Finite Group Databasebeta

← #27296838

Subgroups — #27296838

Order 13568.

388 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 63 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#20275882C211yesmaximal, characteristic
2#20273705C212yesmaximal
3#20272715C212yesmaximal
4#20272645C211yesmaximal, characteristic
5C53×#2817C211yesmaximal, characteristic
6#16811249C2211yescharacteristic
7#16810953—22no—
8#16810486—21no—
9#16810390C2211yescharacteristic
10#16810389—21no—
11#16810359—22no—
12#16810347C2212yes—
13C2×C22⋊Q8×C53—21no—
14C53×#400—22no—
15C53×#397—21no—
16C53×#397C2212yes—
17C53×#395—22no—
18C53×#381C2211yescharacteristic
19#13133196—22no—
20#13133194—42no—
21C2×C22⋊Dic53—21no—
22C2×C22⋊Dic53—21no—
23#13133148—42no—
24#13133140—42no—
25C22⋊#88322—22no—
26#13133124—42no—
27C22×Q8×C53—21no—
28C23×C4×C53D411yescharacteristic
29C22⋊Q8×C53—42no—
30C22⋊Q8×C53—42no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53D412yes—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53D411yescharacteristic
37C2×C22⋊C4×C53C2311yescharacteristic
38#95×C53—22no—
39#95×C53—21no—
40#95×C53—22no—
41#95×C53—22no—
42#95×C53D412yes—
43C24×C53C2×D411yescharacteristic
44C2×Q8×C53—42no—
45C2×Q8×C53—81no—
46C2×Q8×C53—81no—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—41no—
53C22×C4×C53—41no—
54C22×C4×C53SD1612yes—
55C22×C4×C53C2×D412yes—
56C22×Dic53—22no—
57C22×Dic53—41no—
58C2×Dic106—22no—
59C4⋊C4×C53—44no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—81no—
63C22⋊C4×C53—41no—
64C22⋊C4×C53—42no—
65C22⋊C4×C53—42no—
66C22⋊C4×C53—41no—
67C22⋊Dic53—41no—
68C22⋊Dic53—41no—
69C22⋊Dic53—41no—
70C22⋊Dic53—41no—
71C4⋊Dic53—22no—
72#11719919—42no—
73C2×#88322—42no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53—42no—
77C23×C53#12011yescharacteristic
78C23×C53C2×SD1611yescharacteristic
79C23×C53AGL(1,ℤ/8)11yescharacteristic
80Q8×C53—84no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—44no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53—81no—
95C2×Dic53—42no—
96C2×Dic53—82no—
97C2×Dic53—81no—
98C2×Dic53—82no—
99Dic106—82no—
100#88322—82no—
101#12207—531nomaximal, Sylow
102C22×C53—22no—
103C22×C53—21no—
104C22×C53—21no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53—42no—
108C22×C53—81no—
109C22×C53#45011yescharacteristic
110C22×C53#45511yescharacteristic
111C22×C53B(4,𝔽2)11yescharacteristic
112C4×C53—42no—
113C4×C53—42no—
114C4×C53—82no—
115C4×C53—81no—
116C4×C53—82no—
117Dic53—82no—
118Dic53—161no—
119#2817D5311yescharacteristic
120#2030—531no—
121#1608—532no—
122#1344—532no—
123#1317—531no—
124C2×C53—21no—
125C2×C53—21no—
126C2×C53—41no—
127C2×C53—41no—
128C2×C53#160111yescharacteristic
129C2×C53#220112yes—
130#543—531no—
131C2×C22⋊Q8—21no—
132#483—1062no—
133#400—22no—
134#397—21no—
135#397C2×D5312yes—
136#395—22no—
137#381—1061no—
138#381C2×D5311yescharacteristic
139#342—531no—
140#342—1061no—
141#328—1062no—
142#323—532no—
143C53#1220711yescharacteristic, complemented, Sylow
144C22×Q8—21no—
145C23×C4C22⋊D5311yescharacteristic
146C22⋊Q8—42no—
147C22⋊Q8—42no—
148C22⋊Q8—1062no—
149C22⋊Q8—2122no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4C22⋊D5312yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—1061no—
156C2×C22⋊C4—1061no—
157C2×C22⋊C4C22⋊D5311yescharacteristic
158C2×C22⋊C4C22×D5311yescharacteristic
159#106—2122no—
160#103—2122no—
161C22⋊C8—1062no—
162#95—22no—
163#95—21no—
164#95—22no—
165#95—22no—
166#95—2122no—
167#95C22⋊D5312yes—
168C24C2×C22⋊D5311yescharacteristic
169C2×Q8—42no—
170C2×Q8—81no—
171C2×Q8—81no—
172C2×Q8—1062no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—1062no—
181C22×C4—2121no—
182C22×C4#1171992312yes—
183C22×C4C2×C22⋊D5312yes—
184C2×C8—2122no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—44no—
188C4⋊C4—81no—
189C4⋊C4—1062no—
190C4⋊C4—2122no—
191C22⋊C4—41no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—41no—
195C22⋊C4—2121no—
196C22⋊C4—2121no—
197C22⋊C4—2121no—
198C22⋊C4—2121no—
199C23—21no—
200C23—21no—
201C23—42no—
202C23#1313318611yescharacteristic
203C23C2×#1171992311yescharacteristic
204C23#1313324311yescharacteristic
205Q8—84no—
206Q8—4242no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—81no—
221C2×C4—4241no—
222C2×C4—4242no—
223C2×C4—4242no—
224C2×C4—2122no—
225C8—4242no—
226C22—22no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—81no—
233C22#1681077411yescharacteristic
234C22#1681080111yescharacteristic
235C22#1681081011yescharacteristic
236C4—42no—
237C4—42no—
238C4—82no—
239C4—81no—
240C4—82no—
241C4—8481no—
242C4—4242no—
243C2—21no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2#2027366711yescharacteristic
248C2#2027672812yes—
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C22⋊C4×C53C23
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2817C2
socleC22×C53B(4,𝔽2)
prime radicalC24×C53C2×D4
hypercenter#2817D53
hypocenterC53#12207
LCSG ▷ C2×C22⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2817

Complements

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

NH ≅ G/Ntype
143. C53#12207semidirect

Maximal subgroups

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

subgroupindex
#202758822
#202737052
#202727152
#202726452
C53×#28172
#1220753