Finite Group Databasebeta

← #27303774

Subgroups — #27303774

Order 13568.

484 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 79 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#20282661C211yesmaximal, characteristic
2#20274996C212yesmaximal
3#20272840C211yesmaximal, characteristic
4#20272834C212yesmaximal
5C53×#1856C211yesmaximal, characteristic
6C2×#13133132C2212yes—
7#16810562—21no—
8#16810535—22no—
9#16810529—21no—
10#16810503C2212yes—
11#16810498—22no—
12#16810495—22no—
13#16810390—22no—
14C53×#518C2211yescharacteristic
15C53×#428—21no—
16C2×#100×C53C2211yescharacteristic
17C53×#390C2211yescharacteristic
18C53×#339—21no—
19C22×Dic106—21no—
20C23×Dic53D412yes—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—21no—
23C2×#11719919—42no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26C2×C22⋊Dic53D412yes—
27#13133132—22no—
28#13133132Q814yescomplemented
29#13133124—22no—
30#13133124—22no—
31#13133124—21no—
32#13133124—22no—
33#13133124—42no—
34C22×D4×C53C2311yescharacteristic
35C2×M16×C53—21no—
36C4×D4×C53—41no—
37C4×D4×C53—41no—
38#117×C53—21no—
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C42×C53D411yescharacteristic
43#107×C53—41no—
44#106×C53—41no—
45#100×C53—21no—
46#100×C53Q812yescomplemented
47C24×C53C2×D412yes—
48C2×D4×C53—41no—
49C2×D4×C53—41no—
50C2×D4×C53C2×Q811yescharacteristic
51C2×D4×C53C2×Q812yes—
52C2×D4×C53#4111yescharacteristic
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53C2×D411yescharacteristic
58C22×Dic53—22no—
59C22×Dic53—22no—
60C22×Dic53—42no—
61C22×Dic53—42no—
62C22×Dic53—82no—
63C2×Dic106—42no—
64C2×Dic106—81no—
65C2×Dic106—81no—
66M16×C53—22no—
67M16×C53—41no—
68C2×C8×C53—41no—
69C4⋊C4×C53—21no—
70C4⋊C4×C53—21no—
71C4⋊C4×C53—41no—
72C22⋊C4×C53—44no—
73C22⋊C4×C53—42no—
74C22⋊C4×C53—41no—
75C42×C53—41no—
76C42×C53—41no—
77C22⋊Dic53—24no—
78C22⋊Dic53—42no—
79C22⋊Dic53—42no—
80C22⋊Dic53—82no—
81C4⋊Dic53—81no—
82C4⋊Dic53—81no—
83#11719919—84no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53#12011yescharacteristic
89C23×C53C22⋊Q812yes—
90C23×C53C4⋊Q812yes—
91D4×C53—41no—
92D4×C53—42no—
93D4×C53—41no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—42no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—82no—
104C2×C4×C53—82no—
105C2×C4×C53C22⋊Q811yescharacteristic
106C2×C4×C53C4⋊Q811yescharacteristic
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111C2×Dic53—44no—
112C2×Dic53—44no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—82no—
116C2×Dic53—82no—
117C2×Dic53—82no—
118Dic106—84no—
119C8×C53—42no—
120#13463—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—22no—
124C22×C53—22no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—82no—
130C22×C53#39511yescharacteristic
131C22×C53B(4,𝔽2)11yescharacteristic
132C22×C53#45811yescharacteristic
133C4×C53—21no—
134C4×C53—21no—
135C4×C53—42no—
136C4×C53—81no—
137C4×C53—81no—
138C4×C53—81no—
139Dic53—82no—
140Dic53—84no—
141Dic53—82no—
142#2804—531no—
143#1867—532no—
144#1856D5311yescharacteristic
145#1393—531no—
146#1390—532no—
147C2×C53—21no—
148C2×C53—21no—
149C2×C53—42no—
150C2×C53—42no—
151C2×C53#206211yescharacteristic
152C2×C53#219811yescharacteristic
153C2×C53#219911yescharacteristic
154#518C2×D5311yescharacteristic
155#428—21no—
156C2×#100C2×D5311yescharacteristic
157C2×#99—532no—
158#400—1061no—
159#393—1062no—
160#391—1061no—
161#390C2×D5311yescharacteristic
162#386—532no—
163#385—1062no—
164#385—1062no—
165#342—1062no—
166#339—21no—
167C53#1346311yescharacteristic, complemented, Sylow
168C22×Q8—1061no—
169C22×D4C22×D5311yescharacteristic
170C23×C4—532no—
171C2×M16—21no—
172C4×D4—41no—
173C4×D4—41no—
174#117—21no—
175C2×C4⋊C4—1061no—
176C2×C4⋊C4—1061no—
177C2×C4⋊C4—2122no—
178C2×C4⋊C4C22⋊D5311yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—532no—
182C2×C22⋊C4—1062no—
183C2×C22⋊C4—1062no—
184C2×C42C22⋊D5311yescharacteristic
185#107—41no—
186#106—41no—
187#100—21no—
188#100Dic10612yescomplemented
189#99—534no—
190#99—1062no—
191#95—1062no—
192#95—1062no—
193#95—1061no—
194#95—1062no—
195#95—2122no—
196C24D4×D5312yes—
197C2×Q8—4241no—
198C2×Q8—4241no—
199C2×Q8—2122no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4C2×Dic10611yescharacteristic
203C2×D4#1171994511yescharacteristic
204C2×D4Q8×D5312yes—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—1062no—
210C22×C4—1062no—
211C22×C4—4242no—
212C22×C4—2122no—
213C22×C4—2122no—
214C22×C4C2×C22⋊D5311yescharacteristic
215M16—22no—
216M16—41no—
217C2×C8—41no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—41no—
221C4⋊C4—4241no—
222C4⋊C4—4241no—
223C4⋊C4—4244no—
224C22⋊C4—44no—
225C22⋊C4—42no—
226C22⋊C4—41no—
227C22⋊C4—1064no—
228C22⋊C4—4242no—
229C22⋊C4—2122no—
230C22⋊C4—2122no—
231C42—41no—
232C42—41no—
233C23—22no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23#1313318511yescharacteristic
238C23#1313319512yes—
239C23#1313321612yes—
240Q8—4244no—
241D4—41no—
242D4—41no—
243D4—42no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—4242no—
256C2×C4—4242no—
257C2×C4—4242no—
258C2×C4—2124no—
259C2×C4—2122no—
260C2×C4—2122no—
261C2×C4—2122no—
262C2×C4—2122no—
263C2×C4—2124no—
264C2×C4—2122no—
265C2×C4—2122no—
266C2×C4#1313319611yescharacteristic
267C2×C4#1313321511yescharacteristic
268C8—42no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—82no—
278C22#1681054311yescharacteristic
279C22#1681080911yescharacteristic
280C22#1681081211yescharacteristic
281C4—21no—
282C4—21no—
283C4—42no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4—4242no—
288C4—4244no—
289C4—4242no—
290C2—21no—
291C2—21no—
292C2—42no—
293C2—42no—
294C2#2027606811yescharacteristic
295C2#2027671011yescharacteristic
296C2#2027671511yescharacteristic
subgroupG/H
centerC22#16810809
commutator, prime radical, cosocleC22×D4×C53C23
FrattiniC22×D4C22×D53
FittingC53×#1856C2
socleC22×C53B(4,𝔽2)
hypercenter#1856D53
hypocenterC53#13463
LCSG ▷ C22×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1856

Complements

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

NH ≅ G/Ntype
28. #13133132Q8semidirect
46. #100×C53Q8semidirect
167. C53#13463semidirect
188. #100Dic106semidirect

Maximal subgroups

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

subgroupindex
#202826612
#202749962
#202728402
#202728342
C53×#18562
#1346353