Finite Group Databasebeta

← #27271583

Subgroups — #27271583

Order 13568.

268 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 37 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#20275782C211yesmaximal, characteristic
2#20272651C211yesmaximal, characteristic, complemented
3C53×#1298C211yesmaximal, characteristic, complemented
4#16811423C411yescharacteristic
5#16811134—21no—
6#16810639—21no—
7#16810405—21no—
8#16810380C411yescharacteristic
9C53×#407C2211yescharacteristic
10C2×C22⋊C8×C53—21no—
11C2×#11719945—21no—
12#13133250—41no—
13#13133245—41no—
14#13133239—41no—
15#13133200—41no—
16#13133196—41no—
17#13133191—41no—
18#13133186—41no—
19C4×C22⋊D53—42no—
20#13133176—21no—
21#13133157—41no—
22#13133143—42no—
23C23×C4×C53D411yescharacteristic
24C2×M16×C53C2×C411yescharacteristic
25C2×M16×C53D411yescharacteristic
26C22×C8×C53—41no—
27C22⋊C8×C53—22no—
28C22⋊C8×C53—41no—
29C22⋊C8×C53—41no—
30C24×C53SD1611yescharacteristic
31C22×C4×C53—21no—
32C22×C4×C53—21no—
33C22×C4×C53—41no—
34C22×C4×C53—41no—
35C22×C4×C53C22⋊C411yescharacteristic
36C22×C4×C53D811yescharacteristic
37C2×C22⋊D53—42no—
38C2×C22⋊D53—41no—
39#11719945—41no—
40#11719945—41no—
41#11719945—41no—
42#11719945—41no—
43C2×D212—21no—
44C2×C4×D53—41no—
45C2×Dic106—21no—
46M16×C53—41no—
47M16×C53—41no—
48C2×C8×C53—21no—
49C2×C8×C53—21no—
50C2×C8×C53—42no—
51C2×C8×C53—81no—
52C2×C8×C53—81no—
53C22⋊Dic53—42no—
54C22⋊Dic53—41no—
55#11719921—42no—
56C4⋊Dic53—41no—
57#11719919—42no—
58C4×Dic53—22no—
59Dic212—41no—
60D424—41no—
61#11719913—42no—
62#11719908—81no—
63C23×C53—21no—
64C23×C53—41no—
65C23×C53—42no—
66C23×C53#10211yescharacteristic
67C22×D53—41no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—41no—
72C2×C4×C53—41no—
73C2×C4×C53—41no—
74C2×C4×C53—81no—
75C2×C4×C53#9911yescharacteristic
76C2×C4×C53#10411yescharacteristic
77C22⋊D53—81no—
78C22⋊D53—84no—
79C22⋊D53—81no—
80C2×Dic53—41no—
81C2×Dic53—42no—
82D212—41no—
83D212—41no—
84C4×D53—81no—
85C4×D53—81no—
86Dic106—41no—
87Dic106—41no—
88C8×C53—41no—
89C8×C53—41no—
90C8×C53—82no—
91#7397—531nomaximal, Sylow
92C22×C53—22no—
93C22×C53—21no—
94C22×C53—41no—
95C22×C53—41no—
96C22×C53—42no—
97C22×C53—41no—
98C22×C53—81no—
99C22×C53#32711yescharacteristic
100C2×D53—82no—
101C2×D53—81no—
102C4×C53—21no—
103C4×C53—41no—
104C4×C53AΣL(1,𝔽16)11yescharacteristic
105C4×C53#35311yescharacteristic
106Dic53—82no—
107Dic53—81no—
108#2010—531no—
109#1320—531no—
110#1298D5311yescharacteristic, complemented
111C2×C53—21no—
112C2×C53—41no—
113C2×C53—42no—
114C2×C53#134311yescharacteristic
115D53—161no—
116#575—531no—
117#525—1061no—
118#421—1061no—
119#407C2×D5311yescharacteristic
120C2×C22⋊C8—21no—
121#349—1061no—
122H(3,ℤ/4)—531no—
123C53#739711yescharacteristic, complemented, Sylow
124C2×#41—1061no—
125C23×C4C22⋊D5311yescharacteristic
126#137—2121no—
127AGL(1,ℤ/8)—2121no—
128#135—2121no—
129C2×M16C4×D5311yescharacteristic
130C2×M16D21211yescharacteristic
131C22×C8—41no—
132#123—2121no—
133C22⋊Q8—2121no—
134#121—2121no—
135#120—2121no—
136C4×D4—2122no—
137#117—1061no—
138M32—2121no—
139#104—2122no—
140C22⋊C8—22no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C24#1171992411yescharacteristic
144#41—2121no—
145#41—2121no—
146#41—2121no—
147#41—2121no—
148C2×Q8—1061no—
149C2×D4—1061no—
150C2×D4—2122no—
151C2×D4—2121no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—2121no—
157C22×C4#1171992111yescharacteristic
158C22×C4#1171992211yescharacteristic
159Q16—2121no—
160SD16—2122no—
161D8—2121no—
162M16—41no—
163M16—41no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—42no—
167C2×C8—81no—
168C2×C8—81no—
169C4⋊C4—2121no—
170C4⋊C4—2122no—
171C22⋊C4—2122no—
172C22⋊C4—2122no—
173C22⋊C4—2121no—
174C42—1062no—
175C16—4241no—
176C23—21no—
177C23—41no—
178C23—42no—
179C23—2121no—
180C23#1313313711yescharacteristic
181Q8—2121no—
182Q8—2121no—
183D4—4241no—
184D4—4244no—
185D4—4241no—
186D4—2121no—
187D4—2121no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—81no—
195C2×C4—4241no—
196C2×C4—4241no—
197C2×C4—2122no—
198C2×C4—2121no—
199C2×C4#1313313111yescharacteristic
200C2×C4#1313314511yescharacteristic
201C8—41no—
202C8—41no—
203C8—82no—
204C22—22no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—42no—
209C22—41no—
210C22—81no—
211C22—4242no—
212C22—4241no—
213C22#1681035611yescharacteristic
214C4—21no—
215C4—41no—
216C4—4242no—
217C4—4241no—
218C4#1681040911yescharacteristic
219C4#1681041511yescharacteristic
220C2—21no—
221C2—41no—
222C2—42no—
223C2—8481no—
224C2#2027271111yescharacteristic
subgroupG/H
centerC4#16810415
commutatorC2×M16×C53C2×C4
Frattini#407C2×D53
FittingC53×#1298C2
socleC2×C53#1343
prime radical#20275782C2
cosocleC53×#407C22
hypercenter#1298D53
hypocenterC53#7397
LCSG ▷ C2×M16×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #407 ◁ #1298

Complements

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

NH ≅ G/Ntype
2. #20272651C2semidirect
3. C53×#1298C2semidirect
110. #1298D53semidirect
123. C53#7397semidirect

Maximal subgroups

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

subgroupindex
#202757822
#202726512
C53×#12982
#739753