Finite Group Databasebeta

← #27281964

Subgroups — #27281964

Order 13568.

522 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283800C211yesmaximal, characteristic, complemented
2#20279620C211yesmaximal, characteristic
3#20273139C212yesmaximal
4#20272575C212yesmaximal, complemented
5C53×#1801C211yesmaximal, characteristic, complemented
6C2×#13133193—21no—
7C2×C4×D212C2211yescharacteristic
8#16810754—24no—
9#16810680—22no—
10#16810676C2212yes—
11#16810553C412yes—
12#16810540C412yes—
13#16810514—22no—
14#16810489C2211yescharacteristic
15#16810480—21no—
16#16810350—24no—
17C2×C4×Q8×C53C2211yescharacteristic
18C53×#423C2212yes—
19C2×#101×C53—21no—
20C53×#343—21no—
21C22×D212C2×C411yescharacteristic
22C22×C4×D53—21no—
23#13133193—44no—
24C4×D212—22no—
25C4×D212—22no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—22no—
29C2×#11719921—22no—
30C2×#11719921—22no—
31C2×#11719921—21no—
32C4⋊#88322—24no—
33C4⋊#88322—22no—
34C22⋊#88322—22no—
35#13133128—24no—
36C4×#88322D414yes—
37#13133124—22no—
38#13133124—21no—
39#13133124C2×C412yes—
40C22×Q8×C53D411yescharacteristic
41C2×M16×C53—22no—
42C4×Q8×C53—22no—
43C4×Q8×C53—22no—
44C2×C4⋊C4×C53C2×C412yes—
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C42×C53D412yes—
47C2×C42×C53C2311yescharacteristic
48C4⋊C8×C53—22no—
49C4⋊C8×C53—22no—
50#101×C53—42no—
51C23×D53—21no—
52C2×Q8×C53—22no—
53C2×Q8×C53—21no—
54C2×Q8×C53—22no—
55C2×Q8×C53—21no—
56C22×C4×C53C22⋊C412yes—
57C22×C4×C53C22⋊C412yes—
58C22×C4×C53C22×C411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×Dic53—22no—
62C22×Dic53—21no—
63C2×D212—22no—
64C2×D212—42no—
65C2×C4×D53—42no—
66C2×C4×D53—42no—
67C2×C4×D53—41no—
68C2×C4×D53—41no—
69M16×C53—42no—
70M16×C53—42no—
71C2×C8×C53—42no—
72C4⋊C4×C53—24no—
73C4⋊C4×C53—22no—
74C42×C53—22no—
75C42×C53—22no—
76C42×C53C2×D412yes—
77C42×C53#4112yes—
78#11719921—44no—
79#11719921—44no—
80#11719921—41no—
81#11719921—41no—
82C4⋊Dic53—22no—
83C4⋊Dic53—42no—
84#11719919—44no—
85C2×#88322—24no—
86C23×C53C2×C22⋊C411yescharacteristic
87C22×D53—22no—
88C22×D53—41no—
89C22×D53—41no—
90C22×D53—41no—
91C22×D53—41no—
92C22×D53—41no—
93C22×D53—41no—
94Q8×C53—42no—
95Q8×C53—42no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53C4×D412yes—
104C2×C4×C53#12011yescharacteristic
105C2×C4×C53#12112yes—
106C2×C4×C53#12311yescharacteristic
107C2×C4×C53AGL(1,ℤ/8)12yes—
108C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
109C2×C4×C53#13711yescharacteristic
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—42no—
116D212—24no—
117D212—42no—
118C4×D53—82no—
119C4×D53—82no—
120C8×C53—82no—
121#88322—44no—
122#9490—531nomaximal, Sylow
123C22×C53—21no—
124C22×C53—21no—
125C22×C53#38611yescharacteristic
126C22×C53#41711yescharacteristic
127C22×C53#41811yescharacteristic
128C2×D53—42no—
129C2×D53—42no—
130C2×D53—42no—
131C2×D53—81no—
132C2×D53—81no—
133C2×D53—81no—
134C2×D53—81no—
135C4×C53—22no—
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—42no—
139C4×C53—42no—
140C4×C53#46812yes—
141C4×C53#46912yes—
142Dic53—82no—
143Dic53—81no—
144#2983—531no—
145#2536—531no—
146#1801D5311yescharacteristic, complemented
147#1486—532no—
148#1286—532no—
149C2×C53—21no—
150C2×C53—21no—
151C2×C53#189111yescharacteristic
152C2×C53#189511yescharacteristic
153C2×C53#191211yescharacteristic
154D53—82no—
155C2×C22⋊Q8—1061no—
156C2×C4×Q8C2×D5311yescharacteristic
157C2×C4×D4—531no—
158#446—1064no—
159#432—1062no—
160#431—532no—
161#423C2×D5312yes—
162C2×#101—21no—
163#398—532no—
164#394—532no—
165#388—1062no—
166#382—531no—
167#380—1061no—
168#343—21no—
169D4⋊C8—1064no—
170C53#949011yescharacteristic, complemented, Sylow
171C22×Q8C22⋊D5311yescharacteristic
172C22×D4—531no—
173C23×C4—1061no—
174C2×M16—22no—
175C22⋊Q8—2124no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—1062no—
179C4×D4—1062no—
180C2×C4⋊C4—531no—
181C2×C4⋊C4—1061no—
182C2×C4⋊C4—1062no—
183C2×C4⋊C4C4×D5312yes—
184C2×C4⋊C4C22⋊D5311yescharacteristic
185C2×C22⋊C4—1061no—
186C2×C22⋊C4—1062no—
187C2×C22⋊C4—1062no—
188C2×C42D21212yes—
189C2×C42C22×D5311yescharacteristic
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—1064no—
193C4⋊C8—1062no—
194#101—42no—
195C22⋊C8—1062no—
196#97—1064no—
197C4×C8—534no—
198#95—532no—
199#95—1061no—
200#95—1062no—
201C24—1061no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×D4—1062no—
207C2×D4—2122no—
208C22×C4—1062no—
209C22×C4—1061no—
210C22×C4—2122no—
211C22×C4—2122no—
212C22×C4—2121no—
213C22×C4—2121no—
214C22×C4#1171992112yes—
215C22×C4#1171992112yes—
216C22×C4C2×C4×D5311yescharacteristic
217C22×C4C2×D21211yescharacteristic
218C22×C4C2×C22⋊D5311yescharacteristic
219M16—42no—
220M16—42no—
221C2×C8—42no—
222C2×C8—1064no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—1062no—
226C4⋊C4—2124no—
227C4⋊C4—2122no—
228C22⋊C4—2124no—
229C22⋊C4—2124no—
230C22⋊C4—2121no—
231C22⋊C4—2121no—
232C42—22no—
233C42—22no—
234C42D4×D5312yes—
235C42#1171994712yes—
236C23—1062no—
237C23—2121no—
238C23—2121no—
239C23—2121no—
240C23—2121no—
241C23—2121no—
242C23—2121no—
243C23C2×#1171992111yescharacteristic
244Q8—42no—
245Q8—42no—
246D4—1064no—
247D4—2122no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—4242no—
256C2×C4—4242no—
257C2×C4—2122no—
258C2×C4—2122no—
259C2×C4—2121no—
260C2×C4—2121no—
261C2×C4—2122no—
262C2×C4—2121no—
263C2×C4#1313317712yes—
264C2×C4#1313318711yescharacteristic
265C2×C4#1313318912yes—
266C2×C4#1313320111yescharacteristic
267C2×C4#1313324212yes—
268C2×C4#1313324311yescharacteristic
269C2×C4#1313324811yescharacteristic
270C8—82no—
271C8—2124no—
272C22—21no—
273C22—21no—
274C22—4241no—
275C22—4241no—
276C22—4241no—
277C22—4241no—
278C22—2122no—
279C22—2122no—
280C22—2122no—
281C22#1681050011yescharacteristic
282C22#1681062211yescharacteristic
283C22#1681062711yescharacteristic
284C4—21no—
285C4—21no—
286C4—22no—
287C4—42no—
288C4—42no—
289C4—4242no—
290C4—4241no—
291C4#1681087112yes—
292C4#1681087412yes—
293C2—21no—
294C2—21no—
295C2—4242no—
296C2#2027513111yescharacteristic
297C2#2027515111yescharacteristic
298C2#2027525011yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1801C2
socleC22×C53#386
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1801D53
hypocenterC53#9490
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
1. #20283800C2semidirect
4. #20272575C2semidirect
5. C53×#1801C2semidirect
146. #1801D53semidirect
170. C53#9490semidirect

Maximal subgroups

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

subgroupindex
#202838002
#202796202
#202731392
#202725752
C53×#18012
#949053