Finite Group Databasebeta

← #27287588

Subgroups — #27287588

Order 13568.

402 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 93 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#20275685C212yesmaximal
2#20274884C211yesmaximal, characteristic
3C2×#16810365C212yesmaximal
4#20272596C211yesmaximal, characteristic
5C53×#2822C211yesmaximal, characteristic
6C2×#13133150—21no—
7C2×#13133148—21no—
8C2×#13133128C2212yes—
9C2×#13133128C2211yescharacteristic
10#16810493C2211yescharacteristic
11#16810385—22no—
12#16810385—22no—
13#16810365—24no—
14C2×#125×C53C2212yes—
15C53×#400—22no—
16C53×#398—21no—
17C53×#398—21no—
18C53×#398—22no—
19C53×#382C2211yescharacteristic
20C22×#88322—21no—
21C22×#88322—22no—
22C2×C4⋊Dic53—22no—
23C2×C4⋊Dic53—21no—
24#13133150—42no—
25#13133148—42no—
26#13133128—21no—
27#13133128—22no—
28#13133128—22no—
29#13133128D412yes—
30#125×C53—22no—
31#125×C53—22no—
32#125×C53—22no—
33#125×C53—22no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53D412yes—
39C2×C4⋊C4×C53D412yes—
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C42×C53C2311yescharacteristic
43#95×C53—22no—
44#95×C53—22no—
45#95×C53—21no—
46#95×C53—21no—
47#95×C53—22no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53C2×D412yes—
53C22×C4×C53C2×D411yescharacteristic
54C22×Dic53—41no—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—22no—
57C4⋊C4×C53—22no—
58C4⋊C4×C53—22no—
59C4⋊C4×C53—41no—
60C4⋊C4×C53—41no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—42no—
63C4⋊C4×C53—42no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—81no—
66C4⋊C4×C53—81no—
67C42×C53C2×D411yescharacteristic
68C42×C53#4111yescharacteristic
69C42×C53#4112yes—
70C4⋊Dic53—41no—
71C4⋊Dic53—41no—
72C4⋊Dic53—44no—
73C2×#88322—22no—
74C2×#88322—22no—
75C2×#88322—22no—
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—42no—
85C2×C4×C53—42no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53#12111yescharacteristic
95C2×C4×C53#12112yes—
96C2×C4×C53#12312yes—
97C2×C4×C53#12411yescharacteristic
98C2×Dic53—81no—
99C2×Dic53—82no—
100#88322—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—82no—
115C4×C53—82no—
116Dic53—161no—
117#2822D5311yescharacteristic
118#1992—532no—
119#1847—531no—
120C2×#330—532no—
121#1295—531no—
122C2×C53#165711yescharacteristic
123C2×C53#166011yescharacteristic
124C2×C53#166312yes—
125C2×C53#203812yes—
126C2×C53#203911yescharacteristic
127C2×#125C2×D5312yes—
128C2×#107—1061no—
129C2×#106—1061no—
130C2×#97—531no—
131C2×#97—532no—
132#400—22no—
133#398—21no—
134#398—21no—
135#398—22no—
136#384—531no—
137#382C2×D5311yescharacteristic
138#340—1062no—
139#340—1062no—
140#330—1064no—
141C53#1042911yescharacteristic, complemented, Sylow
142C22×C8—1061no—
143C22×C8—1062no—
144#125—22no—
145#125—22no—
146#125—22no—
147#125—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—1062no—
153C2×C4⋊C4—1061no—
154C2×C4⋊C4C22⋊D5312yes—
155C2×C4⋊C4C22⋊D5312yes—
156C2×C4⋊C4C22⋊D5311yescharacteristic
157C2×C4⋊C4C22⋊D5311yescharacteristic
158C2×C42C22×D5311yescharacteristic
159#107—2122no—
160#106—2122no—
161#97—532no—
162#97—1062no—
163#97—1061no—
164#97—1062no—
165#95—22no—
166#95—22no—
167#95—21no—
168#95—21no—
169#95—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—2121no—
175C22×C4C2×C22⋊D5311yescharacteristic
176C22×C4C2×C22⋊D5312yes—
177C2×C8—1062no—
178C2×C8—1062no—
179C2×C8—1062no—
180C2×C8—1064no—
181C2×C8—2122no—
182C2×C8—2122no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4—2121no—
196C4⋊C4—2121no—
197C4⋊C4—2124no—
198C42D4×D5311yescharacteristic
199C42#1171994711yescharacteristic
200C42#1171994712yes—
201C23#1313318611yescharacteristic
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—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—4241no—
217C2×C4—4242no—
218C2×C4#1313318811yescharacteristic
219C2×C4#1313318912yes—
220C2×C4#1313319812yes—
221C2×C4#1313320411yescharacteristic
222C8—1064no—
223C8—2124no—
224C22#1681053711yescharacteristic
225C22#1681079111yescharacteristic
226C22#1681079612yes—
227C22#1681080612yes—
228C22#1681080611yescharacteristic
229C4—22no—
230C4—22no—
231C4—21no—
232C4—21no—
233C4—81no—
234C4—81no—
235C4—82no—
236C4—82no—
237C4—8481no—
238C2#2027401311yescharacteristic
239C2#2027402611yescharacteristic
240C2#2027404312yes—
241C2#2027592912yes—
242C2#2027593611yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2822C2
socle, prime radicalC23×C53#120
hypercenter#2822D53
hypocenterC53#10429
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2822

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
#202756852
#202748842
C2×#168103652
#202725962
C53×#28222
#1042953