Finite Group Databasebeta

← #27293743

Subgroups — #27293743

Order 13568.

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

normal subgroups only

This group has 480 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#20273053C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic
3#20272661C214yesmaximal, complemented
4C53×#2542C211yesmaximal, characteristic
5#16810587—21no—
6C2×C22⋊#88322—21no—
7#16810349C2212yes—
8#16810349C2212yes—
9#16810347—21no—
10#16810347—21no—
11C2×C4⋊Q8×C53C411yescharacteristic
12C2×C22⋊Q8×C53—21no—
13C2×C22⋊Q8×C53—21no—
14C2×C4×D4×C53C2211yescharacteristic
15C53×#399—21no—
16C53×#399—21no—
17C53×#390C412yes—
18C53×#384C2212yes—
19C4×C4⋊C4×C53C411yescharacteristic
20C2×#11719917—41no—
21C22×#88322—41no—
22C22⋊#88322—22no—
23C22⋊#88322—22no—
24C22⋊#88322—41no—
25C22⋊#88322—41no—
26C22×Q8×C53—21no—
27C22×D4×C53C2×C411yescharacteristic
28C23×C4×C53D411yescharacteristic
29C23×C4×C53D411yescharacteristic
30C4⋊Q8×C53—41no—
31C4⋊Q8×C53—41no—
32C22⋊Q8×C53—44no—
33C22⋊Q8×C53—44no—
34C4×D4×C53—41no—
35C4×D4×C53—41no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53C2×C412yes—
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53D411yescharacteristic
45C2×C22⋊C4×C53D411yescharacteristic
46C2×C42×C53C2×C412yes—
47C2×C42×C53C2×C411yescharacteristic
48#95×C53—21no—
49C24×C53C22⋊C411yescharacteristic
50C24×C53C22⋊C411yescharacteristic
51C2×Q8×C53—42no—
52C2×Q8×C53—42no—
53C2×Q8×C53—41no—
54C2×Q8×C53—41no—
55C2×D4×C53—22no—
56C2×D4×C53—22no—
57C2×D4×C53—41no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C4⋊C4×C53—24no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—44no—
74C4⋊C4×C53—44no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—42no—
77C22⋊C4×C53—41no—
78C22⋊C4×C53—41no—
79C22⋊C4×C53—44no—
80C22⋊C4×C53—44no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—24no—
84#11719917—82no—
85C2×#88322—42no—
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
99Q8×C53—82no—
100Q8×C53—82no—
101D4×C53—44no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—22no—
105C2×C4×C53—24no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—81no—
120C2×C4×C53—81no—
121C2×C4×C53#9911yescharacteristic
122C2×C4×C53#9911yescharacteristic
123C2×C4×C53#13112yes—
124#88322—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—44no—
145C4×C53—41no—
146C4×C53—41no—
147C4×C53—41no—
148C4×C53—82no—
149C4×C53#35312yes—
150#2542D5311yescharacteristic
151#1465—531no—
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⋊Q8Dic5311yescharacteristic
162C2×C22⋊Q8—21no—
163C2×C22⋊Q8—21no—
164C2×C4×D4C2×D5311yescharacteristic
165#407—1061no—
166C2×C22⋊C8—1061no—
167#399—21no—
168#399—21no—
169#390Dic5312yes—
170#384C2×D5312yes—
171C4×C4⋊C4Dic5311yescharacteristic
172#324—532no—
173#324—532no—
174#323—1061no—
175#323—1061no—
176C53#1157011yescharacteristic, complemented, Sylow
177C22×Q8—21no—
178C22×D4C2×Dic5311yescharacteristic
179C23×C4C22⋊D5311yescharacteristic
180C23×C4C22⋊D5311yescharacteristic
181C2×M16—2121no—
182C22×C8—2121no—
183C4⋊Q8—41no—
184C4⋊Q8—41no—
185C22⋊Q8—44no—
186C22⋊Q8—44no—
187C4×D4—41no—
188C4×D4—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C2×Dic5312yes—
194C2×C4⋊C4C22×D5311yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4C22⋊D5311yescharacteristic
198C2×C22⋊C4C22⋊D5311yescharacteristic
199C2×C42C2×Dic5312yes—
200C2×C42C2×Dic5311yescharacteristic
201C22⋊C8—1062no—
202C22⋊C8—1062no—
203C22⋊C8—2121no—
204C22⋊C8—2121no—
205#95—21no—
206C24C22⋊Dic5311yescharacteristic
207C24C22⋊Dic5311yescharacteristic
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—41no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4C22⋊Dic5311yescharacteristic
224C22×C4C22⋊Dic5311yescharacteristic
225C22×C4C22×Dic5311yescharacteristic
226C22×C4C2×C22⋊D5311yescharacteristic
227C22×C4C2×C22⋊D5311yescharacteristic
228M16—4242no—
229C2×C8—4242no—
230C2×C8—2122no—
231C2×C8—2122no—
232C4⋊C4—24no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—42no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—44no—
241C22⋊C4—44no—
242C42—21no—
243C42—21no—
244C42—24no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23#1313313612yes—
255C23C2×C22⋊Dic5311yescharacteristic
256Q8—82no—
257Q8—82no—
258D4—44no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4#1313313211yescharacteristic
279C2×C4#1313313211yescharacteristic
280C2×C4#1313322412yes—
281C8—4242no—
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#1681059011yescharacteristic
296C22C2×#1313313211yescharacteristic
297C22C2×#1313313611yescharacteristic
298C4—21no—
299C4—22no—
300C4—44no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—82no—
305C4#1681041612yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2#2027307911yescharacteristic
311C2C2×#1681041611yescharacteristic
312C2#2027641011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2542C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2542D53
hypocenterC53#11570
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
3. #20272661C2semidirect
176. C53#11570semidirect

Maximal subgroups

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

subgroupindex
#202730532
#202730332
#202726612
C53×#25422
#1157053