Finite Group Databasebeta

← #27296939

Subgroups — #27296939

Order 13568.

336 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 59 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#20282703C211yesmaximal, characteristic
2#20273757C212yesmaximal
3#20272741C212yesmaximal
4#20272649C211yesmaximal, characteristic
5C53×#2028C211yesmaximal, characteristic
6#16810956—22no—
7#16810562C2212yes—
8#16810555—22no—
9#16810530—21no—
10#16810521—21no—
11#16810510—22no—
12#16810360—22no—
13#16810349C2212yes—
14C53×#544C2211yescharacteristic
15C53×#419—21no—
16C53×#391C2211yescharacteristic
17C53×#343C2211yescharacteristic
18C53×#339—21no—
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53—21no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53D412yes—
23C2×#11719919—22no—
24C2×C4×Dic53—21no—
25#13133150—42no—
26#13133141—42no—
27C22⋊#88322—22no—
28#13133124—22no—
29#13133124—22no—
30#13133124—22no—
31#13133124D412yes—
32C22×Q8×C53D411yescharacteristic
33C2×M16×C53—21no—
34C4⋊Q8×C53—41no—
35#125×C53—41no—
36C22⋊Q8×C53—22no—
37#117×C53—21no—
38C2×C4⋊C4×C53C2311yescharacteristic
39C2×C42×C53D411yescharacteristic
40#103×C53—41no—
41#103×C53—41no—
42#95×C53—22no—
43C2×Q8×C53—21no—
44C2×Q8×C53—22no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53Q1612yes—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C2×D411yescharacteristic
51C22×C4×C53C2×D412yes—
52C22×Dic53—22no—
53C22×Dic53—22no—
54M16×C53—41no—
55M16×C53—41no—
56C2×C8×C53—41no—
57C4⋊C4×C53—21no—
58C4⋊C4×C53—21no—
59C4⋊C4×C53—22no—
60C4⋊C4×C53—41no—
61C4⋊C4×C53—42no—
62C22⋊C4×C53—41no—
63C42×C53—41no—
64C42×C53—41no—
65C4⋊Dic53—42no—
66C4⋊Dic53—42no—
67C4⋊Dic53—81no—
68C4⋊Dic53—81no—
69#11719919—44no—
70C4×Dic53—81no—
71C2×#88322—42no—
72C23×C53#12011yescharacteristic
73Q8×C53—42no—
74Q8×C53—81no—
75Q8×C53—82no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—41no—
82C2×C4×C53—41no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×C4×C53—42no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53C2×Q1611yescharacteristic
89C2×C4×C53#13711yescharacteristic
90C2×Dic53—42no—
91C2×Dic53—42no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53—42no—
96C8×C53—81no—
97#88322—82no—
98#12225—531nomaximal, Sylow
99C22×C53—21no—
100C22×C53—21no—
101C22×C53#45111yescharacteristic
102C22×C53#45511yescharacteristic
103C22×C53B(4,𝔽2)11yescharacteristic
104C4×C53—21no—
105C4×C53—21no—
106C4×C53—42no—
107C4×C53—42no—
108C4×C53—42no—
109C4×C53—81no—
110C4×C53—81no—
111Dic53—82no—
112Dic53—82no—
113#2809—531no—
114#2028D5311yescharacteristic
115#1617—532no—
116#1353—532no—
117#1319—531no—
118C2×C53—21no—
119C2×C53—21no—
120C2×C53#160211yescharacteristic
121C2×C53#220111yescharacteristic
122C2×C53#220411yescharacteristic
123#544C2×D5311yescharacteristic
124#484—1062no—
125#419—21no—
126#400—532no—
127#398—1062no—
128#391—1061no—
129#391C2×D5311yescharacteristic
130#389—1061no—
131#387—1062no—
132#343C2×D5311yescharacteristic
133#339—21no—
134#328—1062no—
135#324—532no—
136C53#1222511yescharacteristic, complemented, Sylow
137C22×Q8C22⋊D5311yescharacteristic
138C2×M16—21no—
139C4⋊Q8—41no—
140#125—41no—
141C22⋊Q8—22no—
142#117—21no—
143C2×C4⋊C4—532no—
144C2×C4⋊C4—1061no—
145C2×C4⋊C4—1061no—
146C2×C4⋊C4—1062no—
147C2×C4⋊C4—1061no—
148C2×C4⋊C4C22×D5311yescharacteristic
149C2×C42—1061no—
150C2×C42C22⋊D5311yescharacteristic
151#107—2122no—
152#103—41no—
153#103—41no—
154#103—2122no—
155C22⋊C8—1062no—
156#95—22no—
157#95—532no—
158#95—1062no—
159#95—1062no—
160#95—1062no—
161C2×Q8—21no—
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8#1171992512yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—1062no—
169C22×C4—1062no—
170C22×C4D4×D5312yes—
171C22×C4C2×C22⋊D5311yescharacteristic
172M16—41no—
173M16—41no—
174C2×C8—41no—
175C2×C8—2122no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—22no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—4241no—
182C4⋊C4—4241no—
183C4⋊C4—2122no—
184C4⋊C4—2124no—
185C4⋊C4—2122no—
186C22⋊C4—41no—
187C42—41no—
188C42—41no—
189C42—4241no—
190C23#1313318511yescharacteristic
191Q8—42no—
192Q8—81no—
193Q8—82no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—2122no—
207C2×C4—2122no—
208C2×C4—2122no—
209C2×C4—2122no—
210C2×C4—2122no—
211C2×C4—2122no—
212C2×C4C2×#1171992511yescharacteristic
213C2×C4#1313324711yescharacteristic
214C8—81no—
215C8—4242no—
216C22—21no—
217C22—21no—
218C22#1681077811yescharacteristic
219C22#1681080011yescharacteristic
220C22#1681080911yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—81no—
227C4—81no—
228C4—4242no—
229C4—4242no—
230C2—21no—
231C2—21no—
232C2#2027367111yescharacteristic
233C2#2027672711yescharacteristic
234C2#2027674011yescharacteristic
subgroupG/H
centerC22#16810809
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2028C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#2028D53
hypocenterC53#12225
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2028

Complements

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

NH ≅ G/Ntype
136. C53#12225semidirect

Maximal subgroups

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

subgroupindex
#202827032
#202737572
#202727412
#202726492
C53×#20282
#1222553