Finite Group Databasebeta

← #27271601

Subgroups — #27271601

Order 13568.

282 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 41 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#20275797C211yesmaximal, characteristic
2#20272647C211yesmaximal, characteristic
3C53×#1393C211yesmaximal, characteristic
4C22×#11719923—21no—
5C2×#13133217C411yescharacteristic, complemented
6C2×#13133140—21no—
7#16810378C411yescharacteristic, complemented
8#16810349—21no—
9C2×#99×C53—21no—
10C53×#390C2211yescharacteristic
11C22×Dic106—21no—
12C2×#11719923—42no—
13C2×#11719923—42no—
14C2×#11719923—41no—
15C2×#11719923—41no—
16C22×#88322—21no—
17#13133217—42no—
18C2×C4⋊Dic53—41no—
19#13133140—42no—
20C22⋊#88322—41no—
21C22×D4×C53D411yescharacteristic
22C2×C4⋊C4×C53D411yescharacteristic
23C2×C22⋊C4×C53—41no—
24C2×C22⋊C4×C53—41no—
25C2×C42×C53C2×C411yescharacteristic
26#99×C53—22no—
27#99×C53—41no—
28C24×C53—21no—
29C2×D4×C53—21no—
30C2×D4×C53—41no—
31C2×D4×C53—41no—
32C2×D4×C53D811yescharacteristic
33C2×D4×C53SD1611yescharacteristic
34C22×C4×C53—21no—
35C22×C4×C53—21no—
36C22×C4×C53—81no—
37C22×C4×C53C22⋊C411yescharacteristic
38C22×Dic53—41no—
39C2×Dic106—22no—
40C2×Dic106—41no—
41C2×Dic106—41no—
42C2×Dic106—41no—
43C2×Dic106—41no—
44C2×Dic106—41no—
45C4⋊C4×C53—41no—
46C22⋊C4×C53—42no—
47C22⋊C4×C53—82no—
48C22⋊C4×C53—81no—
49C42×C53—41no—
50#11719923—44no—
51#11719923—44no—
52C4⋊Dic53—42no—
53C4⋊Dic53—42no—
54C2×#88322—22no—
55C2×#88322—41no—
56C2×#88322—41no—
57C2×#88322—81no—
58C23×C53—21no—
59C23×C53—21no—
60C23×C53—41no—
61C23×C53—41no—
62C23×C53—81no—
63C23×C53—81no—
64C23×C53#9911yescharacteristic
65D4×C53—41no—
66D4×C53—41no—
67D4×C53—81no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—22no—
71C2×C4×C53—41no—
72C2×C4×C53—41no—
73C2×C4×C53—41no—
74C2×C4×C53—81no—
75C2×C4×C53—82no—
76C2×C4×C53—161no—
77C2×C4×C53—161no—
78C2×C4×C53#10211yescharacteristic
79C2×C4×C53#10411yescharacteristic
80C2×Dic53—42no—
81C2×Dic53—81no—
82C2×Dic53—81no—
83Dic106—42no—
84Dic106—42no—
85Dic106—81no—
86Dic106—81no—
87#88322—42no—
88#88322—161no—
89#7403—531nomaximal, Sylow
90C22×C53—21no—
91C22×C53—21no—
92C22×C53—41no—
93C22×C53—41no—
94C22×C53—81no—
95C22×C53—81no—
96C22×C53—81no—
97C22×C53—81no—
98C22×C53—161no—
99C22×C53#32711yescharacteristic
100C22×C53AΣL(1,𝔽16)11yescharacteristic
101C22×C53#35311yescharacteristic
102C4×C53—21no—
103C4×C53—21no—
104C4×C53—42no—
105C4×C53—81no—
106C4×C53—162no—
107Dic53—82no—
108#2013—531no—
109#1393D5311yescharacteristic
110#1318—531no—
111C2×C53—21no—
112C2×C53—21no—
113C2×C53—81no—
114C2×C53—81no—
115C2×C53#134311yescharacteristic
116C2×C53#140711yescharacteristic
117C2×C53#140911yescharacteristic
118C22×SD16—1061no—
119C2×C4⋊Q8—531no—
120C2×#103—1061no—
121C2×#99—21no—
122#390C2×D5311yescharacteristic
123#336—531no—
124#324—1061no—
125C53#740311yescharacteristic, complemented, Sylow
126C22×Q8—1061no—
127C22×D4C22⋊D5311yescharacteristic
128C2×SD16—2122no—
129C2×SD16—2122no—
130C2×SD16—2121no—
131C2×SD16—2121no—
132C22×C8—1061no—
133C4⋊Q8—2122no—
134C2×C4⋊C4—2121no—
135C2×C4⋊C4D21211yescharacteristic
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C42C4×D5311yescharacteristic
139#103—2122no—
140#99—22no—
141#99—41no—
142C22⋊C8—2121no—
143C24—21no—
144C2×Q8—1062no—
145C2×Q8—2121no—
146C2×Q8—2121no—
147C2×Q8—2121no—
148C2×Q8—2121no—
149C2×Q8—2121no—
150C2×D4—21no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4#1171992211yescharacteristic
154C2×D4#1171992411yescharacteristic
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—81no—
158C22×C4—2121no—
159C22×C4#1171992111yescharacteristic
160SD16—2124no—
161SD16—2124no—
162C2×C8—1062no—
163C2×C8—4241no—
164C2×C8—2121no—
165C2×C8—2121no—
166C4⋊C4—41no—
167C4⋊C4—2122no—
168C4⋊C4—2122no—
169C22⋊C4—42no—
170C22⋊C4—82no—
171C22⋊C4—81no—
172C42—41no—
173C23—21no—
174C23—21no—
175C23—41no—
176C23—41no—
177C23—81no—
178C23—81no—
179C23#1313313111yescharacteristic
180Q8—4241no—
181Q8—4241no—
182Q8—2122no—
183Q8—2122no—
184D4—41no—
185D4—41no—
186D4—81no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—81no—
194C2×C4—82no—
195C2×C4—161no—
196C2×C4—161no—
197C2×C4—4241no—
198C2×C4—4241no—
199C2×C4—2122no—
200C2×C4#1313313711yescharacteristic
201C2×C4#1313314511yescharacteristic
202C8—8481no—
203C8—2122no—
204C22—21no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—81no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22—161no—
213C22#1681035611yescharacteristic
214C22#1681040911yescharacteristic
215C22#1681041511yescharacteristic
216C4—21no—
217C4—21no—
218C4—42no—
219C4—81no—
220C4—162no—
221C4—4242no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—81no—
226C2#2027271111yescharacteristic
227C2#2027287411yescharacteristic
228C2#2027287911yescharacteristic
subgroupG/H
centerC22#16810415
commutatorC2×C42×C53C2×C4
Frattini#390C2×D53
FittingC53×#1393C2
socleC22×C53#353
prime radicalC22×D4×C53D4
cosocleC53×#390C22
hypercenter#1393D53
hypocenterC53#7403
LCSG ▷ C2×C42×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #1393

Complements

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

NH ≅ G/Ntype
5. C2×#13133217C4semidirect
7. #16810378C4semidirect
125. C53#7403semidirect

Maximal subgroups

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

subgroupindex
#202757972
#202726472
C53×#13932
#740353