Finite Group Databasebeta

← #27270771

Subgroups — #27270771

Order 13568.

286 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 51 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#20272639C211yesmaximal, characteristic
2#20272583C211yesmaximal, characteristic
3C53×#2001C211yesmaximal, characteristic
4#16810483—21no—
5C22⋊#11719908—21no—
6C22×Q16×C53C411yescharacteristic, complemented
7C2×C22⋊Q8×C53—21no—
8C2×#103×C53—21no—
9C2×C22⋊C8×C53C2211yescharacteristic
10C53×#340C411yescharacteristic, complemented
11C2×C4⋊Dic53—41no—
12C2×C22⋊Dic53—41no—
13C2×#11719908—41no—
14#13133124—22no—
15C22×Q8×C53—21no—
16C23×C4×C53D411yescharacteristic
17C2×Q16×C53—42no—
18C2×Q16×C53—41no—
19C22×C8×C53C2×C411yescharacteristic
20C22×C8×C53D411yescharacteristic
21C22⋊Q8×C53—42no—
22C22⋊Q8×C53—42no—
23C2×C4⋊C4×C53—21no—
24C2×C4⋊C4×C53—41no—
25C2×C22⋊C4×C53—41no—
26#103×C53—42no—
27C22⋊C8×C53—41no—
28C24×C53D811yescharacteristic
29C2×Q8×C53—22no—
30C2×Q8×C53—41no—
31C2×Q8×C53—41no—
32C2×Q8×C53—41no—
33C2×Q8×C53—41no—
34C2×Q8×C53—41no—
35C22×C4×C53—21no—
36C22×C4×C53—21no—
37C22×C4×C53—41no—
38C22×C4×C53—41no—
39C22×C4×C53—41no—
40C22×C4×C53—41no—
41C22×C4×C53C22⋊C411yescharacteristic
42C22×C4×C53SD1611yescharacteristic
43C22×Dic53—42no—
44Q16×C53—42no—
45Q16×C53—42no—
46C2×C8×C53—21no—
47C2×C8×C53—21no—
48C2×C8×C53—41no—
49C2×C8×C53—41no—
50C4⋊C4×C53—22no—
51C4⋊C4×C53—41no—
52C4⋊C4×C53—44no—
53C22⋊C4×C53—44no—
54C22⋊Dic53—81no—
55C22⋊Dic53—81no—
56C4⋊Dic53—81no—
57C4⋊Dic53—81no—
58#11719908—81no—
59C23×C53—21no—
60C23×C53—21no—
61C23×C53—41no—
62C23×C53—41no—
63C23×C53#10211yescharacteristic
64C23×C53SD3211yescharacteristic
65C23×C53Q3211yescharacteristic
66Q8×C53—42no—
67Q8×C53—42no—
68Q8×C53—81no—
69Q8×C53—81no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—41no—
74C2×C4×C53—41no—
75C2×C4×C53—41no—
76C2×C4×C53—42no—
77C2×C4×C53—42no—
78C2×C4×C53—81no—
79C2×C4×C53—81no—
80C2×C4×C53—81no—
81C2×C4×C53—81no—
82C2×C4×C53—81no—
83C2×C4×C53#9911yescharacteristic
84C2×C4×C53#10411yescharacteristic
85C2×Dic53—82no—
86C2×Dic53—82no—
87C2×Dic53—82no—
88C8×C53—41no—
89C8×C53—41no—
90#7078—531nomaximal, Sylow
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—41no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53—81no—
100C22×C53#32711yescharacteristic
101C22×C53#35811yescharacteristic
102C22×C53#36011yescharacteristic
103C4×C53—21no—
104C4×C53—41no—
105C4×C53—82no—
106C4×C53—82no—
107C4×C53AΣL(1,𝔽16)11yescharacteristic
108C4×C53#35311yescharacteristic
109Dic53—162no—
110#2001D5311yescharacteristic
111#1314—531no—
112#1289—531no—
113C2×C53—21no—
114C2×C53—21no—
115C2×C53—41no—
116C2×C53—41no—
117C2×C53#134011yescharacteristic
118C2×C53#134311yescharacteristic
119C2×C53#134811yescharacteristic
120C22×Q16Dic5311yescharacteristic, complemented
121C2×C22⋊Q8—21no—
122C2×#103—21no—
123C2×C22⋊C8C2×D5311yescharacteristic
124#380—1061no—
125C22⋊C16—1061no—
126#340Dic5311yescharacteristic, complemented
127C53#707811yescharacteristic, complemented, Sylow
128C22×Q8—21no—
129C23×C4C22⋊D5311yescharacteristic
130C2×Q16—41no—
131C2×Q16—42no—
132C22×C8C2×Dic5311yescharacteristic
133C22×C8C22⋊D5311yescharacteristic
134C22⋊Q8—42no—
135C22⋊Q8—42no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—2121no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—2121no—
141C2×C16—2121no—
142#103—42no—
143C22⋊C8—41no—
144#95—1062no—
145C24#1171992211yescharacteristic
146C2×Q8—22no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—41no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—2122no—
159C22×C4#1171992311yescharacteristic
160C22×C4C22⋊Dic5311yescharacteristic
161Q16—42no—
162Q16—42no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—41no—
166C2×C8—41no—
167C4⋊C4—22no—
168C4⋊C4—41no—
169C4⋊C4—44no—
170C4⋊C4—4241no—
171C4⋊C4—4241no—
172C22⋊C4—44no—
173C22⋊C4—4241no—
174C22⋊C4—4241no—
175C16—4241no—
176C23—21no—
177C23—21no—
178C23—41no—
179C23—41no—
180C23#1313313811yescharacteristic
181C23#1313316011yescharacteristic
182C23#1313316311yescharacteristic
183Q8—42no—
184Q8—42no—
185Q8—81no—
186Q8—81no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—4242no—
201C2×C4—4242no—
202C2×C4—4242no—
203C2×C4#1313313211yescharacteristic
204C2×C4#1313314411yescharacteristic
205C8—41no—
206C8—41no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—81no—
216C22#1681035711yescharacteristic
217C22#1681042711yescharacteristic
218C22#1681043311yescharacteristic
219C4—21no—
220C4—41no—
221C4—82no—
222C4—82no—
223C4—8482no—
224C4#1681041011yescharacteristic
225C4#1681041611yescharacteristic
226C2—21no—
227C2—21no—
228C2—41no—
229C2—41no—
230C2#2027270311yescharacteristic
231C2#2027271211yescharacteristic
232C2#2027272711yescharacteristic
subgroupG/H
centerC22#16810357
commutatorC22×C8×C53C2×C4
FrattiniC2×C22⋊C8C2×D53
FittingC53×#2001C2
socleC22×C53#327
prime radicalC24×C53D8
cosocleC2×C22⋊C8×C53C22
hypercenter#2001D53
hypocenterC53#7078
LCSG ▷ C22×C8×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #2001

Complements

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

NH ≅ G/Ntype
6. C22×Q16×C53C4semidirect
10. C53×#340C4semidirect
120. C22×Q16Dic53semidirect
126. #340Dic53semidirect
127. C53#7078semidirect

Maximal subgroups

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

subgroupindex
#202726392
#202725832
C53×#20012
#707853