Finite Group Databasebeta

← #27293744

Subgroups — #27293744

Order 13568.

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

normal subgroups only

This group has 480 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#20279668C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic
3#20272660C214yesmaximal, complemented
4C53×#1465C211yesmaximal, characteristic
5C2×#13133217C411yescharacteristic
6C2×#13133196—21no—
7C2×#13133196—21no—
8C2×C22⋊#88322—21no—
9#16810559—21no—
10#16810559—21no—
11#16810526C412yes—
12#16810492C2212yes—
13C4×C4⋊Dic53C411yescharacteristic
14#16810349C2212yes—
15#16810347—21no—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#407—21no—
18C53×#324C2212yes—
19C53×#323—21no—
20C22×Dic106—21no—
21C22×#88322—41no—
22#13133217—41no—
23#13133217—41no—
24#13133196—44no—
25#13133196—44no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53C2×C412yes—
29C2×#11719919—22no—
30C2×#11719919—22no—
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53—22no—
33C2×C4×Dic53C2×C412yes—
34C22⋊#88322—22no—
35C22⋊#88322—41no—
36#13133124—21no—
37C22×D4×C53C2×C411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C2×M16×C53—41no—
41C4×D4×C53—41no—
42C4×D4×C53—41no—
43C2×C4⋊C4×C53C2311yescharacteristic
44C2×C22⋊C4×C53D411yescharacteristic
45C2×C22⋊C4×C53D411yescharacteristic
46C2×C42×C53C2×C411yescharacteristic
47C22⋊C8×C53—22no—
48C22⋊C8×C53—41no—
49C24×C53C22⋊C411yescharacteristic
50C24×C53C22⋊C411yescharacteristic
51C2×D4×C53—22no—
52C2×D4×C53—22no—
53C2×D4×C53—41no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—42no—
59C22×C4×C53—42no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×Dic53—22no—
66C22×Dic53—22no—
67C2×Dic106—42no—
68C2×Dic106—42no—
69C2×Dic106—41no—
70C2×Dic106—41no—
71M16×C53—82no—
72C2×C8×C53—42no—
73C4⋊C4×C53—41no—
74C22⋊C4×C53—41no—
75C22⋊C4×C53—41no—
76C42×C53—21no—
77C42×C53—21no—
78C22⋊Dic53—44no—
79C22⋊Dic53—44no—
80C4⋊Dic53—24no—
81C4⋊Dic53—42no—
82C4⋊Dic53—42no—
83#11719919—44no—
84#11719919—44no—
85C4×Dic53—24no—
86C2×#88322—42no—
87C2×#88322—82no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53#10112yes—
98C23×C53C2×C22⋊C411yescharacteristic
99D4×C53—44no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—81no—
112C2×C4×C53—81no—
113C2×C4×C53#9911yescharacteristic
114C2×C4×C53#9911yescharacteristic
115C2×C4×C53#13112yes—
116C2×Dic53—24no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—42no—
122Dic106—82no—
123Dic106—82no—
124C8×C53—82no—
125#88322—82no—
126#11570—531nomaximal, Sylow
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—22no—
130C22×C53—42no—
131C22×C53—42no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—41no—
137C22×C53—81no—
138C22×C53—81no—
139C22×C53#40811yescharacteristic
140C22×C53C2×#9911yescharacteristic
141C22×C53C2×#10111yescharacteristic
142C4×C53—21no—
143C4×C53—22no—
144C4×C53—41no—
145C4×C53—41no—
146C4×C53—41no—
147C4×C53#35312yes—
148Dic53—44no—
149Dic53—82no—
150#2542—531no—
151#1465D5311yescharacteristic
152#1459—531no—
153#1325—534no—
154C2×C53—21no—
155C2×C53—21no—
156C2×C53—42no—
157C2×C53—81no—
158C2×C53#147211yescharacteristic
159C2×C53C2×#35311yescharacteristic
160C2×C53#213511yescharacteristic
161C2×C4⋊Q8—531no—
162C2×C22⋊Q8—1061no—
163C2×C22⋊Q8—1061no—
164C2×C4×D4C2×D5311yescharacteristic
165#407—21no—
166C2×C22⋊C8—1061no—
167#399—1061no—
168#399—1061no—
169#390—532no—
170#384—532no—
171C4×C4⋊C4—531no—
172#324—532no—
173#324C2×D5312yes—
174#323—21no—
175#323—1061no—
176C53#1157011yescharacteristic, complemented, Sylow
177C22×Q8—1061no—
178C22×D4C4×D5311yescharacteristic
179C23×C4D21211yescharacteristic
180C23×C4C22⋊D5311yescharacteristic
181C2×M16—41no—
182C22×C8—2121no—
183C4⋊Q8—2121no—
184C4⋊Q8—2121no—
185C22⋊Q8—2124no—
186C22⋊Q8—2124no—
187C4×D4—41no—
188C4×D4—41no—
189C2×C4⋊C4—532no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4—1062no—
192C2×C4⋊C4—1062no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4C22×D5311yescharacteristic
195C2×C22⋊C4—1062no—
196C2×C22⋊C4—1062no—
197C2×C22⋊C4D21211yescharacteristic
198C2×C22⋊C4C22⋊D5311yescharacteristic
199C2×C42—532no—
200C2×C42C4×D5311yescharacteristic
201C22⋊C8—22no—
202C22⋊C8—41no—
203C22⋊C8—1062no—
204C22⋊C8—2121no—
205#95—1061no—
206C24#1171992111yescharacteristic
207C24#1171992111yescharacteristic
208C2×Q8—2122no—
209C2×Q8—2122no—
210C2×Q8—2121no—
211C2×Q8—2121no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—1062no—
222C22×C4—1062no—
223C22×C4#1171992111yescharacteristic
224C22×C4#1171992111yescharacteristic
225C22×C4C2×C4×D5311yescharacteristic
226C22×C4C2×D21211yescharacteristic
227C22×C4C2×C22⋊D5311yescharacteristic
228M16—82no—
229C2×C8—42no—
230C2×C8—4242no—
231C2×C8—2122no—
232C4⋊C4—41no—
233C4⋊C4—1064no—
234C4⋊C4—2122no—
235C4⋊C4—2122no—
236C4⋊C4—2124no—
237C4⋊C4—2124no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—2124no—
241C22⋊C4—2124no—
242C42—21no—
243C42—21no—
244C42—1064no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23#1313313512yes—
255C23C2×#1171992111yescharacteristic
256Q8—4242no—
257Q8—4242no—
258D4—44no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—1064no—
273C2×C4—2122no—
274C2×C4—2122no—
275C2×C4—2122no—
276C2×C4—2122no—
277C2×C4—2122no—
278C2×C4#1313313111yescharacteristic
279C2×C4#1313313111yescharacteristic
280C2×C4#1313322512yes—
281C8—82no—
282C8—4242no—
283C22—21no—
284C22—21no—
285C22—22no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—81no—
295C22#1681059111yescharacteristic
296C22C2×#1313313111yescharacteristic
297C22C2×#1313313511yescharacteristic
298C4—21no—
299C4—22no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—4242no—
304C4—2124no—
305C4#1681041512yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2#2027308011yescharacteristic
311C2C2×#1681041511yescharacteristic
312C2#2027640911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1465C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1465D53
hypocenterC53#11570
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #20272660C2semidirect
176. C53#11570semidirect

Maximal subgroups

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

subgroupindex
#202796682
#202730332
#202726602
C53×#14652
#1157053