Finite Group Databasebeta

← #27270790

Subgroups — #27270790

Order 13568.

316 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 45 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#20275713C211yesmaximal, characteristic
2#20272592C211yesmaximal, characteristic, complemented
3C53×#1314C211yesmaximal, characteristic, complemented
4C22×D424C411yescharacteristic
5C2×#13133191—21no—
6C2×#13133139—21no—
7#16810587—21no—
8#16810386C411yescharacteristic
9C2×C22⋊C8×C53C2211yescharacteristic
10C53×C22⋊C16—21no—
11C22×D212—21no—
12C22×C22⋊D53—41no—
13C2×D424—42no—
14C2×D424—41no—
15C2×#11719917—41no—
16#13133191—42no—
17#13133191—42no—
18C2×C4⋊Dic53—21no—
19C2×#11719921—41no—
20#13133139—42no—
21C22⋊#88322—22no—
22C22⋊#88322—41no—
23C23×C4×C53D411yescharacteristic
24C22×C8×C53C2×C411yescharacteristic
25C22×C8×C53D411yescharacteristic
26C2×C16×C53—41no—
27C22⋊C8×C53—41no—
28C24×C53SD1611yescharacteristic
29C23×D53—41no—
30C22×C4×C53—21no—
31C22×C4×C53—21no—
32C22×C4×C53—41no—
33C22×C4×C53—41no—
34C22×C4×C53C22⋊C411yescharacteristic
35C22×C4×C53D811yescharacteristic
36C2×C22⋊D53—44no—
37C2×C22⋊D53—82no—
38C2×C22⋊D53—82no—
39C22×Dic53—41no—
40C2×D212—22no—
41C2×D212—41no—
42C2×D212—41no—
43C2×D212—41no—
44C2×D212—41no—
45C2×D212—41no—
46C2×C8×C53—21no—
47C2×C8×C53—21no—
48C2×C8×C53—41no—
49C2×C8×C53—41no—
50#11719921—44no—
51C4⋊Dic53—22no—
52C4⋊Dic53—41no—
53#11719917—81no—
54#11719917—81no—
55C2×#88322—42no—
56D424—42no—
57D424—42no—
58C16×C53—81no—
59C23×C53—21no—
60C23×C53—21no—
61C23×C53—21no—
62C23×C53—41no—
63C23×C53—41no—
64C23×C53#10211yescharacteristic
65C22×D53—42no—
66C22×D53—81no—
67C22×D53—81no—
68C22×D53—81no—
69C22×D53—81no—
70C22×D53—81no—
71C22×D53—81no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—41no—
76C2×C4×C53—41no—
77C2×C4×C53—41no—
78C2×C4×C53—81no—
79C2×C4×C53#9911yescharacteristic
80C2×C4×C53#10411yescharacteristic
81C22⋊D53—88no—
82C2×Dic53—42no—
83C2×Dic53—81no—
84C2×Dic53—81no—
85D212—42no—
86D212—42no—
87D212—81no—
88D212—81no—
89C8×C53—41no—
90C8×C53—41no—
91#88322—82no—
92#7084—531nomaximal, Sylow
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—42no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53—81no—
100C22×C53#32711yescharacteristic
101C22×C53#35911yescharacteristic
102C22×C53#36111yescharacteristic
103C2×D53—82no—
104C2×D53—82no—
105C2×D53—82no—
106C2×D53—161no—
107C2×D53—161no—
108C2×D53—161no—
109C2×D53—161no—
110C4×C53—21no—
111C4×C53—41no—
112C4×C53AΣL(1,𝔽16)11yescharacteristic
113C4×C53#35311yescharacteristic
114Dic53—82no—
115#1999—531no—
116#1314D5311yescharacteristic, complemented
117#1293—531no—
118C2×C53—21no—
119C2×C53—21no—
120C2×C53—81no—
121C2×C53#134111yescharacteristic
122C2×C53#134311yescharacteristic
123C2×C53#134911yescharacteristic
124D53—162no—
125C22×D8—531no—
126C2×#121—1061no—
127C2×#102—1061no—
128#407—1061no—
129C2×C22⋊C8C2×D5311yescharacteristic
130C22⋊C16—21no—
131#340—531no—
132C53#708411yescharacteristic, complemented, Sylow
133C22×D4—1061no—
134C22×D4—2121no—
135C23×C4D21211yescharacteristic
136C2×D8—2121no—
137C2×D8—2122no—
138C2×M16—2121no—
139C22×C8C4×D5311yescharacteristic
140C22×C8C22⋊D5311yescharacteristic
141#121—2122no—
142#121—2122no—
143C2×C4⋊C4—1061no—
144C2×C22⋊C4—2121no—
145C2×C16—41no—
146#102—2122no—
147C22⋊C8—41no—
148C22⋊C8—1062no—
149C22⋊C8—2121no—
150C24—2121no—
151C24#1171991311yescharacteristic
152C2×D4—1062no—
153C2×D4—4242no—
154C2×D4—4242no—
155C2×D4—2121no—
156C2×D4—2121no—
157C2×D4—2121no—
158C2×D4—2124no—
159C2×D4—2121no—
160C2×D4—2121no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—2121no—
166C22×C4D42411yescharacteristic
167C22×C4#1171992111yescharacteristic
168D8—2122no—
169D8—2122no—
170M16—4241no—
171M16—4241no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—2122no—
177C4⋊C4—1062no—
178C4⋊C4—2121no—
179C22⋊C4—2124no—
180C16—81no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—41no—
185C23—41no—
186C23—4241no—
187C23—4241no—
188C23—4241no—
189C23—4241no—
190C23—4241no—
191C23—4241no—
192C23—2122no—
193C23#1313313911yescharacteristic
194D4—4241no—
195D4—4248no—
196D4—4241no—
197D4—2122no—
198D4—2122no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—81no—
206C2×C4—4241no—
207C2×C4—4241no—
208C2×C4—2122no—
209C2×C4#1313313111yescharacteristic
210C2×C4#1313314311yescharacteristic
211C8—41no—
212C8—41no—
213C8—4242no—
214C22—21no—
215C22—21no—
216C22—42no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—8481no—
222C22—8481no—
223C22—8481no—
224C22—8481no—
225C22—4242no—
226C22—4242no—
227C22—4242no—
228C22#1681035811yescharacteristic
229C22#1681043111yescharacteristic
230C22#1681043711yescharacteristic
231C4—21no—
232C4—41no—
233C4—4242no—
234C4#1681041111yescharacteristic
235C4#1681041511yescharacteristic
236C2—21no—
237C2—21no—
238C2—81no—
239C2—8482no—
240C2#2027270711yescharacteristic
241C2#2027271311yescharacteristic
242C2#2027273111yescharacteristic
subgroupG/H
centerC22#16810358
commutatorC22×C8×C53C2×C4
FrattiniC2×C22⋊C8C2×D53
FittingC53×#1314C2
socleC22×C53#327
prime radical#20275713C2
cosocleC2×C22⋊C8×C53C22
hypercenter#1314D53
hypocenterC53#7084
LCSG ▷ C22×C8×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #20272592C2semidirect
3. C53×#1314C2semidirect
116. #1314D53semidirect
132. C53#7084semidirect

Maximal subgroups

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

subgroupindex
#202757132
#202725922
C53×#13142
#708453