Finite Group Databasebeta

← #27277725

Subgroups — #27277725

Order 13568.

466 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 95 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#20277688C211yesmaximal, characteristic
2C4×#13133132C211yesmaximal, characteristic
3#20272600C212yesmaximal, complemented
4#20272569C212yesmaximal, complemented
5C53×#1465C211yesmaximal, characteristic
6C2×#13133132—21no—
7#16810526—21no—
8#16810516—21no—
9#16810516—21no—
10#16810489C411yescharacteristic, complemented
11#16810486C411yescharacteristic, complemented
12#16810486C2211yescharacteristic
13C4⋊C4×Dic53—21no—
14C4×C4⋊Dic53C411yescharacteristic, complemented
15C4×C22⋊Dic53C411yescharacteristic, complemented
16C4×C22⋊Dic53C2212yes—
17C4×C22⋊Dic53C2211yescharacteristic
18#16810390—21no—
19C2×C4×D4×C53C2211yescharacteristic
20C53×#407C2212yes—
21C53×#324—21no—
22C53×#323—21no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C22⋊Dic53—21no—
26C2×C22⋊Dic53—21no—
27C2×C22⋊Dic53—21no—
28C2×C22⋊Dic53—21no—
29C2×C22⋊Dic53—22no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53—21no—
33C2×C4×Dic53—22no—
34#13133132—24no—
35#13133124—21no—
36#13133124—21no—
37#13133124—21no—
38#13133124—22no—
39#13133124C2×C411yescharacteristic
40#13133124C2×C411yescharacteristic
41#13133124C2×C411yescharacteristic
42#13133124C2×C411yescharacteristic
43C22×D4×C53D411yescharacteristic
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53C2311yescharacteristic
46C2×M16×C53—22no—
47C4×D4×C53—22no—
48C4×D4×C53—22no—
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C42×C53C2×C411yescharacteristic
53C22⋊C8×C53—22no—
54C22⋊C8×C53—22no—
55C24×C53C22⋊C411yescharacteristic
56C24×C53C2×D411yescharacteristic
57C2×D4×C53—22no—
58C2×D4×C53—22no—
59C2×D4×C53—41no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—81no—
63C22×C4×C53—81no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53#4112yes—
70C22×C4×C53#4112yes—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—42no—
76M16×C53—82no—
77C2×C8×C53—42no—
78C4⋊C4×C53—22no—
79C22⋊C4×C53—22no—
80C22⋊C4×C53—22no—
81C42×C53—21no—
82C42×C53—21no—
83C22⋊Dic53—24no—
84C22⋊Dic53—42no—
85C22⋊Dic53—42no—
86C22⋊Dic53—42no—
87C22⋊Dic53—81no—
88C22⋊Dic53—81no—
89C4⋊Dic53—42no—
90C4⋊Dic53—42no—
91C4×Dic53—42no—
92C4×Dic53—81no—
93C4×Dic53—81no—
94C4×Dic53—82no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53C2×C22⋊C411yescharacteristic
105C23×C53#12312yes—
106D4×C53—44no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—81no—
118C2×C4×C53—81no—
119C2×C4×C53#10412yes—
120C2×C4×C53#11711yescharacteristic
121C2×C4×C53#11711yescharacteristic
122C2×C4×C53#12312yes—
123C2×Dic53—22no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—44no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—82no—
137C2×Dic53—82no—
138C8×C53—82no—
139#8778—531nomaximal, Sylow
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—22no—
143C22×C53—22no—
144C22×C53—22no—
145C22×C53—41no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—81no—
149C22×C53—81no—
150C22×C53—81no—
151C22×C53—81no—
152C22×C53#38111yescharacteristic
153C22×C53C2×#10411yescharacteristic
154C22×C53#42111yescharacteristic
155C4×C53—22no—
156C4×C53—22no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—41no—
160C4×C53—41no—
161Dic53—42no—
162Dic53—81no—
163Dic53—81no—
164Dic53—81no—
165Dic53—84no—
166#2357—531no—
167C4×#99—531no—
168#1465D5311yescharacteristic
169#1297—532no—
170#1284—532no—
171C2×C53—21no—
172C2×C53—21no—
173C2×C53—42no—
174C2×C53—81no—
175C2×C53#182611yescharacteristic
176C2×C53#182711yescharacteristic
177C2×C53#182811yescharacteristic
178C2×C4×D4C2×D5311yescharacteristic
179C2×#99—1061no—
180#407C2×D5312yes—
181#390—1061no—
182#388—1061no—
183#388—1061no—
184#382—531no—
185#381—531no—
186#381—531no—
187C4×C4⋊C4—531no—
188C4×C4⋊C4—1061no—
189C4×C22⋊C4—531no—
190C4×C22⋊C4—531no—
191C4×C22⋊C4—532no—
192#342—1061no—
193#324—21no—
194#323—21no—
195C53#877811yescharacteristic, complemented, Sylow
196C22×D4D21211yescharacteristic
197C23×C4C4×D5311yescharacteristic
198C23×C4C22×D5311yescharacteristic
199C2×M16—22no—
200C4×D4—22no—
201C4×D4—22no—
202C2×C4⋊C4—1061no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4C22⋊D5311yescharacteristic
205C2×C22⋊C4—1062no—
206C2×C22⋊C4—1061no—
207C2×C22⋊C4—1061no—
208C2×C22⋊C4—1061no—
209C2×C22⋊C4—1061no—
210C2×C22⋊C4D21211yescharacteristic
211C2×C22⋊C4C22⋊D5311yescharacteristic
212C2×C42—1061no—
213C2×C42—1061no—
214C2×C42—1062no—
215C2×C42—1061no—
216C2×C42C4×D5311yescharacteristic
217#99—1064no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220#95—531no—
221#95—531no—
222#95—531no—
223#95—531no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1062no—
228C24#1171992111yescharacteristic
229C24C2×D21211yescharacteristic
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—1061no—
238C22×C4—1061no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4—2122no—
242C22×C4#1171992111yescharacteristic
243C22×C4#1171992111yescharacteristic
244C22×C4#1171992111yescharacteristic
245C22×C4C2×C4×D5311yescharacteristic
246C22×C4#1171994712yes—
247C22×C4#1171994712yes—
248C22×C4C2×C22⋊D5311yescharacteristic
249M16—82no—
250C2×C8—42no—
251C4⋊C4—22no—
252C4⋊C4—2122no—
253C4⋊C4—2122no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—1064no—
257C22⋊C4—4241no—
258C22⋊C4—4241no—
259C22⋊C4—2122no—
260C22⋊C4—2122no—
261C22⋊C4—2122no—
262C42—21no—
263C42—21no—
264C42—4242no—
265C42—4241no—
266C42—4241no—
267C42—2122no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23C2×#1171992111yescharacteristic
278C23#1313320112yes—
279D4—44no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—1062no—
293C2×C4—4242no—
294C2×C4—4242no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2124no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4#1313314312yes—
308C2×C4#1313317311yescharacteristic
309C2×C4#1313317311yescharacteristic
310C2×C4#1313319812yes—
311C8—82no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22#1681048411yescharacteristic
325C22C2×#1313314311yescharacteristic
326C22#1681063911yescharacteristic
327C4—22no—
328C4—22no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—4241no—
334C4—4241no—
335C4—4241no—
336C4—4244no—
337C4—2122no—
338C2—21no—
339C2—21no—
340C2—42no—
341C2—81no—
342C2#2027481211yescharacteristic
343C2#2027482011yescharacteristic
344C2#2027482111yescharacteristic
subgroupG/H
centerC22#16810484
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1465C2
socleC22×C53#381
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#1465D53
hypocenterC53#8778
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #20272600C2semidirect
4. #20272569C2semidirect
10. #16810489C4semidirect
11. #16810486C4semidirect
14. C4×C4⋊Dic53C4semidirect
15. C4×C22⋊Dic53C4semidirect
195. C53#8778semidirect

Maximal subgroups

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

subgroupindex
#202776882
C4×#131331322
#202726002
#202725692
C53×#14652
#877853