Finite Group Databasebeta

← #27272657

Subgroups — #27272657

Order 13568.

500 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 119 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#20274783C211yesmaximal, characteristic
2#20274477C211yesmaximal, characteristic
3#20272538C214yesmaximal, complemented
4C53×#2857C211yesmaximal, characteristic
5C2×C22⋊#88322—21no—
6C2×C22⋊#88322—21no—
7C2×#13133128—21no—
8#16810378—21no—
9#16810378C2212yes—
10#16810378C2212yes—
11C2×C4⋊Q8×C53C411yescharacteristic
12C2×C22⋊Q8×C53—21no—
13C22×C42×C53C2211yescharacteristic
14C53×#391—21no—
15C53×#389—21no—
16C53×#389C2212yes—
17C53×#384C411yescharacteristic
18C53×#380—21no—
19C53×#380C412yes—
20C22×#88322—22no—
21C22×#88322—22no—
22C22⋊#88322—24no—
23C22⋊#88322—42no—
24#13133128—42no—
25C22×Q8×C53—21no—
26C23×C4×C53C2×C411yescharacteristic
27C23×C4×C53D411yescharacteristic
28C23×C4×C53D411yescharacteristic
29C4⋊Q8×C53—42no—
30C22⋊Q8×C53—44no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53C2×C412yes—
36C2×C4⋊C4×C53C2×C412yes—
37C2×C22⋊C4×C53—22no—
38C2×C42×C53—42no—
39C2×C42×C53C2×C411yescharacteristic
40C2×C42×C53D411yescharacteristic
41C2×C42×C53D411yescharacteristic
42C2×C42×C53C2311yescharacteristic
43#95×C53—22no—
44#95×C53—22no—
45C24×C53C22⋊C411yescharacteristic
46C2×Q8×C53—42no—
47C2×Q8×C53—42no—
48C2×Q8×C53—42no—
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—41no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C4⋊C4×C53—24no—
67C4⋊C4×C53—42no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—44no—
72C22⋊C4×C53—44no—
73C42×C53—22no—
74C42×C53—22no—
75C42×C53—41no—
76C42×C53—41no—
77C2×#88322—24no—
78C2×#88322—42no—
79C2×#88322—42no—
80C2×#88322—42no—
81C2×#88322—42no—
82C2×#88322—42no—
83C23×C53—21no—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53#10112yes—
90C23×C53C2×C22⋊C411yescharacteristic
91Q8×C53—84no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—44no—
106C2×C4×C53—44no—
107C2×C4×C53—44no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53#9912yes—
115C2×C4×C53#10412yes—
116C2×C4×C53#13112yes—
117C2×C4×C53AGL(1,ℤ/8)12yes—
118C2×C4×C53#13712yes—
119#88322—44no—
120#88322—82no—
121#7705—531nomaximal, Sylow
122C22×C53—22no—
123C22×C53—22no—
124C22×C53—22no—
125C22×C53—41no—
126C22×C53—41no—
127C22×C53—41no—
128C22×C53—41no—
129C22×C53#40811yescharacteristic
130C22×C53C2×#9911yescharacteristic
131C22×C53C2×#10111yescharacteristic
132C22×C53#41811yescharacteristic
133C22×C53#41911yescharacteristic
134C22×C53C2×#10411yescharacteristic
135C22×C53#42111yescharacteristic
136C4×C53—22no—
137C4×C53—22no—
138C4×C53—22no—
139C4×C53—22no—
140C4×C53—22no—
141C4×C53—41no—
142C4×C53—82no—
143C4×C53—82no—
144#2857D5311yescharacteristic
145#1817—531no—
146#1749—531no—
147#1271—534no—
148C2×C53—42no—
149C2×C53#147211yescharacteristic
150C2×C53#148511yescharacteristic
151C2×C53#149611yescharacteristic
152C2×C53#150711yescharacteristic
153C2×C53#151111yescharacteristic
154C2×C53#153111yescharacteristic
155C2×C53#155611yescharacteristic
156C2×C4⋊Q8Dic5311yescharacteristic
157C2×C22⋊Q8—21no—
158C22×C42C2×D5311yescharacteristic
159C2×C22⋊C8—1061no—
160C2×C22⋊C8—1061no—
161C2×#97—1061no—
162#391—21no—
163#389—21no—
164#389C2×D5312yes—
165#384Dic5311yescharacteristic
166#380—21no—
167#380Dic5312yes—
168#336—532no—
169#336—532no—
170#336—1061no—
171C53#770511yescharacteristic, complemented, Sylow
172C22×Q8—21no—
173C23×C4C2×Dic5311yescharacteristic
174C23×C4C22⋊D5311yescharacteristic
175C23×C4C22⋊D5311yescharacteristic
176C22×C8—1062no—
177C22×C8—1062no—
178C4⋊Q8—42no—
179C22⋊Q8—44no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4C2×Dic5312yes—
185C2×C4⋊C4C2×Dic5312yes—
186C2×C22⋊C4—22no—
187C2×C42—42no—
188C2×C42C2×Dic5311yescharacteristic
189C2×C42C22⋊D5311yescharacteristic
190C2×C42C22⋊D5311yescharacteristic
191C2×C42C22×D5311yescharacteristic
192C22⋊C8—1064no—
193C22⋊C8—2122no—
194#97—2122no—
195#95—22no—
196#95—22no—
197C24C22⋊Dic5311yescharacteristic
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4C22⋊Dic5311yescharacteristic
213C22×C4C22⋊Dic5311yescharacteristic
214C22×C4C22⋊Dic5311yescharacteristic
215C22×C4C22×Dic5311yescharacteristic
216C22×C4C2×C22⋊D5311yescharacteristic
217C22×C4C2×C22⋊D5311yescharacteristic
218C2×C8—1064no—
219C2×C8—2122no—
220C2×C8—2122no—
221C2×C8—2122no—
222C2×C8—2122no—
223C2×C8—2122no—
224C4⋊C4—24no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C22⋊C4—44no—
231C42—22no—
232C42—22no—
233C42—41no—
234C42—41no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23#1313313612yes—
242C23C2×C22⋊Dic5311yescharacteristic
243Q8—84no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4#1313313212yes—
267C2×C4#1313314412yes—
268C2×C4#1313322412yes—
269C2×C4#1313324312yes—
270C2×C4#1313324812yes—
271C8—4242no—
272C8—2124no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22#1681059011yescharacteristic
281C22C2×#1313313211yescharacteristic
282C22C2×#1313313611yescharacteristic
283C22#1681062811yescharacteristic
284C22#1681063211yescharacteristic
285C22C2×#1313314411yescharacteristic
286C22#1681064011yescharacteristic
287C4—22no—
288C4—22no—
289C4—22no—
290C4—22no—
291C4—22no—
292C4—41no—
293C4—82no—
294C4—82no—
295C2—42no—
296C2#2027307911yescharacteristic
297C2#2027313611yescharacteristic
298C2#2027318411yescharacteristic
299C2#2027323111yescharacteristic
300C2#2027324811yescharacteristic
301C2#2027333011yescharacteristic
302C2#2027343211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2857C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2857D53
hypocenterC53#7705
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
3. #20272538C2semidirect
171. C53#7705semidirect

Maximal subgroups

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

subgroupindex
#202747832
#202744772
#202725382
C53×#28572
#770553