Finite Group Databasebeta

← #27279418

Subgroups — #27279418

Order 13568.

426 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 107 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#20285040C211yesmaximal, characteristic
2#20273310C211yesmaximal, characteristic, complemented
3#20273131C212yesmaximal
4#20272551C212yesmaximal, complemented
5C53×#2986C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7#16810980—22no—
8#16810975C412yes—
9#16810971C412yes—
10#16810617—22no—
11C4×#11719917C2212yes—
12C2×C4×#88322C2211yescharacteristic
13#16810365—22no—
14#16810350—22no—
15#16810350—22no—
16C53×#533C2211yescharacteristic
17C53×#446—22no—
18C8×Q8×C53—22no—
19C53×#433—22no—
20C53×#431C2212yes—
21C2×#11719945—21no—
22C2×#11719917—22no—
23C22×#88322—21no—
24#13133217C2×C411yescharacteristic
25#13133214C2×C411yescharacteristic
26#13133205—21no—
27#13133191—22no—
28C4×D212—22no—
29#13133140—24no—
30#13133137—24no—
31#13133128—22no—
32C4×#88322—21no—
33C4×#88322—22no—
34C4×#88322C2×C412yes—
35C4⋊Q8×C53C2×C412yes—
36#125×C53D412yes—
37C22⋊Q8×C53—22no—
38C4×Q8×C53—22no—
39#117×C53D412yes—
40C2×C42×C53C2311yescharacteristic
41C4⋊C8×C53—24no—
42C4⋊C8×C53D412yes—
43C4⋊C8×C53D412yes—
44C22⋊C8×C53—22no—
45#97×C53—22no—
46C4×C8×C53—22no—
47C4×C8×C53—22no—
48C2×Q8×C53—22no—
49C22×C4×C53C2×D411yescharacteristic
50C22×C4×C53#4112yes—
51C2×C22⋊D53—41no—
52#11719945—44no—
53C2×D212—21no—
54C2×D212—22no—
55C2×C4×D53—41no—
56C2×Dic106—21no—
57C2×C8×C53—22no—
58C2×C8×C53—22no—
59C4⋊C4×C53—22no—
60C4⋊C4×C53—24no—
61C4⋊C4×C53C22⋊C414yes—
62C22⋊C4×C53—22no—
63C42×C53—22no—
64C42×C53C22×C411yescharacteristic
65C42×C53C2×D411yescharacteristic
66C42×C53C2×D412yes—
67#11719921—42no—
68C4⋊Dic53—22no—
69#11719917—24no—
70#11719917—42no—
71C2×#88322—22no—
72C2×#88322—42no—
73C2×#88322—42no—
74C23×C53#12311yescharacteristic
75C22×D53—41no—
76Q8×C53—42no—
77Q8×C53—42no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—24no—
82C2×C4×C53C2×C22⋊C411yescharacteristic
83C2×C4×C53C4×D412yes—
84C2×C4×C53#12011yescharacteristic
85C2×C4×C53#12112yes—
86C22⋊D53—82no—
87C2×Dic53—41no—
88D212—42no—
89D212—42no—
90D212—42no—
91C4×D53—82no—
92Dic106—42no—
93C8×C53—42no—
94C8×C53—42no—
95#88322—42no—
96#88322—44no—
97#9070—531nomaximal, Sylow
98C22×C53—22no—
99C22×C53—21no—
100C22×C53#38611yescharacteristic
101C2×D53—82no—
102C2×D53—81no—
103C4×C53—22no—
104C4×C53—21no—
105C4×C53—44no—
106C4×C53#44312yes—
107C4×C53#44412yes—
108C4×C53#45412yes—
109Dic53—81no—
110#3131—531no—
111#2986D5311yescharacteristic, complemented
112#1526—531no—
113#1484—532no—
114#1278—532no—
115C2×C53—41no—
116C2×C53#187612yes—
117C2×C53#188811yescharacteristic
118D53—161no—
119#533C2×D5311yescharacteristic
120#532—531no—
121#488—1062no—
122#487—532no—
123#486—532no—
124#446—22no—
125C8×Q8—22no—
126#433—22no—
127#431C2×D5312yes—
128#416—1062no—
129C4×M16—532no—
130C2×C4×C8—531no—
131#330—1062no—
132D4⋊C8—1062no—
133D4⋊C8—1062no—
134C53#907011yescharacteristic, complemented, Sylow
135C2×#41—1061no—
136C2×M16—1062no—
137C22×C8—1061no—
138C4⋊Q8—531no—
139C4⋊Q8C4×D5312yes—
140#127—531no—
141#125D21212yes—
142#124—1061no—
143C22⋊Q8—22no—
144#121—1062no—
145C4×Q8—22no—
146C4×D4—1062no—
147#117C22⋊D5312yes—
148C2×C42C22×D5311yescharacteristic
149C4⋊C8—24no—
150C4⋊C8C22⋊D5312yes—
151C4⋊C8C22⋊D5312yes—
152#103—1064no—
153#102—1064no—
154C22⋊C8—22no—
155#97—22no—
156#97—1062no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—532no—
160C4×C8—1061no—
161C4×C8—1062no—
162#41—2124no—
163C2×Q8—22no—
164C2×Q8—1061no—
165C2×D4—1061no—
166C2×D4—1062no—
167C2×D4—2121no—
168C22×C4—2121no—
169C22×C4#1171994512yes—
170C22×C4C2×C22⋊D5311yescharacteristic
171M16—1064no—
172M16—2122no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—1062no—
176C2×C8—2122no—
177C2×C8—2122no—
178C4⋊C4—22no—
179C4⋊C4—24no—
180C4⋊C4—1062no—
181C4⋊C4#1171992114yes—
182C22⋊C4—22no—
183C22⋊C4—2122no—
184C42—22no—
185C42C2×C4×D5311yescharacteristic
186C42C2×D21211yescharacteristic
187C42C2×C22⋊D5312yes—
188C23—2121no—
189C23#1313320011yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—2122no—
193D4—4242no—
194D4—2122no—
195D4—2122no—
196D4—2122no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—4242no—
202C2×C4—2121no—
203C2×C4C2×#1171992111yescharacteristic
204C2×C4C4×C22⋊D5312yes—
205C2×C4#1313318611yescharacteristic
206C2×C4#1313319112yes—
207C8—42no—
208C8—42no—
209C8—2122no—
210C8—2124no—
211C22—22no—
212C22—21no—
213C22—4242no—
214C22—4241no—
215C22#1681050211yescharacteristic
216C4—22no—
217C4—21no—
218C4—44no—
219C4—4241no—
220C4#1681073812yes—
221C4#1681074312yes—
222C4#1681079612yes—
223C2—41no—
224C2—8481no—
225C2#2027504212yes—
226C2#2027511811yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2986C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#2986D53
hypocenterC53#9070
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2986

Complements

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

NH ≅ G/Ntype
2. #20273310C2semidirect
4. #20272551C2semidirect
5. C53×#2986C2semidirect
111. #2986D53semidirect
134. C53#9070semidirect

Maximal subgroups

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

subgroupindex
#202850402
#202733102
#202731312
#202725512
C53×#29862
#907053