Finite Group Databasebeta

← #27277419

Subgroups — #27277419

Order 13568.

898 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 898 conjugacy classes of subgroups — too many to draw. Only its 159 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×C4×C22⋊Dic53C211yesmaximal, characteristic
2#20274625C211yesmaximal, characteristic
3#20272557C214yesmaximal, complemented
4C53×#1779C211yesmaximal, characteristic
5C22×C22⋊Dic53C411yescharacteristic
6C22×C22⋊Dic53C411yescharacteristic
7C22×C4×Dic53C412yes—
8C2×C22⋊#88322—22no—
9C2×C22⋊#88322C2212yes—
10C4×C22⋊Dic53—22no—
11C4×C22⋊Dic53—22no—
12C2×#13133124C2212yes—
13C24×C4×C53C2211yescharacteristic
14C2×C22⋊C8×C53—22no—
15C2×C22⋊C8×C53C2212yes—
16C23×Dic53C2×C412yes—
17C23×Dic53C2×C412yes—
18C22×#88322—22no—
19C2×C22⋊Dic53—22no—
20C2×C22⋊Dic53—22no—
21C2×C22⋊Dic53—22no—
22C2×C22⋊Dic53—24no—
23C2×C22⋊Dic53C814yescomplemented
24C2×C4×Dic53—22no—
25C2×C4×Dic53—22no—
26C2×C4×Dic53—44no—
27C22⋊#88322—24no—
28C22⋊#88322—44no—
29#13133124—22no—
30#13133124—22no—
31C25×C53C2×C411yescharacteristic
32C23×C4×C53—24no—
33C23×C4×C53—44no—
34C23×C4×C53C2×C411yescharacteristic
35C23×C4×C53D412yes—
36C23×C4×C53D412yes—
37C23×C4×C53C2311yescharacteristic
38C22×C8×C53—22no—
39C22⋊C8×C53—24no—
40C22⋊C8×C53—44no—
41C24×C53—22no—
42C24×C53—22no—
43C24×C53—42no—
44C24×C53—42no—
45C24×C53C22⋊C412yes—
46C24×C53C2×C812yes—
47C24×C53M1612yes—
48C24×C53C22×C411yescharacteristic
49C22×C4×C53—22no—
50C22×C4×C53—24no—
51C22×C4×C53—24no—
52C22×C4×C53—24no—
53C22×C4×C53—24no—
54C22×C4×C53—24no—
55C22×C4×C53—42no—
56C22×C4×C53—42no—
57C22×C4×C53—42no—
58C22×C4×C53—416no—
59C22×C4×C53C22⋊C412yes—
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53#4114yes—
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—22no—
66C22×Dic53—22no—
67C22×Dic53—42no—
68C22×Dic53—42no—
69C22×Dic53—42no—
70C22×Dic53—42no—
71C22×Dic53C2×C814yes—
72C22×Dic53M1614yes—
73C2×C8×C53—44no—
74C2×C8×C53—42no—
75C22⋊Dic53—28no—
76C22⋊Dic53—28no—
77C4×Dic53—44no—
78C4×Dic53—44no—
79C2×#88322—44no—
80C2×#88322—42no—
81C23×C53—22no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—24no—
90C23×C53—24no—
91C23×C53—44no—
92C23×C53—44no—
93C23×C53—44no—
94C23×C53—44no—
95C23×C53—44no—
96C23×C53—44no—
97C23×C53C22⋊C814yes—
98C23×C53C2×C22⋊C411yescharacteristic
99C23×C53#11712yes—
100C23×C53C22×C811yescharacteristic
101C23×C53C2×M1612yes—
102C23×C53C2×M1611yescharacteristic
103C2×C4×C53—28no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—44no—
109C2×C4×C53—48no—
110C2×C4×C53—48no—
111C2×C4×C53#9914yes—
112C2×C4×C53#12312yes—
113C2×C4×C53#12312yes—
114C2×Dic53—24no—
115C2×Dic53—24no—
116C2×Dic53—44no—
117C2×Dic53—44no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—44no—
123C2×Dic53—44no—
124C8×C53—82no—
125#88322—82no—
126#8700—531nomaximal, Sylow
127C22×C53—22no—
128C22×C53—22no—
129C22×C53—22no—
130C22×C53—22no—
131C22×C53—22no—
132C22×C53—22no—
133C22×C53—24no—
134C22×C53—24no—
135C22×C53—24no—
136C22×C53—48no—
137C22×C53—48no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—42no—
142C22×C53#38111yescharacteristic
143C22×C53C2×C22⋊C811yescharacteristic
144C22×C53#40711yescharacteristic
145C22×C53C2×#9912yes—
146C22×C53#41312yes—
147C22×C53#43112yes—
148C22×C53#43212yes—
149C4×C53—22no—
150C4×C53—22no—
151C4×C53—24no—
152Dic53—44no—
153Dic53—44no—
154C2×C4×C22⋊C4—531no—
155#1779—531no—
156#1779D5311yescharacteristic
157#1280—534no—
158C2×C53—22no—
159C2×C53—22no—
160C2×C53—44no—
161C2×C53#145912yes—
162C2×C53#146512yes—
163C2×C53#181711yescharacteristic
164C2×C53#181911yescharacteristic
165C2×C53#182211yescharacteristic
166C24×C4C2×D5311yescharacteristic
167C22×C22⋊C4—531no—
168C22×C22⋊C4—531no—
169C22×C42—532no—
170C2×C22⋊C8—22no—
171C2×C22⋊C8—532no—
172C2×C22⋊C8—1062no—
173C2×C22⋊C8C2×D5312yes—
174C4×C22⋊C4—1062no—
175C4×C22⋊C4—1062no—
176C2×#95—532no—
177C53#870011yescharacteristic, complemented, Sylow
178C25C4×D5311yescharacteristic
179C23×C4—24no—
180C23×C4—44no—
181C23×C4—532no—
182C23×C4—532no—
183C23×C4C4×D5311yescharacteristic
184C23×C4D21212yes—
185C23×C4C22⋊D5312yes—
186C23×C4C22×D5311yescharacteristic
187C22×C8—22no—
188C22×C8—1062no—
189C2×C22⋊C4—534no—
190C2×C22⋊C4—1064no—
191C2×C22⋊C4—1062no—
192C2×C22⋊C4—1062no—
193C2×C22⋊C4—1062no—
194C2×C42—1062no—
195C2×C42—1062no—
196C2×C42—2124no—
197C22⋊C8—24no—
198C22⋊C8—44no—
199C22⋊C8—1064no—
200C22⋊C8—2124no—
201#95—1062no—
202#95—1062no—
203C24—22no—
204C24—22no—
205C24—42no—
206C24—42no—
207C24C8×D5312yes—
208C24#1171991212yes—
209C24#1171992112yes—
210C24C2×C4×D5311yescharacteristic
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—416no—
221C22×C4—534no—
222C22×C4—534no—
223C22×C4—1062no—
224C22×C4—1062no—
225C22×C4—1062no—
226C22×C4—1062no—
227C22×C4—2122no—
228C22×C4—2122no—
229C22×C4—2122no—
230C22×C4—2122no—
231C22×C4#1171992112yes—
232C22×C4C2×D21211yescharacteristic
233C22×C4#1171994714yes—
234C22×C4C2×C22⋊D5311yescharacteristic
235C2×C8—44no—
236C2×C8—42no—
237C2×C8—2124no—
238C2×C8—2122no—
239C22⋊C4—1068no—
240C22⋊C4—1068no—
241C42—2124no—
242C42—2124no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—24no—
251C23—24no—
252C23—22no—
253C23—44no—
254C23—44no—
255C23—44no—
256C23—44no—
257C23—44no—
258C23—44no—
259C23#1313312914yes—
260C23C2×#1171992111yescharacteristic
261C23#1313317312yes—
262C23C2×C8×D5311yescharacteristic
263C23M16×D5312yes—
264C23C2×#1171991211yescharacteristic
265C2×C4—28no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—44no—
271C2×C4—48no—
272C2×C4—48no—
273C2×C4—1064no—
274C2×C4—1064no—
275C2×C4—2124no—
276C2×C4—2124no—
277C2×C4—2122no—
278C2×C4—2122no—
279C2×C4—2122no—
280C2×C4—2122no—
281C2×C4—2124no—
282C2×C4—2124no—
283C2×C4#1313313114yes—
284C2×C4#1313319812yes—
285C2×C4#1313320112yes—
286C8—82no—
287C8—4242no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—24no—
295C22—24no—
296C22—24no—
297C22—48no—
298C22—48no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22#1681048411yescharacteristic
304C22C2×#1313312911yescharacteristic
305C22#1681058611yescharacteristic
306C22C2×#1313313112yes—
307C22#1681060512yes—
308C22#1681067312yes—
309C22#1681067712yes—
310C4—22no—
311C4—22no—
312C4—24no—
313C4—2124no—
314C4—2124no—
315C2—22no—
316C2—22no—
317C2—44no—
318C2#2027303112yes—
319C2#2027305112yes—
320C2#2027478111yescharacteristic
321C2#2027478711yescharacteristic
322C2#2027479611yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC23×C53C22×C8
FrattiniC23×C4C22×D53
FittingC53×#1779C2
socleC23×C53C2×C22⋊C4
prime radicalC25×C53C2×C4
cosocleC23×C4×C53C23
hypercenter#1779D53
hypocenterC53#8700
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
3. #20272557C2semidirect
23. C2×C22⋊Dic53C8semidirect
177. C53#8700semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Dic532
#202746252
#202725572
C53×#17792
#870053