Finite Group Databasebeta

← #27281937

Subgroups — #27281937

Order 13568.

474 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 135 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#20283796C211yesmaximal, characteristic
2#20274711C211yesmaximal, characteristic
3#20273144C212yesmaximal
4#20272573C212yesmaximal
5C53×#2581C211yesmaximal, characteristic
6Q8×#88322—24no—
7#16810676—22no—
8#16810676C2212yes—
9#16810647C2212yes—
10C2×#13133136—21no—
11#16810392—21no—
12Q8⋊#88322—24no—
13C2×C4×Q8×C53C2211yescharacteristic
14C2×C4×Q8×C53C2211yescharacteristic
15C53×#400C412yes—
16C53×#395C412yes—
17C53×#391—21no—
18C53×#389—21no—
19C53×#387—22no—
20C53×#382C2211yescharacteristic
21C2×#11719917—22no—
22C4⋊#88322—24no—
23C4⋊#88322—22no—
24C4⋊#88322—22no—
25C4⋊#88322—22no—
26#13133136—42no—
27C22⋊#88322—22no—
28C4×#88322—24no—
29C4×#88322D414yes—
30C22×Q8×C53C2×C411yescharacteristic
31C22×Q8×C53D411yescharacteristic
32C4×Q8×C53—22no—
33C4×Q8×C53—22no—
34C4×Q8×C53—22no—
35C4×Q8×C53—22no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C412yes—
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C42×C53—21no—
43C2×C42×C53D412yes—
44C2×C42×C53C2311yescharacteristic
45#95×C53—22no—
46#95×C53—22no—
47#95×C53—22no—
48#95×C53—21no—
49#95×C53C2×C412yes—
50C2×Q8×C53—22no—
51C2×Q8×C53—22no—
52C2×Q8×C53—22no—
53C2×Q8×C53—42no—
54C2×Q8×C53SD1612yes—
55C2×Q8×C53Q1612yes—
56C22×C4×C53—21no—
57C22×C4×C53—22no—
58C22×C4×C53—21no—
59C22×C4×C53C22⋊C412yes—
60C22×C4×C53C22⋊C412yes—
61C22×C4×C53C22×C411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C4⋊C4×C53—24no—
65C4⋊C4×C53—22no—
66C4⋊C4×C53—22no—
67C4⋊C4×C53—44no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—42no—
71C42×C53—22no—
72C42×C53—22no—
73C42×C53—42no—
74C42×C53C2×D412yes—
75C42×C53#4112yes—
76#11719917—42no—
77#11719917—42no—
78C2×#88322—24no—
79C2×#88322—42no—
80C23×C53C2×C22⋊C411yescharacteristic
81Q8×C53—24no—
82Q8×C53—24no—
83Q8×C53—24no—
84Q8×C53—42no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—41no—
98C2×C4×C53—42no—
99C2×C4×C53#10314yes—
100C2×C4×C53C4×D412yes—
101C2×C4×C53#12011yescharacteristic
102C2×C4×C53#12112yes—
103C2×C4×C53#12311yescharacteristic
104C2×C4×C53C2×SD1611yescharacteristic
105C2×C4×C53C2×Q1611yescharacteristic
106C2×C4×C53#13512yes—
107#88322—44no—
108#88322—82no—
109#9484—531nomaximal, Sylow
110C22×C53—21no—
111C22×C53—21no—
112C22×C53#38611yescharacteristic
113C22×C53C2×#10311yescharacteristic
114C22×C53#41611yescharacteristic
115C4×C53—22no—
116C4×C53—24no—
117C4×C53—21no—
118C4×C53—21no—
119C4×C53—42no—
120C4×C53—42no—
121C4×C53—82no—
122C4×C53—81no—
123C4×C53#46512yes—
124C4×C53#46712yes—
125#2982—531no—
126#2581D5311yescharacteristic
127#1801—531no—
128#1487—532no—
129#1286—532no—
130C2×C53—21no—
131C2×C53—21no—
132C2×C53#189111yescharacteristic
133C2×C53#189211yescharacteristic
134C2×C53#191211yescharacteristic
135C2×C4×Q8C2×D5311yescharacteristic
136C2×C4×Q8C2×D5311yescharacteristic
137C8×Q8—1064no—
138#431—532no—
139#431—1062no—
140#423—532no—
141C2×#101—1061no—
142#400Dic5312yes—
143#395Dic5312yes—
144#391—21no—
145#389—21no—
146#387—22no—
147#382C2×D5311yescharacteristic
148#343—1061no—
149Q8⋊C8—1064no—
150C53#948411yescharacteristic, complemented, Sylow
151C22×Q8C2×Dic5311yescharacteristic
152C22×Q8C22⋊D5311yescharacteristic
153C2×M16—1062no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4C2×Dic5311yescharacteristic
162C2×C4⋊C4C2×Dic5312yes—
163C2×C4⋊C4C22⋊D5311yescharacteristic
164C2×C42—21no—
165C2×C42C22⋊D5312yes—
166C2×C42C22×D5311yescharacteristic
167C4⋊C8—1064no—
168C4⋊C8—1062no—
169C4⋊C8—1062no—
170C4⋊C8—1062no—
171#101—2122no—
172C22⋊C8—1062no—
173C4×C8—534no—
174C4×C8—1064no—
175#95—21no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95C2×Dic5312yes—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—42no—
184C2×Q8#1171992412yes—
185C2×Q8#1171992512yes—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4C22⋊Dic5312yes—
190C22×C4C22⋊Dic5312yes—
191C22×C4C22×Dic5311yescharacteristic
192C22×C4C2×C22⋊D5311yescharacteristic
193C22×C4C2×C22⋊D5311yescharacteristic
194M16—2122no—
195M16—2122no—
196C2×C8—1064no—
197C2×C8—2122no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—44no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—42no—
205C42—22no—
206C42—22no—
207C42—42no—
208C42D4×D5312yes—
209C42#1171994712yes—
210C23C2×C22⋊Dic5311yescharacteristic
211Q8—24no—
212Q8—24no—
213Q8—24no—
214Q8—42no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4#1313314114yes—
230C2×C4D4×Dic5312yes—
231C2×C4#1313318511yescharacteristic
232C2×C4#1313318912yes—
233C2×C4#1313319811yescharacteristic
234C2×C4C2×#1171992411yescharacteristic
235C2×C4C2×#1171992511yescharacteristic
236C2×C4#1313323712yes—
237C8—4242no—
238C8—2124no—
239C22—21no—
240C22—21no—
241C22#1681050311yescharacteristic
242C22C2×#1313314111yescharacteristic
243C22#1681061911yescharacteristic
244C4—21no—
245C4—21no—
246C4—22no—
247C4—24no—
248C4—42no—
249C4—42no—
250C4—82no—
251C4—81no—
252C4#1681085412yes—
253C4#1681086212yes—
254C2—21no—
255C2—21no—
256C2#2027513411yescharacteristic
257C2#2027513911yescharacteristic
258C2#2027525311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2581C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2581D53
hypocenterC53#9484
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2581

Complements

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

NH ≅ G/Ntype
150. C53#9484semidirect

Maximal subgroups

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

subgroupindex
#202837962
#202747112
#202731442
#202725732
C53×#25812
#948453