Finite Group Databasebeta

← #27284962

Subgroups — #27284962

Order 13568.

528 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 105 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#20284754C212yesmaximal
2#20275696C211yesmaximal, characteristic
3#20273144C212yesmaximal
4#20272544C211yesmaximal, characteristic
5C53×#2399C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16810958—22no—
8#16810918—22no—
9#16810833—22no—
10#16810779—22no—
11C4×#11719925—24no—
12#16810676C2212yes—
13#16810647C2211yescharacteristic
14#16810383—22no—
15Q8⋊#88322—24no—
16C2×C4×Q8×C53C2212yes—
17C53×#398—22no—
18C53×#395—22no—
19C4×C4⋊C4×C53—22no—
20C53×#376C2211yescharacteristic
21C2×#11719925—24no—
22C2×#11719917—21no—
23#13133217D412yes—
24#13133209—21no—
25#13133196—22no—
26C4×Dic106—22no—
27#13133176—21no—
28#13133150—24no—
29C4⋊#88322—22no—
30C4⋊#88322—22no—
31#13133141—24no—
32#13133140—24no—
33C22⋊#88322—22no—
34C4×#88322D414yes—
35C22×Q8×C53D412yes—
36C4×Q8×C53—22no—
37C4×Q8×C53—22no—
38C4×Q8×C53—22no—
39C4×Q8×C53—22no—
40C2×C4⋊C4×C53—24no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53D412yes—
43C2×C42×C53—22no—
44C2×C42×C53D412yes—
45C2×C42×C53C2311yescharacteristic
46#95×C53—22no—
47#95×C53—22no—
48#95×C53—22no—
49C2×Q8×C53—22no—
50C2×Q8×C53—22no—
51C2×Q8×C53—44no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53C2×D412yes—
56C22×C4×C53C2×D411yescharacteristic
57C2×Dic106—22no—
58C4⋊C4×C53—22no—
59C4⋊C4×C53—22no—
60C4⋊C4×C53—48no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—42no—
63C42×C53—22no—
64C42×C53—42no—
65C42×C53—42no—
66C42×C53Q1614yescomplemented
67C42×C53C2×D411yescharacteristic
68C42×C53C2×D412yes—
69C42×C53#4111yescharacteristic
70C22⋊Dic53—41no—
71#11719925—48no—
72C4⋊Dic53—22no—
73C4⋊Dic53—22no—
74#11719919—42no—
75C4×Dic53—41no—
76#11719917—42no—
77C2×#88322—24no—
78C2×#88322—41no—
79C23×C53#12011yescharacteristic
80Q8×C53—24no—
81Q8×C53—24no—
82Q8×C53—42no—
83Q8×C53—42no—
84C2×C4×C53—24no—
85C2×C4×C53—24no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—44no—
89C2×C4×C53—42no—
90C2×C4×C53—44no—
91C2×C4×C53—42no—
92C2×C4×C53#12012yes—
93C2×C4×C53#12111yescharacteristic
94C2×C4×C53#12112yes—
95C2×C4×C53#12711yescharacteristic
96C2×C4×C53C2×Q1612yes—
97C2×C4×C53#13712yes—
98C2×Dic53—42no—
99Dic106—44no—
100#88322—44no—
101#88322—81no—
102#10017—531nomaximal, Sylow
103C22×C53—22no—
104C22×C53#39211yescharacteristic
105C22×C53#45112yes—
106C4×C53—22no—
107C4×C53—22no—
108C4×C53—44no—
109C4×C53—44no—
110C4×C53—82no—
111C4×C53#46212yes—
112C4×C53#49411yescharacteristic
113C4×C53#49711yescharacteristic
114Dic53—82no—
115#3104—532no—
116#2399D5311yescharacteristic
117#1995—531no—
118#1487—532no—
119#1274—531no—
120C2×C53—22no—
121C2×C53#201012yes—
122C2×C53#201611yescharacteristic
123#533—531no—
124C2×C4×Q8C2×D5312yes—
125#484—1062no—
126#477—1062no—
127#462—1062no—
128#451—1062no—
129C4×Q16—1064no—
130#431—532no—
131#423—531no—
132#398—22no—
133#395—22no—
134C4×C4⋊C4—22no—
135#376C2×D5311yescharacteristic
136#339—1062no—
137Q8⋊C8—1064no—
138C53#1001711yescharacteristic, complemented, Sylow
139C22×Q8C22⋊D5312yes—
140C2×Q16—1064no—
141C2×M16—1061no—
142C4⋊Q8—532no—
143#125—1061no—
144C22⋊Q8—1062no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—1062no—
150#117—1061no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C22⋊D5312yes—
154C2×C42—22no—
155C2×C42C22⋊D5312yes—
156C2×C42C22×D5311yescharacteristic
157#107—1064no—
158C4⋊C8—1062no—
159C4⋊C8—1062no—
160#103—1064no—
161#103—1064no—
162C22⋊C8—1062no—
163C4×C8—534no—
164#95—22no—
165#95—22no—
166#95—22no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—44no—
170C2×Q8—1062no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4C2×C22⋊D5312yes—
175C22×C4C2×C22⋊D5311yescharacteristic
176Q16—2128no—
177M16—2122no—
178C2×C8—1064no—
179C2×C8—2121no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—48no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—1062no—
186C4⋊C4—1062no—
187C4⋊C4—2122no—
188C22⋊C4—2121no—
189C42—22no—
190C42—42no—
191C42—42no—
192C42—2121no—
193C42#1171992514yescomplemented
194C42D4×D5311yescharacteristic
195C42D4×D5312yes—
196C42#1171994711yescharacteristic
197C23#1313318611yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—42no—
201Q8—42no—
202Q8—2124no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—2122no—
212C2×C4#1313318512yes—
213C2×C4#1313318811yescharacteristic
214C2×C4#1313318912yes—
215C2×C4#1313321311yescharacteristic
216C2×C4C2×#1171992512yes—
217C2×C4#1313324812yes—
218C8—4241no—
219C8—2124no—
220C22—22no—
221C22#1681053111yescharacteristic
222C22#1681077912yes—
223C4—22no—
224C4—22no—
225C4—44no—
226C4—44no—
227C4—82no—
228C4—4242no—
229C4#1681083012yes—
230C4#1681100211yescharacteristic
231C4#1681101511yescharacteristic
232C2—22no—
233C2#2027578012yes—
234C2#2027581511yescharacteristic
subgroupG/H
centerC2×C4#13133213
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2399C2
socleC22×C53#392
prime radicalC23×C53#120
hypercenter#2399D53
hypocenterC53#10017
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

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
#202847542
#202756962
#202731442
#202725442
C53×#23992
#1001753