Finite Group Databasebeta

← #27284617

Subgroups — #27284617

Order 13568.

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

normal subgroups only

This group has 552 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#20285487C211yesmaximal, characteristic
2C2×#16810953C211yesmaximal, characteristic
3#20275700C211yesmaximal, characteristic
4#20274801C211yesmaximal, characteristic
5#20273287C211yesmaximal, characteristic
6#20272563C211yesmaximal, characteristic
7C53×#2503C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810958—21no—
10#16810956—21no—
11#16810953—22no—
12#16810953—22no—
13#16810862—22no—
14#16810779—21no—
15#16810778—21no—
16C2×#13133148—21no—
17C2×#13133140—21no—
18C2×C22⋊#88322C2211yescharacteristic
19C2×C22⋊#88322C2211yescharacteristic
20#16810483C2211yescharacteristic
21#16810389—22no—
22#16810385—22no—
23#16810360—22no—
24C2×C22⋊Q8×C53C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C53×#400—22no—
27C53×#391—21no—
28C53×#382—21no—
29C53×#381—22no—
30C4×C22⋊C4×C53C2211yescharacteristic
31C22×Dic106D411yescharacteristic
32C2×#11719925—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—22no—
45#13133148—24no—
46#13133141—22no—
47#13133140—22no—
48#13133140—24no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322D412yes—
53#13133124—21no—
54C22×Q8×C53D411yescharacteristic
55C23×C4×C53—21no—
56C23×C4×C53C2311yescharacteristic
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60C22⋊Q8×C53—21no—
61C2×C4⋊C4×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—21no—
70C2×C22⋊C4×C53D412yes—
71C2×C42×C53—21no—
72#95×C53—24no—
73#95×C53—24no—
74#95×C53—21no—
75C24×C53C2×D411yescharacteristic
76C2×Q8×C53—21no—
77C2×Q8×C53—21no—
78C2×Q8×C53—42no—
79C22×C4×C53—22no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—22no—
83C22×C4×C53—42no—
84C22×C4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53—42no—
87C22×C4×C53—42no—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53#4111yescharacteristic
94C22×Dic53—21no—
95C22×Dic53—21no—
96C22×Dic53—41no—
97C2×Dic106—21no—
98C2×Dic106—21no—
99C2×Dic106—42no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C22⋊C4×C53—22no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—41no—
112C42×C53—42no—
113C22⋊Dic53—41no—
114C22⋊Dic53—41no—
115C22⋊Dic53—42no—
116#11719925—82no—
117C4⋊Dic53—21no—
118C4⋊Dic53—21no—
119C4⋊Dic53—42no—
120#11719919—41no—
121#11719919—41no—
122C2×#88322—22no—
123C2×#88322—22no—
124C2×#88322—42no—
125C2×#88322—42no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53#12011yescharacteristic
129C23×C53#12111yescharacteristic
130C23×C53#12711yescharacteristic
131C23×C53#13512yes—
132C23×C53#13712yes—
133Q8×C53—42no—
134Q8×C53—81no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—44no—
149C2×C4×C53—42no—
150C2×C4×C53—82no—
151C2×C4×C53—81no—
152C2×C4×C53—81no—
153C2×C4×C53#12011yescharacteristic
154C2×C4×C53#12011yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—81no—
160C2×Dic53—81no—
161C2×Dic53—81no—
162C2×Dic53—81no—
163C2×Dic53—81no—
164Dic106—42no—
165Dic106—42no—
166#88322—42no—
167#88322—42no—
168#9961—531nomaximal, Sylow
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53#39211yescharacteristic
178C22×C53#45212yes—
179C22×C53#46412yes—
180C22×C53#49511yescharacteristic
181C22×C53#49711yescharacteristic
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—82no—
186C4×C53—81no—
187C4×C53—82no—
188C4×C53—81no—
189C4×C53#45812yes—
190Dic53—81no—
191Dic53—81no—
192Dic53—161no—
193#3189—531no—
194C2×#483—531no—
195#2503D5311yescharacteristic
196#1996—531no—
197#1823—531no—
198#1520—531no—
199#1282—531no—
200C2×C53—21no—
201C2×C53—22no—
202C2×C53—22no—
203C2×C53—21no—
204C2×C53#200911yescharacteristic
205C2×C53#201511yescharacteristic
206C2×C53#202211yescharacteristic
207C2×C22⋊Q8—531no—
208C2×C22⋊Q8C2×D5311yescharacteristic
209C22×C4⋊C4C2×D5311yescharacteristic
210#484—1061no—
211#484—1061no—
212#483—1062no—
213#483—1062no—
214#467—1062no—
215#451—1061no—
216#451—1061no—
217C2×#106—1061no—
218C2×#103—1061no—
219C2×C22⋊C8—531no—
220C2×C22⋊C8—531no—
221#400—22no—
222#391—21no—
223#382—21no—
224#381—22no—
225#380—531no—
226C4×C22⋊C4C2×D5311yescharacteristic
227#342—1062no—
228#340—1062no—
229#328—1062no—
230C53#996111yescharacteristic, complemented, Sylow
231C22×Q8—531no—
232C22×Q8C22⋊D5311yescharacteristic
233C23×C4—21no—
234C23×C4C22×D5311yescharacteristic
235C2×Q16—1062no—
236C22×C8—532no—
237C22×C8—1061no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8—1061no—
243C22⋊Q8—1061no—
244C22⋊Q8—1061no—
245C22⋊Q8—1061no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—531no—
253C2×C4⋊C4—1061no—
254C2×C4⋊C4—1061no—
255C2×C4⋊C4C22⋊D5312yes—
256C2×C4⋊C4C22⋊D5311yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—1061no—
259C2×C22⋊C4—1061no—
260C2×C22⋊C4C22⋊D5312yes—
261C2×C42—21no—
262#107—1062no—
263#106—1064no—
264#103—1064no—
265#103—1062no—
266#103—1062no—
267C22⋊C8—532no—
268C22⋊C8—1061no—
269C22⋊C8—1061no—
270C22⋊C8—1061no—
271#95—24no—
272#95—24no—
273#95—21no—
274#95—1061no—
275C24C2×C22⋊D5311yescharacteristic
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—42no—
279C2×Q8—1061no—
280C2×Q8—1061no—
281C2×Q8—2122no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—1061no—
292C22×C4—1061no—
293C22×C4—2121no—
294C22×C4D4×D5311yescharacteristic
295C22×C4D4×D5311yescharacteristic
296C22×C4D4×D5311yescharacteristic
297C22×C4#1171994711yescharacteristic
298C22×C4C2×C22⋊D5311yescharacteristic
299C22×C4C2×C22⋊D5311yescharacteristic
300Q16—4242no—
301C2×C8—1062no—
302C2×C8—1062no—
303C2×C8—2122no—
304C2×C8—2122no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—1061no—
314C4⋊C4—1061no—
315C4⋊C4—2122no—
316C4⋊C4—2121no—
317C4⋊C4—2121no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—2122no—
323C22⋊C4—2121no—
324C22⋊C4—2121no—
325C42—42no—
326C23—41no—
327C23—41no—
328C23#1313318611yescharacteristic
329C23#1313318811yescharacteristic
330C23#1313321311yescharacteristic
331C23#1313323712yes—
332C23#1313324712yes—
333Q8—42no—
334Q8—81no—
335Q8—2122no—
336Q8—2122no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—4241no—
356C2×C4—4241no—
357C2×C4—4241no—
358C2×C4—4241no—
359C2×C4—4241no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4#1313318511yescharacteristic
363C2×C4#1313318511yescharacteristic
364C2×C4#1313318911yescharacteristic
365C2×C4#1313318911yescharacteristic
366C8—2122no—
367C8—2122no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#1681053111yescharacteristic
377C22#1681078512yes—
378C22#1681084212yes—
379C22#1681100511yescharacteristic
380C22#1681101811yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—81no—
386C4—82no—
387C4—81no—
388C4—8481no—
389C4—4241no—
390C4—4241no—
391C4#1681081212yes—
392C2—22no—
393C2—21no—
394C2—21no—
395C2—22no—
396C2#2027577511yescharacteristic
397C2#2027580911yescharacteristic
398C2#2027584311yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#2503C2
socleC22×C53#392
prime radicalC24×C53C2×D4
hypercenter#2503D53
hypocenterC53#9961
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2503

Complements

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

NH ≅ G/Ntype
230. C53#9961semidirect

Maximal subgroups

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

subgroupindex
#202854872
C2×#168109532
#202757002
#202748012
#202732872
#202725632
C53×#25032
#996153