Finite Group Databasebeta

← #27287225

Subgroups — #27287225

Order 13568.

416 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 99 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#20278008C211yesmaximal, characteristic
2#20273214C212yesmaximal
3#20272616C212yesmaximal
4#20272544C211yesmaximal, characteristic
5C53×#1995C211yesmaximal, characteristic
6C2×C4×Dic106—21no—
7#16810676C2212yes—
8#16810555—22no—
9#16810544—22no—
10C4×C4⋊Dic53—21no—
11C4×C4⋊Dic53C2212yes—
12#16810360—22no—
13#16810360—22no—
14C53×#533C2211yescharacteristic
15C53×#423C2211yescharacteristic
16C53×#376C2211yescharacteristic
17C53×#339—21no—
18C53×#339—21no—
19C22×Dic106—21no—
20C4×Dic106—42no—
21C4×Dic106—42no—
22C2×C4⋊Dic53—22no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—22no—
25C2×#11719919—22no—
26C2×C4×Dic53—22no—
27C4⋊#88322—22no—
28C22⋊#88322—22no—
29C4×#88322—22no—
30#13133124—21no—
31#13133124—21no—
32#13133124D412yes—
33#13133124D412yes—
34C2×M16×C53—21no—
35C4⋊Q8×C53Q812yescomplemented
36#125×C53—21no—
37C22⋊Q8×C53—22no—
38C4×Q8×C53—22no—
39#117×C53—21no—
40C2×C42×C53D411yescharacteristic
41C2×C42×C53D411yescharacteristic
42C2×C42×C53C2311yescharacteristic
43C4⋊C8×C53—22no—
44#95×C53—22no—
45C2×Q8×C53—22no—
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53C2×D412yes—
49C22×C4×C53C2×D411yescharacteristic
50C22×Dic53—22no—
51C22×Dic53—22no—
52C2×Dic106—44no—
53C2×Dic106—42no—
54M16×C53—41no—
55M16×C53—41no—
56C2×C8×C53—41no—
57C4⋊C4×C53—22no—
58C4⋊C4×C53—22no—
59C4⋊C4×C53—42no—
60C22⋊C4×C53—41no—
61C42×C53—21no—
62C42×C53—21no—
63C42×C53—41no—
64C42×C53SD1612yes—
65C42×C53Q1612yes—
66C42×C53C2×Q811yescharacteristic
67C42×C53#4111yescharacteristic
68C42×C53#4112yes—
69C4⋊Dic53—44no—
70C4⋊Dic53—44no—
71C4⋊Dic53—42no—
72#11719919—44no—
73C4×Dic53—42no—
74C4×Dic53—42no—
75C2×#88322—42no—
76C23×C53#12011yescharacteristic
77Q8×C53—82no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—42no—
81C2×C4×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53C22⋊Q812yes—
85C2×C4×C53C22⋊Q811yescharacteristic
86C2×C4×C53#12412yes—
87C2×C4×C53#12411yescharacteristic
88C2×C4×C53C2×SD1611yescharacteristic
89C2×C4×C53C2×Q1611yescharacteristic
90C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
91C2×C4×C53#13711yescharacteristic
92C2×Dic53—24no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53—42no—
96C2×Dic53—42no—
97C2×Dic53—42no—
98Dic106—48no—
99C8×C53—81no—
100#88322—82no—
101#10371—531nomaximal, Sylow
102C22×C53—22no—
103C22×C53#39311yescharacteristic
104C22×C53#45011yescharacteristic
105C22×C53#45111yescharacteristic
106C4×C53—22no—
107C4×C53—22no—
108C4×C53—42no—
109C4×C53—42no—
110C4×C53—82no—
111C4×C53#47511yescharacteristic
112C4×C53#47711yescharacteristic
113C4×C53#48711yescharacteristic
114C4×C53#48911yescharacteristic
115Dic53—44no—
116Dic53—82no—
117#2399—531no—
118#1995D5311yescharacteristic
119#1503—532no—
120#1303—532no—
121#1274—531no—
122C2×C53—21no—
123C2×C53—21no—
124C2×C53#202411yescharacteristic
125C2×C53#203112yes—
126#533C2×D5311yescharacteristic
127C2×C4×Q8—1061no—
128#431—532no—
129#423C2×D5311yescharacteristic
130#398—1062no—
131#395—1062no—
132C4×C4⋊C4—532no—
133C4×C4⋊C4—1061no—
134#376C2×D5311yescharacteristic
135#339—21no—
136#339—21no—
137#328—1062no—
138#328—1062no—
139C53#1037111yescharacteristic, complemented, Sylow
140C22×Q8—1061no—
141C2×M16—21no—
142C4⋊Q8Dic10612yescomplemented
143#125—21no—
144C22⋊Q8—22no—
145C4×Q8—22no—
146C4×Q8—2122no—
147C4×Q8—2122no—
148#117—21no—
149C2×C4⋊C4—1062no—
150C2×C4⋊C4—1061no—
151C2×C4⋊C4—1062no—
152C2×C4⋊C4—1062no—
153C2×C42—1062no—
154C2×C42C22⋊D5311yescharacteristic
155C2×C42C22⋊D5311yescharacteristic
156C2×C42C22×D5311yescharacteristic
157C4⋊C8—22no—
158C4⋊C8—1062no—
159C22⋊C8—1062no—
160C4×C8—1062no—
161#95—22no—
162#95—532no—
163#95—532no—
164#95—1061no—
165#95—1061no—
166C2×Q8—22no—
167C2×Q8—2122no—
168C2×Q8—2124no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—1062no—
172C22×C4—1062no—
173C22×C4D4×D5312yes—
174C22×C4C2×C22⋊D5311yescharacteristic
175M16—41no—
176M16—41no—
177C2×C8—41no—
178C2×C8—2122no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4—2124no—
183C4⋊C4—2122no—
184C4⋊C4—2124no—
185C4⋊C4—2124no—
186C22⋊C4—41no—
187C42—21no—
188C42—21no—
189C42—41no—
190C42—2122no—
191C42—2122no—
192C42#1171992412yes—
193C42#1171992512yes—
194C42C2×Dic10611yescharacteristic
195C42#1171994511yescharacteristic
196C42#1171994712yes—
197C23#1313318511yescharacteristic
198Q8—82no—
199Q8—2128no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—1064no—
207C2×C4—2122no—
208C2×C4—2122no—
209C2×C4—2122no—
210C2×C4—2122no—
211C2×C4—2122no—
212C2×C4#1313319412yes—
213C2×C4#1313319611yescharacteristic
214C2×C4#1313320312yes—
215C2×C4#1313320411yescharacteristic
216C2×C4C2×#1171992411yescharacteristic
217C2×C4C2×#1171992511yescharacteristic
218C2×C4#1313324211yescharacteristic
219C2×C4#1313324711yescharacteristic
220C8—81no—
221C8—4242no—
222C22—22no—
223C22#1681053511yescharacteristic
224C22#1681077311yescharacteristic
225C22#1681077811yescharacteristic
226C4—22no—
227C4—22no—
228C4—42no—
229C4—42no—
230C4—82no—
231C4—4242no—
232C4—2124no—
233C4#1681090811yescharacteristic
234C4#1681091811yescharacteristic
235C4#1681097311yescharacteristic
236C4#1681098211yescharacteristic
237C2—21no—
238C2—21no—
239C2#2027585611yescharacteristic
240C2#2027588512yes—
subgroupG/H
centerC2×C4#13133204
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1995C2
socleC22×C53#393
prime radicalC23×C53#120
hypercenter#1995D53
hypocenterC53#10371
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1995

Complements

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

NH ≅ G/Ntype
35. C4⋊Q8×C53Q8semidirect
139. C53#10371semidirect
142. C4⋊Q8Dic106semidirect

Maximal subgroups

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

subgroupindex
#202780082
#202732142
#202726162
#202725442
C53×#19952
#1037153