Finite Group Databasebeta

← #27288433

Subgroups — #27288433

Order 13568.

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

normal subgroups only

This group has 500 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#20285466C211yesmaximal, characteristic
2#20275178C212yesmaximal, complemented
3#20273230C212yesmaximal, complemented
4#20272592C211yesmaximal, characteristic, complemented
5C53×#1851C211yesmaximal, characteristic, complemented
6C2×#13133191C2212yes—
7#16810937—22no—
8#16810859—21no—
9#16810859—21no—
10#16810759—22no—
11C2×#13133133—22no—
12#16810587C2212yes—
13#16810391—22no—
14#16810356—22no—
15#16810356—22no—
16C22×M16×C53C2211yescharacteristic
17C2×C4⋊C8×C53—21no—
18C2×C22⋊C8×C53C2211yescharacteristic
19C4×C22⋊C4×C53C2211yescharacteristic
20C53×#336—21no—
21C22×D212D412yes—
22C22×C22⋊D53—22no—
23C2×D424—21no—
24C2×D424—21no—
25C2×#11719917—22no—
26#13133191—22no—
27#13133191—22no—
28#13133191—22no—
29#13133191—22no—
30C2×C4⋊Dic53D412yes—
31C2×#11719921—22no—
32#13133151—21no—
33#13133151—21no—
34#13133139—22no—
35#13133139—22no—
36#13133133—44no—
37C22⋊#88322—22no—
38C22⋊#88322—22no—
39C23×C4×C53C2311yescharacteristic
40C2×M16×C53—22no—
41C2×M16×C53—42no—
42C22×C8×C53—21no—
43C22×C8×C53D411yescharacteristic
44C22×C8×C53D411yescharacteristic
45C2×C22⋊C4×C53—21no—
46C2×C42×C53—21no—
47C4⋊C8×C53—42no—
48C22⋊C8×C53—22no—
49#95×C53D411yescharacteristic
50#95×C53D411yescharacteristic
51C24×C53#4111yescharacteristic
52C23×D53—22no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—41no—
56C22×C4×C53—41no—
57C22×C4×C53C2×D412yes—
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53#4112yes—
61C2×C22⋊D53—42no—
62C2×C22⋊D53—42no—
63C2×C22⋊D53—82no—
64C2×C22⋊D53—82no—
65C22×Dic53—22no—
66C2×D212—22no—
67C2×D212—22no—
68C2×D212—42no—
69C2×D212—42no—
70M16×C53—24no—
71M16×C53—42no—
72C2×C8×C53—22no—
73C2×C8×C53—21no—
74C2×C8×C53—21no—
75C2×C8×C53—41no—
76C2×C8×C53—41no—
77C2×C8×C53—41no—
78C2×C8×C53—41no—
79C22⋊C4×C53—42no—
80C42×C53—41no—
81C42×C53—41no—
82#11719921—42no—
83#11719921—42no—
84C4⋊Dic53—22no—
85C4⋊Dic53—22no—
86#11719917—42no—
87#11719917—42no—
88C2×#88322—42no—
89D424—81no—
90D424—81no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53#12111yescharacteristic
96C23×C53#12312yes—
97C22×D53—42no—
98C22×D53—44no—
99C22×D53—82no—
100C22×D53—82no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—41no—
107C2×C4×C53—42no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53#12011yescharacteristic
114C2×C4×C53#12112yes—
115C2×C4×C53#12411yescharacteristic
116C22⋊D53—84no—
117C22⋊D53—82no—
118C22⋊D53—82no—
119C2×Dic53—42no—
120C2×Dic53—82no—
121D212—42no—
122D212—42no—
123D212—82no—
124C8×C53—41no—
125C8×C53—42no—
126C8×C53—41no—
127#88322—82no—
128#10557—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—22no—
132C22×C53—22no—
133C22×C53—22no—
134C22×C53—81no—
135C22×C53—81no—
136C22×C53#39411yescharacteristic
137C22×C53#47211yescharacteristic
138C22×C53#47211yescharacteristic
139C2×D53—82no—
140C2×D53—82no—
141C2×D53—82no—
142C2×D53—164no—
143C4×C53—22no—
144C4×C53—21no—
145C4×C53—42no—
146C4×C53—81no—
147C4×C53B(4,𝔽2)11yescharacteristic
148C4×C53#45811yescharacteristic
149Dic53—82no—
150#3186—531no—
151#1900—532no—
152#1851D5311yescharacteristic, complemented
153#1507—532no—
154#1293—531no—
155C2×C53—21no—
156C2×C53—21no—
157C2×C53—42no—
158C2×C53#204411yescharacteristic
159C2×C53#204511yescharacteristic
160C2×C53#205011yescharacteristic
161D53—162no—
162C22×M16C2×D5311yescharacteristic
163C2×#121—532no—
164#480—1062no—
165#466—1061no—
166#466—1061no—
167#447—1062no—
168C2×C4⋊C8—21no—
169C2×#100—1062no—
170#407—532no—
171C2×C22⋊C8C2×D5311yescharacteristic
172C4×C22⋊C4C2×D5311yescharacteristic
173#343—1062no—
174#336—21no—
175#327—1062no—
176#327—1062no—
177C53#1055711yescharacteristic, complemented, Sylow
178C22×D4—532no—
179C22×D4—1062no—
180C23×C4C22×D5311yescharacteristic
181C2×D8—1061no—
182C2×D8—1061no—
183C2×M16—22no—
184C2×M16—42no—
185C2×M16—1062no—
186C22×C8—21no—
187C22×C8C22⋊D5311yescharacteristic
188C22×C8C22⋊D5311yescharacteristic
189#121—1062no—
190#121—1062no—
191#121—1062no—
192#121—1062no—
193C2×C4⋊C4—532no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—1062no—
196C2×C42—21no—
197#107—1061no—
198#107—1061no—
199C4⋊C8—42no—
200#102—1062no—
201#102—1062no—
202#100—2124no—
203C22⋊C8—22no—
204C22⋊C8—1062no—
205C22⋊C8—1062no—
206#95D21211yescharacteristic
207#95D21211yescharacteristic
208C24—1062no—
209C24#1171994511yescharacteristic
210C2×D4—1062no—
211C2×D4—1062no—
212C2×D4—4242no—
213C2×D4—4242no—
214C2×D4—2122no—
215C2×D4—2122no—
216C2×D4—2122no—
217C2×D4—2122no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—1062no—
223C22×C4C2×D21211yescharacteristic
224C22×C4D4×D5312yes—
225C22×C4#1171994912yes—
226C22×C4C2×C22⋊D5311yescharacteristic
227D8—4241no—
228D8—4241no—
229M16—24no—
230M16—42no—
231M16—2122no—
232M16—2122no—
233C2×C8—22no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—2122no—
241C4⋊C4—1062no—
242C4⋊C4—1062no—
243C22⋊C4—42no—
244C22⋊C4—2122no—
245C22⋊C4—2122no—
246C42—41no—
247C42—41no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—4242no—
253C23—4242no—
254C23—2124no—
255C23—2122no—
256C23#1313319111yescharacteristic
257C23#1313320212yes—
258D4—4242no—
259D4—4244no—
260D4—4242no—
261D4—4242no—
262D4—2122no—
263D4—2122no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—4242no—
277C2×C4—2122no—
278C2×C4#1313318511yescharacteristic
279C2×C4#1313319212yes—
280C2×C4#1313320611yescharacteristic
281C8—42no—
282C8—41no—
283C8—41no—
284C8—4242no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—81no—
291C22—81no—
292C22—8484no—
293C22—4242no—
294C22—4242no—
295C22—4242no—
296C22#1681054011yescharacteristic
297C22#1681089311yescharacteristic
298C22#1681089311yescharacteristic
299C4—22no—
300C4—21no—
301C4—42no—
302C4—81no—
303C4—4242no—
304C4#1681080911yescharacteristic
305C4#1681081211yescharacteristic
306C2—21no—
307C2—21no—
308C2—42no—
309C2—8482no—
310C2#2027596611yescharacteristic
311C2#2027597111yescharacteristic
312C2#2027600211yescharacteristic
subgroupG/H
centerC22#16810540
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1851C2
socleC22×C53#394
prime radical#20285466C2
hypercenter#1851D53
hypocenterC53#10557
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
2. #20275178C2semidirect
3. #20273230C2semidirect
4. #20272592C2semidirect
5. C53×#1851C2semidirect
152. #1851D53semidirect
177. C53#10557semidirect

Maximal subgroups

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

subgroupindex
#202854662
#202751782
#202732302
#202725922
C53×#18512
#1055753