Finite Group Databasebeta

← #27295255

Subgroups — #27295255

Order 13568.

372 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 107 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#20274711C211yesmaximal, characteristic
2#20273039C211yesmaximal, characteristic
3#20272742C214yesmaximal
4C53×#2568C211yesmaximal, characteristic
5#16810676—21no—
6#16810647—21no—
7C2×#13133136—21no—
8#16810392C2212yes—
9#16810349—21no—
10#16810349C2212yes—
11C2×C4⋊Q8×C53C411yescharacteristic
12C2×C4×Q8×C53C2211yescharacteristic
13C53×#400C412yes—
14C53×#400C2212yes—
15C53×#391—21no—
16C53×#391—21no—
17C53×#391—21no—
18C53×#382—21no—
19C4×C4⋊C4×C53C411yescharacteristic
20C2×#11719917—22no—
21C4⋊#88322—41no—
22C4⋊#88322—41no—
23C4⋊#88322—41no—
24#13133136—42no—
25C22⋊#88322—22no—
26C4×#88322—41no—
27C22×Q8×C53—21no—
28C22×Q8×C53D411yescharacteristic
29C4⋊Q8×C53—41no—
30C4⋊Q8×C53—41no—
31C4×Q8×C53—41no—
32C4×Q8×C53—41no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53C2×C412yes—
36C2×C4⋊C4×C53C2×C411yescharacteristic
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C4⋊C4×C53C2311yescharacteristic
39C2×C42×C53—21no—
40C2×C42×C53C2×C411yescharacteristic
41C2×C42×C53D411yescharacteristic
42C2×C42×C53D411yescharacteristic
43#95×C53—22no—
44#95×C53—22no—
45#95×C53—22no—
46#95×C53—22no—
47#95×C53C2×C412yes—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53—44no—
51C2×Q8×C53—41no—
52C2×Q8×C53—41no—
53C2×Q8×C53SD1612yes—
54C2×Q8×C53Q1612yes—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C4⋊C4×C53—41no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—42no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—42no—
70C42×C53—21no—
71C42×C53—21no—
72C42×C53—21no—
73C42×C53—21no—
74C42×C53—21no—
75C42×C53—21no—
76C42×C53—42no—
77#11719917—42no—
78#11719917—42no—
79C2×#88322—42no—
80C2×#88322—42no—
81C23×C53C2×C22⋊C411yescharacteristic
82Q8×C53—42no—
83Q8×C53—42no—
84Q8×C53—82no—
85Q8×C53—82no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—42no—
100C2×C4×C53—44no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53#9911yescharacteristic
105C2×C4×C53#9911yescharacteristic
106C2×C4×C53#10312yes—
107C2×C4×C53#10312yes—
108C2×C4×C53C2×SD1611yescharacteristic
109C2×C4×C53C2×Q1611yescharacteristic
110#88322—82no—
111#88322—82no—
112#11907—531nomaximal, Sylow
113C22×C53—21no—
114C22×C53—21no—
115C22×C53C2×#9911yescharacteristic
116C22×C53C2×#10311yescharacteristic
117C22×C53#42111yescharacteristic
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—21no—
123C4×C53—41no—
124C4×C53—41no—
125C4×C53—41no—
126C4×C53—82no—
127C4×C53—82no—
128C4×C53#35512yes—
129#2568D5311yescharacteristic
130#1801—531no—
131#1461—531no—
132#1353—534no—
133C2×C53—21no—
134C2×C53—21no—
135C2×C53#152011yescharacteristic
136C2×C53#212311yescharacteristic
137C2×C53C2×#35511yescharacteristic
138C2×C4⋊Q8Dic5311yescharacteristic
139C2×C4×Q8C2×D5311yescharacteristic
140#431—1061no—
141#423—1061no—
142C2×#101—1061no—
143#400Dic5312yes—
144#400C2×D5312yes—
145#391—21no—
146#391—21no—
147#391—21no—
148#382—21no—
149C4×C4⋊C4Dic5311yescharacteristic
150#343—532no—
151#324—532no—
152#324—1061no—
153C53#1190711yescharacteristic, complemented, Sylow
154C22×Q8—21no—
155C22×Q8C22⋊D5311yescharacteristic
156C2×M16—1062no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159C4×Q8—41no—
160C4×Q8—41no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C2×Dic5312yes—
164C2×C4⋊C4C2×Dic5311yescharacteristic
165C2×C4⋊C4C22⋊D5311yescharacteristic
166C2×C4⋊C4C22×D5311yescharacteristic
167C2×C42—21no—
168C2×C42C2×Dic5311yescharacteristic
169C2×C42C22⋊D5311yescharacteristic
170C2×C42C22⋊D5311yescharacteristic
171C4⋊C8—2121no—
172C4⋊C8—2121no—
173C4⋊C8—2121no—
174#101—2122no—
175C22⋊C8—1062no—
176C4×C8—2121no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95C2×Dic5312yes—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—44no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8#1171992412yes—
188C2×Q8#1171992512yes—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4C22⋊Dic5311yescharacteristic
192C22×C4C22⋊Dic5311yescharacteristic
193C22×C4C22⋊Dic5311yescharacteristic
194C22×C4C22⋊Dic5311yescharacteristic
195C22×C4C22×Dic5311yescharacteristic
196C22×C4C2×C22⋊D5311yescharacteristic
197C22×C4C2×C22⋊D5311yescharacteristic
198M16—2122no—
199M16—2122no—
200C2×C8—2122no—
201C2×C8—2122no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—42no—
215C23C2×C22⋊Dic5311yescharacteristic
216Q8—42no—
217Q8—42no—
218Q8—82no—
219Q8—82no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4#1313313211yescharacteristic
239C2×C4#1313313211yescharacteristic
240C2×C4#1313314112yes—
241C2×C4#1313314112yes—
242C2×C4C2×#1171992411yescharacteristic
243C2×C4C2×#1171992511yescharacteristic
244C8—4242no—
245C8—4242no—
246C22—21no—
247C22—21no—
248C22C2×#1313313211yescharacteristic
249C22C2×#1313314111yescharacteristic
250C22#1681064011yescharacteristic
251C4—21no—
252C4—22no—
253C4—22no—
254C4—21no—
255C4—21no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—82no—
260C4—82no—
261C4#1681042012yes—
262C2—21no—
263C2—21no—
264C2#2027328711yescharacteristic
265C2#2027636911yescharacteristic
266C2C2×#1681042011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2568C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2568D53
hypocenterC53#11907
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2568

Complements

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

NH ≅ G/Ntype
153. C53#11907semidirect

Maximal subgroups

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

subgroupindex
#202747112
#202730392
#202727422
C53×#25682
#1190753