Finite Group Databasebeta

← #27273361

Subgroups — #27273361

Order 13568.

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

normal subgroups only

This group has 418 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, complemented
2#20276248C211yesmaximal, characteristic
3#20272538C212yesmaximal, complemented
4#20272536C211yesmaximal, characteristic
5C53×#2724C211yesmaximal, characteristic
6C2×#13133148—21no—
7C2×#13133138—21no—
8#16810493C2211yescharacteristic
9#16810385—22no—
10#16810385—22no—
11#16810378C2211yescharacteristic
12#16810378C2212yes—
13C2×#123×C53—21no—
14C53×#398—21no—
15C53×#397—22no—
16C53×#390C2211yescharacteristic
17C53×#389C2212yes—
18C53×#385—22no—
19C22×#88322—21no—
20C22×#88322—22no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—22no—
23#13133148—42no—
24#13133138—42no—
25C22×D4×C53D411yescharacteristic
26C23×C4×C53—21no—
27#123×C53—42no—
28#123×C53—42no—
29C2×C4⋊C4×C53—21no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53D412yes—
32C2×C4⋊C4×C53D412yes—
33C2×C4⋊C4×C53D411yescharacteristic
34C2×C22⋊C4×C53—22no—
35C2×C22⋊C4×C53—21no—
36C2×C22⋊C4×C53—22no—
37C2×C42×C53C2311yescharacteristic
38#95×C53—22no—
39#95×C53—21no—
40#95×C53—22no—
41C24×C53—21no—
42C2×D4×C53—21no—
43C2×D4×C53—41no—
44C2×D4×C53—41no—
45C2×D4×C53SD1612yes—
46C22×C4×C53—22no—
47C22×C4×C53—21no—
48C22×C4×C53—22no—
49C22×C4×C53—42no—
50C22×C4×C53—81no—
51C22×C4×C53—81no—
52C22×C4×C53C2×D411yescharacteristic
53C22×C4×C53C2×D412yes—
54C22×Dic53—41no—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—22no—
57C4⋊C4×C53—42no—
58C4⋊C4×C53—41no—
59C4⋊C4×C53—42no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—81no—
62C22⋊C4×C53—44no—
63C22⋊C4×C53—42no—
64C22⋊C4×C53—42no—
65C22⋊C4×C53—42no—
66C42×C53—41no—
67C4⋊Dic53—41no—
68C4⋊Dic53—41no—
69C4⋊Dic53—44no—
70C2×#88322—22no—
71C2×#88322—41no—
72C2×#88322—41no—
73C2×#88322—42no—
74C2×#88322—42no—
75C2×#88322—42no—
76C23×C53—22no—
77C23×C53—42no—
78C23×C53—41no—
79C23×C53—42no—
80C23×C53—41no—
81C23×C53#12011yescharacteristic
82D4×C53—42no—
83D4×C53—81no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—84no—
97C2×C4×C53C2×SD1611yescharacteristic
98C2×C4×C53#13512yes—
99C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
100C2×C4×C53#13712yes—
101C2×Dic53—81no—
102C2×Dic53—82no—
103#88322—42no—
104#88322—82no—
105#7824—531nomaximal, Sylow
106C22×C53—42no—
107C22×C53—44no—
108C22×C53—82no—
109C22×C53—81no—
110C22×C53—81no—
111C22×C53#45011yescharacteristic
112C22×C53#45212yes—
113C22×C53#45311yescharacteristic
114C22×C53#45412yes—
115C22×C53#45811yescharacteristic
116C4×C53—21no—
117C4×C53—21no—
118C4×C53—22no—
119C4×C53—22no—
120C4×C53—82no—
121C4×C53—81no—
122C4×C53—82no—
123Dic53—161no—
124#2724D5311yescharacteristic
125#2100—532no—
126#2094—531no—
127#1271—532no—
128#1270—531no—
129C2×C53—82no—
130C2×C53#161311yescharacteristic
131C2×C53#161512yes—
132C2×C53#163011yescharacteristic
133C2×C53#163512yes—
134C2×C53#165711yescharacteristic
135C2×#123—21no—
136C2×#106—1061no—
137C2×#102—1061no—
138#398—21no—
139#397—22no—
140#390C2×D5311yescharacteristic
141#389C2×D5312yes—
142#385—22no—
143#384—531no—
144#340—1062no—
145#340—1062no—
146#336—531no—
147#336—532no—
148C53#782411yescharacteristic, complemented, Sylow
149C22×D4C22⋊D5311yescharacteristic
150C23×C4—21no—
151C22×C8—1061no—
152C22×C8—1062no—
153#123—42no—
154#123—42no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—1062no—
158C2×C4⋊C4—1061no—
159C2×C4⋊C4C22⋊D5312yes—
160C2×C4⋊C4C22⋊D5312yes—
161C2×C4⋊C4C22⋊D5311yescharacteristic
162C2×C22⋊C4—22no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—22no—
165C2×C42C22×D5311yescharacteristic
166#106—2122no—
167#102—2122no—
168#95—22no—
169#95—21no—
170#95—22no—
171C24—21no—
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4#1171992312yes—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—42no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—2121no—
183C22×C4C2×C22⋊D5311yescharacteristic
184C22×C4C2×C22⋊D5312yes—
185C2×C8—1062no—
186C2×C8—2122no—
187C2×C8—2122no—
188C2×C8—2122no—
189C2×C8—2121no—
190C2×C8—2121no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—81no—
198C4⋊C4—2121no—
199C4⋊C4—2121no—
200C4⋊C4—2124no—
201C22⋊C4—42no—
202C22⋊C4—44no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C42—41no—
206C23—22no—
207C23—42no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23#1313318611yescharacteristic
212D4—42no—
213D4—81no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—84no—
227C2×C4—4241no—
228C2×C4—4242no—
229C2×C4C2×#1171992311yescharacteristic
230C2×C4#1313323712yes—
231C2×C4#1313324311yescharacteristic
232C2×C4#1313324712yes—
233C8—4242no—
234C8—2122no—
235C22—42no—
236C22—44no—
237C22—82no—
238C22—81no—
239C22—81no—
240C22#1681077411yescharacteristic
241C22#1681078512yes—
242C22#1681079111yescharacteristic
243C22#1681079612yes—
244C22#1681081311yescharacteristic
245C4—21no—
246C4—21no—
247C4—22no—
248C4—22no—
249C4—82no—
250C4—81no—
251C4—82no—
252C4—8481no—
253C2—82no—
254C2#2027373511yescharacteristic
255C2#2027374612yes—
256C2#2027383611yescharacteristic
257C2#2027386612yes—
258C2#2027401311yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2724C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#2724D53
hypocenterC53#7824
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2724

Complements

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

NH ≅ G/Ntype
1. #20276278C2semidirect
3. #20272538C2semidirect
148. C53#7824semidirect

Maximal subgroups

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

subgroupindex
#202762782
#202762482
#202725382
#202725362
C53×#27242
#782453