Finite Group Databasebeta

← #27287591

Subgroups — #27287591

Order 13568.

402 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 91 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#20275683C212yesmaximal
3C2×#16810364C212yesmaximal
4#20272596C211yesmaximal, characteristic
5C53×#1847C211yesmaximal, characteristic
6C2×#13133209C2212yes—
7C2×#13133128C2212yes—
8#16810563—22no—
9#16810555—22no—
10#16810553—21no—
11#16810553—21no—
12#16810385—22no—
13#16810385—22no—
14#16810364—24no—
15C2×#107×C53—21no—
16C2×#106×C53—21no—
17C2×#97×C53C2211yescharacteristic
18C53×#384C2211yescharacteristic
19C53×#382C2211yescharacteristic
20C22×#88322—22no—
21#13133209—22no—
22#13133209—22no—
23#13133209—22no—
24#13133209—22no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53D412yes—
28C2×C4⋊Dic53D412yes—
29C2×#11719919—22no—
30C2×#11719919—22no—
31#13133128—22no—
32#13133128—22no—
33#13133124—22no—
34#13133124—22no—
35#13133124—21no—
36#13133124—21no—
37C22×C8×C53—21no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C42×C53C2311yescharacteristic
43#107×C53—42no—
44#106×C53—42no—
45#97×C53—21no—
46#97×C53D412yes—
47#95×C53—22no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—41no—
51C22×C4×C53C2×D412yes—
52C22×C4×C53C2×D411yescharacteristic
53C22×Dic53—22no—
54C22×Dic53—22no—
55C2×C8×C53—22no—
56C2×C8×C53—22no—
57C2×C8×C53—22no—
58C4⋊C4×C53—41no—
59C4⋊C4×C53—41no—
60C4⋊C4×C53—41no—
61C4⋊C4×C53—41no—
62C4⋊C4×C53—44no—
63C42×C53C2×D411yescharacteristic
64C42×C53#4111yescharacteristic
65C42×C53#4112yes—
66C4⋊Dic53—22no—
67C4⋊Dic53—22no—
68C4⋊Dic53—22no—
69C4⋊Dic53—22no—
70C4⋊Dic53—81no—
71C4⋊Dic53—81no—
72#11719919—42no—
73#11719919—42no—
74#11719919—42no—
75#11719919—42no—
76C2×#88322—24no—
77C2×#88322—42no—
78C2×#88322—42no—
79C23×C53#12011yescharacteristic
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—21no—
84C2×C4×C53—41no—
85C2×C4×C53—41no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—81no—
89C2×C4×C53—82no—
90C2×C4×C53#12111yescharacteristic
91C2×C4×C53#12112yes—
92C2×C4×C53#12312yes—
93C2×C4×C53#12411yescharacteristic
94C2×Dic53—42no—
95C2×Dic53—42no—
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C8×C53—24no—
101#88322—44no—
102#10429—531nomaximal, Sylow
103C22×C53#39411yescharacteristic
104C22×C53#45311yescharacteristic
105C22×C53#45412yes—
106C22×C53#45612yes—
107C22×C53#45611yescharacteristic
108C4×C53—22no—
109C4×C53—22no—
110C4×C53—21no—
111C4×C53—21no—
112C4×C53—81no—
113C4×C53—81no—
114C4×C53—161no—
115Dic53—82no—
116Dic53—82no—
117#2822—531no—
118#1992—532no—
119#1847D5311yescharacteristic
120C2×#330—532no—
121#1295—531no—
122C2×C53#165711yescharacteristic
123C2×C53#166011yescharacteristic
124C2×C53#166312yes—
125C2×C53#203812yes—
126C2×C53#203911yescharacteristic
127C2×#125—532no—
128C2×#107—21no—
129C2×#106—21no—
130C2×#97—532no—
131C2×#97C2×D5311yescharacteristic
132#400—1062no—
133#398—1062no—
134#398—1061no—
135#398—1061no—
136#384C2×D5311yescharacteristic
137#382C2×D5311yescharacteristic
138#340—1062no—
139#340—1062no—
140#330—1064no—
141C53#1042911yescharacteristic, complemented, Sylow
142C22×C8—21no—
143C22×C8—1062no—
144#125—1062no—
145#125—1062no—
146#125—1062no—
147#125—1062no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—532no—
151C2×C4⋊C4—532no—
152C2×C4⋊C4—1061no—
153C2×C4⋊C4—1061no—
154C2×C4⋊C4—1062no—
155C2×C4⋊C4—1062no—
156C2×C4⋊C4C22⋊D5311yescharacteristic
157C2×C4⋊C4C22⋊D5311yescharacteristic
158C2×C42C22×D5311yescharacteristic
159#107—42no—
160#106—42no—
161#97—21no—
162#97—1062no—
163#97—1062no—
164#97D21212yes—
165#95—22no—
166#95—1062no—
167#95—1062no—
168#95—1061no—
169#95—1061no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173C22×C4—1062no—
174C22×C4—1062no—
175C22×C4D4×D5312yes—
176C22×C4C2×C22⋊D5311yescharacteristic
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—1064no—
181C2×C8—2122no—
182C2×C8—2122no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—44no—
188C4⋊C4—1062no—
189C4⋊C4—1062no—
190C4⋊C4—1062no—
191C4⋊C4—1062no—
192C4⋊C4—4241no—
193C4⋊C4—4241no—
194C4⋊C4—2122no—
195C4⋊C4—2122no—
196C4⋊C4—2122no—
197C4⋊C4—2122no—
198C42C2×D21211yescharacteristic
199C42#1171994511yescharacteristic
200C42#1171994912yes—
201C23#1313318511yescharacteristic
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—81no—
211C2×C4—82no—
212C2×C4—2122no—
213C2×C4—2122no—
214C2×C4—2122no—
215C2×C4—2122no—
216C2×C4—2122no—
217C2×C4—2122no—
218C2×C4#1313319111yescharacteristic
219C2×C4#1313319212yes—
220C2×C4#1313320212yes—
221C2×C4#1313320611yescharacteristic
222C8—24no—
223C8—2124no—
224C22#1681054011yescharacteristic
225C22#1681079011yescharacteristic
226C22#1681079412yes—
227C22#1681080412yes—
228C22#1681080511yescharacteristic
229C4—22no—
230C4—22no—
231C4—21no—
232C4—21no—
233C4—81no—
234C4—81no—
235C4—161no—
236C4—4242no—
237C4—4242no—
238C2#2027401111yescharacteristic
239C2#2027402411yescharacteristic
240C2#2027404212yes—
241C2#2027593412yes—
242C2#2027593911yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1847C2
socle, prime radicalC23×C53#120
hypercenter#1847D53
hypocenterC53#10429
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
141. C53#10429semidirect

Maximal subgroups

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

subgroupindex
#202827832
#202756832
C2×#168103642
#202725962
C53×#18472
#1042953