Finite Group Databasebeta

← #27300103

Subgroups — #27300103

Order 13568.

432 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#16810422C212yesmaximal
2#20274711C212yesmaximal
3#20272842C212yesmaximal
4C53×#2327C211yesmaximal, characteristic
5#16810647—22no—
6C2×#13133136C2214yes—
7#16810422—24no—
8#16810392—22no—
9C2×C4⋊Q8×C53C2211yescharacteristic
10C2×C4×Q8×C53C411yescharacteristic
11C2×C4×Q8×C53C2211yescharacteristic
12C53×#391C411yescharacteristic
13C53×#391C2211yescharacteristic
14C53×#387—22no—
15C53×#387—22no—
16C53×#382C411yescharacteristic
17C53×#376C411yescharacteristic
18C2×#11719917—24no—
19C4⋊#88322—44no—
20#13133136—24no—
21#13133136—24no—
22C22×Q8×C53C2×C411yescharacteristic
23C22×Q8×C53C2311yescharacteristic
24C4⋊Q8×C53—22no—
25C4⋊Q8×C53D412yes—
26C4⋊Q8×C53Q812yes—
27C4×Q8×C53—42no—
28C4×Q8×C53—42no—
29C2×C4⋊C4×C53—21no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53C2×C411yescharacteristic
32C2×C4⋊C4×C53C2×C411yescharacteristic
33C2×C4⋊C4×C53D412yes—
34C2×C42×C53—21no—
35C2×C42×C53C2×C411yescharacteristic
36C2×C42×C53C2×C411yescharacteristic
37C2×C42×C53C2×C411yescharacteristic
38C2×C42×C53D412yes—
39#95×C53—21no—
40#95×C53—21no—
41#95×C53—22no—
42#95×C53—22no—
43#95×C53—22no—
44C2×Q8×C53—21no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53C4⋊C412yes—
50C2×Q8×C53C2×D411yescharacteristic
51C2×Q8×C53C2×Q811yescharacteristic
52C2×Q8×C53#4112yes—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53C22⋊C412yes—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×C4×C53C2×D412yes—
62C4⋊C4×C53—24no—
63C4⋊C4×C53—24no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—42no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—42no—
68C42×C53—22no—
69C42×C53—24no—
70C42×C53—21no—
71C42×C53—42no—
72C42×C53—41no—
73C42×C53C4⋊C412yes—
74#11719917—44no—
75#11719917—44no—
76C2×#88322—44no—
77C23×C53C2×C22⋊C411yescharacteristic
78Q8×C53—41no—
79Q8×C53—41no—
80Q8×C53—42no—
81Q8×C53—42no—
82Q8×C53—42no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—42no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—42no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53#9912yes—
104C2×C4×C53C2×C4⋊C411yescharacteristic
105C2×C4×C53#11711yescharacteristic
106C2×C4×C53#12112yes—
107C2×C4×C53C22⋊Q812yes—
108#88322—84no—
109#12804—531nomaximal, Sylow
110C22×C53—21no—
111C22×C53—21no—
112C22×C53#38011yescharacteristic
113C22×C53C2×#9911yescharacteristic
114C22×C53#41011yescharacteristic
115C4×C53—22no—
116C4×C53—21no—
117C4×C53—21no—
118C4×C53—22no—
119C4×C53—22no—
120C4×C53—42no—
121C4×C53—42no—
122C4×C53—41no—
123C4×C53—42no—
124C4×C53—81no—
125C4×C53—81no—
126#2327D5311yescharacteristic
127C2×#356—532no—
128#1801—532no—
129#1394—532no—
130C2×C53—21no—
131C2×C53—21no—
132C2×C53#179911yescharacteristic
133C2×C53#213311yescharacteristic
134C2×C53#213511yescharacteristic
135C2×C4⋊Q8C2×D5311yescharacteristic
136C2×C4×Q8Dic5311yescharacteristic
137C2×C4×Q8C2×D5311yescharacteristic
138#423—1062no—
139C2×#101—534no—
140#391Dic5311yescharacteristic
141#391C2×D5311yescharacteristic
142#387—22no—
143#387—22no—
144#382Dic5311yescharacteristic
145#376Dic5311yescharacteristic
146#356—1064no—
147#343—1062no—
148C53#1280411yescharacteristic, complemented, Sylow
149C22×Q8C2×Dic5311yescharacteristic
150C22×Q8C22×D5311yescharacteristic
151C2×M16—1064no—
152C4⋊Q8—22no—
153C4⋊Q8Dic10612yes—
154C4⋊Q8D21212yes—
155C4×Q8—42no—
156C4×Q8—42no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4C2×Dic5311yescharacteristic
160C2×C4⋊C4C2×Dic5311yescharacteristic
161C2×C4⋊C4C22⋊D5312yes—
162C2×C42—21no—
163C2×C42C2×Dic5311yescharacteristic
164C2×C42C2×Dic5311yescharacteristic
165C2×C42C2×Dic5311yescharacteristic
166C2×C42C22⋊D5312yes—
167C4⋊C8—2124no—
168#101—1064no—
169#101—1064no—
170#95—22no—
171#95—21no—
172#95—22no—
173#95—22no—
174#95—21no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8C4⋊Dic5312yes—
181C2×Q8C2×Dic10611yescharacteristic
182C2×Q8C2×D21211yescharacteristic
183C2×Q8#1171994512yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4C22⋊Dic5312yes—
189C22×C4C22⋊Dic5311yescharacteristic
190C22×C4C22⋊Dic5311yescharacteristic
191C22×C4C22×Dic5311yescharacteristic
192C22×C4C2×C22⋊D5312yes—
193M16—2124no—
194M16—2124no—
195C2×C8—2124no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C42—24no—
203C42—22no—
204C42—21no—
205C42—41no—
206C42—42no—
207C42C4⋊Dic5312yes—
208C23C2×C22⋊Dic5311yescharacteristic
209Q8—41no—
210Q8—41no—
211Q8—42no—
212Q8—42no—
213Q8—42no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4#1313313212yes—
235C2×C4C2×C4⋊Dic5311yescharacteristic
236C2×C4#1313317611yescharacteristic
237C2×C4#1313319112yes—
238C2×C4#1313319612yes—
239C8—4244no—
240C22—21no—
241C22—21no—
242C22#1681048311yescharacteristic
243C22C2×#1313313211yescharacteristic
244C22#1681059711yescharacteristic
245C4—22no—
246C4—21no—
247C4—21no—
248C4—22no—
249C4—22no—
250C4—42no—
251C4—42no—
252C4—41no—
253C4—42no—
254C4—81no—
255C4—81no—
256C2—21no—
257C2—21no—
258C2#2027470311yescharacteristic
259C2#2027640411yescharacteristic
260C2#2027641011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#2327C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Q8×C53C23
hypercenter#2327D53
hypocenterC53#12804
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2327

Complements

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

NH ≅ G/Ntype
148. C53#12804semidirect

Maximal subgroups

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

subgroupindex
C2×#168104222
#202747112
#202728422
C53×#23272
#1280453