Finite Group Databasebeta

← #27281512

Subgroups — #27281512

Order 13568.

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

normal subgroups only

This group has 450 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#20279927C211yesmaximal, characteristic
2#20274711C211yesmaximal, characteristic
3#20273399C212yesmaximal
4#20272617C212yesmaximal
5C53×#2983C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7#16810647C2212yes—
8C2×#13133136—21no—
9#16810562C414yes—
10#16810530—21no—
11#16810529—22no—
12#16810488—21no—
13C4×C4⋊Dic53C2211yescharacteristic
14#16810392—21no—
15#16810370—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—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53C2×C412yes—
27C2×C4×Dic53—21no—
28C4⋊#88322—22no—
29C4⋊#88322—22no—
30#13133136—42no—
31#13133124—24no—
32#13133124—24no—
33#13133124C2×C412yes—
34C22×Q8×C53D411yescharacteristic
35C4×Q8×C53—22no—
36C4×Q8×C53—22no—
37C2×C4⋊C4×C53C2×C412yes—
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C42×C53D412yes—
40C2×C42×C53C2311yescharacteristic
41C4⋊C8×C53—24no—
42C4⋊C8×C53D414yes—
43C22⋊C8×C53—22no—
44#97×C53—24no—
45C4×C8×C53—22no—
46C2×Q8×C53—22no—
47C2×Q8×C53—21no—
48C2×Q8×C53—21no—
49C2×Q8×C53—22no—
50C22×C4×C53C22⋊C412yes—
51C22×C4×C53C22⋊C412yes—
52C22×C4×C53C22×C411yescharacteristic
53C22×C4×C53C2×D411yescharacteristic
54C22×C4×C53C2×D411yescharacteristic
55C22×Dic53—22no—
56C22×Dic53—22no—
57C2×Dic106—44no—
58C2×Dic106—41no—
59C2×Dic106—41no—
60C2×C8×C53—24no—
61C4⋊C4×C53—24no—
62C4⋊C4×C53—22no—
63C42×C53—22no—
64C42×C53—22no—
65C42×C53C2×D412yes—
66C42×C53#4112yes—
67C4⋊Dic53—24no—
68C4⋊Dic53—42no—
69C4⋊Dic53—42no—
70C4×Dic53—41no—
71C4×Dic53—41no—
72#11719917—42no—
73#11719917—42no—
74C2×#88322—42no—
75C23×C53C2×C22⋊C411yescharacteristic
76Q8×C53—42no—
77Q8×C53—42no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—41no—
84C2×C4×C53—42no—
85C2×C4×C53C4×D412yes—
86C2×C4×C53#12011yescharacteristic
87C2×C4×C53#12112yes—
88C2×C4×C53#12311yescharacteristic
89C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
90C2×C4×C53#13712yes—
91C2×C4×C53#13711yescharacteristic
92C2×Dic53—44no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53—42no—
96C2×Dic53—42no—
97Dic106—82no—
98Dic106—82no—
99C8×C53—44no—
100#88322—82no—
101#9404—531nomaximal, Sylow
102C22×C53—21no—
103C22×C53—21no—
104C22×C53#38611yescharacteristic
105C22×C53#41711yescharacteristic
106C22×C53#41911yescharacteristic
107C4×C53—21no—
108C4×C53—22no—
109C4×C53—42no—
110C4×C53—42no—
111C4×C53#45412yes—
112C4×C53#46812yes—
113C4×C53#47012yes—
114Dic53—82no—
115Dic53—82no—
116#2983D5311yescharacteristic
117#2568—531no—
118#1801—531no—
119#1549—532no—
120#1303—532no—
121C2×C53—21no—
122C2×C53—21no—
123C2×C53#188811yescharacteristic
124C2×C53#189411yescharacteristic
125C2×C53#191511yescharacteristic
126C2×C4⋊Q8—531no—
127C2×C4×Q8C2×D5311yescharacteristic
128#446—24no—
129#432—22no—
130#431C2×D5312yes—
131#423—532no—
132C2×#101—1061no—
133#400—534no—
134#391—1061no—
135#391—1062no—
136#382—1061no—
137C4×C4⋊C4—531no—
138#343—1061no—
139#332—1064no—
140C53#940411yescharacteristic, complemented, Sylow
141C22×Q8—1061no—
142C22×Q8D21211yescharacteristic
143C2×M16—1062no—
144C4⋊Q8—1062no—
145C4⋊Q8—1062no—
146C4×Q8—22no—
147C4×Q8—22no—
148C2×C4⋊C4—532no—
149C2×C4⋊C4—1061no—
150C2×C4⋊C4—1061no—
151C2×C4⋊C4C4×D5312yes—
152C2×C4⋊C4D21211yescharacteristic
153C2×C42—1061no—
154C2×C42C22⋊D5312yes—
155C2×C42C22×D5311yescharacteristic
156C4⋊C8—24no—
157C4⋊C8—1062no—
158C4⋊C8—1062no—
159C4⋊C8C22⋊D5314yes—
160#101—2122no—
161C22⋊C8—22no—
162#97—24no—
163C4×C8—22no—
164#95—532no—
165#95—1064no—
166#95—1064no—
167C2×Q8—22no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—22no—
171C2×Q8—2124no—
172C2×Q8—2121no—
173C2×Q8—2121no—
174C22×C4—1062no—
175C22×C4—1062no—
176C22×C4#1171992112yes—
177C22×C4#1171992112yes—
178C22×C4C2×C4×D5311yescharacteristic
179C22×C4C2×D21211yescharacteristic
180C22×C4C2×C22⋊D5311yescharacteristic
181M16—2122no—
182M16—2122no—
183C2×C8—24no—
184C2×C8—2122no—
185C4⋊C4—24no—
186C4⋊C4—22no—
187C4⋊C4—1064no—
188C4⋊C4—2122no—
189C4⋊C4—2122no—
190C42—22no—
191C42—22no—
192C42—2121no—
193C42—2121no—
194C42#1171994512yes—
195C42C2×C22⋊D5312yes—
196C23C2×#1171992111yescharacteristic
197Q8—42no—
198Q8—42no—
199Q8—4242no—
200Q8—4242no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—2122no—
209C2×C4—2122no—
210C2×C4—2124no—
211C2×C4—2122no—
212C2×C4—2122no—
213C2×C4C4×C22⋊D5312yes—
214C2×C4#1313318611yescharacteristic
215C2×C4#1313319112yes—
216C2×C4#1313320011yescharacteristic
217C2×C4#1313324511yescharacteristic
218C2×C4#1313325012yes—
219C2×C4#1313325011yescharacteristic
220C8—44no—
221C8—4242no—
222C22—21no—
223C22—21no—
224C22#1681050211yescharacteristic
225C22#1681062511yescharacteristic
226C22#1681063311yescharacteristic
227C4—22no—
228C4—21no—
229C4—42no—
230C4—42no—
231C4—4242no—
232C4—4242no—
233C4#1681079612yes—
234C4#1681086912yes—
235C4#1681088112yes—
236C2—21no—
237C2—21no—
238C2#2027511811yescharacteristic
239C2#2027514811yescharacteristic
240C2#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#9404
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
140. C53#9404semidirect

Maximal subgroups

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

subgroupindex
#202799272
#202747112
#202733992
#202726172
C53×#29832
#940453