Finite Group Databasebeta

← #27273402

Subgroups — #27273402

Order 13568.

350 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 87 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#20276278C212yesmaximal
2#20276253C211yesmaximal, characteristic
3#20272541C211yesmaximal, characteristic
4#20272538C212yesmaximal
5C53×#2809C211yesmaximal, characteristic
6C2×#13133150—21no—
7C2×#13133141—21no—
8#16810493C2211yescharacteristic
9#16810385—22no—
10#16810385—22no—
11#16810378C2211yescharacteristic
12#16810378C2212yes—
13C53×#400—21no—
14C53×#398—22no—
15C53×#391—21no—
16C53×#391C2211yescharacteristic
17C53×#389C2212yes—
18C53×#387—22no—
19C22×#88322—21no—
20C22×#88322—22no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—22no—
23#13133150—42no—
24#13133141—42no—
25C22×Q8×C53D411yescharacteristic
26C2×C4⋊C4×C53—21no—
27C2×C4⋊C4×C53—21no—
28C2×C4⋊C4×C53—22no—
29C2×C4⋊C4×C53D412yes—
30C2×C4⋊C4×C53D412yes—
31C2×C4⋊C4×C53D411yescharacteristic
32C2×C42×C53—21no—
33C2×C42×C53C2311yescharacteristic
34#95×C53—21no—
35#95×C53—22no—
36#95×C53—22no—
37#95×C53—22no—
38#95×C53—22no—
39C2×Q8×C53—21no—
40C2×Q8×C53—41no—
41C2×Q8×C53—41no—
42C2×Q8×C53Q1612yes—
43C22×C4×C53—21no—
44C22×C4×C53—21no—
45C22×C4×C53—22no—
46C22×C4×C53—22no—
47C22×C4×C53C2×D411yescharacteristic
48C22×C4×C53C2×D412yes—
49C22×Dic53—41no—
50C4⋊C4×C53—22no—
51C4⋊C4×C53—22no—
52C4⋊C4×C53—42no—
53C4⋊C4×C53—41no—
54C4⋊C4×C53—42no—
55C4⋊C4×C53—42no—
56C4⋊C4×C53—81no—
57C4⋊C4×C53—81no—
58C42×C53—41no—
59C42×C53—81no—
60C4⋊Dic53—41no—
61C4⋊Dic53—41no—
62C4⋊Dic53—44no—
63C2×#88322—22no—
64C2×#88322—41no—
65C2×#88322—41no—
66C2×#88322—42no—
67C2×#88322—42no—
68C2×#88322—42no—
69C23×C53#12011yescharacteristic
70Q8×C53—42no—
71Q8×C53—81no—
72C2×C4×C53—21no—
73C2×C4×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—41no—
78C2×C4×C53—41no—
79C2×C4×C53—42no—
80C2×C4×C53—42no—
81C2×C4×C53—42no—
82C2×C4×C53—41no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53C2×Q1611yescharacteristic
88C2×C4×C53#13512yes—
89C2×C4×C53#13712yes—
90C2×C4×C53#13711yescharacteristic
91C2×Dic53—81no—
92C2×Dic53—82no—
93#88322—42no—
94#88322—82no—
95#7831—531nomaximal, Sylow
96C22×C53#45111yescharacteristic
97C22×C53#45212yes—
98C22×C53#45311yescharacteristic
99C22×C53#45412yes—
100C22×C53#45811yescharacteristic
101C4×C53—21no—
102C4×C53—21no—
103C4×C53—22no—
104C4×C53—22no—
105C4×C53—42no—
106C4×C53—82no—
107C4×C53—82no—
108C4×C53—81no—
109Dic53—161no—
110#2809D5311yescharacteristic
111#2100—532no—
112#2095—531no—
113#1272—531no—
114#1271—532no—
115C2×C53#161411yescharacteristic
116C2×C53#161512yes—
117C2×C53#163411yescharacteristic
118C2×C53#163612yes—
119C2×C53#165711yescharacteristic
120C2×#107—1061no—
121C2×#103—1061no—
122#400—21no—
123#398—22no—
124#391—21no—
125#391C2×D5311yescharacteristic
126#389C2×D5312yes—
127#387—22no—
128#384—531no—
129#340—1062no—
130#340—1062no—
131#336—531no—
132#336—532no—
133C53#783111yescharacteristic, complemented, Sylow
134C22×Q8C22⋊D5311yescharacteristic
135C22×C8—1061no—
136C22×C8—1062no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—1062no—
141C2×C4⋊C4—1061no—
142C2×C4⋊C4C22⋊D5312yes—
143C2×C4⋊C4C22⋊D5312yes—
144C2×C4⋊C4C22⋊D5311yescharacteristic
145C2×C42—21no—
146C2×C42C22×D5311yescharacteristic
147#107—2122no—
148#103—2122no—
149#95—21no—
150#95—22no—
151#95—22no—
152#95—22no—
153#95—22no—
154C2×Q8—21no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8#1171992512yes—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—2121no—
163C22×C4C2×C22⋊D5311yescharacteristic
164C22×C4C2×C22⋊D5312yes—
165C2×C8—1062no—
166C2×C8—2122no—
167C2×C8—2122no—
168C2×C8—2122no—
169C2×C8—2121no—
170C2×C8—2121no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C4⋊C4—2121no—
180C4⋊C4—2121no—
181C4⋊C4—2124no—
182C42—41no—
183C42—81no—
184C23#1313318611yescharacteristic
185Q8—42no—
186Q8—81no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—4241no—
203C2×C4—4242no—
204C2×C4C2×#1171992511yescharacteristic
205C2×C4#1313323712yes—
206C2×C4#1313324712yes—
207C2×C4#1313324811yescharacteristic
208C8—4242no—
209C8—2122no—
210C22#1681077911yescharacteristic
211C22#1681078512yes—
212C22#1681079111yescharacteristic
213C22#1681079612yes—
214C22#1681081311yescharacteristic
215C4—21no—
216C4—21no—
217C4—22no—
218C4—22no—
219C4—42no—
220C4—82no—
221C4—81no—
222C4—82no—
223C4—8481no—
224C2#2027374011yescharacteristic
225C2#2027374612yes—
226C2#2027386011yescharacteristic
227C2#2027387312yes—
228C2#2027401311yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2809C2
socle, prime radicalC23×C53#120
hypercenter#2809D53
hypocenterC53#7831
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2809

Complements

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

NH ≅ G/Ntype
133. C53#7831semidirect

Maximal subgroups

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

subgroupindex
#202762782
#202762532
#202725412
#202725382
C53×#28092
#783153