescharacteristic
189C22×C53#45012yes—
190C22×C53#46012yes—
191C22×C53#49211yescharacteristic
192C22×C53#49611yescharacteristic
193C2×D53—84no—
194C2×D53—81no—
195C2×D53—81no—
196C2×D53—81no—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—81no—
201C4×C53—82no—
202C4×C53—81no—
203C4×C53#45812yes—
204Dic53—44no—
205Dic53—81no—
206Dic53—161no—
207#3187—531no—
208C2×#483D5311yescharacteristic, complemented
209#2501—531no—
210#1996—531no—
211#1823—531no—
212#1519—531no—
213#1282—531no—
214C2×C53—22no—
215C2×C53—21no—
216C2×C53—22no—
217C2×C53—21no—
218C2×C53#200911yescharacteristic
219C2×C53#201411yescharacteristic
220C2×C53#202211yescharacteristic
221D53—161no—
222C2×#123—1062no—
223C2×C22⋊Q8C2×D5311yescharacteristic
224C2×#121—531no—
225C22×C4⋊C4C2×D5311yescharacteristic
226#483—22no—
227#483—22no—
228#483—1061no—
229#481—1061no—
230#465—1062no—
231#450—1061no—
232#448—1061no—
233C2×#106—21no—
234C2×#103—21no—
235C2×C22⋊C8—531no—
236C2×C22⋊C8C2×D5311yescharacteristic
237#399—1062no—
238#390—1061no—
239#384—1061no—
240#380—531no—
241C4×C22⋊C4—531no—
242#342—1062no—
243#340—1062no—
244#327—1062no—
245C53#996011yescharacteristic, complemented, Sylow
246C22×Q8C22⋊D5311yescharacteristic
247C22×D4—531no—
248C22×D4—1061no—
249C23×C4—21no—
250C23×C4C22×D5311yescharacteristic
251C2×SD16—1062no—
252C22×C8—21no—
253C22×C8—532no—
254#123—2124no—
255#123—2124no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260#121—1061no—
261#121—1061no—
262#121—1061no—
263#121—1061no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—531no—
271C2×C4⋊C4—1064no—
272C2×C4⋊C4—1061no—
273C2×C4⋊C4C22⋊D5312yes—
274C2×C4⋊C4C22⋊D5311yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—532no—
277C2×C22⋊C4—1064no—
278C2×C22⋊C4—1061no—
279C2×C22⋊C4—1061no—
280C2×C42—1061no—
281#106—24no—
282#106—1062no—
283#103—24no—
284#103—1062no—
285#102—1062no—
286C22⋊C8—21no—
287C22⋊C8—1061no—
288C22⋊C8—1061no—
289C22⋊C8C22⋊D5312yes—
290#95—1061no—
291#95—1061no—
292C24—1061no—
293C24D4×D5311yescharacteristic
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—42no—
297C2×D4—1061no—
298C2×D4—1061no—
299C2×D4—2122no—
300C2×D4—2121no—
301C2×D4—2121no—
302C2×D4—2122no—
303C2×D4—2122no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—1062no—
313C22×C4—1061no—
314C22×C4—2121no—
315C22×C4D4×D5311yescharacteristic
316C22×C4D4×D5311yescharacteristic
317C22×C4#1171994711yescharacteristic
318C22×C4C2×C22⋊D5311yescharacteristic
319C22×C4C2×C22⋊D5311yescharacteristic
320C22×C4C2×C22⋊D5311yescharacteristic
321SD16—4242no—
322C2×C8—22no—
323C2×C8—42no—
324C2×C8—1062no—
325C2×C8—2122no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—1061no—
335C4⋊C4—1061no—
336C4⋊C4—2122no—
337C4⋊C4—2128no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—534no—
341C22⋊C4—1062no—
342C22⋊C4—2128no—
343C22⋊C4—2122no—
344C22⋊C4—2121no—
345C22⋊C4—2121no—
346C42—2122no—
347C23—41no—
348C23—41no—
349C23—2122no—
350C23—2122no—
351C23—2122no—
352C23—2121no—
353C23#1313318511yescharacteristic
354C23#1313318911yescharacteristic
355C23#1313321311yescharacteristic
356C23SD16×D5312yes—
357C23#1313324412yes—
358Q8—42no—
359Q8—42no—
360D4—4241no—
361D4—4241no—
362D4—4242no—
363D4—4241no—
364D4—2122no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—44no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—1064no—
381C2×C4—4241no—
382C2×C4—4241no—
383C2×C4—4241no—
384C2×C4—2122no—
385C2×C4—2121no—
386C2×C4—2122no—
387C2×C4—2122no—
388C2×C4#1313318511yescharacteristic
389C2×C4#1313318611yescharacteristic
390C2×C4#1313318811yescharacteristic
391C2×C4#1313318911yescharacteristic
392C8—42no—
393C8—2122no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—42no—
401C22—42no—
402C22—4244no—
403C22—4241no—
404C22—4241no—
405C22—4241no—
406C22#1681053111yescharacteristic
407C22#1681077212yes—
408C22#1681082112yes—
409C22#1681099411yescharacteristic
410C22#1681100911yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—81no—
415C4—82no—
416C4—81no—
417C4—8481no—
418C4—4241no—
419C4—2124no—
420C4#1681081212yes—
421C2—21no—
422C2—22no—
423C2—22no—
424C2—21no—
425C2—8481no—
426C2#2027577311yescharacteristic
427C2#2027580111yescharacteristic
428C2#2027584311yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC2×C53×#483C2
socleC22×C53#392
prime radicalC2×#13133191C22
hypercenterC2×#483D53
hypocenterC53#9960
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
3. #20275701C2semidirect
4. #20274801C2semidirect
6. #20272561C2semidirect
7. C2×C53×#483C2semidirect
120. C22⋊Dic53SD16semidirect
208. C2×#483D53semidirect
245. C53#9960semidirect

Maximal subgroups

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

subgroupindex
#202854682
#202791822
#202757012
#202748012
#202732842
#202725612
C2×C53×#4832
#996053