Finite Group Databasebeta

← #27287441

Subgroups — #27287441

Order 13568.

620 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 113 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#20280829C211yesmaximal, characteristic
2#20275020C212yesmaximal
3C2×#16810361C212yesmaximal
4#20272979C211yesmaximal, characteristic
5C53×#1280C211yesmaximal, characteristic
6C2×#13133196C2212yes—
7C22×C4⋊Dic53C2211yescharacteristic
8C22×#11719919—21no—
9C2×#13133142—22no—
10#16810555—22no—
11#16810544—22no—
12#16810483—21no—
13#16810483—21no—
14C2×#13133124—21no—
15C2×#13133124C2211yescharacteristic
16#16810386—22no—
17#16810361—24no—
18C2×C22⋊C8×C53C2212yes—
19C2×#95×C53C2211yescharacteristic
20C22×Dic106D412yes—
21C23×Dic53—21no—
22C23×Dic53—21no—
23#13133196—22no—
24#13133196—22no—
25#13133196—22no—
26#13133196—22no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53—42no—
33C2×C4⋊Dic53D412yes—
34C2×#11719919—22no—
35C2×#11719919—42no—
36C2×#11719919—42no—
37C2×#11719919—42no—
38C2×C22⋊Dic53—22no—
39#13133142—44no—
40#13133124—22no—
41#13133124—22no—
42#13133124—21no—
43#13133124—42no—
44#13133124D412yes—
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C23×C4×C53C2311yescharacteristic
48C22×C8×C53—22no—
49C22⋊C8×C53—22no—
50C22⋊C8×C53—22no—
51#95×C53—22no—
52C24×C53C2×D411yescharacteristic
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—41no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53—41no—
62C22×C4×C53—41no—
63C22×C4×C53SD1612yes—
64C22×C4×C53Q1612yes—
65C22×C4×C53C2×D412yes—
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53#4112yes—
68C22×C4×C53#4111yescharacteristic
69C22×Dic53—22no—
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—22no—
74C22×Dic53—22no—
75C22×Dic53—41no—
76C22×Dic53—41no—
77C22×Dic53—41no—
78C22×Dic53—41no—
79C22×Dic53—41no—
80C22×Dic53—41no—
81C22×Dic53—41no—
82C22×Dic53—41no—
83C22×Dic53—41no—
84C22×Dic53—41no—
85C2×Dic106—22no—
86C2×Dic106—22no—
87C2×Dic106—42no—
88C2×Dic106—42no—
89C2×C8×C53—42no—
90C2×C8×C53—42no—
91C2×C8×C53—42no—
92C22⋊Dic53—42no—
93C22⋊Dic53—42no—
94C4⋊Dic53—24no—
95C4⋊Dic53—22no—
96C4⋊Dic53—22no—
97C4⋊Dic53—42no—
98C4⋊Dic53—81no—
99C4⋊Dic53—81no—
100C4⋊Dic53—81no—
101C4⋊Dic53—81no—
102#11719919—48no—
103#11719919—42no—
104#11719919—42no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53#12011yescharacteristic
110C23×C53#12111yescharacteristic
111C23×C53#12411yescharacteristic
112C23×C53C2×SD1611yescharacteristic
113C23×C53C2×Q1611yescharacteristic
114C23×C53#13711yescharacteristic
115C23×C53#13711yescharacteristic
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—41no—
121C2×C4×C53—42no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—81no—
130C2×C4×C53—81no—
131C2×C4×C53#12112yes—
132C2×C4×C53#12312yes—
133C2×Dic53—24no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C2×Dic53—42no—
144C2×Dic53—42no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—42no—
148C2×Dic53—42no—
149C2×Dic53—81no—
150C2×Dic53—81no—
151C2×Dic53—81no—
152C2×Dic53—81no—
153Dic106—44no—
154Dic106—82no—
155C8×C53—82no—
156#10406—531nomaximal, Sylow
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53#39411yescharacteristic
170C22×C53#44811yescharacteristic
171C22×C53#45111yescharacteristic
172C22×C53#45311yescharacteristic
173C22×C53#45611yescharacteristic
174C22×C53B(4,𝔽2)11yescharacteristic
175C22×C53#45811yescharacteristic
176C22×C53#46111yescharacteristic
177C22×C53#46211yescharacteristic
178C22×C53#48711yescharacteristic
179C22×C53#49111yescharacteristic
180C4×C53—22no—
181C4×C53—22no—
182C4×C53—42no—
183C4×C53—42no—
184Dic53—44no—
185Dic53—82no—
186Dic53—82no—
187Dic53—82no—
188#2659—531no—
189#1872—532no—
190C2×#328—532no—
191#1444—531no—
192#1280D5311yescharacteristic
193C2×C53—21no—
194C2×C53—21no—
195C2×C53—21no—
196C2×C53—21no—
197C2×C53#159611yescharacteristic
198C2×C53#160511yescharacteristic
199C2×C53#161111yescharacteristic
200C2×C53#161411yescharacteristic
201C2×C53#203911yescharacteristic
202C2×C53#204311yescharacteristic
203C2×C53#204411yescharacteristic
204C2×C22⋊Q8—532no—
205C22×C4⋊C4—531no—
206C22×C4⋊C4—1061no—
207C2×#103—1062no—
208C2×C22⋊C8C2×D5312yes—
209#398—1062no—
210#395—1062no—
211#380—1061no—
212#380—1061no—
213C2×#95—531no—
214C2×#95—1061no—
215C2×#95C2×D5311yescharacteristic
216#340—1062no—
217#328—1064no—
218C53#1040611yescharacteristic, complemented, Sylow
219C22×Q8—532no—
220C23×C4—1061no—
221C23×C4—1061no—
222C23×C4D21211yescharacteristic
223C23×C4D21211yescharacteristic
224C23×C4C22×D5311yescharacteristic
225C22×C8—22no—
226C22⋊Q8—1062no—
227C22⋊Q8—1062no—
228C22⋊Q8—1062no—
229C22⋊Q8—1062no—
230C2×C4⋊C4—532no—
231C2×C4⋊C4—1062no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4—1062no—
237C2×C4⋊C4—2122no—
238C2×C4⋊C4—2122no—
239C2×C4⋊C4—2122no—
240C2×C4⋊C4—2122no—
241C2×C22⋊C4—1062no—
242#103—2124no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245#95—22no—
246#95—532no—
247#95—1062no—
248#95—1062no—
249#95—1061no—
250#95—2122no—
251C24C2×D21211yescharacteristic
252C2×Q8—1062no—
253C2×Q8—1062no—
254C2×Q8—2122no—
255C2×Q8—2122no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—1062no—
267C22×C4—1062no—
268C22×C4—1062no—
269C22×C4—1062no—
270C22×C4—1062no—
271C22×C4—1062no—
272C22×C4—2121no—
273C22×C4—2121no—
274C22×C4—2121no—
275C22×C4—2121no—
276C22×C4—2121no—
277C22×C4—2121no—
278C22×C4—2121no—
279C22×C4—2121no—
280C22×C4—2121no—
281C22×C4—2121no—
282C22×C4#1171991312yes—
283C22×C4Dic21212yes—
284C22×C4#1171994512yes—
285C22×C4D4×D5312yes—
286C22×C4D4×D5311yescharacteristic
287C22×C4#1171994911yescharacteristic
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C4⋊C4—1064no—
292C4⋊C4—1062no—
293C4⋊C4—1062no—
294C4⋊C4—4241no—
295C4⋊C4—4241no—
296C4⋊C4—4241no—
297C4⋊C4—4241no—
298C4⋊C4—2122no—
299C4⋊C4—2128no—
300C4⋊C4—2122no—
301C4⋊C4—2122no—
302C22⋊C4—2122no—
303C22⋊C4—2122no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23#1313318711yescharacteristic
309C23#1313319211yescharacteristic
310C23#1313320511yescharacteristic
311C23C2×#1171991311yescharacteristic
312C23C2×Dic21211yescharacteristic
313C23#1313325011yescharacteristic
314C23#1313325011yescharacteristic
315Q8—4242no—
316Q8—2124no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—1064no—
333C2×C4—4241no—
334C2×C4—4241no—
335C2×C4—4241no—
336C2×C4—4241no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4—2122no—
347C2×C4—2122no—
348C2×C4—2122no—
349C2×C4—2122no—
350C2×C4—2122no—
351C2×C4—2122no—
352C2×C4#1313319012yes—
353C2×C4#1313320212yes—
354C8—82no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22#1681054111yescharacteristic
368C22#1681076411yescharacteristic
369C22#1681078111yescharacteristic
370C22#1681079311yescharacteristic
371C22#1681080811yescharacteristic
372C22#1681081111yescharacteristic
373C22#1681081411yescharacteristic
374C22#1681082911yescharacteristic
375C22#1681083411yescharacteristic
376C22#1681097411yescharacteristic
377C22#1681099211yescharacteristic
378C4—22no—
379C4—22no—
380C4—42no—
381C4—42no—
382C4—4242no—
383C4—4242no—
384C4—4242no—
385C4—2124no—
386C2—21no—
387C2—21no—
388C2—21no—
389C2—21no—
390C2#2027364011yescharacteristic
391C2#2027369111yescharacteristic
392C2#2027372511yescharacteristic
393C2#2027374211yescharacteristic
394C2#2027594011yescharacteristic
395C2#2027596211yescharacteristic
396C2#2027596711yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1280C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenter#1280D53
hypocenterC53#10406
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
218. C53#10406semidirect

Maximal subgroups

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

subgroupindex
#202808292
#202750202
C2×#168103612
#202729792
C53×#12802
#1040653