Finite Group Databasebeta

← #27273222

Subgroups — #27273222

Order 13568.

366 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 83 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#20276187C212yesmaximal
3#20272541C211yesmaximal, characteristic
4#20272537C212yesmaximal
5C53×#2096C211yesmaximal, characteristic
6C2×#13133150—22no—
7#16810562—21no—
8#16810555—22no—
9#16810530—21no—
10#16810521C2212yes—
11#16810510—22no—
12#16810385—22no—
13#16810378C2212yes—
14C2×#106×C53—21no—
15C2×#103×C53—21no—
16C53×#391C2211yescharacteristic
17C53×#384C2211yescharacteristic
18C53×#336C2211yescharacteristic
19C22×#88322—22no—
20C2×C4⋊Dic53—21no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53D412yes—
23C2×C4⋊Dic53D412yes—
24C2×#11719919—22no—
25C2×C4×Dic53—21no—
26#13133150—44no—
27#13133124—21no—
28#13133124—22no—
29#13133124—22no—
30#13133124—22no—
31C22×Q8×C53D411yescharacteristic
32C22×C8×C53—21no—
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53D411yescharacteristic
36C2×C42×C53C2311yescharacteristic
37#106×C53—42no—
38#103×C53—42no—
39#95×C53—22no—
40C2×Q8×C53—21no—
41C2×Q8×C53—41no—
42C2×Q8×C53—41no—
43C2×Q8×C53Q1612yes—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—41no—
47C22×C4×C53C2×D411yescharacteristic
48C22×C4×C53C2×D412yes—
49C22×Dic53—22no—
50C22×Dic53—22no—
51C2×C8×C53—22no—
52C2×C8×C53—41no—
53C2×C8×C53—41no—
54C4⋊C4×C53—41no—
55C4⋊C4×C53—41no—
56C4⋊C4×C53—41no—
57C4⋊C4×C53—42no—
58C4⋊C4×C53—42no—
59C42×C53—41no—
60C4⋊Dic53—22no—
61C4⋊Dic53—22no—
62C4⋊Dic53—42no—
63C4⋊Dic53—81no—
64C4⋊Dic53—81no—
65#11719919—42no—
66#11719919—42no—
67C4×Dic53—81no—
68C2×#88322—24no—
69C2×#88322—42no—
70C2×#88322—42no—
71C23×C53#12011yescharacteristic
72Q8×C53—42no—
73Q8×C53—81no—
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—41no—
80C2×C4×C53—41no—
81C2×C4×C53—41no—
82C2×C4×C53—42no—
83C2×C4×C53—82no—
84C2×C4×C53—81no—
85C2×C4×C53C2×Q1611yescharacteristic
86C2×C4×C53#13512yes—
87C2×C4×C53AGL(1,ℤ/8)12yes—
88C2×C4×C53#13711yescharacteristic
89C2×Dic53—42no—
90C2×Dic53—42no—
91C2×Dic53—42no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C8×C53—42no—
96#88322—44no—
97#7802—531nomaximal, Sylow
98C22×C53#44912yes—
99C22×C53#45111yescharacteristic
100C22×C53#45311yescharacteristic
101C22×C53#45612yes—
102C22×C53#45811yescharacteristic
103C4×C53—22no—
104C4×C53—22no—
105C4×C53—21no—
106C4×C53—21no—
107C4×C53—42no—
108C4×C53—81no—
109C4×C53—161no—
110Dic53—82no—
111Dic53—82no—
112#2809—531no—
113#2096D5311yescharacteristic
114#2083—532no—
115#1272—531no—
116#1271—532no—
117C2×C53#160912yes—
118C2×C53#161411yescharacteristic
119C2×C53#162411yescharacteristic
120C2×C53#164812yes—
121C2×C53#166011yescharacteristic
122C2×#107—1062no—
123C2×#106—21no—
124C2×#103—21no—
125#400—1061no—
126#398—1062no—
127#391—1061no—
128#391C2×D5311yescharacteristic
129#389—532no—
130#387—1062no—
131#384C2×D5311yescharacteristic
132#340—1062no—
133#336—532no—
134#336C2×D5311yescharacteristic
135C53#780211yescharacteristic, complemented, Sylow
136C22×Q8C22⋊D5311yescharacteristic
137C22×C8—21no—
138C22×C8—1062no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4—532no—
143C2×C4⋊C4—1061no—
144C2×C4⋊C4—1061no—
145C2×C4⋊C4—1062no—
146C2×C4⋊C4C22⋊D5311yescharacteristic
147C2×C42—1061no—
148C2×C42C22×D5311yescharacteristic
149#107—2124no—
150#106—42no—
151#103—42no—
152#95—22no—
153#95—1061no—
154#95—1062no—
155#95—1062no—
156#95—1062no—
157C2×Q8—21no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8#1171992512yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—1062no—
165C22×C4—1062no—
166C22×C4D4×D5312yes—
167C22×C4C2×C22⋊D5311yescharacteristic
168C2×C8—22no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—1064no—
172C2×C8—2122no—
173C2×C8—2122no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—1062no—
180C4⋊C4—1062no—
181C4⋊C4—4241no—
182C4⋊C4—4241no—
183C4⋊C4—2122no—
184C4⋊C4—2122no—
185C4⋊C4—2122no—
186C42—41no—
187C42—4241no—
188C23#1313318511yescharacteristic
189Q8—42no—
190Q8—81no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—82no—
201C2×C4—81no—
202C2×C4—2122no—
203C2×C4—2122no—
204C2×C4—2122no—
205C2×C4—2122no—
206C2×C4—2122no—
207C2×C4—2122no—
208C2×C4C2×#1171992511yescharacteristic
209C2×C4#1313323812yes—
210C2×C4#1313324412yes—
211C2×C4#1313324711yescharacteristic
212C8—42no—
213C8—2124no—
214C22#1681076712yes—
215C22#1681077811yescharacteristic
216C22#1681079011yescharacteristic
217C22#1681080412yes—
218C22#1681081211yescharacteristic
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—42no—
224C4—81no—
225C4—161no—
226C4—4242no—
227C4—4242no—
228C2#2027371112yes—
229C2#2027373911yescharacteristic
230C2#2027379611yescharacteristic
231C2#2027395612yes—
232C2#2027402411yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2096C2
socle, prime radicalC23×C53#120
hypercenter#2096D53
hypocenterC53#7802
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2096

Complements

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

NH ≅ G/Ntype
135. C53#7802semidirect

Maximal subgroups

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

subgroupindex
#202827032
#202761872
#202725412
#202725372
C53×#20962
#780253