Finite Group Databasebeta

← #27293560

Subgroups — #27293560

Order 13568.

596 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 99 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#20273053C212yesmaximal, complemented
2#20272661C212yesmaximal, complemented
3#20272647C212yesmaximal, complemented
4C4×#127×C53C211yesmaximal, characteristic
5#16810587—22no—
6#16810349C2214yes—
7#16810347—22no—
8C2×#127×C53C411yescharacteristic
9C2×C4×D4×C53—22no—
10C2×C4×D4×C53C411yescharacteristic
11C2×C4×D4×C53C2211yescharacteristic
12C53×#390—22no—
13C53×#390C411yescharacteristic
14C53×#390C2211yescharacteristic
15C53×#384C2211yescharacteristic
16C43×C53C411yescharacteristic
17C2×#11719917—42no—
18C22⋊#88322—24no—
19C22⋊#88322—42no—
20C22×D4×C53—22no—
21C22×D4×C53C2×C411yescharacteristic
22C22×D4×C53C2×C411yescharacteristic
23C23×C4×C53—21no—
24C23×C4×C53D412yes—
25#127×C53—22no—
26#127×C53—22no—
27C4×D4×C53—41no—
28C4×D4×C53—41no—
29C4×D4×C53—48no—
30C4×D4×C53—41no—
31C4×D4×C53—41no—
32C2×C4⋊C4×C53—22no—
33C2×C4⋊C4×C53C2×C411yescharacteristic
34C2×C4⋊C4×C53C2311yescharacteristic
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—21no—
37C2×C22⋊C4×C53—22no—
38C2×C22⋊C4×C53D412yes—
39C2×C42×C53—21no—
40C2×C42×C53—21no—
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C2×C42×C53C2×C411yescharacteristic
44C24×C53—21no—
45C24×C53C22⋊C412yes—
46C2×D4×C53—28no—
47C2×D4×C53—21no—
48C2×D4×C53—21no—
49C2×D4×C53—21no—
50C2×D4×C53—41no—
51C2×D4×C53—41no—
52C2×D4×C53—44no—
53C2×D4×C53—44no—
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C2×D4×C53M1612yescomplemented
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—44no—
63C22×C4×C53—44no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D412yes—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—44no—
72C4⋊C4×C53—41no—
73C22⋊C4×C53—42no—
74C22⋊C4×C53—44no—
75C22⋊C4×C53—44no—
76C22⋊C4×C53—42no—
77C42×C53—24no—
78C42×C53—21no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—42no—
82C42×C53—41no—
83C42×C53M1612yescomplemented
84#11719917—84no—
85C2×#88322—44no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53C2×C22⋊C411yescharacteristic
97D4×C53—42no—
98D4×C53—416no—
99D4×C53—42no—
100D4×C53—42no—
101D4×C53—42no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—44no—
114C2×C4×C53—42no—
115C2×C4×C53—41no—
116C2×C4×C53—42no—
117C2×C4×C53—41no—
118C2×C4×C53—82no—
119C2×C4×C53—82no—
120C2×C4×C53—82no—
121C2×C4×C53#9911yescharacteristic
122C2×C4×C53#9911yescharacteristic
123C2×C4×C53C2×M1611yescharacteristic
124C2×C4×C53C2×M1611yescharacteristic
125#88322—84no—
126#11529—531nomaximal, Sylow
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—44no—
130C22×C53—44no—
131C22×C53—42no—
132C22×C53—42no—
133C22×C53—42no—
134C22×C53—42no—
135C22×C53—82no—
136C22×C53—82no—
137C22×C53—81no—
138C22×C53—81no—
139C22×C53#40711yescharacteristic
140C22×C53C2×#9911yescharacteristic
141C22×C53#41311yescharacteristic
142C4×C53—24no—
143C4×C53—21no—
144C4×C53—21no—
145C4×C53—21no—
146C4×C53—42no—
147C4×C53—42no—
148C4×C53—41no—
149C4×C53—81no—
150C4×C53#35312yes—
151C4×#127D5311yescharacteristic
152#1465—532no—
153#1325—532no—
154#1318—532no—
155C2×C53—21no—
156C2×C53—21no—
157C2×C53—82no—
158C2×C53—82no—
159C2×C53#146511yescharacteristic
160C2×C53C2×#35311yescharacteristic
161C2×C53#213211yescharacteristic
162C2×#127Dic5311yescharacteristic
163C2×C4×D4—22no—
164C2×C4×D4Dic5311yescharacteristic
165C2×C4×D4C2×D5311yescharacteristic
166#407—1062no—
167#390—22no—
168#390Dic5311yescharacteristic
169#390C2×D5311yescharacteristic
170#384C2×D5311yescharacteristic
171C43Dic5311yescharacteristic
172#324—534no—
173#323—1062no—
174C53#1152911yescharacteristic, complemented, Sylow
175C22×D4—22no—
176C22×D4C2×Dic5311yescharacteristic
177C22×D4C2×Dic5311yescharacteristic
178C23×C4—21no—
179C23×C4C22⋊D5312yes—
180C2×M16—2122no—
181#127—22no—
182#127—22no—
183C4×D4—41no—
184C4×D4—41no—
185C4×D4—48no—
186C4×D4—41no—
187C4×D4—41no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C2×Dic5311yescharacteristic
190C2×C4⋊C4C22×D5311yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4C22⋊D5312yes—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42C2×Dic5311yescharacteristic
198C2×C42C2×Dic5311yescharacteristic
199C2×C42C2×Dic5311yescharacteristic
200C22⋊C8—1064no—
201C22⋊C8—2122no—
202C24—21no—
203C24C22⋊Dic5312yes—
204C2×D4—28no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—44no—
213C2×D4—44no—
214C2×D4#1171991712yescomplemented
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4C22⋊Dic5311yescharacteristic
225C22×C4C22⋊Dic5311yescharacteristic
226C22×C4C22×Dic5311yescharacteristic
227C22×C4C2×C22⋊D5312yes—
228M16—4244no—
229C2×C8—2124no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—44no—
233C22⋊C4—42no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—42no—
237C42—24no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—41no—
242C42—42no—
243C42#1171991712yescomplemented
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—41no—
253C23—41no—
254C23C2×C22⋊Dic5311yescharacteristic
255D4—42no—
256D4—416no—
257D4—42no—
258D4—42no—
259D4—42no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4#1313313211yescharacteristic
280C2×C4#1313313211yescharacteristic
281C2×C4C2×#1171991711yescharacteristic
282C2×C4C2×#1171991711yescharacteristic
283C8—4244no—
284C22—21no—
285C22—21no—
286C22—44no—
287C22—44no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—82no—
293C22—82no—
294C22—81no—
295C22—81no—
296C22#1681058711yescharacteristic
297C22C2×#1313313211yescharacteristic
298C22#1681060711yescharacteristic
299C4—24no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—42no—
304C4—42no—
305C4—41no—
306C4—81no—
307C4#1681041612yes—
308C2—21no—
309C2—21no—
310C2—82no—
311C2—82no—
312C2#2027305311yescharacteristic
313C2C2×#1681041611yescharacteristic
314C2#2027640011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC4×#127×C53C2
socleC22×C53C2×#99
prime radicalC2×#127×C53C4
cosocleC2×C4⋊C4×C53C23
hypercenterC4×#127D53
hypocenterC53#11529
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
1. #20273053C2semidirect
2. #20272661C2semidirect
3. #20272647C2semidirect
56. C2×D4×C53M16semidirect
83. C42×C53M16semidirect
174. C53#11529semidirect
214. C2×D4#11719917semidirect
243. C42#11719917semidirect

Maximal subgroups

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

subgroupindex
#202730532
#202726612
#202726472
C4×#127×C532
#1152953