Finite Group Databasebeta

← #27271621

Subgroups — #27271621

Order 13568.

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

normal subgroups only

This group has 268 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#20275761C211yesmaximal, characteristic
2#20272842C211yesmaximal, characteristic, complemented
3C53×#1272C211yesmaximal, characteristic, complemented
4C2×#13133248—21no—
5#16811204C411yescharacteristic
6C2×#13133145—21no—
7C2×#13133136—21no—
8#16810392C411yescharacteristic
9C53×#391C2211yescharacteristic
10C53×#336—21no—
11C2×#11719945—21no—
12#13133248—42no—
13#13133248—41no—
14#13133248—41no—
15C2×#11719925—41no—
16C2×#11719924—41no—
17C2×#11719917—21no—
18C2×#11719917—41no—
19#13133197—41no—
20#13133192—41no—
21#13133191—41no—
22#13133145—42no—
23#13133136—22no—
24#13133136—41no—
25C22×Q8×C53D411yescharacteristic
26C22×C8×C53—41no—
27C2×C4⋊C4×C53C2×C411yescharacteristic
28C2×C42×C53D411yescharacteristic
29#95×C53—41no—
30C2×Q8×C53—21no—
31C2×Q8×C53—41no—
32C2×Q8×C53—41no—
33C2×Q8×C53D811yescharacteristic
34C2×Q8×C53SD1611yescharacteristic
35C22×C4×C53—21no—
36C22×C4×C53—21no—
37C22×C4×C53—21no—
38C22×C4×C53C22⋊C411yescharacteristic
39C2×C22⋊D53—41no—
40#11719945—41no—
41#11719945—41no—
42#11719945—41no—
43#11719945—41no—
44C2×D212—21no—
45C2×D212—41no—
46C2×C4×D53—41no—
47C2×Dic106—21no—
48C2×C8×C53—42no—
49C2×C8×C53—81no—
50C2×C8×C53—81no—
51C4⋊C4×C53—41no—
52C42×C53—41no—
53#11719925—42no—
54#11719925—42no—
55#11719924—42no—
56#11719924—42no—
57#11719921—42no—
58C4⋊Dic53—41no—
59#11719917—22no—
60#11719917—41no—
61#11719917—42no—
62#11719917—81no—
63C2×#88322—41no—
64C2×#88322—81no—
65C23×C53#9911yescharacteristic
66C22×D53—41no—
67Q8×C53—41no—
68Q8×C53—41no—
69Q8×C53—81no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—41no—
77C2×C4×C53—41no—
78C2×C4×C53—41no—
79C2×C4×C53—41no—
80C2×C4×C53—41no—
81C2×C4×C53—81no—
82C2×C4×C53#10211yescharacteristic
83C2×C4×C53#10411yescharacteristic
84C22⋊D53—81no—
85C22⋊D53—81no—
86C2×Dic53—41no—
87D212—41no—
88D212—41no—
89D212—161no—
90C4×D53—81no—
91C4×D53—81no—
92Dic106—41no—
93Dic106—41no—
94C8×C53—82no—
95#88322—42no—
96#88322—82no—
97#7409—531nomaximal, Sylow
98C22×C53—21no—
99C22×C53—21no—
100C22×C53#32711yescharacteristic
101C22×C53AΣL(1,𝔽16)11yescharacteristic
102C22×C53#35311yescharacteristic
103C2×D53—81no—
104C2×D53—81no—
105C2×D53—81no—
106C4×C53—21no—
107C4×C53—21no—
108C4×C53—41no—
109C4×C53—41no—
110C4×C53—41no—
111C4×C53—41no—
112C4×C53—81no—
113Dic53—81no—
114#2007—531no—
115#1394—531no—
116#1272D5311yescharacteristic, complemented
117C2×C53—21no—
118C2×C53—21no—
119C2×C53#134311yescharacteristic
120C2×C53#140311yescharacteristic
121C2×C53#140511yescharacteristic
122D53—161no—
123C2×#137—1061no—
124#537—531no—
125C2×#104—1061no—
126C2×#101—1061no—
127#391C2×D5311yescharacteristic
128#343—531no—
129#336—21no—
130C53#740911yescharacteristic, complemented, Sylow
131C2×#41—1061no—
132C22×Q8D21211yescharacteristic
133#137—2122no—
134#137—2121no—
135#137—2121no—
136C2×Q16—2121no—
137C2×SD16—2121no—
138C2×M16—1061no—
139C2×M16—2121no—
140C22×C8—41no—
141C22⋊Q8—2121no—
142#121—2121no—
143#121—2121no—
144C2×C4⋊C4C4×D5311yescharacteristic
145C2×C42C22⋊D5311yescharacteristic
146#104—2122no—
147#101—1062no—
148#101—2121no—
149#95—41no—
150#41—2121no—
151#41—2121no—
152#41—2121no—
153#41—2121no—
154C2×Q8—21no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—1061no—
158C2×Q8#1171991311yescharacteristic
159C2×Q8D42411yescharacteristic
160C2×D4—1061no—
161C2×D4—2121no—
162C2×D4—2121no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—2121no—
167C22×C4#1171992111yescharacteristic
168Q16—2122no—
169Q16—2122no—
170SD16—2122no—
171SD16—2122no—
172M16—1062no—
173M16—4241no—
174M16—2122no—
175M16—2121no—
176C2×C8—42no—
177C2×C8—81no—
178C2×C8—81no—
179C2×C8—4241no—
180C2×C8—2121no—
181C4⋊C4—41no—
182C4⋊C4—2121no—
183C22⋊C4—2122no—
184C42—41no—
185C23—2121no—
186C23#1313313111yescharacteristic
187Q8—41no—
188Q8—41no—
189Q8—81no—
190Q8—2121no—
191Q8—2121no—
192D4—8481no—
193D4—4241no—
194D4—4241no—
195D4—2121no—
196D4—2121no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—81no—
209C2×C4—4241no—
210C2×C4—4241no—
211C2×C4—2121no—
212C2×C4#1313313911yescharacteristic
213C2×C4#1313314311yescharacteristic
214C8—82no—
215C8—4242no—
216C8—2122no—
217C22—21no—
218C22—21no—
219C22—4241no—
220C22—4241no—
221C22—4241no—
222C22#1681035811yescharacteristic
223C22#1681041111yescharacteristic
224C22#1681041511yescharacteristic
225C4—21no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—41no—
231C4—81no—
232C4—4241no—
233C2—21no—
234C2—21no—
235C2—8481no—
236C2#2027271311yescharacteristic
237C2#2027286411yescharacteristic
238C2#2027287011yescharacteristic
subgroupG/H
centerC22#16810411
commutatorC2×C4⋊C4×C53C2×C4
Frattini#391C2×D53
FittingC53×#1272C2
socleC22×C53AΣL(1,𝔽16)
prime radical#16811204C4
cosocleC53×#391C22
hypercenter#1272D53
hypocenterC53#7409
LCSG ▷ C2×C4⋊C4×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #20272842C2semidirect
3. C53×#1272C2semidirect
116. #1272D53semidirect
130. C53#7409semidirect

Maximal subgroups

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

subgroupindex
#202757612
#202728422
C53×#12722
#740953