Finite Group Databasebeta

← #27271645

Subgroups — #27271645

Order 13568.

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

normal subgroups only

This group has 228 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#20275939C211yesmaximal, characteristic
2#20272598C211yesmaximal, characteristic, complemented
3C53×#1325C211yesmaximal, characteristic, complemented
4#16811204C411yescharacteristic
5C2×C4⋊#88322—21no—
6C2×#13133145—21no—
7#16810622—21no—
8#16810392C411yescharacteristic
9C53×#384C2211yescharacteristic
10C53×#324—21no—
11C2×#11719945—21no—
12C2×#11719917—21no—
13C22×#88322—41no—
14#13133197—41no—
15#13133192—41no—
16#13133191—41no—
17C4⋊#88322—22no—
18C4⋊#88322—41no—
19#13133145—41no—
20#13133145—41no—
21#13133140—41no—
22#13133137—41no—
23C2×C4⋊C4×C53—41no—
24C2×C4⋊C4×C53C2×C411yescharacteristic
25C2×C4⋊C4×C53D411yescharacteristic
26C2×C42×C53D411yescharacteristic
27C22⋊C8×C53—41no—
28C22×C4×C53—21no—
29C22×C4×C53—21no—
30C22×C4×C53—21no—
31C22×C4×C53C22⋊C411yescharacteristic
32C2×C22⋊D53—41no—
33#11719945—41no—
34#11719945—41no—
35#11719945—41no—
36#11719945—41no—
37C2×D212—21no—
38C2×D212—41no—
39C2×C4×D53—41no—
40C2×Dic106—21no—
41C2×C8×C53—81no—
42C4⋊C4×C53—41no—
43C4⋊C4×C53—41no—
44C4⋊C4×C53—81no—
45C4⋊C4×C53—81no—
46C42×C53—21no—
47C42×C53D811yescharacteristic
48C42×C53SD1611yescharacteristic
49#11719921—42no—
50C4⋊Dic53—41no—
51#11719917—41no—
52#11719917—41no—
53C2×#88322—41no—
54C2×#88322—42no—
55C2×#88322—81no—
56C2×#88322—81no—
57C23×C53#9911yescharacteristic
58C22×D53—41no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—41no—
68C2×C4×C53—41no—
69C2×C4×C53—81no—
70C2×C4×C53—81no—
71C2×C4×C53#10211yescharacteristic
72C2×C4×C53#10411yescharacteristic
73C22⋊D53—81no—
74C22⋊D53—81no—
75C2×Dic53—41no—
76D212—41no—
77D212—41no—
78D212—161no—
79C4×D53—81no—
80C4×D53—81no—
81Dic106—41no—
82Dic106—41no—
83C8×C53—161no—
84#88322—82no—
85#88322—81no—
86#7417—531nomaximal, Sylow
87C22×C53—21no—
88C22×C53—21no—
89C22×C53#32711yescharacteristic
90C22×C53AΣL(1,𝔽16)11yescharacteristic
91C22×C53#35511yescharacteristic
92C2×D53—81no—
93C2×D53—81no—
94C2×D53—81no—
95C4×C53—21no—
96C4×C53—21no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—21no—
101C4×C53—81no—
102C4×C53—81no—
103Dic53—81no—
104#2039—531no—
105#1325D5311yescharacteristic, complemented
106#1296—531no—
107C2×C53—21no—
108C2×C53—21no—
109C2×C53#134511yescharacteristic
110C2×C53#140211yescharacteristic
111C2×C53#140511yescharacteristic
112D53—161no—
113#537—531no—
114C2×C4⋊C8—1061no—
115C2×#104—1061no—
116#417—1061no—
117#384C2×D5311yescharacteristic
118#343—531no—
119#324—21no—
120C53#741711yescharacteristic, complemented, Sylow
121C2×#41—1061no—
122C2×M16—1061no—
123C22×C8—2121no—
124C22⋊Q8—2121no—
125#121—2121no—
126#121—2121no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4C4×D5311yescharacteristic
129C2×C4⋊C4C22⋊D5311yescharacteristic
130C2×C42D21211yescharacteristic
131C4⋊C8—1062no—
132C4⋊C8—2121no—
133#104—2121no—
134#104—2121no—
135#103—2121no—
136#102—2121no—
137C22⋊C8—41no—
138#41—2121no—
139#41—2121no—
140#41—2121no—
141#41—2121no—
142C2×Q8—1061no—
143C2×D4—1061no—
144C2×D4—2121no—
145C2×D4—2121no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—2121no—
150C22×C4#1171992111yescharacteristic
151M16—2121no—
152M16—2121no—
153C2×C8—81no—
154C2×C8—4241no—
155C2×C8—4241no—
156C2×C8—2121no—
157C2×C8—2122no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—81no—
161C4⋊C4—81no—
162C4⋊C4—2121no—
163C22⋊C4—2122no—
164C42—21no—
165C42#1171991311yescharacteristic
166C42D42411yescharacteristic
167C23—2121no—
168C23#1313313111yescharacteristic
169Q8—2121no—
170Q8—2121no—
171D4—8481no—
172D4—4241no—
173D4—4241no—
174D4—2121no—
175D4—2121no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—4241no—
189C2×C4—4241no—
190C2×C4—2121no—
191C2×C4#1313313911yescharacteristic
192C2×C4#1313314311yescharacteristic
193C8—161no—
194C8—4242no—
195C8—4241no—
196C22—21no—
197C22—21no—
198C22—4241no—
199C22—4241no—
200C22—4241no—
201C22#1681035811yescharacteristic
202C22#1681040911yescharacteristic
203C22#1681041911yescharacteristic
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—21no—
210C4—81no—
211C4—81no—
212C4—4241no—
213C2—21no—
214C2—21no—
215C2—8481no—
216C2#2027271911yescharacteristic
217C2#2027285911yescharacteristic
218C2#2027286811yescharacteristic
subgroupG/H
centerC22#16810409
commutatorC2×C4⋊C4×C53C2×C4
Frattini#384C2×D53
FittingC53×#1325C2
socleC22×C53AΣL(1,𝔽16)
prime radical#16811204C4
cosocleC53×#384C22
hypercenter#1325D53
hypocenterC53#7417
LCSG ▷ C2×C4⋊C4×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #384 ◁ #1325

Complements

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

NH ≅ G/Ntype
2. #20272598C2semidirect
3. C53×#1325C2semidirect
105. #1325D53semidirect
120. C53#7417semidirect

Maximal subgroups

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

subgroupindex
#202759392
#202725982
C53×#13252
#741753