Finite Group Databasebeta

← #27287336

Subgroups — #27287336

Order 13568.

406 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 77 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#20274815C212yesmaximal
2#20274790C211yesmaximal, characteristic
3#20272623C211yesmaximal, characteristic
4#20272568C212yesmaximal
5C53×#2608C211yesmaximal, characteristic
6C2×#13133176C2211yescharacteristic
7#16810587C2212yes—
8#16810486C2211yescharacteristic
9#16810389—22no—
10#16810383—22no—
11#16810383—22no—
12C53×#396—22no—
13C53×#389—22no—
14C53×#387—22no—
15C53×#381C2211yescharacteristic
16C4×C22⋊C4×C53C2212yes—
17C2×#11719917—22no—
18#13133176—22no—
19#13133176—22no—
20C2×C4⋊Dic53Q812yes—
21C2×C22⋊Dic53—21no—
22C2×C22⋊Dic53—21no—
23C2×C4×Dic53—21no—
24C22⋊#88322—22no—
25C22⋊#88322—22no—
26#13133124—22no—
27C23×C4×C53C2311yescharacteristic
28C2×C4⋊C4×C53—22no—
29C2×C4⋊C4×C53—22no—
30C2×C4⋊C4×C53—22no—
31C2×C22⋊C4×C53—22no—
32C2×C22⋊C4×C53D412yes—
33C2×C42×C53D412yes—
34C2×C42×C53D412yes—
35#95×C53—22no—
36#95×C53—22no—
37#95×C53—22no—
38C24×C53C2×D411yescharacteristic
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53—22no—
42C22×C4×C53—42no—
43C22×C4×C53C2×D412yes—
44C22×C4×C53C2×Q811yescharacteristic
45C22×C4×C53#4112yes—
46C22×C4×C53#4111yescharacteristic
47C22×Dic53—22no—
48C22×Dic53—41no—
49C4⋊C4×C53—44no—
50C4⋊C4×C53—44no—
51C4⋊C4×C53—82no—
52C22⋊C4×C53—44no—
53C22⋊C4×C53—42no—
54C42×C53—42no—
55C42×C53—42no—
56C22⋊Dic53—41no—
57C22⋊Dic53—41no—
58C22⋊Dic53—42no—
59C4⋊Dic53—24no—
60C4×Dic53—42no—
61#11719917—42no—
62#11719917—42no—
63C2×#88322—42no—
64C23×C53—22no—
65C23×C53—42no—
66C23×C53#12011yescharacteristic
67C23×C53C22⋊Q811yescharacteristic
68C23×C53#12411yescharacteristic
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—24no—
72C2×C4×C53—41no—
73C2×C4×C53—41no—
74C2×C4×C53—42no—
75C2×C4×C53—42no—
76C2×C4×C53—42no—
77C2×C4×C53—42no—
78C2×C4×C53—42no—
79C2×C4×C53—42no—
80C2×C4×C53—42no—
81C2×C4×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53C22⋊Q812yes—
84C2×C4×C53#12412yes—
85C2×Dic53—42no—
86C2×Dic53—82no—
87C2×Dic53—82no—
88C2×Dic53—81no—
89#88322—82no—
90#10389—531nomaximal, Sylow
91C22×C53—22no—
92C22×C53—21no—
93C22×C53—21no—
94C22×C53—22no—
95C22×C53—22no—
96C22×C53—82no—
97C22×C53#39311yescharacteristic
98C22×C53#45812yes—
99C4×C53—22no—
100C4×C53—22no—
101C4×C53—44no—
102C4×C53—82no—
103C4×C53—82no—
104Dic53—82no—
105Dic53—161no—
106#2608D5311yescharacteristic
107#1826—532no—
108#1820—531no—
109#1306—531no—
110#1284—532no—
111C2×C53—22no—
112C2×C53—41no—
113C2×C53—41no—
114C2×C53#203011yescharacteristic
115C2×C53#203112yes—
116C2×#117—531no—
117#407—532no—
118#396—22no—
119#389—22no—
120#387—22no—
121#381—531no—
122#381C2×D5311yescharacteristic
123C4×C22⋊C4C2×D5312yes—
124#342—1062no—
125#339—1062no—
126#339—1062no—
127C53#1038911yescharacteristic, complemented, Sylow
128C23×C4C22×D5311yescharacteristic
129C2×M16—1062no—
130#117—1062no—
131#117—1062no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—532no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—1061no—
138C2×C22⋊C4—1061no—
139C2×C22⋊C4C22⋊D5312yes—
140C2×C42—1061no—
141C2×C42C22⋊D5312yes—
142C2×C42C22⋊D5312yes—
143C22⋊C8—1062no—
144C22⋊C8—1062no—
145#95—22no—
146#95—22no—
147#95—22no—
148#95—1062no—
149C24C2×C22⋊D5311yescharacteristic
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—42no—
154C22×C4—1062no—
155C22×C4—2121no—
156C22×C4Q8×D5311yescharacteristic
157C22×C4#1171994912yes—
158C22×C4#1171994911yescharacteristic
159C22×C4C2×C22⋊D5312yes—
160M16—2122no—
161M16—2122no—
162C2×C8—2122no—
163C4⋊C4—44no—
164C4⋊C4—44no—
165C4⋊C4—82no—
166C4⋊C4—1064no—
167C22⋊C4—44no—
168C22⋊C4—42no—
169C22⋊C4—2122no—
170C22⋊C4—2121no—
171C22⋊C4—2121no—
172C42—42no—
173C42—42no—
174C42—2122no—
175C23—22no—
176C23—42no—
177C23#1313318611yescharacteristic
178C23#1313319311yescharacteristic
179C23#1313320611yescharacteristic
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—4241no—
195C2×C4—4242no—
196C2×C4—4242no—
197C2×C4—2122no—
198C2×C4#1313319312yes—
199C2×C4#1313320612yes—
200C8—4242no—
201C22—22no—
202C22—21no—
203C22—21no—
204C22—22no—
205C22—22no—
206C22—82no—
207C22#1681053411yescharacteristic
208C22#1681081312yes—
209C4—22no—
210C4—22no—
211C4—44no—
212C4—82no—
213C4—82no—
214C4—8481no—
215C4—4242no—
216C2—22no—
217C2—41no—
218C2—41no—
219C2#2027588111yescharacteristic
220C2#2027588412yes—
subgroupG/H
centerC22#16810534
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#2608C2
socleC22×C53#393
prime radicalC24×C53C2×D4
hypercenter#2608D53
hypocenterC53#10389
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2608

Complements

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

NH ≅ G/Ntype
127. C53#10389semidirect

Maximal subgroups

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

subgroupindex
#202748152
#202747902
#202726232
#202725682
C53×#26082
#1038953