Finite Group Databasebeta

← #27291958

Subgroups — #27291958

Order 13568.

1042 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 1042 conjugacy classes of subgroups — too many to draw. Only its 227 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×#16810587C212yesmaximal
2C2×#16810392C214yesmaximal, complemented
3C2×C53×#380C211yesmaximal, characteristic
4C22×#11719917C2214yes—
5#16810587—28no—
6C2×C22⋊#88322—22no—
7C2×C22⋊#88322—22no—
8#16810392—28no—
9C24×C4×C53C2211yescharacteristic
10C22×C4⋊C4×C53C2212yes—
11C22×C22⋊C4×C53C412yes—
12C53×#380—22no—
13C53×#380—22no—
14C2×#95×C53C412yes—
15C2×#11719917—24no—
16C2×#11719917—24no—
17C2×#11719917—24no—
18C2×#11719917—24no—
19C2×#11719917—24no—
20C2×#11719917D418yes—
21C22×#88322—24no—
22C22⋊#88322—28no—
23C22⋊#88322—28no—
24C25×C53C2×C411yescharacteristic
25C23×C4×C53—24no—
26C23×C4×C53—24no—
27C23×C4×C53—24no—
28C23×C4×C53C2×C411yescharacteristic
29C23×C4×C53C2×C414yes—
30C23×C4×C53D412yes—
31C23×C4×C53D412yes—
32C23×C4×C53C2311yescharacteristic
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53—24no—
36C2×C4⋊C4×C53—24no—
37C2×C22⋊C4×C53—22no—
38C2×C22⋊C4×C53—22no—
39C2×C22⋊C4×C53—44no—
40#95×C53—24no—
41C24×C53—22no—
42C24×C53—22no—
43C24×C53—22no—
44C24×C53—24no—
45C24×C53—42no—
46C24×C53C22⋊C411yescharacteristic
47C24×C53C22⋊C411yescharacteristic
48C24×C53C22×C411yescharacteristic
49C22×C4×C53—22no—
50C22×C4×C53—24no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—24no—
54C22×C4×C53—24no—
55C22×C4×C53—24no—
56C22×C4×C53—24no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—24no—
60C22×C4×C53—28no—
61C22×C4×C53—44no—
62C22×C4×C53—44no—
63C22×C4×C53—44no—
64C22×C4×C53—44no—
65C22×C4×C53—44no—
66C22×C4×C53—44no—
67C22×C4×C53C22⋊C412yes—
68C22×C4×C53C2×D414yes—
69C22×C4×C53C2×D412yes—
70C22×C4×C53#4114yes—
71C22×C4×C53#4114yes—
72C4⋊C4×C53—42no—
73C4⋊C4×C53—44no—
74C4⋊C4×C53—42no—
75C22⋊C4×C53—44no—
76C22⋊C4×C53—44no—
77#11719917—28no—
78#11719917—28no—
79#11719917—44no—
80#11719917—44no—
81C2×#88322—28no—
82C2×#88322—44no—
83C2×#88322—44no—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53—22no—
87C23×C53—28no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—22no—
95C23×C53—22no—
96C23×C53—48no—
97C23×C53—48no—
98C23×C53—44no—
99C23×C53—44no—
100C23×C53#10012yes—
101C23×C53#10112yes—
102C23×C53C2×C22⋊C411yescharacteristic
103C23×C53#11712yes—
104C23×C53C4×D414yes—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—44no—
109C2×C4×C53—44no—
110C2×C4×C53—44no—
111C2×C4×C53—44no—
112C2×C4×C53—44no—
113C2×C4×C53—44no—
114C2×C4×C53—44no—
115C2×C4×C53—44no—
116C2×C4×C53—88no—
117C2×C4×C53#12012yes—
118C2×C4×C53#12114yes—
119C2×C4×C53#12114yes—
120C2×C4×C53#12312yes—
121C2×C4×C53#12314yes—
122C2×C4×C53#12314yes—
123C2×C4×C53#12414yes—
124C2×C4×C53#12614yes—
125#88322—48no—
126#11198—531nomaximal, Sylow
127C22×C53—22no—
128C22×C53—22no—
129C22×C53—24no—
130C22×C53—24no—
131C22×C53—22no—
132C22×C53—22no—
133C22×C53—24no—
134C22×C53—24no—
135C22×C53—416no—
136C22×C53—41no—
137C22×C53—41no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53#38111yescharacteristic
143C22×C53#38612yes—
144C22×C53#38814yes—
145C22×C53C2×#10011yescharacteristic
146C22×C53C2×#10111yescharacteristic
147C22×C53#41312yes—
148C4×C53—42no—
149C4×C53—84no—
150C4×C53#39414yes—
151C4×C53#39714yes—
152C2×#407—532no—
153C2×#380D5311yescharacteristic
154C2×#343—534no—
155C2×C53—24no—
156C2×C53—42no—
157C2×C53—42no—
158C2×C53#144011yescharacteristic
159C2×C53#182111yescharacteristic
160C2×C53#182211yescharacteristic
161C2×C53#190012yes—
162C2×C53#190112yes—
163C24×C4C2×D5311yescharacteristic
164C22×M16—534no—
165C22×C4⋊C4C2×D5312yes—
166C22×C22⋊C4Dic5312yes—
167#407—1068no—
168C2×C22⋊C8—1062no—
169C2×C22⋊C8—1062no—
170#380—22no—
171#380—22no—
172C2×#95Dic5312yes—
173#343—1068no—
174C53#1119811yescharacteristic, complemented, Sylow
175C25C2×Dic5311yescharacteristic
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4C2×Dic5311yescharacteristic
180C23×C4C2×Dic5314yes—
181C23×C4C22⋊D5312yes—
182C23×C4C22⋊D5312yes—
183C23×C4C22×D5311yescharacteristic
184C2×M16—538no—
185C2×M16—1064no—
186C2×M16—1064no—
187C2×M16—1064no—
188C2×M16—1064no—
189C2×M16—1064no—
190C22×C8—1064no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—44no—
198C22⋊C8—1068no—
199C22⋊C8—1068no—
200#95—24no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—24no—
205C24—42no—
206C24C22⋊Dic5311yescharacteristic
207C24C22⋊Dic5311yescharacteristic
208C24C22×Dic5311yescharacteristic
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—28no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4C22⋊Dic5312yes—
228C22×C4#1171994514yes—
229C22×C4D4×D5314yes—
230C22×C4#1171994714yes—
231C22×C4C2×C22⋊D5312yes—
232M16—1068no—
233M16—1068no—
234M16—2124no—
235M16—2124no—
236C2×C8—1068no—
237C2×C8—2124no—
238C2×C8—2124no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—28no—
248C23—22no—
249C23—22no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—21no—
256C23—48no—
257C23—48no—
258C23—44no—
259C23—44no—
260C23#1313313412yes—
261C23#1313313612yes—
262C23C2×C22⋊Dic5311yescharacteristic
263C23#1313317612yes—
264C23D4×Dic5314yes—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—88no—
277C2×C4#1313318512yes—
278C2×C4#1313318914yes—
279C2×C4#1313319014yes—
280C2×C4#1313319812yes—
281C2×C4#1313319914yes—
282C2×C4#1313320014yes—
283C2×C4#1313320314yes—
284C2×C4#1313321014yes—
285C8—2128no—
286C22—22no—
287C22—22no—
288C22—24no—
289C22—24no—
290C22—22no—
291C22—22no—
292C22—24no—
293C22—24no—
294C22—416no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22#1681048611yescharacteristic
302C22#1681050312yes—
303C22#1681051614yes—
304C22C2×#1313313411yescharacteristic
305C22C2×#1313313611yescharacteristic
306C22#1681060712yes—
307C4—42no—
308C4—84no—
309C4#1681053814yes—
310C4#1681055114yes—
311C2—24no—
312C2—42no—
313C2—42no—
314C2#2027296411yescharacteristic
315C2#2027479511yescharacteristic
316C2#2027479811yescharacteristic
317C2#2027517912yes—
318C2#2027518412yes—
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#380C2
socleC23×C53C2×C22⋊C4
prime radicalC25×C53C2×C4
cosocleC23×C4×C53C23
hypercenterC2×#380D53
hypocenterC53#11198
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C2×#16810392C2semidirect
174. C53#11198semidirect

Maximal subgroups

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

subgroupindex
C2×#168105872
C2×#168103922
C2×C53×#3802
#1119853