Finite Group Databasebeta

← #27277663

Subgroups — #27277663

Order 13568.

498 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 131 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#20274707C211yesmaximal, characteristic
2#20273036C211yesmaximal, characteristic
3#20272618C212yesmaximal, complemented
4#20272573C212yesmaximal, complemented
5C53×#2349C211yesmaximal, characteristic
6#16810676C2212yes—
7#16810647C2212yes—
8C2×#13133134—21no—
9#16810392—21no—
10#16810347—22no—
11C2×C4×D4×C53—21no—
12C2×C4×D4×C53C2211yescharacteristic
13C53×#389—21no—
14C53×#385—22no—
15C53×#382C2211yescharacteristic
16C53×#380C412yes—
17C4×C4⋊C4×C53C2211yescharacteristic
18C2×#95×C53C412yes—
19C2×#11719917—22no—
20C4⋊#88322—22no—
21C4⋊#88322—22no—
22C4⋊#88322—22no—
23#13133134—42no—
24C22⋊#88322—22no—
25C4×#88322—22no—
26C22×D4×C53D411yescharacteristic
27C23×C4×C53—22no—
28C23×C4×C53C2×C412yes—
29C4×D4×C53—22no—
30C4×D4×C53—22no—
31C4×D4×C53—44no—
32C2×C4⋊C4×C53—22no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53D411yescharacteristic
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53D412yes—
38C2×C42×C53—21no—
39C2×C42×C53C2311yescharacteristic
40#95×C53—22no—
41#95×C53—21no—
42#95×C53C2×C411yescharacteristic
43#95×C53C2×C412yes—
44#95×C53C2×C411yescharacteristic
45C24×C53C22⋊C412yes—
46C2×D4×C53—22no—
47C2×D4×C53—22no—
48C2×D4×C53SD1612yes—
49C2×D4×C53Q1612yes—
50C22×C4×C53—21no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—21no—
55C22×C4×C53—44no—
56C22×C4×C53—44no—
57C22×C4×C53—82no—
58C22×C4×C53—82no—
59C22×C4×C53C22⋊C412yes—
60C22×C4×C53C22×C411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C4⋊C4×C53—22no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—42no—
66C4⋊C4×C53—82no—
67C22⋊C4×C53—24no—
68C22⋊C4×C53—44no—
69C42×C53—42no—
70C42×C53#4112yes—
71C42×C53#4112yes—
72#11719917—42no—
73#11719917—42no—
74C2×#88322—42no—
75C2×#88322—42no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53—42no—
79C23×C53—42no—
80C23×C53#10314yes—
81C23×C53C2×C22⋊C411yescharacteristic
82D4×C53—24no—
83D4×C53—24no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—42no—
96C2×C4×C53—41no—
97C2×C4×C53—82no—
98C2×C4×C53—82no—
99C2×C4×C53—88no—
100C2×C4×C53#11711yescharacteristic
101C2×C4×C53#11711yescharacteristic
102C2×C4×C53#12312yes—
103C2×C4×C53#12312yes—
104C2×C4×C53C2×SD1611yescharacteristic
105C2×C4×C53C2×Q1611yescharacteristic
106C2×C4×C53AGL(1,ℤ/8)12yes—
107#88322—82no—
108#88322—82no—
109#8765—531nomaximal, Sylow
110C22×C53—21no—
111C22×C53—21no—
112C22×C53—24no—
113C22×C53—44no—
114C22×C53—44no—
115C22×C53—42no—
116C22×C53—42no—
117C22×C53#38111yescharacteristic
118C22×C53C2×#10311yescharacteristic
119C22×C53#41811yescharacteristic
120C4×C53—22no—
121C4×C53—21no—
122C4×C53—21no—
123C4×C53—42no—
124C4×C53—42no—
125C4×C53—81no—
126C4×C53—82no—
127C4×C53#48112yes—
128C4×C53#48412yes—
129#2349D5311yescharacteristic
130#1800—531no—
131#1460—531no—
132#1303—532no—
133#1286—532no—
134C2×C53—21no—
135C2×C53—21no—
136C2×C53—44no—
137C2×C53#182411yescharacteristic
138C2×C53#182711yescharacteristic
139C2×C53#182911yescharacteristic
140C2×C4×D4—21no—
141C2×C4×D4C2×D5311yescharacteristic
142#431—532no—
143#423—532no—
144C2×#100—1061no—
145#389—21no—
146#385—22no—
147#382C2×D5311yescharacteristic
148#380Dic5312yes—
149C4×C4⋊C4C2×D5311yescharacteristic
150C2×#95Dic5312yes—
151#343—1061no—
152#323—1062no—
153C53#876511yescharacteristic, complemented, Sylow
154C22×D4C22⋊D5311yescharacteristic
155C23×C4—22no—
156C23×C4C2×Dic5312yes—
157C2×M16—1062no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—44no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C22⋊D5311yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4C22⋊D5312yes—
167C2×C42—21no—
168C2×C42C22×D5311yescharacteristic
169C4⋊C8—1062no—
170C4⋊C8—1062no—
171C4⋊C8—1062no—
172#100—2122no—
173C22⋊C8—1062no—
174C4×C8—1062no—
175#95—21no—
176#95—22no—
177#95C2×Dic5311yescharacteristic
178#95C2×Dic5312yes—
179#95C2×Dic5311yescharacteristic
180C24C22⋊Dic5312yes—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4#1171992412yes—
184C2×D4#1171992512yes—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4C22⋊Dic5312yes—
195C22×C4C22×Dic5311yescharacteristic
196C22×C4C2×C22⋊D5311yescharacteristic
197C22×C4C2×C22⋊D5311yescharacteristic
198M16—2122no—
199M16—2122no—
200C2×C8—2122no—
201C2×C8—2122no—
202C4⋊C4—22no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—82no—
206C22⋊C4—24no—
207C22⋊C4—44no—
208C42—42no—
209C42#1171994512yes—
210C42#1171994512yes—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23#1313314114yes—
216C23C2×C22⋊Dic5311yescharacteristic
217D4—24no—
218D4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—88no—
235C2×C4#1313317611yescharacteristic
236C2×C4#1313317611yescharacteristic
237C2×C4#1313320012yes—
238C2×C4#1313320012yes—
239C2×C4C2×#1171992411yescharacteristic
240C2×C4C2×#1171992511yescharacteristic
241C2×C4#1313324312yes—
242C8—4242no—
243C8—4242no—
244C22—21no—
245C22—21no—
246C22—24no—
247C22—44no—
248C22—44no—
249C22—42no—
250C22—42no—
251C22#1681048611yescharacteristic
252C22C2×#1313314111yescharacteristic
253C22#1681062811yescharacteristic
254C4—22no—
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—82no—
260C4—81no—
261C4#1681094112yes—
262C4#1681095812yes—
263C2—21no—
264C2—21no—
265C2—44no—
266C2#2027480511yescharacteristic
267C2#2027481911yescharacteristic
268C2#2027482611yescharacteristic
subgroupG/H
centerC22#16810486
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2349C2
socleC22×C53#381
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#2349D53
hypocenterC53#8765
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #20272618C2semidirect
4. #20272573C2semidirect
153. C53#8765semidirect

Maximal subgroups

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

subgroupindex
#202747072
#202730362
#202726182
#202725732
C53×#23492
#876553