Finite Group Databasebeta

← #27296894

Subgroups — #27296894

Order 13568.

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

normal subgroups only

This group has 484 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#20275879C211yesmaximal, characteristic
2#20273671C212yesmaximal, complemented
3#20272736C212yesmaximal, complemented
4#20272643C211yesmaximal, characteristic
5C53×#2840C211yesmaximal, characteristic
6#16811249C2211yescharacteristic
7#16810779—22no—
8#16810479—21no—
9#16810390C2211yescharacteristic
10#16810389—21no—
11#16810359—22no—
12#16810347C2212yes—
13C53×#393—22no—
14C53×#393—21no—
15C53×#393—22no—
16C53×#393C2212yes—
17C53×#379—21no—
18C53×#379C2211yescharacteristic
19C2×#11719925—42no—
20#13133196—22no—
21#13133194—42no—
22C2×C22⋊Dic53—21no—
23C2×C22⋊Dic53—21no—
24C2×C22⋊Dic53—42no—
25#13133141—42no—
26C22⋊#88322—22no—
27C25×C53D411yescharacteristic
28C22×Q8×C53—21no—
29C22×Q8×C53D412yes—
30C2×C22⋊C4×C53—21no—
31C2×C22⋊C4×C53—22no—
32C2×C22⋊C4×C53—21no—
33C2×C22⋊C4×C53—22no—
34C2×C22⋊C4×C53—22no—
35C2×C22⋊C4×C53D411yescharacteristic
36C2×C22⋊C4×C53C2311yescharacteristic
37#95×C53—21no—
38#95×C53—22no—
39#95×C53—22no—
40#95×C53D412yes—
41C24×C53—21no—
42C24×C53—21no—
43C24×C53—41no—
44C24×C53—41no—
45C24×C53—82no—
46C24×C53Q1612yescomplemented
47C24×C53C2×D411yescharacteristic
48C2×Q8×C53—42no—
49C2×Q8×C53—82no—
50C2×Q8×C53—82no—
51C2×Q8×C53—81no—
52C22×C4×C53—22no—
53C22×C4×C53—21no—
54C22×C4×C53—22no—
55C22×C4×C53C2×D412yes—
56C22×Dic53—22no—
57C22×Dic53—41no—
58C2×Dic106—22no—
59C22⋊C4×C53—41no—
60C22⋊C4×C53—41no—
61C22⋊C4×C53—82no—
62C22⋊C4×C53—82no—
63C22⋊C4×C53—82no—
64C22⋊C4×C53—81no—
65C22⋊C4×C53—81no—
66C22⋊Dic53—41no—
67C22⋊Dic53—41no—
68C22⋊Dic53—41no—
69C22⋊Dic53—41no—
70C22⋊Dic53—84no—
71#11719925—84no—
72C4⋊Dic53—22no—
73#11719919—42no—
74C2×#88322—42no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53—22no—
78C23×C53—42no—
79C23×C53—42no—
80C23×C53—42no—
81C23×C53—42no—
82C23×C53—44no—
83C23×C53—41no—
84C23×C53—41no—
85C23×C53—81no—
86C23×C53—82no—
87C23×C53—82no—
88C23×C53—82no—
89C23×C53—84no—
90C23×C53#12011yescharacteristic
91C23×C53C2×Q1611yescharacteristic
92C23×C53#13711yescharacteristic
93Q8×C53—44no—
94Q8×C53—82no—
95Q8×C53—84no—
96C2×C4×C53—22no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×Dic53—42no—
107C2×Dic53—82no—
108C2×Dic53—81no—
109C2×Dic53—82no—
110Dic106—82no—
111#88322—82no—
112#12217—531nomaximal, Sylow
113C22×C53—22no—
114C22×C53—21no—
115C22×C53—21no—
116C22×C53—41no—
117C22×C53—41no—
118C22×C53—42no—
119C22×C53—42no—
120C22×C53—42no—
121C22×C53—44no—
122C22×C53—42no—
123C22×C53—81no—
124C22×C53—82no—
125C22×C53—82no—
126C22×C53—84no—
127C22×C53—82no—
128C22×C53#45111yescharacteristic
129C22×C53#45511yescharacteristic
130C22×C53B(4,𝔽2)11yescharacteristic
131C4×C53—42no—
132C4×C53—81no—
133C4×C53—82no—
134C4×C53—82no—
135Dic53—82no—
136Dic53—161no—
137#2840D5311yescharacteristic
138#2029—531no—
139#1602—532no—
140#1351—532no—
141#1316—531no—
142C2×C53—21no—
143C2×C53—21no—
144C2×C53—41no—
145C2×C53—41no—
146C2×C53—82no—
147C2×C53#160211yescharacteristic
148C2×C53#219912yes—
149#543—531no—
150#451—1062no—
151#393—22no—
152#393—21no—
153#393—22no—
154#393C2×D5312yes—
155#379—21no—
156#379—1061no—
157#379C2×D5311yescharacteristic
158#342—531no—
159#342—1061no—
160#328—1062no—
161#323—532no—
162C53#1221711yescharacteristic, complemented, Sylow
163C25C22⋊D5311yescharacteristic
164C22×Q8—21no—
165C22×Q8C22⋊D5312yes—
166C2×Q16—2122no—
167C22⋊Q8—1062no—
168C22⋊Q8—2122no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—1061no—
175C2×C22⋊C4—1061no—
176C2×C22⋊C4—2122no—
177C2×C22⋊C4C22⋊D5311yescharacteristic
178C2×C22⋊C4C22×D5311yescharacteristic
179#103—2122no—
180C22⋊C8—1062no—
181#95—21no—
182#95—22no—
183#95—22no—
184#95C22⋊D5312yes—
185C24—21no—
186C24—21no—
187C24—41no—
188C24—41no—
189C24—82no—
190C24#1171992512yescomplemented
191C24C2×C22⋊D5311yescharacteristic
192C2×Q8—42no—
193C2×Q8—82no—
194C2×Q8—82no—
195C2×Q8—81no—
196C2×Q8—1062no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—1062no—
201C22×C4—2121no—
202C22×C4C2×C22⋊D5312yes—
203Q16—4244no—
204C2×C8—2122no—
205C4⋊C4—1062no—
206C4⋊C4—2122no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C22⋊C4—4244no—
215C22⋊C4—2121no—
216C22⋊C4—2121no—
217C22⋊C4—2121no—
218C22⋊C4—2121no—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—44no—
227C23—41no—
228C23—41no—
229C23—81no—
230C23—82no—
231C23—82no—
232C23—82no—
233C23—84no—
234C23#1313318611yescharacteristic
235C23C2×#1171992511yescharacteristic
236C23#1313324811yescharacteristic
237Q8—44no—
238Q8—82no—
239Q8—84no—
240Q8—4242no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—4241no—
252C2×C4—4242no—
253C2×C4—4242no—
254C2×C4—2122no—
255C8—4242no—
256C22—22no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—44no—
265C22—42no—
266C22—81no—
267C22—82no—
268C22—82no—
269C22—84no—
270C22—82no—
271C22#1681077911yescharacteristic
272C22#1681080111yescharacteristic
273C22#1681081011yescharacteristic
274C4—42no—
275C4—81no—
276C4—82no—
277C4—82no—
278C4—8481no—
279C4—4242no—
280C2—21no—
281C2—21no—
282C2—41no—
283C2—41no—
284C2—82no—
285C2#2027367211yescharacteristic
286C2#2027671612yes—
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C22⋊C4×C53C23
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2840C2
socleC22×C53B(4,𝔽2)
prime radicalC25×C53D4
hypercenter#2840D53
hypocenterC53#12217
LCSG ▷ C2×C22⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2840

Complements

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

NH ≅ G/Ntype
2. #20273671C2semidirect
3. #20272736C2semidirect
46. C24×C53Q16semidirect
162. C53#12217semidirect
190. C24#11719925semidirect

Maximal subgroups

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

subgroupindex
#202758792
#202736712
#202727362
#202726432
C53×#28402
#1221753