Finite Group Databasebeta

← #27281502

Subgroups — #27281502

Order 13568.

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

normal subgroups only

This group has 486 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
1#20279700C211yesmaximal, characteristic
2#20274711C211yesmaximal, characteristic
3#20273386C212yesmaximal
4#20272617C212yesmaximal
5C53×#2983C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7C2×#13133215—21no—
8#16810647C2212yes—
9C2×#13133136—21no—
10#16810553C414yes—
11#16810528—21no—
12#16810521—22no—
13C4×C4⋊Dic53C2211yescharacteristic
14#16810392—21no—
15#16810368—24no—
16C2×C4×Q8×C53C2211yescharacteristic
17C53×#446—24no—
18C53×#432—22no—
19C53×#431C2212yes—
20C22×Dic106—21no—
21C2×#11719917—22no—
22#13133217—22no—
23#13133217—22no—
24#13133215—44no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53C2×C412yes—
28C2×#11719919—24no—
29C2×C4×Dic53—21no—
30C4⋊#88322—22no—
31C4⋊#88322—22no—
32#13133136—42no—
33#13133124—24no—
34#13133124C2×C412yes—
35C22×Q8×C53D411yescharacteristic
36C4×Q8×C53—22no—
37C4×Q8×C53—22no—
38C2×C4⋊C4×C53C2×C412yes—
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C42×C53D412yes—
41C2×C42×C53C2311yescharacteristic
42C4⋊C8×C53—24no—
43C4⋊C8×C53D414yes—
44C22⋊C8×C53—22no—
45#97×C53—24no—
46C4×C8×C53—22no—
47C2×Q8×C53—22no—
48C2×Q8×C53—21no—
49C2×Q8×C53—22no—
50C2×Q8×C53—21no—
51C22×C4×C53C22⋊C412yes—
52C22×C4×C53C22⋊C412yes—
53C22×C4×C53C22×C411yescharacteristic
54C22×C4×C53C2×D411yescharacteristic
55C22×C4×C53C2×D411yescharacteristic
56C22×Dic53—22no—
57C22×Dic53—22no—
58C2×Dic106—42no—
59C2×Dic106—42no—
60C2×Dic106—41no—
61C2×Dic106—41no—
62C2×C8×C53—24no—
63C4⋊C4×C53—24no—
64C4⋊C4×C53—22no—
65C42×C53—22no—
66C42×C53—22no—
67C42×C53C2×D412yes—
68C42×C53#4112yes—
69C4⋊Dic53—24no—
70C4⋊Dic53—24no—
71C4⋊Dic53—41no—
72C4⋊Dic53—41no—
73#11719919—48no—
74C4×Dic53—42no—
75#11719917—42no—
76#11719917—42no—
77C2×#88322—42no—
78C23×C53C2×C22⋊C411yescharacteristic
79Q8×C53—42no—
80Q8×C53—42no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—41no—
87C2×C4×C53—42no—
88C2×C4×C53C4×D412yes—
89C2×C4×C53#12011yescharacteristic
90C2×C4×C53#12112yes—
91C2×C4×C53#12311yescharacteristic
92C2×C4×C53AGL(1,ℤ/8)12yes—
93C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
94C2×C4×C53#13711yescharacteristic
95C2×Dic53—24no—
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101Dic106—82no—
102Dic106—82no—
103C8×C53—44no—
104#88322—82no—
105#9402—531nomaximal, Sylow
106C22×C53—21no—
107C22×C53—21no—
108C22×C53#38611yescharacteristic
109C22×C53#41711yescharacteristic
110C22×C53#41811yescharacteristic
111C4×C53—21no—
112C4×C53—22no—
113C4×C53—42no—
114C4×C53—42no—
115C4×C53#45612yes—
116C4×C53#46812yes—
117C4×C53#46912yes—
118Dic53—44no—
119Dic53—82no—
120#2983D5311yescharacteristic
121#2546—531no—
122#1801—531no—
123#1546—532no—
124#1303—532no—
125C2×C53—21no—
126C2×C53—21no—
127C2×C53#188611yescharacteristic
128C2×C53#189511yescharacteristic
129C2×C53#191511yescharacteristic
130C2×C4⋊Q8—531no—
131C2×C4⋊Q8—1061no—
132C2×C4×Q8C2×D5311yescharacteristic
133#446—24no—
134#432—22no—
135#431C2×D5312yes—
136#423—532no—
137C2×#101—1061no—
138#398—534no—
139#391—1061no—
140#389—1062no—
141C4×C4⋊C4—531no—
142#343—1061no—
143#332—1064no—
144C53#940211yescharacteristic, complemented, Sylow
145C22×Q8—1061no—
146C22×Q8D21211yescharacteristic
147C2×M16—1062no—
148C4⋊Q8—1062no—
149C4⋊Q8—1062no—
150C4⋊Q8—2124no—
151C4×Q8—22no—
152C4×Q8—22no—
153C2×C4⋊C4—532no—
154C2×C4⋊C4—1061no—
155C2×C4⋊C4—1061no—
156C2×C4⋊C4—1064no—
157C2×C4⋊C4C4×D5312yes—
158C2×C4⋊C4D21211yescharacteristic
159C2×C42—1061no—
160C2×C42C22⋊D5312yes—
161C2×C42C22×D5311yescharacteristic
162C4⋊C8—24no—
163C4⋊C8—1062no—
164C4⋊C8—1062no—
165C4⋊C8C22⋊D5314yes—
166#101—2122no—
167C22⋊C8—22no—
168#97—24no—
169C4×C8—22no—
170#95—532no—
171#95—1064no—
172C2×Q8—22no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—21no—
176C2×Q8—2122no—
177C2×Q8—2122no—
178C2×Q8—2121no—
179C2×Q8—2121no—
180C22×C4—1062no—
181C22×C4—1062no—
182C22×C4#1171992112yes—
183C22×C4#1171992112yes—
184C22×C4C2×C4×D5311yescharacteristic
185C22×C4C2×D21211yescharacteristic
186C22×C4C2×C22⋊D5311yescharacteristic
187M16—2122no—
188M16—2122no—
189C2×C8—24no—
190C2×C8—2122no—
191C4⋊C4—24no—
192C4⋊C4—22no—
193C4⋊C4—1064no—
194C4⋊C4—1064no—
195C4⋊C4—2128no—
196C4⋊C4—2121no—
197C4⋊C4—2121no—
198C42—22no—
199C42—22no—
200C42—2122no—
201C42#1171994512yes—
202C42C2×C22⋊D5312yes—
203C23C2×#1171992111yescharacteristic
204Q8—42no—
205Q8—42no—
206Q8—4242no—
207Q8—4242no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—1064no—
216C2×C4—2122no—
217C2×C4—2122no—
218C2×C4—2122no—
219C2×C4—2122no—
220C2×C4—2122no—
221C2×C4C4×C22⋊D5312yes—
222C2×C4#1313318611yescharacteristic
223C2×C4#1313319112yes—
224C2×C4#1313320011yescharacteristic
225C2×C4#1313324512yes—
226C2×C4#1313324511yescharacteristic
227C2×C4#1313325011yescharacteristic
228C8—44no—
229C8—4242no—
230C22—21no—
231C22—21no—
232C22#1681050211yescharacteristic
233C22#1681062511yescharacteristic
234C22#1681062911yescharacteristic
235C4—22no—
236C4—21no—
237C4—42no—
238C4—42no—
239C4—4242no—
240C4—2124no—
241C4#1681080612yes—
242C4#1681086912yes—
243C4#1681087612yes—
244C2—21no—
245C2—21no—
246C2#2027510811yescharacteristic
247C2#2027515311yescharacteristic
248C2#2027526711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2983C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2983D53
hypocenterC53#9402
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
144. C53#9402semidirect

Maximal subgroups

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

subgroupindex
#202797002
#202747112
#202733862
#202726172
C53×#29832
#940253