Finite Group Databasebeta

← #27284719

Subgroups — #27284719

Order 13568.

492 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 85 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#20284659C212yesmaximal
2#20274784C211yesmaximal, characteristic
3#20273079C212yesmaximal
4#20272592C211yesmaximal, characteristic
5C53×#3189C211yesmaximal, characteristic
6C2×#13133176C2211yescharacteristic
7#16810961—24no—
8#16810956—22no—
9#16810951—22no—
10#16810663—22no—
11#16810632—22no—
12#16810587C2212yes—
13C2×C22⋊#88322C2211yescharacteristic
14#16810377—22no—
15#16810349—24no—
16C2×C22⋊Q8×C53C2212yes—
17C53×#484—22no—
18C53×#467—22no—
19C53×#451—22no—
20C2×C22⋊C8×C53C2211yescharacteristic
21C2×#11719917—22no—
22C22×#88322—21no—
23#13133176—22no—
24#13133176—22no—
25C2×C4⋊Dic53D411yescharacteristic
26C2×C4⋊Dic53D411yescharacteristic
27C2×C22⋊Dic53—21no—
28C2×C4×Dic53—21no—
29#13133150—24no—
30#13133148—24no—
31#13133141—24no—
32#13133141—24no—
33C22⋊#88322—22no—
34C22⋊#88322—22no—
35C22⋊#88322D414yes—
36C22×Q8×C53D412yes—
37C23×C4×C53C2311yescharacteristic
38C2×Q16×C53—22no—
39C22×C8×C53D412yes—
40C22⋊Q8×C53—22no—
41C22⋊Q8×C53—22no—
42C22⋊Q8×C53—22no—
43C22⋊Q8×C53—22no—
44C2×C4⋊C4×C53—22no—
45C2×C4⋊C4×C53D412yes—
46C2×C22⋊C4×C53—22no—
47#107×C53—22no—
48#103×C53—24no—
49C22⋊C8×C53—22no—
50C24×C53#4111yescharacteristic
51C2×Q8×C53—22no—
52C2×Q8×C53—22no—
53C2×Q8×C53—44no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—42no—
57C22×C4×C53C2×D412yes—
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D412yes—
61C22×Dic53—21no—
62C22×Dic53—21no—
63Q16×C53—82no—
64C2×C8×C53—22no—
65C2×C8×C53—42no—
66C4⋊C4×C53—22no—
67C4⋊C4×C53—22no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—42no—
70C22⋊C4×C53—42no—
71C22⋊C4×C53—42no—
72C22⋊Dic53—42no—
73C4⋊Dic53—22no—
74C4⋊Dic53—22no—
75C4×Dic53—41no—
76C4×Dic53—41no—
77#11719917—44no—
78C2×#88322—24no—
79C2×#88322—42no—
80C2×#88322—41no—
81C23×C53—22no—
82C23×C53—41no—
83C23×C53—41no—
84C23×C53#12111yescharacteristic
85C23×C53#12112yes—
86Q8×C53—44no—
87Q8×C53—44no—
88C2×C4×C53—24no—
89C2×C4×C53—22no—
90C2×C4×C53—41no—
91C2×C4×C53—42no—
92C2×C4×C53—44no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—82no—
96C2×C4×C53#12012yes—
97C2×C4×C53#12011yescharacteristic
98C2×C4×C53#12711yescharacteristic
99C2×Dic53—42no—
100C2×Dic53—41no—
101C2×Dic53—41no—
102C2×Dic53—42no—
103C8×C53—42no—
104#88322—44no—
105#88322—81no—
106#9979—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—22no—
110C22×C53—22no—
111C22×C53—22no—
112C22×C53—82no—
113C22×C53#39211yescharacteristic
114C22×C53#47312yes—
115C4×C53—22no—
116C4×C53—21no—
117C4×C53—82no—
118C4×C53—82no—
119C4×C53#45812yes—
120Dic53—81no—
121Dic53—81no—
122#3189D5311yescharacteristic
123#3095—532no—
124#1818—531no—
125#1472—532no—
126#1293—531no—
127C2×C53—22no—
128C2×C53—42no—
129C2×C53#201711yescharacteristic
130C2×C53#202011yescharacteristic
131C2×C53#202211yescharacteristic
132C2×C22⋊Q8C2×D5312yes—
133C2×#117—531no—
134#485—1064no—
135#484—22no—
136#484—1062no—
137#483—1062no—
138#467—22no—
139#451—22no—
140#428—1062no—
141#419—1062no—
142#407—532no—
143C2×C22⋊C8—531no—
144C2×C22⋊C8C2×D5311yescharacteristic
145#336—1062no—
146#324—1064no—
147C53#997911yescharacteristic, complemented, Sylow
148C22×Q8C22⋊D5312yes—
149C23×C4C22×D5311yescharacteristic
150C2×Q16—22no—
151C2×M16—1062no—
152C22×C8—1061no—
153C22×C8C22⋊D5312yes—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158#117—1062no—
159#117—1062no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—531no—
162C2×C4⋊C4—531no—
163C2×C4⋊C4C22⋊D5312yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—1061no—
166C2×C42—1061no—
167#107—22no—
168#107—1064no—
169#106—1064no—
170#103—24no—
171#103—1064no—
172#103—1064no—
173C22⋊C8—22no—
174C22⋊C8—534no—
175C22⋊C8—1062no—
176C22⋊C8—1062no—
177C24#1171994711yescharacteristic
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—44no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—1061no—
185C22×C4—1061no—
186C22×C4D4×D5311yescharacteristic
187C22×C4D4×D5312yes—
188C22×C4C2×C22⋊D5312yes—
189C22×C4C2×C22⋊D5311yescharacteristic
190Q16—82no—
191M16—2124no—
192C2×C8—22no—
193C2×C8—42no—
194C2×C8—1064no—
195C2×C8—2122no—
196C2×C8—2121no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—1062no—
202C4⋊C4—1062no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—2122no—
206C42—2121no—
207C42—2121no—
208C23—22no—
209C23—41no—
210C23—41no—
211C23#1313318811yescharacteristic
212C23#1313318912yes—
213Q8—44no—
214Q8—44no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—82no—
223C2×C4—2122no—
224C2×C4—2121no—
225C2×C4—2121no—
226C2×C4—2122no—
227C2×C4#1313318512yes—
228C2×C4#1313318611yescharacteristic
229C2×C4#1313321311yescharacteristic
230C8—42no—
231C8—4241no—
232C8—2124no—
233C22—21no—
234C22—21no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—82no—
239C22#1681053111yescharacteristic
240C22#1681089512yes—
241C4—21no—
242C4—22no—
243C4—82no—
244C4—82no—
245C4—4241no—
246C4—4241no—
247C4#1681081312yes—
248C2—22no—
249C2—42no—
250C2#2027581811yescharacteristic
251C2#2027583411yescharacteristic
252C2#2027584411yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#3189C2
socleC22×C53#392
prime radicalC24×C53#41
hypercenter#3189D53
hypocenterC53#9979
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3189

Complements

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

NH ≅ G/Ntype
147. C53#9979semidirect

Maximal subgroups

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

subgroupindex
#202846592
#202747842
#202730792
#202725922
C53×#31892
#997953