Finite Group Databasebeta

← #27416032

Subgroups — #27416032

Order 13568.

1286 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1286 conjugacy classes of subgroups — too many to draw. Only its 195 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×#16811228C211yesmaximal, characteristic
2C2×#16810762C212yesmaximal, complemented
3#20275866C212yesmaximal, complemented
4#20275802C212yesmaximal, complemented
5#20275052C214yesmaximal, complemented
6#20275037C212yesmaximal, complemented
7#20274786C211yesmaximal, characteristic, complemented
8C53×#2310C211yesmaximal, characteristic, complemented
9C22×#11719924C2214yes—
10#16811228—28no—
11C2×#13133214C2211yescharacteristic
12C2×#13133209C2211yescharacteristic
13C2×#13133202—22no—
14C2×#13133191C2212yes—
15C2×#13133190—22no—
16C2×C4×D212C2212yes—
17#16810762—28no—
18C2×#13133141C2214yes—
19C2×#13133137C2214yes—
20C2×C22⋊#88322C2212yes—
21#16810385C2214yes—
22#16810378C2212yes—
23#16810378C2212yes—
24C23×Q8×C53C2211yescharacteristic
25C2×#117×C53C2211yescharacteristic
26C2×#117×C53C2211yescharacteristic
27C53×#391—22no—
28C53×#391C2212yes—
29C53×#391C2212yes—
30C53×#381—22no—
31C22×D212—22no—
32C22×D212C2312yes—
33C22×C22⋊D53—22no—
34C22×C4×D53—22no—
35C2×#11719924—28no—
36C2×#11719924—28no—
37C2×#11719924—44no—
38C22×#88322C2314yes—
39#13133214—24no—
40#13133209—24no—
41#13133202—216no—
42#13133191—24no—
43#13133191D418yes—
44#13133190—216no—
45C4×D212—24no—
46C4×D212—24no—
47C2×C4⋊Dic53C2312yes—
48C2×#11719919—22no—
49C2×#11719921—22no—
50C2×#11719921—22no—
51#13133141—28no—
52#13133137—44no—
53C22⋊#88322—24no—
54C22×Q8×C53—24no—
55C22×Q8×C53—21no—
56C22×Q8×C53—24no—
57C22×Q8×C53—42no—
58C22×Q8×C53C2312yes—
59C23×C4×C53—21no—
60C23×C4×C53C2311yescharacteristic
61#117×C53—24no—
62#117×C53—42no—
63C2×C4⋊C4×C53D412yes—
64C2×C4⋊C4×C53C2312yes—
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53D412yes—
67C2×C42×C53—21no—
68C2×C42×C53C2311yescharacteristic
69C2×C42×C53C2311yescharacteristic
70#95×C53—24no—
71#95×C53—24no—
72C24×C53C2×D411yescharacteristic
73C23×D53—22no—
74C2×Q8×C53—24no—
75C2×Q8×C53—22no—
76C2×Q8×C53—24no—
77C2×Q8×C53—24no—
78C2×Q8×C53—416no—
79C2×Q8×C53—42no—
80C2×Q8×C53—42no—
81C2×Q8×C53#4114yes—
82C22×C4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—24no—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53—42no—
88C22×C4×C53—42no—
89C22×C4×C53C2×D412yes—
90C22×C4×C53C2×D412yes—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D414yes—
93C22×C4×C53C2411yescharacteristic
94C2×C22⋊D53—28no—
95C2×C22⋊D53—84no—
96C22×Dic53—22no—
97C2×D212—24no—
98C2×D212—24no—
99C2×D212—22no—
100C2×D212—28no—
101C2×D212—84no—
102C2×D212C2×D414yes—
103C2×C4×D53—28no—
104C2×C4×D53—84no—
105C4⋊C4×C53—24no—
106C4⋊C4×C53—24no—
107C22⋊C4×C53—24no—
108C22⋊C4×C53—42no—
109C42×C53—22no—
110C42×C53—22no—
111C42×C53—42no—
112#11719924—416no—
113#11719921—28no—
114#11719921—28no—
115C4⋊Dic53—22no—
116C4⋊Dic53C2×D414yes—
117#11719919—28no—
118C2×#88322—24no—
119C2×#88322—24no—
120C2×#88322—24no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—24no—
124C23×C53#12012yes—
125C23×C53C22×D411yescharacteristic
126C22×D53—24no—
127C22×D53—42no—
128C22×D53—42no—
129C22×D53—44no—
130C22×D53—44no—
131Q8×C53—28no—
132Q8×C53—48no—
133Q8×C53—44no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—28no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—24no—
140C2×C4×C53—24no—
141C2×C4×C53—24no—
142C2×C4×C53—48no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—44no—
147C2×C4×C53#12012yes—
148C2×C4×C53C22×D412yes—
149C2×C4×C53C2×#4112yes—
150C2×C4×C53H(5,𝔽2)12yes—
151C22⋊D53—88no—
152C2×Dic53—24no—
153C2×Dic53—44no—
154D212—28no—
155D212—44no—
156D212—88no—
157C4×D53—88no—
158#88322—44no—
159#23940—531nomaximal, Sylow
160C22×C53—22no—
161C22×C53—24no—
162C22×C53—42no—
163C22×C53—42no—
164C22×C53C2×#12011yescharacteristic
165C22×C53#52511yescharacteristic
166C22×C53#53511yescharacteristic
167C22×C53#57714yes—
168C2×D53—44no—
169C2×D53—44no—
170C2×D53—44no—
171C2×D53—84no—
172C2×D53—84no—
173C4×C53—22no—
174C4×C53—44no—
175C4×C53—42no—
176C4×C53—42no—
177C4×C53—82no—
178C4×C53#54614yes—
179Dic53—44no—
180C2×#540—531no—
181C2×#448—532no—
182#2310D5311yescharacteristic, complemented
183#2026—532no—
184#2014—532no—
185#1877—534no—
186#1875—532no—
187#1818—531no—
188C2×C53—42no—
189C2×C53#240411yescharacteristic
190C2×C53#300912yes—
191C2×C53#312612yes—
192C2×C53#322112yes—
193D53—84no—
194C23×Q8C2×D5311yescharacteristic
195C22×SD16—534no—
196#540—1068no—
197C2×#127—531no—
198C2×#125—531no—
199C2×#123—1062no—
200C2×#121—532no—
201C2×#121—1062no—
202C2×C4×D4—532no—
203C2×#117C2×D5311yescharacteristic
204C2×#117C2×D5311yescharacteristic
205#448—1068no—
206C2×#103—534no—
207C2×#102—534no—
208C2×C22⋊C8—532no—
209#391—22no—
210#391C2×D5312yes—
211#391C2×D5312yes—
212#381—22no—
213#340—534no—
214#336—532no—
215#336—532no—
216C53#2394011yescharacteristic, complemented, Sylow
217C22×Q8—24no—
218C22×Q8—21no—
219C22×Q8—24no—
220C22×Q8—42no—
221C22×Q8C22×D5312yes—
222C22×D4—532no—
223C22×D4—1062no—
224C22×D4—1062no—
225C23×C4—21no—
226C23×C4—1062no—
227C23×C4C22×D5311yescharacteristic
228C2×SD16—1068no—
229C2×SD16—1068no—
230C2×SD16—2124no—
231C22×C8—534no—
232#127—1064no—
233#125—1064no—
234#123—10616no—
235#121—538no—
236#121—10616no—
237#121—1064no—
238C4×D4—1064no—
239C4×D4—1064no—
240#117—24no—
241#117—42no—
242C2×C4⋊C4—532no—
243C2×C4⋊C4—1062no—
244C2×C4⋊C4C22⋊D5312yes—
245C2×C4⋊C4C22×D5312yes—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—1062no—
248C2×C22⋊C4—1062no—
249C2×C22⋊C4C22⋊D5312yes—
250C2×C42—21no—
251C2×C42C22×D5311yescharacteristic
252C2×C42C22×D5311yescharacteristic
253#103—1068no—
254#102—2124no—
255C22⋊C8—1064no—
256#95—24no—
257#95—24no—
258C24—1062no—
259C24C2×C22⋊D5311yescharacteristic
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—416no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8#1171994514yes—
268C2×D4—534no—
269C2×D4—1068no—
270C2×D4—1068no—
271C2×D4—1062no—
272C2×D4—1064no—
273C2×D4—1064no—
274C2×D4—4244no—
275C2×D4—4244no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—1068no—
284C22×C4—1062no—
285C22×C4—4244no—
286C22×C4D4×D5314yes—
287C22×C4C2×C22⋊D5312yes—
288C22×C4C2×C22⋊D5312yes—
289C22×C4C2×C22⋊D5311yescharacteristic
290C22×C4C23×D5311yescharacteristic
291SD16—21216no—
292C2×C8—1064no—
293C2×C8—1064no—
294C2×C8—1064no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—534no—
298C4⋊C4—1068no—
299C4⋊C4—1062no—
300C22⋊C4—24no—
301C22⋊C4—42no—
302C22⋊C4—1068no—
303C22⋊C4—1068no—
304C42—22no—
305C42—22no—
306C42—42no—
307C23—21no—
308C23—24no—
309C23—21no—
310C23—1064no—
311C23—2122no—
312C23—2124no—
313C23—2124no—
314C23—2122no—
315C23#1313318512yes—
316C23C22×C22⋊D5311yescharacteristic
317Q8—28no—
318Q8—48no—
319Q8—44no—
320D4—1068no—
321D4—4248no—
322D4—4248no—
323D4—2124no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—28no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—48no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—1064no—
338C2×C4—4248no—
339C2×C4—2124no—
340C2×C4#1313318512yes—
341C2×C4C2×D4×D5312yes—
342C2×C4C2×#1171994512yes—
343C2×C4#1313326212yes—
344C8—2124no—
345C22—22no—
346C22—24no—
347C22—42no—
348C22—42no—
349C22—4244no—
350C22—4244no—
351C22—2124no—
352C22—2124no—
353C22—2124no—
354C22C2×#1313318511yescharacteristic
355C22#1681113411yescharacteristic
356C22#1681119011yescharacteristic
357C22#1681143114yes—
358C4—22no—
359C4—44no—
360C4—42no—
361C4—42no—
362C4—82no—
363C4—2124no—
364C4#1681126314yes—
365C2—42no—
366C2—4244no—
367C2#2027803411yescharacteristic
368C2#2028399012yes—
369C2#2028500612yes—
370C2#2028577612yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2310C2
socleC23×C53C22×D4
prime radicalC2×#16811228C2
hypercenter#2310D53
hypocenterC53#23940
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. C2×#16810762C2semidirect
3. #20275866C2semidirect
4. #20275802C2semidirect
5. #20275052C2semidirect
6. #20275037C2semidirect
7. #20274786C2semidirect
8. C53×#2310C2semidirect
182. #2310D53semidirect
216. C53#23940semidirect

Maximal subgroups

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

subgroupindex
C2×#168112282
C2×#168107622
#202758662
#202758022
#202750522
#202750372
#202747862
C53×#23102
#2394053