Finite Group Databasebeta

← #27294621

Subgroups — #27294621

Order 13568.

462 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 87 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#20280095C211yesmaximal, characteristic
2#20273045C211yesmaximal, characteristic
3#20272744C212yesmaximal, complemented
4#20272720C212yesmaximal, complemented
5C2×C53×#343C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic
7#16810587—21no—
8#16810553C411yescharacteristic
9#16810553C2211yescharacteristic
10#16810543C411yescharacteristic
11#16810543C2211yescharacteristic
12#16810518—21no—
13#16810518—21no—
14#16810496—21no—
15#16810496—21no—
16C4×C22⋊Dic53C411yescharacteristic
17#16810349—21no—
18#16810349C2212yes—
19C22×M16×C53—21no—
20C22×C4⋊C4×C53C2211yescharacteristic
21C53×#343—21no—
22C53×#343C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C2×#11719917—41no—
25#13133196—41no—
26#13133196—41no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C4×Dic53—21no—
36C22⋊#88322—22no—
37C22⋊#88322—41no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C2×M16×C53—24no—
48C2×M16×C53—22no—
49C2×M16×C53—41no—
50C2×M16×C53—42no—
51C22×C8×C53—41no—
52C2×C4⋊C4×C53—41no—
53C2×C4⋊C4×C53—41no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C4⋊C4×C53C2311yescharacteristic
58C24×C53C22⋊C411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—81no—
69C22×C4×C53—81no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C2×Dic106—41no—
81C2×Dic106—81no—
82M16×C53—24no—
83M16×C53—44no—
84M16×C53—42no—
85C2×C8×C53—44no—
86C2×C8×C53—42no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C22⋊Dic53—24no—
92C22⋊Dic53—41no—
93C22⋊Dic53—41no—
94C4⋊Dic53—41no—
95#11719919—42no—
96#11719919—42no—
97#11719919—41no—
98#11719919—41no—
99#11719919—81no—
100#11719919—81no—
101C4×Dic53—81no—
102#11719917—82no—
103C2×#88322—42no—
104C23×C53—22no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—41no—
108C23×C53#9911yescharacteristic
109C23×C53#9911yescharacteristic
110C23×C53C2×C22⋊C411yescharacteristic
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—42no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—81no—
129C2×C4×C53—81no—
130C2×C4×C53—81no—
131C2×C4×C53#10412yes—
132C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C53#13711yescharacteristic
134C2×Dic53—22no—
135C2×Dic53—22no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146Dic106—82no—
147C8×C53—44no—
148#88322—82no—
149#11768—531nomaximal, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—42no—
158C22×C53—42no—
159C22×C53#35612yes—
160C22×C53C2×#9911yescharacteristic
161C22×C53#41711yescharacteristic
162C22×C53C2×#10411yescharacteristic
163C4×C53—21no—
164C4×C53—22no—
165C4×C53—21no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170Dic53—42no—
171Dic53—42no—
172Dic53—81no—
173Dic53—81no—
174#2583—531no—
175C2×#343D5311yescharacteristic
176#1463—531no—
177#1354—532no—
178#1346—532no—
179C2×C53—21no—
180C2×C53—21no—
181C2×C53—22no—
182C2×C53—41no—
183C2×C53#150411yescharacteristic
184C2×C53#212111yescharacteristic
185C2×C53C2×#35611yescharacteristic
186C22×M16—21no—
187C2×C22⋊Q8—531no—
188C22×C4⋊C4C2×D5311yescharacteristic
189#407—1061no—
190#398—531no—
191#398—531no—
192#395—531no—
193#395—531no—
194#389—1061no—
195#389—1061no—
196#385—1061no—
197#385—1061no—
198C4×C22⋊C4—531no—
199#343—21no—
200#343C2×D5312yes—
201#324—532no—
202#324—1061no—
203C53#1176811yescharacteristic, complemented, Sylow
204C22×Q8—531no—
205C23×C4C4×D5311yescharacteristic
206C23×C4D21211yescharacteristic
207C23×C4C22⋊D5311yescharacteristic
208C2×M16—24no—
209C2×M16—22no—
210C2×M16—41no—
211C2×M16—42no—
212C2×M16—2121no—
213C22×C8—41no—
214C22⋊Q8—2121no—
215C22⋊Q8—2121no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—531no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4C4×D5311yescharacteristic
225C2×C4⋊C4D21211yescharacteristic
226C2×C4⋊C4C22⋊D5311yescharacteristic
227C2×C4⋊C4C22×D5311yescharacteristic
228C2×C22⋊C4—1061no—
229C2×C22⋊C4—1061no—
230C2×C42—1061no—
231C22⋊C8—1062no—
232C22⋊C8—2121no—
233#95—531no—
234#95—531no—
235#95—1061no—
236#95—1061no—
237#95—1061no—
238#95—1061no—
239C24#1171992111yescharacteristic
240C2×Q8—4241no—
241C2×Q8—2121no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4#1171992111yescharacteristic
258C22×C4#1171992111yescharacteristic
259C22×C4#1171992111yescharacteristic
260C22×C4C2×C4×D5311yescharacteristic
261C22×C4C2×D21211yescharacteristic
262C22×C4C2×C22⋊D5311yescharacteristic
263M16—24no—
264M16—44no—
265M16—42no—
266M16—4242no—
267C2×C8—44no—
268C2×C8—42no—
269C2×C8—2122no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—4241no—
275C4⋊C4—4241no—
276C4⋊C4—2121no—
277C4⋊C4—2122no—
278C4⋊C4—2122no—
279C4⋊C4—2121no—
280C4⋊C4—2121no—
281C22⋊C4—1064no—
282C22⋊C4—2121no—
283C22⋊C4—2121no—
284C42—4241no—
285C23—21no—
286C23—21no—
287C23—22no—
288C23—41no—
289C23#1313313111yescharacteristic
290C23#1313313111yescharacteristic
291C23C2×#1171992111yescharacteristic
292Q8—4242no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—1062no—
314C2×C4—1062no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4#1313314512yes—
326C2×C4#1313324311yescharacteristic
327C2×C4#1313324811yescharacteristic
328C8—44no—
329C8—4242no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—42no—
339C22#1681042112yes—
340C22C2×#1313313111yescharacteristic
341C22#1681062211yescharacteristic
342C22C2×#1313314511yescharacteristic
343C4—21no—
344C4—22no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—4241no—
351C4—4241no—
352C4—2122no—
353C4—2122no—
354C2—21no—
355C2—21no—
356C2—22no—
357C2—41no—
358C2#2027321711yescharacteristic
359C2#2027636211yescharacteristic
360C2C2×#1681042111yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC2×C53×#343C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenterC2×#343D53
hypocenterC53#11768
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
3. #20272744C2semidirect
4. #20272720C2semidirect
203. C53#11768semidirect

Maximal subgroups

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

subgroupindex
#202800952
#202730452
#202727442
#202727202
C2×C53×#3432
#1176853