Finite Group Databasebeta

← #27284596

Subgroups — #27284596

Order 13568.

594 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 101 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#20285472C211yesmaximal, characteristic
2C2×#16810958C211yesmaximal, characteristic, complemented
3#20275700C211yesmaximal, characteristic, complemented
4#20274805C211yesmaximal, characteristic, complemented
5#20273283C211yesmaximal, characteristic
6#20272563C211yesmaximal, characteristic, complemented
7C53×#2504C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810958—22no—
10#16810958—22no—
11#16810953—21no—
12#16810939—21no—
13#16810850—22no—
14#16810774—21no—
15#16810761—21no—
16C2×#13133150—21no—
17C2×#13133140—21no—
18C2×C22⋊#88322C2211yescharacteristic
19C2×C22⋊#88322C2211yescharacteristic
20#16810483C2211yescharacteristic
21#16810389—22no—
22#16810385—22no—
23#16810357—22no—
24C2×#121×C53C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C53×#397—22no—
27C53×#390—21no—
28C53×#386—22no—
29C53×#382—21no—
30C4×C22⋊C4×C53C2211yescharacteristic
31C22×Dic106D411yescharacteristic
32C2×#11719923—22no—
33C22×#88322—21no—
34C22×#88322D412yes—
35#13133196—21no—
36#13133196—21no—
37#13133196—21no—
38#13133196—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53D411yescharacteristic
41C2×#11719919—21no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53—21no—
44#13133150—24no—
45#13133148—22no—
46#13133140—22no—
47#13133140—24no—
48#13133138—22no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322D412yes—
53#13133124—21no—
54C22×D4×C53—21no—
55C22×D4×C53D411yescharacteristic
56C23×C4×C53—21no—
57C23×C4×C53C2311yescharacteristic
58#121×C53—21no—
59#121×C53—21no—
60#121×C53—21no—
61#121×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53D412yes—
68C2×C4⋊C4×C53D411yescharacteristic
69C2×C22⋊C4×C53—24no—
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53D412yes—
72C2×C42×C53—21no—
73#95×C53—24no—
74#95×C53—21no—
75C24×C53—21no—
76C24×C53C2×D411yescharacteristic
77C2×D4×C53—21no—
78C2×D4×C53—21no—
79C2×D4×C53—42no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—42no—
83C2×D4×C53—42no—
84C22×C4×C53—21no—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53—42no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53—82no—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53#4111yescharacteristic
97C22×Dic53—21no—
98C22×Dic53—21no—
99C22×Dic53—41no—
100C2×Dic106—21no—
101C2×Dic106—21no—
102C2×Dic106—42no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—22no—
111C22⋊C4×C53—41no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53—84no—
114C42×C53—42no—
115C22⋊Dic53—41no—
116C22⋊Dic53—41no—
117C22⋊Dic53—42no—
118#11719923—82no—
119C4⋊Dic53—21no—
120C4⋊Dic53—21no—
121C4⋊Dic53—42no—
122#11719919—41no—
123#11719919—41no—
124C2×#88322—22no—
125C2×#88322—22no—
126C2×#88322—42no—
127C2×#88322—42no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—42no—
131C23×C53—42no—
132C23×C53—41no—
133C23×C53—42no—
134C23×C53#12011yescharacteristic
135C23×C53#12111yescharacteristic
136C23×C53#12711yescharacteristic
137C23×C53#13512yes—
138C23×C53#13712yes—
139D4×C53—42no—
140D4×C53—81no—
141D4×C53—82no—
142D4×C53—81no—
143D4×C53—81no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—44no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—41no—
155C2×C4×C53—42no—
156C2×C4×C53—42no—
157C2×C4×C53—82no—
158C2×C4×C53—82no—
159C2×C4×C53#12011yescharacteristic
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—81no—
166C2×Dic53—81no—
167C2×Dic53—81no—
168C2×Dic53—81no—
169C2×Dic53—81no—
170Dic106—42no—
171Dic106—42no—
172#88322—42no—
173#88322—42no—
174#9958—531nomaximal, Sylow
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—42no—
182C22×C53—42no—
183C22×C53—84no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53#39211yescharacteristic
188C22×C53#45212yes—
189C22×C53#46412yes—
190C22×C53#49511yescharacteristic
191C22×C53#49711yescharacteristic
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—82no—
196C4×C53—82no—
197C4×C53—81no—
198C4×C53#45812yes—
199Dic53—81no—
200Dic53—81no—
201Dic53—161no—
202#3187—531no—
203C2×#484—531no—
204#2504D5311yescharacteristic
205#1996—531no—
206#1824—531no—
207#1519—531no—
208#1282—531no—
209C2×C53—21no—
210C2×C53—22no—
211C2×C53—22no—
212C2×C53—21no—
213C2×C53—161no—
214C2×C53#200811yescharacteristic
215C2×C53#201511yescharacteristic
216C2×C53#202211yescharacteristic
217C2×C22⋊Q8—531no—
218C2×#121C2×D5311yescharacteristic
219C22×C4⋊C4C2×D5311yescharacteristic
220#484—1062no—
221#484—1062no—
222#483—1061no—
223#481—1061no—
224#465—1062no—
225#450—1061no—
226#448—1061no—
227C2×#107—1061no—
228C2×#103—1061no—
229C2×C22⋊C8—531no—
230C2×C22⋊C8—531no—
231#397—22no—
232#390—21no—
233#386—22no—
234#382—21no—
235#380—531no—
236C4×C22⋊C4C2×D5311yescharacteristic
237#342—1062no—
238#340—1062no—
239#327—1062no—
240C53#995811yescharacteristic, complemented, Sylow
241C22×Q8—531no—
242C22×D4—21no—
243C22×D4C22⋊D5311yescharacteristic
244C23×C4—21no—
245C23×C4C22×D5311yescharacteristic
246C2×SD16—1062no—
247C22×C8—532no—
248C22×C8—1061no—
249C22⋊Q8—1061no—
250C22⋊Q8—1061no—
251C22⋊Q8—1061no—
252C22⋊Q8—1061no—
253#121—21no—
254#121—21no—
255#121—21no—
256#121—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—531no—
263C2×C4⋊C4—1061no—
264C2×C4⋊C4—1061no—
265C2×C4⋊C4C22⋊D5312yes—
266C2×C4⋊C4C22⋊D5311yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—1061no—
270C2×C22⋊C4—1061no—
271C2×C22⋊C4C22⋊D5312yes—
272C2×C42—21no—
273#107—1064no—
274#106—1062no—
275#103—1064no—
276#103—1062no—
277#102—1062no—
278C22⋊C8—532no—
279C22⋊C8—1061no—
280C22⋊C8—1061no—
281C22⋊C8—1061no—
282#95—24no—
283#95—21no—
284#95—1061no—
285C24—21no—
286C24C2×C22⋊D5311yescharacteristic
287C2×Q8—1061no—
288C2×Q8—1061no—
289C2×Q8—2122no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—42no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—82no—
304C22×C4—1061no—
305C22×C4—1061no—
306C22×C4—2121no—
307C22×C4D4×D5311yescharacteristic
308C22×C4D4×D5311yescharacteristic
309C22×C4D4×D5311yescharacteristic
310C22×C4#1171994711yescharacteristic
311C22×C4C2×C22⋊D5311yescharacteristic
312C22×C4C2×C22⋊D5311yescharacteristic
313SD16—4242no—
314C2×C8—1062no—
315C2×C8—1062no—
316C2×C8—2122no—
317C2×C8—2122no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—1061no—
325C4⋊C4—1061no—
326C4⋊C4—2122no—
327C4⋊C4—2121no—
328C4⋊C4—2121no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—84no—
334C22⋊C4—2122no—
335C22⋊C4—2121no—
336C22⋊C4—2121no—
337C42—42no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23#1313318611yescharacteristic
345C23#1313318811yescharacteristic
346C23#1313321311yescharacteristic
347C23#1313323712yes—
348C23#1313324712yes—
349Q8—2122no—
350Q8—2122no—
351D4—42no—
352D4—81no—
353D4—81no—
354D4—82no—
355D4—81no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—4241no—
372C2×C4—4241no—
373C2×C4—4241no—
374C2×C4—4241no—
375C2×C4—4241no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4#1313318511yescharacteristic
379C2×C4#1313318511yescharacteristic
380C2×C4#1313318911yescharacteristic
381C2×C4#1313318911yescharacteristic
382C8—2122no—
383C8—2122no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—42no—
391C22—42no—
392C22—84no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22#1681053111yescharacteristic
397C22#1681078512yes—
398C22#1681084212yes—
399C22#1681100511yescharacteristic
400C22#1681101811yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—82no—
405C4—82no—
406C4—81no—
407C4—8481no—
408C4—4241no—
409C4—4241no—
410C4#1681081212yes—
411C2—22no—
412C2—21no—
413C2—21no—
414C2—22no—
415C2—161no—
416C2#2027576811yescharacteristic
417C2#2027580911yescharacteristic
418C2#2027584311yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#2504C2
socleC22×C53#392
prime radicalC2×#121×C53C22
hypercenter#2504D53
hypocenterC53#9958
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2504

Complements

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

NH ≅ G/Ntype
2. C2×#16810958C2semidirect
3. #20275700C2semidirect
4. #20274805C2semidirect
6. #20272563C2semidirect
240. C53#9958semidirect

Maximal subgroups

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

subgroupindex
#202854722
C2×#168109582
#202757002
#202748052
#202732832
#202725632
C53×#25042
#995853