Finite Group Databasebeta

← #27296934

Subgroups — #27296934

Order 13568.

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

normal subgroups only

This group has 380 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#20282708C211yesmaximal, characteristic
2#20273689C212yesmaximal
3#20272744C212yesmaximal
4#20272649C211yesmaximal, characteristic
5C53×#2028C211yesmaximal, characteristic
6#16810778—22no—
7#16810544—22no—
8#16810543C2212yes—
9#16810530—21no—
10#16810529—21no—
11#16810527—22no—
12#16810360—22no—
13#16810349C2212yes—
14C53×#544C2211yescharacteristic
15C53×#419—21no—
16C53×#391C2211yescharacteristic
17C53×#343C2211yescharacteristic
18C53×#339—21no—
19C22×Dic106—21no—
20C22×Dic106D412yes—
21C2×#11719925—42no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—21no—
24C2×#11719919—22no—
25C2×C4×Dic53—21no—
26#13133140—42no—
27C22⋊#88322—22no—
28#13133124—22no—
29#13133124—22no—
30#13133124—22no—
31#13133124D412yes—
32C22×Q8×C53D411yescharacteristic
33C2×M16×C53—21no—
34C4⋊Q8×C53—41no—
35#125×C53—41no—
36C22⋊Q8×C53—22no—
37#117×C53—21no—
38C2×C4⋊C4×C53C2311yescharacteristic
39C2×C42×C53D411yescharacteristic
40#103×C53—41no—
41#103×C53—41no—
42#95×C53—22no—
43C2×Q8×C53—21no—
44C2×Q8×C53—22no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53Q1612yescomplemented
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C2×D411yescharacteristic
51C22×C4×C53C2×D412yes—
52C22×Dic53—22no—
53C22×Dic53—22no—
54C2×Dic106—24no—
55C2×Dic106—42no—
56C2×Dic106—82no—
57C2×Dic106—81no—
58C2×Dic106—81no—
59M16×C53—41no—
60M16×C53—41no—
61C2×C8×C53—41no—
62C4⋊C4×C53—21no—
63C4⋊C4×C53—21no—
64C4⋊C4×C53—22no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—42no—
67C22⋊C4×C53—41no—
68C42×C53—41no—
69C42×C53—41no—
70#11719925—84no—
71C4⋊Dic53—81no—
72C4⋊Dic53—81no—
73#11719919—82no—
74C4×Dic53—81no—
75C2×#88322—42no—
76C23×C53#12011yescharacteristic
77Q8×C53—42no—
78Q8×C53—81no—
79Q8×C53—82no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—42no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53C2×Q1611yescharacteristic
93C2×C4×C53#13711yescharacteristic
94C2×Dic53—24no—
95C2×Dic53—42no—
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100Dic106—44no—
101Dic106—84no—
102Dic106—82no—
103C8×C53—81no—
104#88322—82no—
105#12224—531nomaximal, Sylow
106C22×C53—21no—
107C22×C53—21no—
108C22×C53#45111yescharacteristic
109C22×C53#45511yescharacteristic
110C22×C53B(4,𝔽2)11yescharacteristic
111C4×C53—21no—
112C4×C53—21no—
113C4×C53—42no—
114C4×C53—42no—
115C4×C53—42no—
116C4×C53—81no—
117C4×C53—81no—
118Dic53—44no—
119Dic53—82no—
120#2811—531no—
121#2028D5311yescharacteristic
122#1605—532no—
123#1354—532no—
124#1319—531no—
125C2×C53—21no—
126C2×C53—21no—
127C2×C53#160211yescharacteristic
128C2×C53#219911yescharacteristic
129C2×C53#220511yescharacteristic
130#544C2×D5311yescharacteristic
131#451—1062no—
132#419—21no—
133#395—532no—
134#395—1062no—
135#391—1062no—
136#391—1061no—
137#391—1061no—
138#391C2×D5311yescharacteristic
139#343C2×D5311yescharacteristic
140#339—21no—
141#328—1062no—
142#324—532no—
143C53#1222411yescharacteristic, complemented, Sylow
144C22×Q8—532no—
145C22×Q8—1061no—
146C22×Q8C22⋊D5311yescharacteristic
147C2×Q16—2122no—
148C2×M16—21no—
149C4⋊Q8—41no—
150#125—41no—
151C22⋊Q8—22no—
152#117—21no—
153C2×C4⋊C4—1061no—
154C2×C4⋊C4—1061no—
155C2×C4⋊C4—1062no—
156C2×C4⋊C4C22×D5311yescharacteristic
157C2×C42—1061no—
158C2×C42C22⋊D5311yescharacteristic
159#103—41no—
160#103—41no—
161#103—2122no—
162C22⋊C8—1062no—
163#95—22no—
164#95—532no—
165#95—1062no—
166#95—1062no—
167#95—1062no—
168C2×Q8—21no—
169C2×Q8—22no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—1064no—
173C2×Q8—4241no—
174C2×Q8—4241no—
175C2×Q8—4242no—
176C2×Q8—2122no—
177C2×Q8#1171992512yescomplemented
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—1062no—
181C22×C4—1062no—
182C22×C4D4×D5312yes—
183C22×C4C2×C22⋊D5311yescharacteristic
184Q16—4244no—
185M16—41no—
186M16—41no—
187C2×C8—41no—
188C2×C8—2122no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—4242no—
195C4⋊C4—4241no—
196C4⋊C4—4241no—
197C22⋊C4—41no—
198C42—41no—
199C42—41no—
200C42—4241no—
201C23#1313318511yescharacteristic
202Q8—42no—
203Q8—81no—
204Q8—82no—
205Q8—4242no—
206Q8—4244no—
207Q8—2124no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—1064no—
221C2×C4—2122no—
222C2×C4—2122no—
223C2×C4—2122no—
224C2×C4—2122no—
225C2×C4—2122no—
226C2×C4C2×#1171992511yescharacteristic
227C2×C4#1313324711yescharacteristic
228C8—81no—
229C8—4242no—
230C22—21no—
231C22—21no—
232C22#1681077811yescharacteristic
233C22#1681080011yescharacteristic
234C22#1681080911yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—42no—
239C4—42no—
240C4—81no—
241C4—81no—
242C4—4242no—
243C4—2124no—
244C2—21no—
245C2—21no—
246C2#2027367111yescharacteristic
247C2#2027671511yescharacteristic
248C2#2027674411yescharacteristic
subgroupG/H
centerC22#16810809
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2028C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#2028D53
hypocenterC53#12224
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2028

Complements

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

NH ≅ G/Ntype
47. C2×Q8×C53Q16semidirect
143. C53#12224semidirect
177. C2×Q8#11719925semidirect

Maximal subgroups

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

subgroupindex
#202827082
#202736892
#202727442
#202726492
C53×#20282
#1222453