Finite Group Databasebeta

← #27294831

Subgroups — #27294831

Order 13568.

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

normal subgroups only

This group has 586 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
1C2×#16810543C211yesmaximal, characteristic
2#20274795C211yesmaximal, characteristic
3#20272746C214yesmaximal, complemented
4C53×#1463C211yesmaximal, characteristic
5C23×Dic106C411yescharacteristic
6C22×#11719919—21no—
7#16810587—21no—
8#16810543—21no—
9#16810543—21no—
10#16810543C412yes—
11#16810543C2212yes—
12C2×#13133124—21no—
13C2×#13133124C411yescharacteristic
14#16810392—21no—
15#16810392C2212yes—
16C22×C4⋊C4×C53C2211yescharacteristic
17C53×#407—21no—
18C53×#324—21no—
19C53×#324C2212yes—
20C22×Dic106—21no—
21C22×Dic106—41no—
22C22×Dic106—41no—
23C22×Dic106—44no—
24C22×Dic106C2×C412yes—
25C23×Dic53—21no—
26C23×Dic53—21no—
27C2×#11719917—22no—
28C2×#11719919—22no—
29C2×#11719919—22no—
30C2×#11719919—44no—
31C22⋊#88322—41no—
32C22⋊#88322—41no—
33#13133124—21no—
34#13133124—22no—
35#13133124—22no—
36#13133124C2×C412yes—
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C2×M16×C53—41no—
41C2×C4⋊C4×C53—41no—
42C2×C4⋊C4×C53—41no—
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C4⋊C4×C53C2311yescharacteristic
47C22⋊C8×C53—22no—
48C22⋊C8×C53—41no—
49C24×C53C22⋊C411yescharacteristic
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—22no—
57C22×C4×C53—41no—
58C22×C4×C53—42no—
59C22×C4×C53—42no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×Dic53—24no—
67C22×Dic53—24no—
68C22×Dic53—22no—
69C22×Dic53—22no—
70C22×Dic53—42no—
71C22×Dic53—42no—
72C22×Dic53—42no—
73C22×Dic53—41no—
74C22×Dic53—41no—
75C2×Dic106—44no—
76C2×Dic106—42no—
77C2×Dic106—44no—
78C2×Dic106—48no—
79C2×Dic106—42no—
80C2×Dic106—44no—
81C2×Dic106—44no—
82M16×C53—82no—
83C2×C8×C53—42no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—41no—
88#11719919—44no—
89#11719919—44no—
90#11719917—42no—
91#11719917—42no—
92C2×#88322—42no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53#9911yescharacteristic
98C23×C53#9911yescharacteristic
99C23×C53C2×C22⋊C411yescharacteristic
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
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—42no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—81no—
119C2×C4×C53#13712yes—
120C2×Dic53—24no—
121C2×Dic53—24no—
122C2×Dic53—42no—
123C2×Dic53—44no—
124C2×Dic53—48no—
125C2×Dic53—42no—
126C2×Dic53—44no—
127C2×Dic53—42no—
128C2×Dic53—42no—
129Dic106—48no—
130Dic106—44no—
131Dic106—44no—
132C8×C53—82no—
133#88322—82no—
134#11815—531nomaximal, Sylow
135C22×C53—24no—
136C22×C53—24no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—21no—
143C22×C53AΣL(1,𝔽16)12yes—
144C22×C53#35612yes—
145C22×C53C2×#9911yescharacteristic
146C22×C53#41911yescharacteristic
147C22×C53#42111yescharacteristic
148C4×C53—22no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53—41no—
153C4×C53—41no—
154Dic53—44no—
155Dic53—44no—
156C2×#395—531no—
157#1821—531no—
158#1463D5311yescharacteristic
159#1354—534no—
160C2×C53—21no—
161C2×C53—22no—
162C2×C53—22no—
163C2×C53—21no—
164C2×C53#151111yescharacteristic
165C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
166C2×C53C2×#35611yescharacteristic
167C23×Q8—531no—
168C22×C4⋊C4—1061no—
169C22×C4⋊C4C2×D5311yescharacteristic
170#407—21no—
171#407—1061no—
172#395—532no—
173#395—532no—
174#395—1061no—
175#395—1061no—
176C2×#95—531no—
177C2×#95—1061no—
178#343—532no—
179#343—1061no—
180#324—21no—
181#324C2×D5312yes—
182C53#1181511yescharacteristic, complemented, Sylow
183C22×Q8—532no—
184C22×Q8—1061no—
185C22×Q8—2121no—
186C22×Q8—2121no—
187C22×Q8—2124no—
188C23×C4—1061no—
189C23×C4—1061no—
190C23×C4C4×D5311yescharacteristic
191C23×C4D21211yescharacteristic
192C23×C4C22⋊D5311yescharacteristic
193C2×M16—41no—
194C2×M16—1062no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—1062no—
198C2×C4⋊C4—1062no—
199C2×C4⋊C4—2124no—
200C2×C4⋊C4C4×D5311yescharacteristic
201C2×C4⋊C4D21211yescharacteristic
202C2×C4⋊C4C22⋊D5311yescharacteristic
203C2×C4⋊C4C22×D5311yescharacteristic
204C22⋊C8—22no—
205C22⋊C8—41no—
206C22⋊C8—2121no—
207C22⋊C8—2121no—
208#95—532no—
209#95—1062no—
210#95—1061no—
211#95—1062no—
212C24#1171992111yescharacteristic
213C2×Q8—2122no—
214C2×Q8—2122no—
215C2×Q8—2124no—
216C2×Q8—2124no—
217C2×Q8—2128no—
218C2×Q8—2124no—
219C2×Q8—2124no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—1064no—
231C22×C4—1064no—
232C22×C4—1062no—
233C22×C4—1062no—
234C22×C4—2122no—
235C22×C4—2122no—
236C22×C4—2122no—
237C22×C4—2121no—
238C22×C4—2121no—
239C22×C4#1171992111yescharacteristic
240C22×C4#1171992111yescharacteristic
241C22×C4#1171992111yescharacteristic
242C22×C4C2×C4×D5311yescharacteristic
243C22×C4C2×D21211yescharacteristic
244C22×C4C2×C22⋊D5311yescharacteristic
245M16—82no—
246M16—2122no—
247M16—2122no—
248C2×C8—42no—
249C2×C8—2122no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—2124no—
255C4⋊C4—2124no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23#1313313111yescharacteristic
261C23#1313313111yescharacteristic
262C23C2×#1171992111yescharacteristic
263Q8—2124no—
264Q8—2128no—
265Q8—2124no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—1064no—
286C2×C4—1064no—
287C2×C4—2122no—
288C2×C4—2128no—
289C2×C4—2122no—
290C2×C4—2124no—
291C2×C4—2122no—
292C2×C4—2122no—
293C2×C4—2124no—
294C2×C4#1313325012yes—
295C8—82no—
296C8—4242no—
297C22—24no—
298C22—24no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22#1681040912yes—
306C22#1681042112yes—
307C22C2×#1313313111yescharacteristic
308C22#1681063311yescharacteristic
309C22#1681063911yescharacteristic
310C4—22no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—2124no—
317C4—2124no—
318C2—21no—
319C2—21no—
320C2—22no—
321C2—22no—
322C2#2027324911yescharacteristic
323C2C2×#1681040911yescharacteristic
324C2C2×#1681042111yescharacteristic
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#11815
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. #20272746C2semidirect
182. C53#11815semidirect

Maximal subgroups

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

subgroupindex
C2×#168105432
#202747952
#202727462
C53×#14632
#1181553