Finite Group Databasebeta

← #27284656

Subgroups — #27284656

Order 13568.

772 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 121 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×#16810939C212yesmaximal, complemented
2C2×#16810357C212yesmaximal, complemented
3#20272984C211yesmaximal, characteristic, complemented
4#20272558C211yesmaximal, characteristic, complemented
5C53×#2656C211yesmaximal, characteristic
6C22×C4⋊Dic53C2211yescharacteristic
7#16810939—24no—
8#16810939—24no—
9C2×#13133148—22no—
10C2×#13133138—22no—
11C2×C22⋊#88322C2212yes—
12C2×#13133124—22no—
13C2×#13133124C2211yescharacteristic
14#16810357—24no—
15C2×#121×C53—22no—
16C2×#121×C53C2212yes—
17C53×#399—22no—
18C53×#392—22no—
19C2×#95×C53C2211yescharacteristic
20C23×Dic53—21no—
21C23×Dic53—21no—
22C22×#88322—22no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53D412yescomplemented
27#13133148—28no—
28#13133138—28no—
29C22⋊#88322—22no—
30C22⋊#88322D414yescomplemented
31#13133124—22no—
32#13133124—44no—
33C22×D4×C53—22no—
34C22×D4×C53—22no—
35C22×D4×C53D412yes—
36C23×C4×C53D412yes—
37C23×C4×C53C2311yescharacteristic
38#121×C53—22no—
39#121×C53—22no—
40#121×C53—22no—
41#121×C53—22no—
42#121×C53—44no—
43#121×C53—44no—
44C2×C4⋊C4×C53—22no—
45C2×C4⋊C4×C53D412yes—
46C2×C22⋊C4×C53—24no—
47C2×C22⋊C4×C53—22no—
48#95×C53—22no—
49C24×C53—22no—
50C24×C53C2×D411yescharacteristic
51C2×D4×C53—22no—
52C2×D4×C53—22no—
53C2×D4×C53—44no—
54C2×D4×C53—44no—
55C2×D4×C53—42no—
56C2×D4×C53—42no—
57C2×D4×C53—44no—
58C2×D4×C53—44no—
59C2×D4×C53—82no—
60C2×D4×C53—82no—
61C22×C4×C53—22no—
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53—42no—
67C22×C4×C53SD1614yes—
68C22×C4×C53C2×D412yes—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D412yes—
71C22×C4×C53#4111yescharacteristic
72C22×Dic53—22no—
73C22×Dic53—22no—
74C22×Dic53—42no—
75C22×Dic53—41no—
76C22×Dic53—42no—
77C22×Dic53—42no—
78C22×Dic53—41no—
79C22×Dic53—41no—
80C22×Dic53—41no—
81C22×Dic53—42no—
82C4⋊C4×C53—22no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—44no—
85C22⋊C4×C53—48no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—42no—
88C4⋊Dic53—24no—
89C4⋊Dic53—42no—
90C2×#88322—24no—
91C2×#88322—44no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—22no—
95C23×C53—42no—
96C23×C53—44no—
97C23×C53—42no—
98C23×C53—42no—
99C23×C53—44no—
100C23×C53#12011yescharacteristic
101C23×C53#12111yescharacteristic
102C23×C53#12711yescharacteristic
103C23×C53C2×SD1612yes—
104C23×C53AGL(1,ℤ/8)12yes—
105D4×C53—44no—
106D4×C53—82no—
107D4×C53—88no—
108D4×C53—84no—
109D4×C53—82no—
110D4×C53—82no—
111C2×C4×C53—24no—
112C2×C4×C53—24no—
113C2×C4×C53—41no—
114C2×C4×C53—42no—
115C2×C4×C53—41no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—44no—
119C2×C4×C53—42no—
120C2×C4×C53—44no—
121C2×C4×C53—82no—
122C2×C4×C53#12012yes—
123C2×C4×C53#12112yes—
124C2×Dic53—44no—
125C2×Dic53—42no—
126C2×Dic53—44no—
127C2×Dic53—42no—
128C2×Dic53—82no—
129C2×Dic53—81no—
130C2×Dic53—82no—
131C2×Dic53—81no—
132C2×Dic53—81no—
133C2×Dic53—81no—
134#88322—44no—
135#9968—531nomaximal, Sylow
136C22×C53—22no—
137C22×C53—22no—
138C22×C53—22no—
139C22×C53—22no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—84no—
145C22×C53—84no—
146C22×C53—82no—
147C22×C53—82no—
148C22×C53—82no—
149C22×C53#39211yescharacteristic
150C22×C53#45012yes—
151C22×C53#45311yescharacteristic
152C22×C53#45411yescharacteristic
153C22×C53#45812yes—
154C22×C53#46012yes—
155C22×C53#49211yescharacteristic
156C22×C53#49611yescharacteristic
157C4×C53—22no—
158C4×C53—22no—
159C4×C53—44no—
160C4×C53—82no—
161Dic53—82no—
162Dic53—82no—
163C2×#481—532no—
164#2656D5311yescharacteristic
165C2×#327—532no—
166#1445—531no—
167#1280—531no—
168C2×C53—21no—
169C2×C53—22no—
170C2×C53—21no—
171C2×C53—162no—
172C2×C53#161312yes—
173C2×C53#161612yes—
174C2×C53#200611yescharacteristic
175C2×C53#201411yescharacteristic
176C2×C53#202211yescharacteristic
177C2×#121—22no—
178C2×#121C2×D5312yes—
179C22×C4⋊C4—531no—
180#481—1064no—
181#481—1064no—
182C2×#106—1062no—
183C2×#102—1062no—
184C2×C22⋊C8—532no—
185#399—22no—
186#392—22no—
187C2×#95—531no—
188C2×#95—1062no—
189C2×#95C2×D5311yescharacteristic
190#327—1064no—
191C53#996811yescharacteristic, complemented, Sylow
192C22×D4—22no—
193C22×D4—22no—
194C22×D4C22⋊D5312yes—
195C23×C4—1061no—
196C23×C4—1061no—
197C23×C4C22⋊D5312yes—
198C23×C4C22×D5311yescharacteristic
199C22×C8—1062no—
200#121—22no—
201#121—22no—
202#121—22no—
203#121—22no—
204#121—44no—
205#121—44no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—532no—
208C2×C4⋊C4—1062no—
209C2×C4⋊C4—1062no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4C22⋊D5312yes—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214#106—1068no—
215#102—1068no—
216C22⋊C8—534no—
217C22⋊C8—1062no—
218#95—22no—
219#95—1062no—
220#95—2124no—
221C24—22no—
222C24C2×C22⋊D5311yescharacteristic
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4—82no—
232C2×D4—82no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—1062no—
240C22×C4—1062no—
241C22×C4—2122no—
242C22×C4—2121no—
243C22×C4—2122no—
244C22×C4—2122no—
245C22×C4—2121no—
246C22×C4—2121no—
247C22×C4—2121no—
248C22×C4—2122no—
249C22×C4#1171992314yes—
250C22×C4D4×D5311yescharacteristic
251C22×C4D4×D5312yes—
252C22×C4#1171994711yescharacteristic
253C22×C4C2×C22⋊D5312yes—
254C2×C8—1064no—
255C2×C8—2124no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—44no—
259C4⋊C4—1064no—
260C4⋊C4—2122no—
261C22⋊C4—48no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—44no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—44no—
272C23#1313318611yescharacteristic
273C23#1313318811yescharacteristic
274C23#1313321311yescharacteristic
275C23C2×#1171992312yes—
276C23#1313324312yes—
277D4—44no—
278D4—82no—
279D4—88no—
280D4—82no—
281D4—84no—
282D4—82no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—82no—
294C2×C4—4242no—
295C2×C4—4241no—
296C2×C4—4242no—
297C2×C4—4241no—
298C2×C4—4241no—
299C2×C4—4241no—
300C2×C4—2122no—
301C2×C4—2124no—
302C2×C4—2124no—
303C2×C4—2122no—
304C2×C4#1313318512yes—
305C2×C4#1313318912yes—
306C8—2124no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—82no—
316C22—84no—
317C22—84no—
318C22—82no—
319C22—82no—
320C22#1681053111yescharacteristic
321C22#1681077412yes—
322C22#1681079111yescharacteristic
323C22#1681079611yescharacteristic
324C22#1681081312yes—
325C22#1681082012yes—
326C22#1681099511yescharacteristic
327C22#1681101011yescharacteristic
328C4—22no—
329C4—22no—
330C4—44no—
331C4—82no—
332C4—4242no—
333C4—4242no—
334C2—21no—
335C2—22no—
336C2—21no—
337C2—162no—
338C2#2027373512yes—
339C2#2027375212yes—
340C2#2027575711yescharacteristic
341C2#2027580311yescharacteristic
342C2#2027584411yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#2656C2
socleC23×C53#120
hypercenter#2656D53
hypocenterC53#9968
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2656

Complements

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

NH ≅ G/Ntype
1. C2×#16810939C2semidirect
2. C2×#16810357C2semidirect
3. #20272984C2semidirect
4. #20272558C2semidirect
26. C2×C4⋊Dic53D4semidirect
30. C22⋊#88322D4semidirect
191. C53#9968semidirect

Maximal subgroups

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

subgroupindex
C2×#168109392
C2×#168103572
#202729842
#202725582
C53×#26562
#996853