Finite Group Databasebeta

← #27271617

Subgroups — #27271617

Order 13568.

298 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 43 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#20272840C211yesmaximal, characteristic, complemented
2#20272536C211yesmaximal, characteristic, complemented
3C53×#2006C211yesmaximal, characteristic
4C2×#13133132—21no—
5#16810378—21no—
6C2×AGL(1,ℤ/8)×C53—21no—
7C53×#537C411yescharacteristic, complemented
8C2×#104×C53—21no—
9C53×#390C2211yescharacteristic
10C53×#343C411yescharacteristic, complemented
11C22×#88322—41no—
12C2×C22⋊Dic53—41no—
13#13133132—22no—
14#13133132—41no—
15C2×#41×C53—21no—
16C22×D4×C53D411yescharacteristic
17AGL(1,ℤ/8)×C53—42no—
18AGL(1,ℤ/8)×C53—41no—
19AGL(1,ℤ/8)×C53—41no—
20C2×SD16×C53—41no—
21C2×D8×C53—41no—
22C2×M16×C53—21no—
23C22⋊Q8×C53—41no—
24#121×C53—41no—
25#121×C53—41no—
26C2×C4⋊C4×C53C2×C411yescharacteristic
27C2×C22⋊C4×C53—41no—
28C2×C42×C53D411yescharacteristic
29#104×C53—42no—
30C24×C53—21no—
31#41×C53—41no—
32#41×C53—41no—
33#41×C53—41no—
34#41×C53—41no—
35C2×Q8×C53—21no—
36C2×D4×C53—21no—
37C2×D4×C53—21no—
38C2×D4×C53—41no—
39C2×D4×C53—41no—
40C2×D4×C53—41no—
41C2×D4×C53—41no—
42C2×D4×C53D811yescharacteristic
43C2×D4×C53SD1611yescharacteristic
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—41no—
47C22×C4×C53C22⋊C411yescharacteristic
48C22×Dic53—81no—
49SD16×C53—42no—
50SD16×C53—42no—
51D8×C53—42no—
52D8×C53—42no—
53M16×C53—22no—
54M16×C53—41no—
55C2×C8×C53—41no—
56C4⋊C4×C53—41no—
57C4⋊C4×C53—41no—
58C22⋊C4×C53—42no—
59C22⋊C4×C53—81no—
60C22⋊C4×C53—81no—
61C42×C53—41no—
62C22⋊Dic53—42no—
63C22⋊Dic53—81no—
64C2×#88322—42no—
65C2×#88322—81no—
66C2×#88322—81no—
67C23×C53—21no—
68C23×C53—21no—
69C23×C53—41no—
70C23×C53—41no—
71C23×C53—41no—
72C23×C53—81no—
73C23×C53—81no—
74C23×C53#9911yescharacteristic
75Q8×C53—41no—
76Q8×C53—41no—
77D4×C53—41no—
78D4×C53—41no—
79D4×C53—41no—
80D4×C53—41no—
81D4×C53—81no—
82D4×C53—81no—
83D4×C53—81no—
84D4×C53—161no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—81no—
94C2×C4×C53—81no—
95C2×C4×C53—81no—
96C2×C4×C53#10211yescharacteristic
97C2×C4×C53#10411yescharacteristic
98C2×Dic53—82no—
99C2×Dic53—161no—
100C2×Dic53—161no—
101C8×C53—42no—
102#88322—82no—
103#7408—531nomaximal, Sylow
104C22×C53—21no—
105C22×C53—21no—
106C22×C53—41no—
107C22×C53—41no—
108C22×C53—81no—
109C22×C53—81no—
110C22×C53—81no—
111C22×C53—81no—
112C22×C53—81no—
113C22×C53—81no—
114C22×C53—81no—
115C22×C53—161no—
116C22×C53#32711yescharacteristic
117C22×C53AΣL(1,𝔽16)11yescharacteristic
118C22×C53#35311yescharacteristic
119C4×C53—21no—
120C4×C53—21no—
121C4×C53—41no—
122C4×C53—41no—
123C4×C53—81no—
124C4×C53—81no—
125Dic53—162no—
126#2006D5311yescharacteristic
127#1393—531no—
128#1270—531no—
129C2×C53—21no—
130C2×C53—21no—
131C2×C53—81no—
132C2×C53—81no—
133C2×C53—161no—
134C2×C53#134311yescharacteristic
135C2×C53#140211yescharacteristic
136C2×C53#140411yescharacteristic
137C2×AGL(1,ℤ/8)—21no—
138#537Dic5311yescharacteristic, complemented
139C2×#104—21no—
140C2×#99—1061no—
141#390C2×D5311yescharacteristic
142#343Dic5311yescharacteristic, complemented
143#336—1061no—
144C53#740811yescharacteristic, complemented, Sylow
145C2×#41—21no—
146C22×D4C22⋊D5311yescharacteristic
147AGL(1,ℤ/8)—42no—
148AGL(1,ℤ/8)—41no—
149AGL(1,ℤ/8)—41no—
150C2×SD16—41no—
151C2×D8—41no—
152C2×M16—21no—
153C22×C8—2121no—
154C22⋊Q8—41no—
155#121—41no—
156#121—41no—
157C2×C4⋊C4C2×Dic5311yescharacteristic
158C2×C22⋊C4—41no—
159C2×C22⋊C4—2121no—
160C2×C42C22⋊D5311yescharacteristic
161#104—42no—
162#99—1062no—
163#99—2121no—
164C24—21no—
165#41—41no—
166#41—41no—
167#41—41no—
168#41—41no—
169C2×Q8—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4#1171992211yescharacteristic
177C2×D4#1171992311yescharacteristic
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—41no—
181C22×C4—4241no—
182C22×C4C22⋊Dic5311yescharacteristic
183SD16—42no—
184SD16—42no—
185D8—42no—
186D8—42no—
187M16—22no—
188M16—41no—
189C2×C8—41no—
190C2×C8—4241no—
191C2×C8—4241no—
192C2×C8—2122no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C22⋊C4—42no—
196C22⋊C4—81no—
197C22⋊C4—81no—
198C22⋊C4—4241no—
199C22⋊C4—2122no—
200C42—41no—
201C23—21no—
202C23—21no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—81no—
207C23—81no—
208C23#1313313211yescharacteristic
209Q8—41no—
210Q8—41no—
211D4—41no—
212D4—41no—
213D4—41no—
214D4—41no—
215D4—81no—
216D4—81no—
217D4—81no—
218D4—161no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—8481no—
231C2×C4—8481no—
232C2×C4—4242no—
233C2×C4#1313313811yescharacteristic
234C2×C4#1313314411yescharacteristic
235C8—42no—
236C8—4242no—
237C22—21no—
238C22—21no—
239C22—41no—
240C22—41no—
241C22—81no—
242C22—81no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22—161no—
249C22#1681035711yescharacteristic
250C22#1681041011yescharacteristic
251C22#1681041611yescharacteristic
252C4—21no—
253C4—21no—
254C4—41no—
255C4—41no—
256C4—81no—
257C4—81no—
258C4—8482no—
259C2—21no—
260C2—21no—
261C2—81no—
262C2—81no—
263C2—161no—
264C2#2027271211yescharacteristic
265C2#2027286011yescharacteristic
266C2#2027286611yescharacteristic
subgroupG/H
centerC22#16810410
commutatorC2×C4⋊C4×C53C2×C4
Frattini#390C2×D53
Fitting, prime radicalC53×#2006C2
socleC22×C53AΣL(1,𝔽16)
cosocleC53×#390C22
hypercenter#2006D53
hypocenterC53#7408
LCSG ▷ C2×C4⋊C4×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #2006

Complements

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

NH ≅ G/Ntype
1. #20272840C2semidirect
2. #20272536C2semidirect
7. C53×#537C4semidirect
10. C53×#343C4semidirect
138. #537Dic53semidirect
142. #343Dic53semidirect
144. C53#7408semidirect

Maximal subgroups

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

subgroupindex
#202728402
#202725362
C53×#20062
#740853