Finite Group Databasebeta

← #27287587

Subgroups — #27287587

Order 13568.

386 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 89 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#20282783C211yesmaximal, characteristic
2#20275684C212yesmaximal
3#20274848C211yesmaximal, characteristic
4C2×#16810364C212yesmaximal
5C53×#1295C211yesmaximal, characteristic
6C2×#13133209C2212yes—
7C2×#13133128C2211yescharacteristic
8#16810563—22no—
9#16810555—22no—
10#16810553—22no—
11#16810493C2211yescharacteristic
12#16810386—22no—
13#16810386—22no—
14#16810385—22no—
15#16810364—24no—
16C2×#97×C53C2212yes—
17C53×#382C2211yescharacteristic
18C22×#88322—21no—
19#13133209—22no—
20#13133209—22no—
21#13133209—22no—
22#13133209—22no—
23C2×C4⋊Dic53—22no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53D412yes—
27C2×C4⋊Dic53D412yes—
28C2×#11719919—22no—
29C2×#11719919—22no—
30#13133128—21no—
31#13133128D412yes—
32#13133124—24no—
33#13133124—22no—
34C22×C8×C53—22no—
35C2×C4⋊C4×C53D412yes—
36C2×C42×C53C2311yescharacteristic
37#97×C53—22no—
38#97×C53—22no—
39#95×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53C2×D412yes—
42C22×C4×C53C2×D411yescharacteristic
43C22×Dic53—22no—
44C22×Dic53—22no—
45C22×Dic53—41no—
46C2×C8×C53—24no—
47C2×C8×C53—42no—
48C2×C8×C53—42no—
49C4⋊C4×C53—42no—
50C42×C53C2×D411yescharacteristic
51C42×C53#4111yescharacteristic
52C42×C53#4112yes—
53C4⋊Dic53—22no—
54C4⋊Dic53—22no—
55C4⋊Dic53—22no—
56C4⋊Dic53—22no—
57C4⋊Dic53—42no—
58C4⋊Dic53—44no—
59C4⋊Dic53—82no—
60#11719919—42no—
61#11719919—42no—
62#11719919—42no—
63#11719919—42no—
64C2×#88322—22no—
65C2×#88322—41no—
66C2×#88322—41no—
67C23×C53#12011yescharacteristic
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—21no—
71C2×C4×C53—22no—
72C2×C4×C53—42no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53#12112yes—
76C2×C4×C53#12111yescharacteristic
77C2×C4×C53#12312yes—
78C2×C4×C53#12411yescharacteristic
79C2×Dic53—42no—
80C2×Dic53—42no—
81C2×Dic53—42no—
82C2×Dic53—42no—
83C2×Dic53—42no—
84C2×Dic53—42no—
85C2×Dic53—82no—
86C2×Dic53—81no—
87C8×C53—44no—
88#88322—42no—
89#10428—531nomaximal, Sylow
90C22×C53#39411yescharacteristic
91C22×C53#45412yes—
92C22×C53#45411yescharacteristic
93C22×C53#45511yescharacteristic
94C22×C53#45612yes—
95C4×C53—22no—
96C4×C53—22no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—82no—
100Dic53—82no—
101Dic53—82no—
102Dic53—161no—
103#2822—531no—
104#1992—532no—
105#1836—531no—
106C2×#330—532no—
107#1295D5311yescharacteristic
108C2×C53#165811yescharacteristic
109C2×C53#166311yescharacteristic
110C2×C53#166412yes—
111C2×C53#203812yes—
112C2×C53#204011yescharacteristic
113C2×#125—532no—
114C2×#97—531no—
115C2×#97C2×D5312yes—
116#400—1062no—
117#398—1062no—
118#398—1062no—
119#384—531no—
120#382C2×D5311yescharacteristic
121#340—1062no—
122#340—1062no—
123#340—1062no—
124#330—1064no—
125C53#1042811yescharacteristic, complemented, Sylow
126C22×C8—22no—
127C22×C8—1061no—
128#125—1062no—
129#125—1062no—
130#125—1062no—
131#125—1062no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4—1062no—
135C2×C4⋊C4—1062no—
136C2×C4⋊C4—1062no—
137C2×C4⋊C4—1061no—
138C2×C4⋊C4—1062no—
139C2×C4⋊C4D21212yes—
140C2×C42C22×D5311yescharacteristic
141#97—22no—
142#97—22no—
143#97—532no—
144#97—1061no—
145#95—22no—
146#95—1064no—
147#95—1062no—
148C22×C4—22no—
149C22×C4—1062no—
150C22×C4—1062no—
151C22×C4—2121no—
152C22×C4C2×D21211yescharacteristic
153C22×C4D4×D5312yes—
154C2×C8—24no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—1062no—
158C2×C8—2121no—
159C2×C8—2121no—
160C4⋊C4—42no—
161C4⋊C4—1062no—
162C4⋊C4—1062no—
163C4⋊C4—1062no—
164C4⋊C4—1062no—
165C4⋊C4—4242no—
166C4⋊C4—2124no—
167C4⋊C4—2122no—
168C4⋊C4—2122no—
169C4⋊C4—2122no—
170C4⋊C4—2122no—
171C4⋊C4—2122no—
172C42#1171994512yes—
173C42D4×D5311yescharacteristic
174C42#1171994911yescharacteristic
175C23#1313318711yescharacteristic
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—4242no—
184C2×C4—4241no—
185C2×C4—2122no—
186C2×C4—2122no—
187C2×C4—2122no—
188C2×C4—2122no—
189C2×C4—2122no—
190C2×C4—2122no—
191C2×C4#1313319012yes—
192C2×C4#1313319211yescharacteristic
193C2×C4#1313320212yes—
194C2×C4#1313320511yescharacteristic
195C8—44no—
196C8—2122no—
197C22#1681054111yescharacteristic
198C22#1681079712yes—
199C22#1681079811yescharacteristic
200C22#1681080311yescharacteristic
201C22#1681080712yes—
202C4—22no—
203C4—22no—
204C4—21no—
205C4—21no—
206C4—82no—
207C4—8481no—
208C4—4242no—
209C4—4242no—
210C2#2027401811yescharacteristic
211C2#2027404411yescharacteristic
212C2#2027405212yes—
213C2#2027593512yes—
214C2#2027594511yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1295C2
socle, prime radicalC23×C53#120
hypercenter#1295D53
hypocenterC53#10428
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
125. C53#10428semidirect

Maximal subgroups

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

subgroupindex
#202827832
#202756842
#202748482
C2×#168103642
C53×#12952
#1042853