Finite Group Databasebeta

← #27294604

Subgroups — #27294604

Order 13568.

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

normal subgroups only

This group has 446 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#20279002C211yesmaximal, characteristic
2C2×#16810392C211yesmaximal, characteristic
3#20272743C212yesmaximal, complemented
4#20272722C212yesmaximal, complemented
5C53×#1463C211yesmaximal, characteristic
6C22×#11719917—21no—
7C2×#13133176C411yescharacteristic
8#16810562C411yescharacteristic
9#16810562C2211yescharacteristic
10#16810553C411yescharacteristic
11#16810553C2211yescharacteristic
12#16810504—21no—
13#16810504—21no—
14#16810496—21no—
15#16810486—21no—
16#16810486C411yescharacteristic
17#16810392—21no—
18#16810392C2212yes—
19C22×C4⋊C4×C53C2211yescharacteristic
20C53×#407—21no—
21C53×#324—21no—
22C53×#324C2212yes—
23C2×#11719917—24no—
24C2×#11719917—22no—
25C2×#11719917—41no—
26C2×#11719917—42no—
27C22×#88322—41no—
28#13133176—41no—
29#13133176—41no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C4×Dic53—21no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124C2×C411yescharacteristic
45#13133124C2×C411yescharacteristic
46C23×C4×C53C2×C411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C2×M16×C53—41no—
50C2×C4⋊C4×C53—41no—
51C2×C4⋊C4×C53—41no—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C4⋊C4×C53C2311yescharacteristic
56C22⋊C8×C53—22no—
57C22⋊C8×C53—41no—
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—42no—
69C22×C4×C53—42no—
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—
80M16×C53—82no—
81C2×C8×C53—42no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C22⋊Dic53—42no—
87C22⋊Dic53—41no—
88C22⋊Dic53—41no—
89C4⋊Dic53—41no—
90C4⋊Dic53—41no—
91#11719919—42no—
92#11719919—41no—
93#11719919—41no—
94C4×Dic53—42no—
95#11719917—24no—
96#11719917—44no—
97#11719917—42no—
98C2×#88322—44no—
99C2×#88322—42no—
100C23×C53—22no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—41no—
104C23×C53#9911yescharacteristic
105C23×C53#9911yescharacteristic
106C23×C53C2×C22⋊C411yescharacteristic
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—81no—
127C2×C4×C53#10412yes—
128C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C53#13711yescharacteristic
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—42no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C8×C53—82no—
142#88322—44no—
143#11764—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53#35212yes—
154C22×C53C2×#9911yescharacteristic
155C22×C53#41711yescharacteristic
156C22×C53C2×#10411yescharacteristic
157C4×C53—21no—
158C4×C53—22no—
159C4×C53—21no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164Dic53—81no—
165Dic53—81no—
166Dic53—81no—
167Dic53—81no—
168#2484—531no—
169C2×#343—531no—
170#1463D5311yescharacteristic
171#1353—532no—
172#1346—532no—
173C2×C53—21no—
174C2×C53—21no—
175C2×C53—22no—
176C2×C53—41no—
177C2×C53#150411yescharacteristic
178C2×C53C2×#35211yescharacteristic
179C2×C53#213311yescharacteristic
180C22×M16—1061no—
181C2×#117—531no—
182C22×C4⋊C4C2×D5311yescharacteristic
183#407—21no—
184#400—531no—
185#400—531no—
186#398—531no—
187#398—531no—
188#387—1061no—
189#387—1061no—
190#385—1061no—
191#381—531no—
192#381—1061no—
193#343—532no—
194#343—1061no—
195#324—21no—
196#324C2×D5312yes—
197C53#1176411yescharacteristic, complemented, Sylow
198C23×C4C4×D5311yescharacteristic
199C23×C4D21211yescharacteristic
200C23×C4C22⋊D5311yescharacteristic
201C2×M16—41no—
202C2×M16—1064no—
203C2×M16—1062no—
204C2×M16—2121no—
205C2×M16—2122no—
206C22×C8—2121no—
207#117—2121no—
208#117—2121no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—531no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4C4×D5311yescharacteristic
216C2×C4⋊C4D21211yescharacteristic
217C2×C4⋊C4C22⋊D5311yescharacteristic
218C2×C4⋊C4C22×D5311yescharacteristic
219C2×C22⋊C4—1061no—
220C2×C22⋊C4—1061no—
221C2×C42—1061no—
222C22⋊C8—22no—
223C22⋊C8—41no—
224#95—531no—
225#95—531no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233C24#1171992111yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—1061no—
246C22×C4—1061no—
247C22×C4—1061no—
248C22×C4—1061no—
249C22×C4#1171992111yescharacteristic
250C22×C4#1171992111yescharacteristic
251C22×C4#1171992111yescharacteristic
252C22×C4C2×C4×D5311yescharacteristic
253C22×C4C2×D21211yescharacteristic
254C22×C4C2×C22⋊D5311yescharacteristic
255M16—82no—
256M16—1064no—
257M16—2124no—
258M16—2122no—
259C2×C8—42no—
260C2×C8—2124no—
261C2×C8—2122no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—2121no—
267C4⋊C4—2121no—
268C4⋊C4—2122no—
269C4⋊C4—2121no—
270C4⋊C4—2121no—
271C22⋊C4—2122no—
272C22⋊C4—2121no—
273C22⋊C4—2121no—
274C42—2122no—
275C23—21no—
276C23—21no—
277C23—22no—
278C23—41no—
279C23#1313313111yescharacteristic
280C23#1313313111yescharacteristic
281C23C2×#1171992111yescharacteristic
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2122no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4#1313314312yes—
314C2×C4#1313324511yescharacteristic
315C2×C4#1313325011yescharacteristic
316C8—82no—
317C8—2124no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22#1681041212yes—
328C22C2×#1313313111yescharacteristic
329C22#1681062511yescharacteristic
330C22C2×#1313314311yescharacteristic
331C4—21no—
332C4—22no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—4241no—
339C4—4241no—
340C4—4241no—
341C4—4241no—
342C2—21no—
343C2—21no—
344C2—22no—
345C2—41no—
346C2#2027322011yescharacteristic
347C2C2×#1681041211yescharacteristic
348C2#2027640211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1463C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1463D53
hypocenterC53#11764
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #20272743C2semidirect
4. #20272722C2semidirect
197. C53#11764semidirect

Maximal subgroups

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

subgroupindex
#202790022
C2×#168103922
#202727432
#202727222
C53×#14632
#1176453