Finite Group Databasebeta

← #27284608

Subgroups — #27284608

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
1C2×#16810951C211yesmaximal, characteristic, complemented
2#20279187C211yesmaximal, characteristic
3#20275701C211yesmaximal, characteristic, complemented
4#20274802C211yesmaximal, characteristic, complemented
5#20273283C211yesmaximal, characteristic
6#20272562C211yesmaximal, characteristic, complemented
7C53×#3187C211yesmaximal, characteristic
8C2×#13133198—22no—
9C22×C4⋊Dic53C2211yescharacteristic
10#16810951—22no—
11#16810951—22no—
12C2×#13133148—21no—
13C2×#13133141—21no—
14C2×C22⋊#88322C2211yescharacteristic
15#16810559—22no—
16#16810526—21no—
17#16810492—21no—
18#16810483C2211yescharacteristic
19C4×C22⋊Dic53C2211yescharacteristic
20#16810390—22no—
21#16810386—22no—
22#16810357—22no—
23C2×C22⋊Q8×C53C2211yescharacteristic
24C2×#121×C53C2211yescharacteristic
25C53×#483—21no—
26C53×#481—21no—
27C53×#465—22no—
28C53×#450—21no—
29C53×#448—21no—
30C2×C22⋊C8×C53C2211yescharacteristic
31C23×Dic53—21no—
32C22×#88322—21no—
33#13133198—44no—
34#13133198—44no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53D412yes—
42C2×#11719919—24no—
43C2×C22⋊Dic53—24no—
44C2×C22⋊Dic53—21no—
45C2×C22⋊Dic53D412yes—
46C2×C4×Dic53—21no—
47#13133148—24no—
48#13133141—24no—
49C22⋊#88322—21no—
50C22⋊#88322D412yes—
51#13133124—21no—
52#13133124—21no—
53C22×Q8×C53D411yescharacteristic
54C22×D4×C53—21no—
55C22×D4×C53D411yescharacteristic
56C23×C4×C53C2311yescharacteristic
57C2×SD16×C53—22no—
58C22×C8×C53D412yes—
59C22⋊Q8×C53—21no—
60C22⋊Q8×C53—21no—
61C22⋊Q8×C53—21no—
62C22⋊Q8×C53—21no—
63#121×C53—21no—
64#121×C53—21no—
65#121×C53—21no—
66#121×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53D411yescharacteristic
69C2×C4⋊C4×C53D411yescharacteristic
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53—21no—
72#106×C53—22no—
73#103×C53—22no—
74#102×C53—22no—
75C22⋊C8×C53—21no—
76C22⋊C8×C53—21no—
77C24×C53—21no—
78C24×C53C2×D411yescharacteristic
79C2×Q8×C53—21no—
80C2×Q8×C53—21no—
81C2×Q8×C53—42no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—42no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—42no—
88C2×D4×C53—42no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—42no—
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53#4111yescharacteristic
99C22×Dic53—22no—
100C22×Dic53—22no—
101C22×Dic53—41no—
102C22×Dic53—41no—
103C22×Dic53—42no—
104C22×Dic53—42no—
105C22×Dic53—41no—
106SD16×C53—82no—
107C2×C8×C53—22no—
108C2×C8×C53—42no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53—41no—
117C22⋊C4×C53—41no—
118C22⋊C4×C53—41no—
119C22⋊Dic53—22no—
120C22⋊Dic53—48no—
121C22⋊Dic53—42no—
122C22⋊Dic53SD1614yescomplemented
123C4⋊Dic53—22no—
124C4⋊Dic53—22no—
125C4⋊Dic53—41no—
126C4⋊Dic53—41no—
127C4⋊Dic53—42no—
128#11719919—48no—
129C4×Dic53—42no—
130C2×#88322—22no—
131C2×#88322—42no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—42no—
135C23×C53—42no—
136C23×C53—41no—
137C23×C53—42no—
138C23×C53#12011yescharacteristic
139C23×C53#12111yescharacteristic
140C23×C53#12711yescharacteristic
141C23×C53C2×SD1612yes—
142C23×C53AGL(1,ℤ/8)12yes—
143Q8×C53—42no—
144Q8×C53—42no—
145D4×C53—42no—
146D4×C53—81no—
147D4×C53—82no—
148D4×C53—81no—
149D4×C53—81no—
150C2×C4×C53—22no—
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—41no—
158C2×C4×C53—41no—
159C2×C4×C53—42no—
160C2×C4×C53—81no—
161C2×C4×C53—81no—
162C2×C4×C53#12011yescharacteristic
163C2×C4×C53#12011yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×Dic53—24no—
167C2×Dic53—42no—
168C2×Dic53—42no—
169C2×Dic53—42no—
170C2×Dic53—44no—
171C2×Dic53—82no—
172C2×Dic53—81no—
173C2×Dic53—81no—
174C2×Dic53—81no—
175C2×Dic53—81no—
176C2×Dic53—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—84no—
189C22×C53—81no—
190C22×C53—81no—
191C22×C53—81no—
192C22×C53#39211yescharacteristic
193C22×C53#45012yes—
194C22×C53#46012yes—
195C22×C53#49211yescharacteristic
196C22×C53#49611yescharacteristic
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—81no—
201C4×C53—81no—
202C4×C53—81no—
203C4×C53#45812yes—
204Dic53—44no—
205Dic53—82no—
206Dic53—161no—
207#3187D5311yescharacteristic
208C2×#483—531no—
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—161no—
219C2×C53#200911yescharacteristic
220C2×C53#201411yescharacteristic
221C2×C53#202211yescharacteristic
222C2×#123—1062no—
223C2×C22⋊Q8C2×D5311yescharacteristic
224C2×#121C2×D5311yescharacteristic
225C22×C4⋊C4—531no—
226#483—21no—
227#483—1062no—
228#483—1062no—
229#481—21no—
230#465—22no—
231#450—21no—
232#448—21no—
233C2×#106—1061no—
234C2×#103—1061no—
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—21no—
248C22×D4C22⋊D5311yescharacteristic
249C23×C4—1061no—
250C23×C4C22×D5311yescharacteristic
251C2×SD16—22no—
252C22×C8—1061no—
253C22×C8C22⋊D5312yes—
254#123—2124no—
255#123—2124no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260#121—21no—
261#121—21no—
262#121—21no—
263#121—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—532no—
266C2×C4⋊C4—1061no—
267C2×C4⋊C4—1061no—
268C2×C4⋊C4—1061no—
269C2×C4⋊C4—1061no—
270C2×C4⋊C4—1064no—
271C2×C4⋊C4—1061no—
272C2×C4⋊C4—1061no—
273C2×C4⋊C4C22⋊D5311yescharacteristic
274C2×C4⋊C4C22⋊D5311yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—532no—
278C2×C22⋊C4—1064no—
279C2×C22⋊C4—1061no—
280C2×C42—1061no—
281#106—22no—
282#106—1064no—
283#103—22no—
284#103—1064no—
285#102—22no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—532no—
289C22⋊C8—1061no—
290#95—1061no—
291#95—1061no—
292C24—21no—
293C24D4×D5311yescharacteristic
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—42no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—42no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—1062no—
309C22×C4—1062no—
310C22×C4—2121no—
311C22×C4—2121no—
312C22×C4—2121no—
313C22×C4—2122no—
314C22×C4—2122no—
315C22×C4D4×D5311yescharacteristic
316C22×C4D4×D5311yescharacteristic
317C22×C4#1171994711yescharacteristic
318C22×C4C2×C22⋊D5311yescharacteristic
319C22×C4C2×C22⋊D5311yescharacteristic
320C22×C4C2×C22⋊D5311yescharacteristic
321SD16—82no—
322C2×C8—22no—
323C2×C8—42no—
324C2×C8—1062no—
325C2×C8—2122no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—1062no—
333C4⋊C4—1062no—
334C4⋊C4—2122no—
335C4⋊C4—2128no—
336C4⋊C4—2121no—
337C4⋊C4—2121no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—534no—
343C22⋊C4—1062no—
344C22⋊C4—2128no—
345C22⋊C4—2122no—
346C42—2122no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23#1313318511yescharacteristic
354C23#1313318911yescharacteristic
355C23#1313321311yescharacteristic
356C23SD16×D5312yes—
357C23#1313324412yes—
358Q8—42no—
359Q8—42no—
360D4—42no—
361D4—81no—
362D4—81no—
363D4—82no—
364D4—81no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—1064no—
378C2×C4—4241no—
379C2×C4—4241no—
380C2×C4—4241no—
381C2×C4—4241no—
382C2×C4—4242no—
383C2×C4—4241no—
384C2×C4—2124no—
385C2×C4—2122no—
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—84no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22#1681053111yescharacteristic
407C22#1681077212yes—
408C22#1681082112yes—
409C22#1681099411yescharacteristic
410C22#1681100911yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—81no—
415C4—81no—
416C4—81no—
417C4—8481no—
418C4—4242no—
419C4—2124no—
420C4#1681081312yes—
421C2—21no—
422C2—22no—
423C2—22no—
424C2—21no—
425C2—161no—
426C2#2027577211yescharacteristic
427C2#2027580111yescharacteristic
428C2#2027584411yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#3187C2
socleC22×C53#392
prime radicalC2×#121×C53C22
hypercenter#3187D53
hypocenterC53#9960
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
1. C2×#16810951C2semidirect
3. #20275701C2semidirect
4. #20274802C2semidirect
6. #20272562C2semidirect
122. C22⋊Dic53SD16semidirect
245. C53#9960semidirect

Maximal subgroups

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

subgroupindex
C2×#168109512
#202791872
#202757012
#202748022
#202732832
#202725622
C53×#31872
#996053