Finite Group Databasebeta

← #27296867

Subgroups — #27296867

Order 13568.

460 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 63 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#20275876C211yesmaximal, characteristic, complemented
2#20273654C212yesmaximal, complemented
3#20272718C212yesmaximal, complemented
4#20272649C211yesmaximal, characteristic, complemented
5C53×#2733C211yesmaximal, characteristic
6#16811254C2211yescharacteristic
7#16810774—22no—
8#16810632—21no—
9#16810392C2211yescharacteristic
10#16810383—21no—
11#16810359—22no—
12#16810349C2212yes—
13C2×#124×C53—21no—
14C53×#394C2212yes—
15C53×#392—22no—
16C53×#391C2211yescharacteristic
17C53×#390—21no—
18C53×#388—22no—
19C2×#11719923—42no—
20C2×#11719917—21no—
21#13133216—41no—
22#13133208—41no—
23#13133196—22no—
24#13133176—21no—
25#13133141—41no—
26#13133141—41no—
27#13133138—42no—
28C22⋊#88322—22no—
29C22×Q8×C53D411yescharacteristic
30C22×D4×C53—21no—
31C22×D4×C53—21no—
32C22×D4×C53D412yes—
33#124×C53—42no—
34#124×C53—42no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53C2311yescharacteristic
37C2×C22⋊C4×C53—22no—
38C2×C22⋊C4×C53—22no—
39C2×C22⋊C4×C53—22no—
40C2×C22⋊C4×C53—22no—
41C2×C42×C53—21no—
42C2×C42×C53D411yescharacteristic
43#95×C53—22no—
44#95×C53D412yes—
45C24×C53—22no—
46C2×Q8×C53—21no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53SD1612yescomplemented
50C2×D4×C53—42no—
51C2×D4×C53—42no—
52C2×D4×C53—42no—
53C2×D4×C53—82no—
54C2×D4×C53—81no—
55C2×D4×C53—81no—
56C2×D4×C53—81no—
57C2×D4×C53—81no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—22no—
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D412yes—
63C2×Dic106—22no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—81no—
66C22⋊C4×C53—44no—
67C22⋊C4×C53—44no—
68C22⋊C4×C53—42no—
69C22⋊C4×C53—42no—
70C22⋊C4×C53—82no—
71C42×C53—42no—
72C42×C53—41no—
73C22⋊Dic53—41no—
74#11719923—84no—
75C4⋊Dic53—22no—
76C4⋊Dic53—21no—
77C4⋊Dic53—21no—
78#11719919—42no—
79C4×Dic53—41no—
80#11719917—41no—
81#11719917—41no—
82C2×#88322—42no—
83C2×#88322—41no—
84C23×C53—42no—
85C23×C53—42no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53#12011yescharacteristic
92Q8×C53—42no—
93Q8×C53—81no—
94D4×C53—44no—
95D4×C53—82no—
96D4×C53—84no—
97D4×C53—84no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53C2×SD1611yescharacteristic
112C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
113C2×Dic53—41no—
114C2×Dic53—41no—
115Dic106—82no—
116#88322—81no—
117#88322—82no—
118#12212—531nomaximal, Sylow
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—82no—
122C22×C53—82no—
123C22×C53—82no—
124C22×C53—82no—
125C22×C53—82no—
126C22×C53—82no—
127C22×C53—82no—
128C22×C53#45011yescharacteristic
129C22×C53#45511yescharacteristic
130C22×C53B(4,𝔽2)11yescharacteristic
131C4×C53—21no—
132C4×C53—21no—
133C4×C53—42no—
134C4×C53—42no—
135C4×C53—42no—
136C4×C53—82no—
137Dic53—81no—
138Dic53—81no—
139#2733D5311yescharacteristic
140#2028—531no—
141#1599—532no—
142#1345—532no—
143#1319—531no—
144C2×C53—21no—
145C2×C53—21no—
146C2×C53—162no—
147C2×C53#160111yescharacteristic
148C2×C53#219911yescharacteristic
149C2×C53#220411yescharacteristic
150#544—531no—
151C2×#124—21no—
152#450—1062no—
153#419—1061no—
154#394C2×D5312yes—
155#392—22no—
156#391C2×D5311yescharacteristic
157#390—21no—
158#388—22no—
159#343—531no—
160#339—1061no—
161#328—1062no—
162#324—532no—
163C53#1221211yescharacteristic, complemented, Sylow
164C22×Q8C22⋊D5311yescharacteristic
165C22×D4—21no—
166C22×D4—21no—
167C22×D4C22⋊D5312yes—
168C2×SD16—2122no—
169C2×M16—1061no—
170C4⋊Q8—2121no—
171#125—2121no—
172#124—42no—
173#124—42no—
174C22⋊Q8—1062no—
175#117—1061no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4C22×D5311yescharacteristic
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C42—21no—
183C2×C42C22⋊D5311yescharacteristic
184#103—2121no—
185#103—2121no—
186#102—2122no—
187C22⋊C8—1062no—
188#95—22no—
189#95C22⋊D5312yes—
190C24—22no—
191C2×Q8—21no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—1062no—
195C2×Q8#1171992312yescomplemented
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—82no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4—81no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4C2×C22⋊D5311yescharacteristic
208C22×C4C2×C22⋊D5312yes—
209SD16—4244no—
210M16—2121no—
211M16—2121no—
212C2×C8—2122no—
213C2×C8—2121no—
214C4⋊C4—41no—
215C4⋊C4—81no—
216C4⋊C4—1062no—
217C4⋊C4—1061no—
218C4⋊C4—1061no—
219C4⋊C4—2122no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—82no—
225C22⋊C4—2121no—
226C42—41no—
227C42—42no—
228C42—2121no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23#1313318611yescharacteristic
237Q8—42no—
238Q8—81no—
239Q8—4242no—
240D4—44no—
241D4—82no—
242D4—84no—
243D4—84no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—2121no—
258C2×C4—2121no—
259C2×C4C2×#1171992311yescharacteristic
260C2×C4#1313324311yescharacteristic
261C8—4241no—
262C8—4242no—
263C22—21no—
264C22—21no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22—82no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22#1681077411yescharacteristic
273C22#1681080111yescharacteristic
274C22#1681081011yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—42no—
280C4—82no—
281C4—4241no—
282C4—4241no—
283C2—21no—
284C2—21no—
285C2—162no—
286C2#2027366711yescharacteristic
287C2#2027671611yescharacteristic
288C2#2027674111yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
Fitting, prime radicalC53×#2733C2
socleC22×C53B(4,𝔽2)
hypercenter#2733D53
hypocenterC53#12212
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2733

Complements

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

NH ≅ G/Ntype
1. #20275876C2semidirect
2. #20273654C2semidirect
3. #20272718C2semidirect
4. #20272649C2semidirect
49. C2×Q8×C53SD16semidirect
163. C53#12212semidirect
195. C2×Q8#11719923semidirect

Maximal subgroups

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

subgroupindex
#202758762
#202736542
#202727182
#202726492
C53×#27332
#1221253