Finite Group Databasebeta

← #27285051

Subgroups — #27285051

Order 13568.

468 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 85 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#20284841C212yesmaximal
2#20275654C211yesmaximal, characteristic
3#20273330C212yesmaximal
4#20272585C211yesmaximal, characteristic
5C53×#3249C211yesmaximal, characteristic
6C2×C4×Dic106C2211yescharacteristic
7#16810951—22no—
8#16810905—22no—
9#16810830—22no—
10#16810778—22no—
11#16810728—24no—
12#16810680C2212yes—
13C2×#13133128C2211yescharacteristic
14#16810377—22no—
15#16810365—24no—
16C53×#533C2212yes—
17C53×#477—22no—
18C53×#467—22no—
19C53×#462—22no—
20C2×C4⋊C8×C53C2211yescharacteristic
21C22×Dic106D411yescharacteristic
22C2×#11719925—24no—
23C22×#88322—21no—
24C4×Dic106—22no—
25C4×Dic106—22no—
26C2×C4⋊Dic53D411yescharacteristic
27C2×#11719919—21no—
28C2×C4×Dic53—21no—
29#13133148—24no—
30C4⋊#88322—22no—
31#13133141—24no—
32#13133140—24no—
33C22⋊#88322—22no—
34#13133128—22no—
35#13133128D414yes—
36C2×Q16×C53—22no—
37C22×C8×C53D412yes—
38C4⋊Q8×C53D412yes—
39C4⋊Q8×C53D412yes—
40#125×C53—22no—
41C22⋊Q8×C53—22no—
42C22⋊Q8×C53—22no—
43C4×Q8×C53—22no—
44C4×Q8×C53—22no—
45#117×C53—22no—
46C2×C42×C53C2311yescharacteristic
47#107×C53—22no—
48C4⋊C8×C53—22no—
49#103×C53—24no—
50C2×Q8×C53—22no—
51C2×Q8×C53—22no—
52C22×C4×C53C2×D411yescharacteristic
53C22×C4×C53C2×D411yescharacteristic
54C22×C4×C53#4111yescharacteristic
55C22×Dic53—21no—
56C22×Dic53—21no—
57C2×Dic106—22no—
58C2×Dic106—42no—
59Q16×C53—82no—
60C2×C8×C53—22no—
61C2×C8×C53—42no—
62C4⋊C4×C53—22no—
63C4⋊C4×C53—22no—
64C4⋊C4×C53—22no—
65C4⋊C4×C53—22no—
66C4⋊C4×C53—42no—
67C4⋊C4×C53—42no—
68C22⋊C4×C53—42no—
69C42×C53—42no—
70C42×C53C2×D412yes—
71C42×C53C2×D412yes—
72#11719925—48no—
73C4⋊Dic53—22no—
74#11719919—42no—
75C4×Dic53—41no—
76C4×Dic53—41no—
77C2×#88322—22no—
78C2×#88322—24no—
79C2×#88322—41no—
80C2×#88322—41no—
81C23×C53#12111yescharacteristic
82Q8×C53—44no—
83Q8×C53—42no—
84Q8×C53—42no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—41no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53#12012yes—
92C2×C4×C53#12011yescharacteristic
93C2×C4×C53#12112yes—
94C2×C4×C53#12711yescharacteristic
95C2×Dic53—22no—
96C2×Dic53—41no—
97C2×Dic53—41no—
98C2×Dic53—41no—
99C2×Dic53—42no—
100Dic106—24no—
101Dic106—42no—
102C8×C53—42no—
103#88322—42no—
104#88322—44no—
105#10031—531nomaximal, Sylow
106C22×C53—21no—
107C22×C53—21no—
108C22×C53#39211yescharacteristic
109C22×C53#47312yes—
110C4×C53—21no—
111C4×C53—22no—
112C4×C53—22no—
113C4×C53—82no—
114C4×C53—82no—
115C4×C53#45511yescharacteristic
116C4×C53#45611yescharacteristic
117Dic53—42no—
118Dic53—81no—
119#3249D5311yescharacteristic
120#3113—532no—
121#1985—531no—
122#1531—532no—
123#1290—531no—
124C2×C53—22no—
125C2×C53#200711yescharacteristic
126C2×C53#201011yescharacteristic
127C2×C53#202011yescharacteristic
128#533C2×D5312yes—
129C2×C4×Q8—531no—
130#483—1062no—
131#477—22no—
132#475—1062no—
133#467—22no—
134#462—22no—
135#462—1062no—
136#451—1062no—
137#441—1064no—
138#432—532no—
139C2×C4⋊C8C2×D5311yescharacteristic
140C2×#97—531no—
141#336—1062no—
142#330—1064no—
143C53#1003111yescharacteristic, complemented, Sylow
144C22×Q8—531no—
145C2×Q16—22no—
146C2×Q16—1064no—
147C22×C8—1061no—
148C22×C8C22⋊D5312yes—
149C4⋊Q8C22⋊D5312yes—
150C4⋊Q8C22⋊D5312yes—
151#125—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—1062no—
157C4×Q8—1062no—
158#117—22no—
159C2×C4⋊C4—531no—
160C2×C4⋊C4—1061no—
161C2×C42—1061no—
162C2×C42C22×D5311yescharacteristic
163#107—22no—
164#106—1064no—
165C4⋊C8—22no—
166C4⋊C8—1062no—
167#103—24no—
168#103—1064no—
169#103—1064no—
170C22⋊C8—1062no—
171#97—534no—
172#97—1062no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—1062no—
176C2×Q8—2122no—
177C22×C4—1061no—
178C22×C4—1061no—
179C22×C4D4×D5311yescharacteristic
180C22×C4#1171994711yescharacteristic
181C22×C4C2×C22⋊D5311yescharacteristic
182Q16—82no—
183Q16—2128no—
184C2×C8—22no—
185C2×C8—42no—
186C2×C8—1062no—
187C2×C8—1064no—
188C2×C8—2121no—
189C2×C8—2121no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—1062no—
197C4⋊C4—2122no—
198C22⋊C4—42no—
199C42—42no—
200C42—2121no—
201C42—2121no—
202C42D4×D5312yes—
203C42C2×C22⋊D5312yes—
204C23#1313318811yescharacteristic
205Q8—44no—
206Q8—42no—
207Q8—42no—
208Q8—1064no—
209Q8—2122no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—1062no—
217C2×C4—2121no—
218C2×C4—2122no—
219C2×C4—2121no—
220C2×C4—2121no—
221C2×C4#1313318512yes—
222C2×C4#1313318611yescharacteristic
223C2×C4#1313318912yes—
224C2×C4#1313321311yescharacteristic
225C8—42no—
226C8—2122no—
227C8—2124no—
228C22—21no—
229C22—21no—
230C22#1681053111yescharacteristic
231C22#1681089512yes—
232C4—21no—
233C4—22no—
234C4—22no—
235C4—82no—
236C4—82no—
237C4—4241no—
238C4—2122no—
239C4#1681080111yescharacteristic
240C4#1681080611yescharacteristic
241C2—22no—
242C2#2027576211yescharacteristic
243C2#2027577911yescharacteristic
244C2#2027583411yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#3249C2
socleC22×C53#392
prime radicalC23×C53#121
hypercenter#3249D53
hypocenterC53#10031
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
143. C53#10031semidirect

Maximal subgroups

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

subgroupindex
#202848412
#202756542
#202733302
#202725852
C53×#32492
#1003153