Finite Group Databasebeta

← #27296883

Subgroups — #27296883

Order 13568.

400 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 59 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#20280380C211yesmaximal, characteristic
2#20273672C212yesmaximal
3#20272735C212yesmaximal
4#20272645C211yesmaximal, characteristic
5C53×#2030C211yesmaximal, characteristic
6#16810778—22no—
7#16810559—21no—
8#16810535C2212yes—
9#16810527—22no—
10#16810510—22no—
11C4×C22⋊Dic53—21no—
12#16810360—22no—
13#16810347C2212yes—
14C53×#543C2211yescharacteristic
15C53×#381—21no—
16C53×#381C2211yescharacteristic
17C53×#342—21no—
18C53×#342C2211yescharacteristic
19C22×Dic106D412yes—
20C2×#11719925—42no—
21C2×C4⋊Dic53—21no—
22C2×#11719919—22no—
23C2×C22⋊Dic53—22no—
24C2×C4×Dic53—21no—
25C2×C4×Dic53—21no—
26#13133140—42no—
27C22⋊#88322—22no—
28#13133124—22no—
29#13133124—21no—
30#13133124—22no—
31#13133124D412yes—
32C23×C4×C53D411yescharacteristic
33C22⋊Q8×C53—22no—
34C22⋊Q8×C53—42no—
35C2×C22⋊C4×C53—21no—
36C2×C22⋊C4×C53—21no—
37C2×C22⋊C4×C53D411yescharacteristic
38C2×C22⋊C4×C53C2311yescharacteristic
39#95×C53—22no—
40#95×C53—42no—
41C24×C53C2×D411yescharacteristic
42C2×Q8×C53—22no—
43C22×C4×C53—21no—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—22no—
47C22×C4×C53—41no—
48C22×C4×C53—41no—
49C22×C4×C53—41no—
50C22×C4×C53Q1612yescomplemented
51C22×C4×C53C2×D412yes—
52C22×Dic53—22no—
53C22×Dic53—22no—
54C2×Dic106—42no—
55C2×Dic106—44no—
56C4⋊C4×C53—22no—
57C4⋊C4×C53—42no—
58C22⋊C4×C53—41no—
59C22⋊C4×C53—41no—
60C22⋊C4×C53—41no—
61C22⋊C4×C53—41no—
62C22⋊C4×C53—41no—
63C22⋊C4×C53—41no—
64C22⋊Dic53—44no—
65#11719925—84no—
66C4⋊Dic53—81no—
67#11719919—44no—
68C4×Dic53—81no—
69C4×Dic53—81no—
70C2×#88322—42no—
71C23×C53—21no—
72C23×C53—21no—
73C23×C53—42no—
74C23×C53#12011yescharacteristic
75C23×C53C2×Q1611yescharacteristic
76C23×C53#13711yescharacteristic
77Q8×C53—82no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—41no—
81C2×C4×C53—41no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53—41no—
86C2×C4×C53—42no—
87C2×C4×C53—81no—
88C2×C4×C53—82no—
89C2×C4×C53—81no—
90C2×C4×C53—82no—
91C2×Dic53—24no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53—42no—
96C2×Dic53—42no—
97Dic106—44no—
98Dic106—82no—
99#88322—82no—
100#12215—531nomaximal, Sylow
101C22×C53—22no—
102C22×C53—21no—
103C22×C53—21no—
104C22×C53—41no—
105C22×C53—41no—
106C22×C53—42no—
107C22×C53—81no—
108C22×C53#45111yescharacteristic
109C22×C53#45511yescharacteristic
110C22×C53B(4,𝔽2)11yescharacteristic
111C4×C53—42no—
112C4×C53—42no—
113C4×C53—81no—
114C4×C53—82no—
115C4×C53—161no—
116Dic53—44no—
117Dic53—82no—
118#2607—531no—
119#2030D5311yescharacteristic
120#1602—532no—
121#1351—532no—
122#1317—531no—
123C2×C53—21no—
124C2×C53—21no—
125C2×C53—41no—
126C2×C53—41no—
127C2×C53#160211yescharacteristic
128C2×C53#219712yes—
129#543C2×D5311yescharacteristic
130#451—1062no—
131#399—1061no—
132#393—532no—
133#391—1062no—
134#387—1062no—
135#381—21no—
136#381C2×D5311yescharacteristic
137C4×C22⋊C4—1061no—
138#342—21no—
139#342C2×D5311yescharacteristic
140#328—1062no—
141#323—532no—
142C53#1221511yescharacteristic, complemented, Sylow
143C22×Q8—532no—
144C23×C4C22⋊D5311yescharacteristic
145C2×Q16—2122no—
146C22⋊Q8—22no—
147C22⋊Q8—42no—
148C2×C4⋊C4—1061no—
149C2×C4⋊C4—1062no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—1062no—
153C2×C22⋊C4C22⋊D5311yescharacteristic
154C2×C22⋊C4C22×D5311yescharacteristic
155C2×C42—1061no—
156C2×C42—1061no—
157#103—2122no—
158C22⋊C8—1062no—
159#95—22no—
160#95—42no—
161#95—532no—
162#95—1062no—
163#95—1061no—
164#95—1062no—
165C24C2×C22⋊D5311yescharacteristic
166C2×Q8—22no—
167C2×Q8—2122no—
168C2×Q8—2124no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—1062no—
177C22×C4—1062no—
178C22×C4#1171992512yescomplemented
179C22×C4D4×D5312yes—
180Q16—4244no—
181C2×C8—2122no—
182C4⋊C4—22no—
183C4⋊C4—42no—
184C4⋊C4—4241no—
185C4⋊C4—2124no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C22⋊C4—41no—
190C22⋊C4—41no—
191C22⋊C4—41no—
192C22⋊C4—2124no—
193C42—4241no—
194C42—4241no—
195C23—21no—
196C23—21no—
197C23—42no—
198C23#1313318511yescharacteristic
199C23C2×#1171992511yescharacteristic
200C23#1313324711yescharacteristic
201Q8—82no—
202Q8—4242no—
203Q8—2124no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—81no—
214C2×C4—82no—
215C2×C4—81no—
216C2×C4—82no—
217C2×C4—1064no—
218C2×C4—2122no—
219C2×C4—2122no—
220C2×C4—2122no—
221C2×C4—2122no—
222C2×C4—2122no—
223C8—4242no—
224C22—22no—
225C22—21no—
226C22—21no—
227C22—41no—
228C22—41no—
229C22—42no—
230C22—81no—
231C22#1681077811yescharacteristic
232C22#1681080011yescharacteristic
233C22#1681080911yescharacteristic
234C4—42no—
235C4—42no—
236C4—81no—
237C4—82no—
238C4—161no—
239C4—4242no—
240C4—2124no—
241C2—21no—
242C2—21no—
243C2—41no—
244C2—41no—
245C2#2027367111yescharacteristic
246C2#2027670212yes—
subgroupG/H
centerC22#16810809
commutator, cosocleC2×C22⋊C4×C53C23
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2030C2
socleC22×C53B(4,𝔽2)
prime radicalC24×C53C2×D4
hypercenter#2030D53
hypocenterC53#12215
LCSG ▷ C2×C22⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2030

Complements

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

NH ≅ G/Ntype
50. C22×C4×C53Q16semidirect
142. C53#12215semidirect
178. C22×C4#11719925semidirect

Maximal subgroups

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

subgroupindex
#202803802
#202736722
#202727352
#202726452
C53×#20302
#1221553