Finite Group Databasebeta

← #27281520

Subgroups — #27281520

Order 13568.

554 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 115 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#20279714C211yesmaximal, characteristic
3#20273384C212yesmaximal
4#20272616C212yesmaximal, complemented
5C53×#1801C211yesmaximal, characteristic, complemented
6C2×#13133214C2211yescharacteristic
7C2×#13133206—21no—
8#16810754—24no—
9#16810680—22no—
10#16810676C2212yes—
11#16810540C414yes—
12#16810525—22no—
13#16810524—21no—
14C4×C4⋊Dic53C2211yescharacteristic
15#16810366—24no—
16C2×C4×Q8×C53C2211yescharacteristic
17C53×#423C2212yes—
18C2×#101×C53—21no—
19C53×#343—21no—
20C22×D212—21no—
21C22×D212—21no—
22C22×D212C2×C412yes—
23#13133214—22no—
24#13133214—22no—
25#13133206—44no—
26C2×C4⋊Dic53—21no—
27C2×#11719921—24no—
28C2×#11719921—24no—
29C2×C4×Dic53—21no—
30C4⋊#88322—24no—
31C4⋊#88322D414yes—
32C22⋊#88322—22no—
33#13133128—24no—
34C4×#88322—22no—
35#13133124C2×C412yes—
36C22×Q8×C53D411yescharacteristic
37C2×M16×C53—22no—
38C4×Q8×C53—22no—
39C4×Q8×C53—22no—
40C2×C4⋊C4×C53C2×C412yes—
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C42×C53D412yes—
43C2×C42×C53C2311yescharacteristic
44C4⋊C8×C53—22no—
45C4⋊C8×C53—22no—
46#101×C53—42no—
47C23×D53—22no—
48C2×Q8×C53—22no—
49C2×Q8×C53—21no—
50C2×Q8×C53—22no—
51C2×Q8×C53—21no—
52C22×C4×C53C22⋊C412yes—
53C22×C4×C53C22⋊C412yes—
54C22×C4×C53C22×C411yescharacteristic
55C22×C4×C53C2×D411yescharacteristic
56C22×C4×C53C2×D411yescharacteristic
57C22×Dic53—22no—
58C2×D212—24no—
59C2×D212—44no—
60C2×D212—41no—
61C2×D212—42no—
62C2×D212—41no—
63C2×D212—42no—
64C2×D212—41no—
65C2×D212—41no—
66C2×D212—82no—
67M16×C53—42no—
68M16×C53—42no—
69C2×C8×C53—42no—
70C4⋊C4×C53—24no—
71C4⋊C4×C53—22no—
72C42×C53—22no—
73C42×C53—22no—
74C42×C53C2×D412yes—
75C42×C53#4112yes—
76#11719921—48no—
77#11719921—84no—
78C4⋊Dic53—42no—
79C4×Dic53—42no—
80C2×#88322—24no—
81C23×C53C2×C22⋊C411yescharacteristic
82C22×D53—44no—
83C22×D53—44no—
84C22×D53—42no—
85C22×D53—42no—
86C22×D53—42no—
87Q8×C53—42no—
88Q8×C53—42no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—41no—
95C2×C4×C53—42no—
96C2×C4×C53C4×D412yes—
97C2×C4×C53#12011yescharacteristic
98C2×C4×C53#12112yes—
99C2×C4×C53#12311yescharacteristic
100C2×C4×C53AGL(1,ℤ/8)12yes—
101C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
102C2×C4×C53#13711yescharacteristic
103C2×Dic53—42no—
104C2×Dic53—42no—
105C2×Dic53—42no—
106D212—84no—
107D212—82no—
108D212—82no—
109D212—82no—
110D212—82no—
111C8×C53—82no—
112#88322—44no—
113#9406—531nomaximal, Sylow
114C22×C53—21no—
115C22×C53—21no—
116C22×C53#38611yescharacteristic
117C22×C53#41711yescharacteristic
118C22×C53#41811yescharacteristic
119C2×D53—84no—
120C2×D53—84no—
121C2×D53—82no—
122C2×D53—82no—
123C2×D53—82no—
124C4×C53—21no—
125C4×C53—22no—
126C4×C53—42no—
127C4×C53—42no—
128C4×C53#45412yes—
129C4×C53#46812yes—
130C4×C53#46912yes—
131Dic53—82no—
132#2983—531no—
133#2548—531no—
134#1801D5311yescharacteristic, complemented
135#1545—532no—
136#1303—532no—
137C2×C53—21no—
138C2×C53—21no—
139C2×C53#188811yescharacteristic
140C2×C53#189511yescharacteristic
141C2×C53#191411yescharacteristic
142D53—162no—
143C2×#127—531no—
144C2×#124—1061no—
145C2×C4×Q8C2×D5311yescharacteristic
146#446—1064no—
147#432—1062no—
148#431—532no—
149#423C2×D5312yes—
150C2×#101—21no—
151#394—534no—
152#390—1061no—
153#390—1062no—
154C4×C4⋊C4—531no—
155#343—21no—
156#331—1064no—
157C53#940611yescharacteristic, complemented, Sylow
158C22×Q8C22⋊D5311yescharacteristic
159C22×D4—532no—
160C22×D4—1061no—
161C22×D4—1061no—
162C2×M16—22no—
163#127—1062no—
164#127—1062no—
165#124—2124no—
166C4×Q8—22no—
167C4×Q8—22no—
168C2×C4⋊C4—1061no—
169C2×C4⋊C4C4×D5312yes—
170C2×C4⋊C4C22⋊D5311yescharacteristic
171C2×C22⋊C4—1064no—
172C2×C22⋊C4—1064no—
173C2×C42—1061no—
174C2×C42D21212yes—
175C2×C42C22×D5311yescharacteristic
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—534no—
179C4⋊C8—1064no—
180#101—42no—
181C22⋊C8—1062no—
182#97—1064no—
183C4×C8—1062no—
184#95—532no—
185C24—1062no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×D4—1064no—
191C2×D4—4242no—
192C2×D4—2124no—
193C2×D4—2121no—
194C2×D4—2122no—
195C2×D4—2121no—
196C2×D4—2122no—
197C2×D4—2121no—
198C2×D4—2121no—
199C22×C4—1062no—
200C22×C4#1171992112yes—
201C22×C4#1171992112yes—
202C22×C4C2×C4×D5311yescharacteristic
203C22×C4C2×D21211yescharacteristic
204C22×C4C2×C22⋊D5311yescharacteristic
205M16—42no—
206M16—42no—
207C2×C8—42no—
208C2×C8—1064no—
209C4⋊C4—24no—
210C4⋊C4—22no—
211C4⋊C4—2122no—
212C22⋊C4—4244no—
213C22⋊C4—2128no—
214C42—22no—
215C42—22no—
216C42—2122no—
217C42D4×D5312yes—
218C42#1171994712yes—
219C23—2124no—
220C23—2124no—
221C23—2122no—
222C23—2122no—
223C23—2122no—
224C23C2×#1171992111yescharacteristic
225Q8—42no—
226Q8—42no—
227D4—4244no—
228D4—4242no—
229D4—4242no—
230D4—4242no—
231D4—4242no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—2122no—
240C2×C4—2122no—
241C2×C4—2122no—
242C2×C4#1313317712yes—
243C2×C4#1313318711yescharacteristic
244C2×C4#1313318912yes—
245C2×C4#1313320111yescharacteristic
246C2×C4#1313324212yes—
247C2×C4#1313324311yescharacteristic
248C2×C4#1313324811yescharacteristic
249C8—82no—
250C8—2124no—
251C22—21no—
252C22—21no—
253C22—4244no—
254C22—4244no—
255C22—4242no—
256C22—4242no—
257C22—4242no—
258C22#1681050011yescharacteristic
259C22#1681062211yescharacteristic
260C22#1681062711yescharacteristic
261C4—22no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—4242no—
266C4#1681079812yes—
267C4#1681087112yes—
268C4#1681087412yes—
269C2—21no—
270C2—21no—
271C2—8482no—
272C2#2027511611yescharacteristic
273C2#2027515111yescharacteristic
274C2#2027526011yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1801C2
socleC22×C53#386
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenter#1801D53
hypocenterC53#9406
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. #20272616C2semidirect
5. C53×#1801C2semidirect
134. #1801D53semidirect
157. C53#9406semidirect

Maximal subgroups

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

subgroupindex
#202838002
#202797142
#202733842
#202726162
C53×#18012
#940653