Finite Group Databasebeta

← #27416028

Subgroups — #27416028

Order 13568.

1286 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1286 conjugacy classes of subgroups — too many to draw. Only its 207 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×#16810761C212yesmaximal, complemented
2#20277299C211yesmaximal, characteristic, complemented
3#20275863C212yesmaximal, complemented
4#20275803C212yesmaximal, complemented
5#20275049C214yesmaximal, complemented
6#20275038C212yesmaximal, complemented
7#20274786C211yesmaximal, characteristic, complemented
8C2×C53×#540C211yesmaximal, characteristic
9C23×Dic106C2211yescharacteristic
10C22×#11719923C2214yes—
11C2×#13133176C2211yescharacteristic
12#16810761—28no—
13C2×#13133140C2214yes—
14C2×#13133138C2214yes—
15C2×C22⋊#88322C2212yes—
16#16810529C2212yes—
17#16810529C2212yes—
18#16810528—22no—
19#16810484—22no—
20#16810385C2214yes—
21#16810378C2212yes—
22#16810378C2212yes—
23C53×#540—28no—
24C2×#127×C53C2211yescharacteristic
25C2×#125×C53C2211yescharacteristic
26C2×#123×C53—22no—
27C2×#121×C53—22no—
28C2×#121×C53C2212yes—
29C2×C4×D4×C53C2212yes—
30C2×#117×C53C2211yescharacteristic
31C22×Dic106—21no—
32C22×Dic106—24no—
33C22×Dic106—24no—
34C22×Dic106—42no—
35C22×Dic106C2312yes—
36C23×Dic53—21no—
37C2×#11719923—28no—
38C2×#11719923—28no—
39C2×#11719923—44no—
40C22×#88322C2314yes—
41#13133176—42no—
42C2×C4⋊Dic53C2312yes—
43C2×C22⋊Dic53—21no—
44C2×C4×Dic53—21no—
45#13133140—28no—
46#13133138—44no—
47C22⋊#88322—24no—
48#13133124—24no—
49#13133124—24no—
50C22×D4×C53—22no—
51C22×D4×C53—22no—
52C22×D4×C53C2312yes—
53C23×C4×C53—22no—
54C23×C4×C53C2311yescharacteristic
55#127×C53—24no—
56#125×C53—24no—
57#123×C53—216no—
58#121×C53—216no—
59#121×C53—24no—
60#121×C53D418yes—
61C4×D4×C53—24no—
62C4×D4×C53—24no—
63#117×C53—24no—
64C2×C4⋊C4×C53—22no—
65C2×C4⋊C4×C53D412yes—
66C2×C4⋊C4×C53C2312yes—
67C2×C22⋊C4×C53—22no—
68C2×C22⋊C4×C53—22no—
69C2×C22⋊C4×C53D412yes—
70C2×C42×C53C2311yescharacteristic
71C2×C42×C53C2311yescharacteristic
72C24×C53—22no—
73C24×C53C2×D411yescharacteristic
74C2×D4×C53—28no—
75C2×D4×C53—28no—
76C2×D4×C53—22no—
77C2×D4×C53—24no—
78C2×D4×C53—24no—
79C2×D4×C53—84no—
80C2×D4×C53—84no—
81C2×D4×C53C2×D414yes—
82C22×C4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—22no—
85C22×C4×C53—28no—
86C22×C4×C53—84no—
87C22×C4×C53C2×D412yes—
88C22×C4×C53C2×D412yes—
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D414yes—
91C22×C4×C53C2411yescharacteristic
92C22×Dic53—24no—
93C22×Dic53—22no—
94C22×Dic53—22no—
95C22×Dic53—42no—
96C22×Dic53—42no—
97C2×Dic106—24no—
98C2×Dic106—24no—
99C2×Dic106—24no—
100C2×Dic106—22no—
101C2×Dic106—416no—
102C2×Dic106—42no—
103C2×Dic106—42no—
104C2×Dic106#4114yes—
105C4⋊C4×C53—28no—
106C4⋊C4×C53—24no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53C2×D414yes—
109C22⋊C4×C53—28no—
110C22⋊C4×C53—24no—
111C22⋊C4×C53—28no—
112C42×C53—22no—
113C42×C53—22no—
114C22⋊Dic53—42no—
115#11719923—416no—
116C4⋊Dic53—24no—
117C4×Dic53—42no—
118C2×#88322—24no—
119C2×#88322—24no—
120C2×#88322—24no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—24no—
124C23×C53—24no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53—44no—
128C23×C53—44no—
129C23×C53#12012yes—
130C23×C53C22×D411yescharacteristic
131D4×C53—28no—
132D4×C53—44no—
133D4×C53—88no—
134D4×C53—88no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—28no—
138C2×C4×C53—22no—
139C2×C4×C53—24no—
140C2×C4×C53—22no—
141C2×C4×C53—24no—
142C2×C4×C53—24no—
143C2×C4×C53—44no—
144C2×C4×C53—88no—
145C2×C4×C53#12012yes—
146C2×C4×C53C22×D412yes—
147C2×C4×C53C2×#4112yes—
148C2×C4×C53H(5,𝔽2)12yes—
149C2×Dic53—24no—
150C2×Dic53—48no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—44no—
155Dic106—28no—
156Dic106—48no—
157Dic106—44no—
158#88322—44no—
159#23940—531nomaximal, Sylow
160C22×C53—22no—
161C22×C53—24no—
162C22×C53—42no—
163C22×C53—42no—
164C22×C53—44no—
165C22×C53—44no—
166C22×C53—44no—
167C22×C53—84no—
168C22×C53—84no—
169C22×C53C2×#12011yescharacteristic
170C22×C53#52511yescharacteristic
171C22×C53#53511yescharacteristic
172C22×C53#57714yes—
173C4×C53—22no—
174C4×C53—42no—
175C4×C53—42no—
176C4×C53—44no—
177C4×C53#54614yes—
178Dic53—44no—
179Dic53—82no—
180C2×#540D5311yescharacteristic
181C2×#448—532no—
182#2310—531no—
183#2026—532no—
184#2014—532no—
185#1877—534no—
186#1875—532no—
187#1818—531no—
188C2×C53—42no—
189C2×C53—84no—
190C2×C53#240411yescharacteristic
191C2×C53#300912yes—
192C2×C53#312612yes—
193C2×C53#322112yes—
194C23×Q8—531no—
195C22×SD16—534no—
196#540—28no—
197C2×#127C2×D5311yescharacteristic
198C2×#125C2×D5311yescharacteristic
199C2×#123—22no—
200C2×#121—22no—
201C2×#121C2×D5312yes—
202C2×C4×D4C2×D5312yes—
203C2×#117—531no—
204C2×#117C2×D5311yescharacteristic
205#448—1068no—
206C2×#103—534no—
207C2×#102—534no—
208C2×C22⋊C8—532no—
209#391—532no—
210#391—532no—
211#391—1062no—
212#381—1062no—
213#340—534no—
214#336—532no—
215#336—532no—
216C53#2394011yescharacteristic, complemented, Sylow
217C22×Q8—532no—
218C22×Q8—1064no—
219C22×Q8—1061no—
220C22×Q8—1064no—
221C22×Q8—2122no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4C22×D5312yes—
225C23×C4—22no—
226C23×C4—1061no—
227C23×C4C22×D5311yescharacteristic
228C2×SD16—1068no—
229C2×SD16—1068no—
230C2×SD16—2124no—
231C22×C8—534no—
232#127—24no—
233#125—24no—
234#123—216no—
235#121—216no—
236#121—24no—
237#121C22⋊D5318yes—
238C4×D4—24no—
239C4×D4—24no—
240#117—24no—
241#117—2122no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—532no—
244C2×C4⋊C4C22⋊D5312yes—
245C2×C4⋊C4C22×D5312yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—1061no—
249C2×C22⋊C4C22⋊D5312yes—
250C2×C42—1061no—
251C2×C42C22×D5311yescharacteristic
252C2×C42C22×D5311yescharacteristic
253#103—1068no—
254#102—2124no—
255C22⋊C8—1064no—
256#95—1064no—
257#95—1064no—
258C24—22no—
259C24C2×C22⋊D5311yescharacteristic
260C2×Q8—534no—
261C2×Q8—1064no—
262C2×Q8—1062no—
263C2×Q8—1064no—
264C2×Q8—1064no—
265C2×Q8—21216no—
266C2×Q8—2122no—
267C2×Q8—2122no—
268C2×D4—28no—
269C2×D4—28no—
270C2×D4—22no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—84no—
274C2×D4—84no—
275C2×D4C2×C22⋊D5314yes—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—28no—
280C22×C4—84no—
281C22×C4—1064no—
282C22×C4—1062no—
283C22×C4—1062no—
284C22×C4—2122no—
285C22×C4—2122no—
286C22×C4C2×C22⋊D5312yes—
287C22×C4C2×C22⋊D5312yes—
288C22×C4C2×C22⋊D5311yescharacteristic
289C22×C4C2×C22⋊D5314yes—
290C22×C4C23×D5311yescharacteristic
291SD16—21216no—
292C2×C8—1064no—
293C2×C8—1064no—
294C2×C8—1064no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—1064no—
299C4⋊C4C2×C22⋊D5314yes—
300C22⋊C4—28no—
301C22⋊C4—24no—
302C22⋊C4—28no—
303C22⋊C4—2122no—
304C42—22no—
305C42—22no—
306C42—2122no—
307C23—21no—
308C23—24no—
309C23—21no—
310C23—24no—
311C23—42no—
312C23—44no—
313C23—44no—
314C23—42no—
315C23#1313318612yes—
316C23C22×C22⋊D5311yescharacteristic
317Q8—1068no—
318Q8—2128no—
319Q8—2124no—
320D4—28no—
321D4—44no—
322D4—88no—
323D4—88no—
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—44no—
333C2×C4—88no—
334C2×C4—1064no—
335C2×C4—2124no—
336C2×C4—2122no—
337C2×C4—2128no—
338C2×C4—2122no—
339C2×C4—2122no—
340C2×C4#1313318612yes—
341C2×C4C22×C22⋊D5312yes—
342C2×C4#41×D5312yes—
343C2×C4#1313326312yes—
344C8—2124no—
345C22—22no—
346C22—24no—
347C22—42no—
348C22—42no—
349C22—44no—
350C22—44no—
351C22—44no—
352C22—84no—
353C22—84no—
354C22C2×#1313318611yescharacteristic
355C22#1681113011yescharacteristic
356C22#1681118711yescharacteristic
357C22#1681143414yes—
358C4—22no—
359C4—42no—
360C4—42no—
361C4—44no—
362C4—4242no—
363C4—2124no—
364C4#1681126114yes—
365C2—42no—
366C2—84no—
367C2#2027803011yescharacteristic
368C2#2028399212yes—
369C2#2028500112yes—
370C2#2028577012yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#540C2
socleC23×C53C22×D4
hypercenterC2×#540D53
hypocenterC53#23940
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. C2×#16810761C2semidirect
2. #20277299C2semidirect
3. #20275863C2semidirect
4. #20275803C2semidirect
5. #20275049C2semidirect
6. #20275038C2semidirect
7. #20274786C2semidirect
216. C53#23940semidirect

Maximal subgroups

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

subgroupindex
C2×#168107612
#202772992
#202758632
#202758032
#202750492
#202750382
#202747862
C2×C53×#5402
#2394053