Finite Group Databasebeta

← #27271581

Subgroups — #27271581

Order 13568.

296 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 39 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#20272651C211yesmaximal, characteristic, complemented
2#20272600C211yesmaximal, characteristic, complemented
3C53×#2010C211yesmaximal, characteristic
4C4×C22⋊Dic53—21no—
5#16810405—21no—
6C53×#575C411yescharacteristic, complemented
7C53×#525—21no—
8C53×#421—21no—
9C53×#407C2211yescharacteristic
10C53×H(3,ℤ/4)C411yescharacteristic, complemented
11C2×C22⋊Dic53—41no—
12C2×C4×Dic53—41no—
13#13133157—41no—
14#13133124—22no—
15C2×#41×C53—21no—
16C23×C4×C53D411yescharacteristic
17#137×C53—41no—
18AGL(1,ℤ/8)×C53—41no—
19#135×C53—41no—
20C2×M16×C53C2×C411yescharacteristic
21C2×M16×C53D411yescharacteristic
22#123×C53—41no—
23C22⋊Q8×C53—41no—
24#121×C53—41no—
25#120×C53—41no—
26C4×D4×C53—42no—
27#117×C53—21no—
28#104×C53—42no—
29C22⋊C8×C53—41no—
30C24×C53D811yescharacteristic
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—42no—
38C2×D4×C53—41no—
39C22×C4×C53—21no—
40C22×C4×C53—21no—
41C22×C4×C53—41no—
42C22×C4×C53—41no—
43C22×C4×C53—41no—
44C22×C4×C53C22⋊C411yescharacteristic
45C22×C4×C53SD1611yescharacteristic
46C22×Dic53—42no—
47Q16×C53—41no—
48SD16×C53—42no—
49D8×C53—41no—
50M16×C53—41no—
51M16×C53—41no—
52C2×C8×C53—21no—
53C2×C8×C53—21no—
54C4⋊C4×C53—41no—
55C4⋊C4×C53—42no—
56C22⋊C4×C53—42no—
57C22⋊C4×C53—42no—
58C22⋊C4×C53—41no—
59C42×C53—22no—
60C22⋊Dic53—44no—
61C4×Dic53—81no—
62C4×Dic53—81no—
63#11719908—81no—
64C23×C53—21no—
65C23×C53—41no—
66C23×C53—42no—
67C23×C53—41no—
68C23×C53#10211yescharacteristic
69Q8×C53—41no—
70Q8×C53—41no—
71D4×C53—41no—
72D4×C53—41no—
73D4×C53—81no—
74D4×C53—84no—
75D4×C53—81no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—41no—
80C2×C4×C53—41no—
81C2×C4×C53—41no—
82C2×C4×C53—41no—
83C2×C4×C53—42no—
84C2×C4×C53—81no—
85C2×C4×C53—81no—
86C2×C4×C53—81no—
87C2×C4×C53#9911yescharacteristic
88C2×C4×C53#10411yescharacteristic
89C2×Dic53—44no—
90C2×Dic53—82no—
91C2×Dic53—82no—
92C8×C53—41no—
93C8×C53—41no—
94#7396—531nomaximal, Sylow
95C22×C53—22no—
96C22×C53—21no—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53—42no—
100C22×C53—41no—
101C22×C53—81no—
102C22×C53—82no—
103C22×C53—81no—
104C22×C53#32711yescharacteristic
105C4×C53—21no—
106C4×C53—41no—
107C4×C53—82no—
108C4×C53—81no—
109C4×C53AΣL(1,𝔽16)11yescharacteristic
110C4×C53#35311yescharacteristic
111Dic53—84no—
112#2010D5311yescharacteristic
113#1320—531no—
114#1297—531no—
115C2×C53—21no—
116C2×C53—42no—
117C2×C53—41no—
118C2×C53—161no—
119C2×C53#134311yescharacteristic
120#575Dic5311yescharacteristic, complemented
121#525—21no—
122#421—21no—
123#407C2×D5311yescharacteristic
124C4×C22⋊C4—1061no—
125#349—1061no—
126H(3,ℤ/4)Dic5311yescharacteristic, complemented
127C53#739611yescharacteristic, complemented, Sylow
128C2×#41—21no—
129C23×C4C22⋊D5311yescharacteristic
130#137—41no—
131AGL(1,ℤ/8)—41no—
132#135—41no—
133C2×M16C2×Dic5311yescharacteristic
134C2×M16C22⋊D5311yescharacteristic
135#123—41no—
136C22⋊Q8—41no—
137#121—41no—
138#120—41no—
139C4×D4—42no—
140#117—21no—
141C2×C22⋊C4—2121no—
142C2×C42—2121no—
143M32—2121no—
144#104—42no—
145C22⋊C8—41no—
146#95—1062no—
147C24#1171992211yescharacteristic
148#41—41no—
149#41—41no—
150#41—41no—
151#41—41no—
152C2×Q8—21no—
153C2×D4—21no—
154C2×D4—42no—
155C2×D4—41no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—2122no—
162C22×C4#1171992311yescharacteristic
163C22×C4C22⋊Dic5311yescharacteristic
164Q16—41no—
165SD16—42no—
166D8—41no—
167M16—41no—
168M16—41no—
169C2×C8—21no—
170C2×C8—21no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—41no—
176C22⋊C4—2124no—
177C42—22no—
178C42—4241no—
179C42—4241no—
180C16—4241no—
181C23—21no—
182C23—41no—
183C23—42no—
184C23—41no—
185C23#1313313811yescharacteristic
186Q8—41no—
187Q8—41no—
188D4—41no—
189D4—41no—
190D4—81no—
191D4—84no—
192D4—81no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—4242no—
205C2×C4—4242no—
206C2×C4—2124no—
207C2×C4#1313313211yescharacteristic
208C2×C4#1313314411yescharacteristic
209C8—41no—
210C8—41no—
211C22—22no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—42no—
216C22—41no—
217C22—81no—
218C22—82no—
219C22—81no—
220C22#1681035711yescharacteristic
221C4—21no—
222C4—41no—
223C4—82no—
224C4—81no—
225C4—4244no—
226C4#1681041011yescharacteristic
227C4#1681041611yescharacteristic
228C2—21no—
229C2—41no—
230C2—42no—
231C2—161no—
232C2#2027271211yescharacteristic
subgroupG/H
centerC4#16810416
commutatorC2×M16×C53C2×C4
Frattini#407C2×D53
Fitting, prime radicalC53×#2010C2
socleC2×C53#1343
cosocleC53×#407C22
hypercenter#2010D53
hypocenterC53#7396
LCSG ▷ C2×M16×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #407 ◁ #2010

Complements

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

NH ≅ G/Ntype
1. #20272651C2semidirect
2. #20272600C2semidirect
6. C53×#575C4semidirect
10. C53×H(3,ℤ/4)C4semidirect
120. #575Dic53semidirect
126. H(3,ℤ/4)Dic53semidirect
127. C53#7396semidirect

Maximal subgroups

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

subgroupindex
#202726512
#202726002
C53×#20102
#739653