Finite Group Databasebeta

← #27296935

Subgroups — #27296935

Order 13568.

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

normal subgroups only

This group has 380 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#20275876C211yesmaximal, characteristic
2#20273688C212yesmaximal
3#20272745C212yesmaximal
4#20272649C211yesmaximal, characteristic
5C53×#2811C211yesmaximal, characteristic
6#16811254C2211yescharacteristic
7#16810779—22no—
8#16810632—21no—
9#16810392C2211yescharacteristic
10#16810383—21no—
11#16810359—22no—
12#16810349C2212yes—
13C53×#395—22no—
14C53×#395C2212yes—
15C53×#391—21no—
16C53×#391—21no—
17C53×#391—22no—
18C53×#391C2211yescharacteristic
19C2×#11719925—42no—
20C2×#11719917—21no—
21#13133216—41no—
22#13133208—41no—
23#13133196—22no—
24#13133176—21no—
25#13133141—42no—
26#13133141—41no—
27#13133141—41no—
28C22⋊#88322—22no—
29C22×Q8×C53—21no—
30C22×Q8×C53D411yescharacteristic
31C22×Q8×C53D412yes—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53C2311yescharacteristic
36C2×C42×C53—21no—
37C2×C42×C53D411yescharacteristic
38#95×C53—22no—
39#95×C53—22no—
40#95×C53—22no—
41#95×C53—22no—
42#95×C53D412yes—
43C2×Q8×C53—24no—
44C2×Q8×C53—21no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53—42no—
48C2×Q8×C53—82no—
49C2×Q8×C53—81no—
50C2×Q8×C53—81no—
51C2×Q8×C53Q1612yescomplemented
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D412yes—
58C2×Dic106—22no—
59C4⋊C4×C53—41no—
60C4⋊C4×C53—82no—
61C4⋊C4×C53—81no—
62C4⋊C4×C53—81no—
63C42×C53—41no—
64C42×C53—81no—
65C22⋊Dic53—41no—
66#11719925—84no—
67C4⋊Dic53—21no—
68C4⋊Dic53—21no—
69C4⋊Dic53—22no—
70#11719919—42no—
71C4×Dic53—41no—
72#11719917—41no—
73#11719917—41no—
74C2×#88322—42no—
75C2×#88322—41no—
76C23×C53#12011yescharacteristic
77Q8×C53—42no—
78Q8×C53—44no—
79Q8×C53—81no—
80Q8×C53—84no—
81Q8×C53—82no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—24no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—41no—
89C2×C4×C53—42no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—42no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53C2×Q1611yescharacteristic
99C2×C4×C53#13711yescharacteristic
100C2×Dic53—41no—
101C2×Dic53—41no—
102Dic106—82no—
103#88322—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—44no—
115C4×C53—42no—
116C4×C53—42no—
117C4×C53—82no—
118Dic53—81no—
119Dic53—81no—
120#2811D5311yescharacteristic
121#2028—531no—
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#544—531no—
131#451—1062no—
132#419—1061no—
133#395—22no—
134#395C2×D5312yes—
135#391—21no—
136#391—21no—
137#391—22no—
138#391C2×D5311yescharacteristic
139#343—531no—
140#339—1061no—
141#328—1062no—
142#324—532no—
143C53#1222411yescharacteristic, complemented, Sylow
144C22×Q8—21no—
145C22×Q8C22⋊D5311yescharacteristic
146C22×Q8C22⋊D5312yes—
147C2×Q16—2122no—
148C2×M16—1061no—
149C4⋊Q8—2121no—
150#125—2121no—
151C22⋊Q8—1062no—
152#117—1061no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4C22×D5311yescharacteristic
157C2×C42—21no—
158C2×C42C22⋊D5311yescharacteristic
159#103—2121no—
160#103—2121no—
161#103—2122no—
162C22⋊C8—1062no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—22no—
167#95C22⋊D5312yes—
168C2×Q8—24no—
169C2×Q8—21no—
170C2×Q8—41no—
171C2×Q8—42no—
172C2×Q8—41no—
173C2×Q8—82no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8—1062no—
177C2×Q8#1171992512yescomplemented
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4C2×C22⋊D5311yescharacteristic
183C22×C4C2×C22⋊D5312yes—
184Q16—4244no—
185M16—2121no—
186M16—2121no—
187C2×C8—2122no—
188C2×C8—2121no—
189C4⋊C4—41no—
190C4⋊C4—82no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—1061no—
194C4⋊C4—1061no—
195C4⋊C4—1062no—
196C4⋊C4—2122no—
197C22⋊C4—2121no—
198C42—41no—
199C42—81no—
200C42—2121no—
201C23#1313318611yescharacteristic
202Q8—42no—
203Q8—44no—
204Q8—81no—
205Q8—84no—
206Q8—82no—
207Q8—4242no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—2121no—
225C2×C4—2121no—
226C2×C4C2×#1171992511yescharacteristic
227C2×C4#1313324811yescharacteristic
228C8—4241no—
229C8—4242no—
230C22—21no—
231C22—21no—
232C22#1681077911yescharacteristic
233C22#1681080111yescharacteristic
234C22#1681081011yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—82no—
242C4—4241no—
243C4—4241no—
244C2—21no—
245C2—21no—
246C2#2027367211yescharacteristic
247C2#2027671611yescharacteristic
248C2#2027674511yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2811C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#2811D53
hypocenterC53#12224
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2811

Complements

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

NH ≅ G/Ntype
51. 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
#202758762
#202736882
#202727452
#202726492
C53×#28112
#1222453