Finite Group Databasebeta

← #27273189

Subgroups — #27273189

Order 13568.

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

normal subgroups only

This group has 478 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#20281087C211yesmaximal, characteristic
2#20276187C212yesmaximal, complemented
3#20272537C212yesmaximal, complemented
4#20272536C211yesmaximal, characteristic
5C53×#2093C211yesmaximal, characteristic
6C2×#13133194—22no—
7C2×#13133150—22no—
8#16810546—22no—
9#16810543—21no—
10#16810521C2212yes—
11#16810503—21no—
12#16810385—22no—
13#16810378C2212yes—
14C2×#107×C53—21no—
15C2×#102×C53—21no—
16C53×#390C2211yescharacteristic
17C53×#384C2211yescharacteristic
18C53×#336C2211yescharacteristic
19C22×Dic106—21no—
20C23×Dic53—21no—
21C22×#88322—22no—
22#13133194—44no—
23#13133194—44no—
24C2×C4⋊Dic53D412yes—
25C2×C4⋊Dic53D412yes—
26C2×#11719919—22no—
27C2×#11719919—22no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—21no—
30#13133150—44no—
31#13133124—21no—
32#13133124—22no—
33C22×D4×C53D411yescharacteristic
34C22×C8×C53—21no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C22⋊C4×C53—22no—
39C2×C42×C53C2311yescharacteristic
40#107×C53—42no—
41#102×C53—42no—
42C24×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53—41no—
45C2×D4×C53—41no—
46C2×D4×C53Q1612yes—
47C22×C4×C53—21no—
48C22×C4×C53—41no—
49C22×C4×C53C2×D412yes—
50C22×C4×C53C2×D411yescharacteristic
51C22×Dic53—22no—
52C22×Dic53—22no—
53C22×Dic53—44no—
54C22×Dic53—81no—
55C2×Dic106—44no—
56C2×Dic106—81no—
57C2×C8×C53—22no—
58C2×C8×C53—41no—
59C2×C8×C53—41no—
60C4⋊C4×C53—41no—
61C4⋊C4×C53—41no—
62C4⋊C4×C53—41no—
63C4⋊C4×C53—42no—
64C4⋊C4×C53—42no—
65C22⋊C4×C53—42no—
66C22⋊C4×C53—42no—
67C42×C53—41no—
68C22⋊Dic53—44no—
69C22⋊Dic53—81no—
70C4⋊Dic53—22no—
71C4⋊Dic53—42no—
72C4⋊Dic53D814yes—
73#11719919—44no—
74#11719919—42no—
75#11719919—42no—
76C2×#88322—24no—
77C2×#88322—42no—
78C2×#88322—42no—
79C23×C53—22no—
80C23×C53—42no—
81C23×C53—41no—
82C23×C53—42no—
83C23×C53—41no—
84C23×C53#12011yescharacteristic
85D4×C53—42no—
86D4×C53—81no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—42no—
92C2×C4×C53—41no—
93C2×C4×C53—82no—
94C2×C4×C53—81no—
95C2×C4×C53C2×D812yes—
96C2×C4×C53C2×Q1611yescharacteristic
97C2×C4×C53AGL(1,ℤ/8)12yes—
98C2×C4×C53#13711yescharacteristic
99C2×Dic53—24no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53—42no—
105C2×Dic53—84no—
106Dic106—84no—
107C8×C53—42no—
108#88322—44no—
109#7797—531nomaximal, Sylow
110C22×C53—42no—
111C22×C53—44no—
112C22×C53—82no—
113C22×C53—81no—
114C22×C53—81no—
115C22×C53#44712yes—
116C22×C53#45111yescharacteristic
117C22×C53#45311yescharacteristic
118C22×C53#45612yes—
119C22×C53#45811yescharacteristic
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—21no—
123C4×C53—21no—
124C4×C53—81no—
125C4×C53—161no—
126Dic53—44no—
127Dic53—82no—
128#2683—531no—
129#2093D5311yescharacteristic
130#2083—532no—
131#1271—532no—
132#1270—531no—
133C2×C53—82no—
134C2×C53#161012yes—
135C2×C53#161411yescharacteristic
136C2×C53#161911yescharacteristic
137C2×C53#164912yes—
138C2×C53#166011yescharacteristic
139C2×C22⋊Q8—1062no—
140C2×#107—21no—
141C2×#107—1062no—
142C2×#102—21no—
143#396—1062no—
144#395—1061no—
145#390C2×D5311yescharacteristic
146#389—532no—
147#386—1061no—
148#384C2×D5311yescharacteristic
149#340—1062no—
150#336—532no—
151#336C2×D5311yescharacteristic
152C53#779711yescharacteristic, complemented, Sylow
153C22×Q8—1061no—
154C22×D4C22⋊D5311yescharacteristic
155C23×C4—1061no—
156C22×C8—21no—
157C22×C8—1062no—
158C22⋊Q8—2124no—
159C22⋊Q8—2124no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—532no—
163C2×C4⋊C4—532no—
164C2×C4⋊C4—1062no—
165C2×C4⋊C4—1062no—
166C2×C4⋊C4C22⋊D5311yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—1061no—
169C2×C22⋊C4—1062no—
170C2×C42C22×D5311yescharacteristic
171#107—42no—
172#107—2124no—
173#102—42no—
174#95—1061no—
175#95—1062no—
176C24—21no—
177C2×Q8—4241no—
178C2×Q8—2124no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4#1171992512yes—
183C22×C4—21no—
184C22×C4—41no—
185C22×C4—1062no—
186C22×C4—1062no—
187C22×C4—4241no—
188C22×C4—2124no—
189C22×C4D4×D5312yes—
190C22×C4C2×C22⋊D5311yescharacteristic
191C2×C8—22no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—1064no—
195C2×C8—2122no—
196C2×C8—2122no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—534no—
203C4⋊C4—1062no—
204C4⋊C4—2122no—
205C4⋊C4—2124no—
206C4⋊C4—2122no—
207C4⋊C4—2122no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—4241no—
211C22⋊C4—2124no—
212C42—41no—
213C23—22no—
214C23—42no—
215C23—41no—
216C23—42no—
217C23—41no—
218C23#1313318511yescharacteristic
219Q8—4244no—
220D4—42no—
221D4—81no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—82no—
229C2×C4—81no—
230C2×C4—1064no—
231C2×C4—4244no—
232C2×C4—2122no—
233C2×C4—2122no—
234C2×C4—2122no—
235C2×C4—2122no—
236C2×C4—2122no—
237C2×C4D8×D5312yes—
238C2×C4C2×#1171992511yescharacteristic
239C2×C4#1313324112yes—
240C2×C4#1313324711yescharacteristic
241C8—42no—
242C8—2124no—
243C22—44no—
244C22—42no—
245C22—82no—
246C22—81no—
247C22—81no—
248C22#1681075512yes—
249C22#1681077811yescharacteristic
250C22#1681079011yescharacteristic
251C22#1681080412yes—
252C22#1681081211yescharacteristic
253C4—22no—
254C4—22no—
255C4—21no—
256C4—21no—
257C4—81no—
258C4—161no—
259C4—4242no—
260C4—2124no—
261C2—82no—
262C2#2027371612yes—
263C2#2027373911yescharacteristic
264C2#2027376711yescharacteristic
265C2#2027396312yes—
266C2#2027402411yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2093C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#2093D53
hypocenterC53#7797
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2093

Complements

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

NH ≅ G/Ntype
2. #20276187C2semidirect
3. #20272537C2semidirect
152. C53#7797semidirect

Maximal subgroups

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

subgroupindex
#202810872
#202761872
#202725372
#202725362
C53×#20932
#779753