Finite Group Databasebeta

← #27284961

Subgroups — #27284961

Order 13568.

528 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 103 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#20284750C212yesmaximal
2#20278008C211yesmaximal, characteristic
3#20273143C212yesmaximal
4#20272544C211yesmaximal, characteristic
5C53×#1995C211yesmaximal, characteristic
6C2×C4×Dic106C2212yes—
7#16810956—22no—
8#16810915—22no—
9#16810830—22no—
10#16810778—22no—
11C4×#11719925—24no—
12#16810676C2212yes—
13#16810555—22no—
14#16810544—22no—
15C4×C4⋊Dic53—22no—
16#16810353—24no—
17C53×#533C2211yescharacteristic
18C53×#423C2211yescharacteristic
19C53×#376C2211yescharacteristic
20C53×#339—22no—
21C22×Dic106D412yes—
22C2×#11719925—24no—
23C4×Dic106—22no—
24C4×Dic106—22no—
25C4×Dic106—22no—
26C4×Dic106—22no—
27C2×C4⋊Dic53—24no—
28C2×C4⋊Dic53D412yes—
29C2×#11719919—22no—
30C2×C4×Dic53—22no—
31#13133150—24no—
32C4⋊#88322—22no—
33#13133141—24no—
34#13133140—24no—
35C22⋊#88322—22no—
36C4×#88322D414yes—
37#13133124—22no—
38#13133124—22no—
39C2×M16×C53—21no—
40C4⋊Q8×C53D412yes—
41#125×C53—21no—
42C22⋊Q8×C53—22no—
43C4×Q8×C53—22no—
44#117×C53—21no—
45C2×C42×C53D412yes—
46C2×C42×C53C2311yescharacteristic
47C4⋊C8×C53—22no—
48#95×C53—22no—
49C2×Q8×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53C2×D412yes—
52C22×C4×C53C2×D411yescharacteristic
53C22×Dic53—22no—
54C22×Dic53—22no—
55C2×Dic106—22no—
56C2×Dic106—22no—
57C2×Dic106—44no—
58M16×C53—42no—
59C2×C8×C53—41no—
60C4⋊C4×C53—22no—
61C4⋊C4×C53—22no—
62C4⋊C4×C53—42no—
63C22⋊C4×C53—41no—
64C42×C53—22no—
65C42×C53—41no—
66C42×C53Q1614yescomplemented
67C42×C53C2×D411yescharacteristic
68C42×C53C2×D412yes—
69C42×C53#4111yescharacteristic
70#11719925—48no—
71C4⋊Dic53—22no—
72C4⋊Dic53—22no—
73C4⋊Dic53—48no—
74#11719919—42no—
75#11719919—42no—
76C4×Dic53—42no—
77C4×Dic53—42no—
78C2×#88322—24no—
79C23×C53#12011yescharacteristic
80Q8×C53—44no—
81C2×C4×C53—24no—
82C2×C4×C53—42no—
83C2×C4×C53—44no—
84C2×C4×C53—42no—
85C2×C4×C53#12012yes—
86C2×C4×C53#12112yes—
87C2×C4×C53#12111yescharacteristic
88C2×C4×C53#12711yescharacteristic
89C2×C4×C53C2×Q1612yes—
90C2×C4×C53#13712yes—
91C2×Dic53—24no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53—44no—
95C2×Dic53—42no—
96Dic106—24no—
97Dic106—24no—
98Dic106—42no—
99Dic106—42no—
100C8×C53—81no—
101#88322—44no—
102#10017—531nomaximal, Sylow
103C22×C53—22no—
104C22×C53#39211yescharacteristic
105C22×C53#45112yes—
106C4×C53—22no—
107C4×C53—22no—
108C4×C53—44no—
109C4×C53—82no—
110C4×C53#46212yes—
111C4×C53#49411yescharacteristic
112C4×C53#49711yescharacteristic
113Dic53—44no—
114Dic53—82no—
115#3104—532no—
116#2399—531no—
117#1995D5311yescharacteristic
118#1487—532no—
119#1274—531no—
120C2×C53—22no—
121C2×C53#201012yes—
122C2×C53#201611yescharacteristic
123#533C2×D5311yescharacteristic
124C2×C4×Q8—532no—
125#484—1062no—
126#477—1062no—
127#462—1062no—
128#451—1062no—
129C4×Q16—1064no—
130#431—532no—
131#423C2×D5311yescharacteristic
132#398—1062no—
133#395—1062no—
134C4×C4⋊C4—1062no—
135#376C2×D5311yescharacteristic
136#339—22no—
137Q8⋊C8—1064no—
138C53#1001711yescharacteristic, complemented, Sylow
139C22×Q8—532no—
140C2×Q16—1064no—
141C2×M16—21no—
142C4⋊Q8D21212yes—
143#125—21no—
144C22⋊Q8—22no—
145C4×Q8—22no—
146C4×Q8—1062no—
147C4×Q8—1062no—
148C4×Q8—1062no—
149C4×Q8—1062no—
150#117—21no—
151C2×C4⋊C4—532no—
152C2×C4⋊C4—1064no—
153C2×C4⋊C4—1062no—
154C2×C42—1062no—
155C2×C42C22⋊D5312yes—
156C2×C42C22×D5311yescharacteristic
157#107—1064no—
158C4⋊C8—22no—
159C4⋊C8—1062no—
160#103—1064no—
161#103—1064no—
162C22⋊C8—1062no—
163C4×C8—534no—
164#95—22no—
165#95—1062no—
166#95—1062no—
167C2×Q8—22no—
168C2×Q8—1062no—
169C2×Q8—1062no—
170C2×Q8—2124no—
171C22×C4—22no—
172C22×C4—1062no—
173C22×C4—1062no—
174C22×C4D4×D5312yes—
175C22×C4C2×C22⋊D5311yescharacteristic
176Q16—2128no—
177M16—42no—
178C2×C8—41no—
179C2×C8—1064no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—1062no—
184C4⋊C4—1062no—
185C4⋊C4—2128no—
186C4⋊C4—2122no—
187C4⋊C4—2122no—
188C22⋊C4—41no—
189C42—22no—
190C42—41no—
191C42—2122no—
192C42—2122no—
193C42#1171992514yescomplemented
194C42C2×D21211yescharacteristic
195C42#1171994511yescharacteristic
196C42D4×D5312yes—
197C23#1313318511yescharacteristic
198Q8—44no—
199Q8—1064no—
200Q8—1064no—
201Q8—2122no—
202Q8—2122no—
203C2×C4—24no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—1064no—
208C2×C4—2124no—
209C2×C4—2122no—
210C2×C4—2122no—
211C2×C4—2122no—
212C2×C4#1313318712yes—
213C2×C4#1313319012yes—
214C2×C4#1313319111yescharacteristic
215C2×C4#1313321311yescharacteristic
216C2×C4C2×#1171992512yes—
217C2×C4#1313324712yes—
218C8—81no—
219C8—2124no—
220C22—22no—
221C22#1681053211yescharacteristic
222C22#1681077812yes—
223C4—22no—
224C4—22no—
225C4—44no—
226C4—82no—
227C4—4242no—
228C4—2124no—
229C4#1681083312yes—
230C4#1681100211yescharacteristic
231C4#1681101811yescharacteristic
232C2—22no—
233C2#2027578112yes—
234C2#2027581411yescharacteristic
subgroupG/H
centerC2×C4#13133213
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1995C2
socleC22×C53#392
prime radicalC23×C53#120
hypercenter#1995D53
hypocenterC53#10017
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
66. C42×C53Q16semidirect
138. C53#10017semidirect
193. C42#11719925semidirect

Maximal subgroups

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

subgroupindex
#202847502
#202780082
#202731432
#202725442
C53×#19952
#1001753