Finite Group Databasebeta

← #27281699

Subgroups — #27281699

Order 13568.

556 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 103 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#20285168C211yesmaximal, characteristic
2#20283782C211yesmaximal, characteristic, complemented
3#20273384C212yesmaximal, complemented
4#20272604C212yesmaximal, complemented
5C53×#1560C211yesmaximal, characteristic, complemented
6C2×#13133242—22no—
7C2×#13133214C2211yescharacteristic
8#16811013C414yes—
9#16810999—21no—
10#16810997—21no—
11#16810701—22no—
12D4×#88322—22no—
13#16810676C2212yes—
14#16810590—22no—
15C4×#11719917C2211yescharacteristic
16#16810366—22no—
17#16810366—22no—
18C53×#532C2211yescharacteristic
19C53×#423C2212yes—
20C53×#333—21no—
21C53×#331—21no—
22C22×D212—21no—
23C22×D212—22no—
24#13133242—44no—
25#13133242—44no—
26C2×#11719924—24no—
27C2×#11719922—24no—
28C2×#11719917—22no—
29C2×#11719917—21no—
30C22×#88322—22no—
31#13133214—22no—
32#13133214—21no—
33#13133214C2×C412yes—
34C4⋊#88322D412yes—
35C4⋊#88322D412yes—
36C22⋊#88322—24no—
37C22⋊#88322—22no—
38#13133128C2×C412yes—
39C4×#88322—21no—
40C4×#88322—22no—
41C2×#41×C53D412yes—
42C2×M16×C53—22no—
43C4⋊Q8×C53D411yescharacteristic
44#127×C53D411yescharacteristic
45#124×C53C2×C412yes—
46#121×C53—22no—
47C4×D4×C53—22no—
48C2×C42×C53C2311yescharacteristic
49C4⋊C8×C53—22no—
50C4⋊C8×C53—22no—
51C23×D53—41no—
52#41×C53—42no—
53#41×C53—42no—
54C2×Q8×C53C22⋊C412yes—
55C2×D4×C53—22no—
56C2×D4×C53—22no—
57C2×D4×C53C22⋊C412yes—
58C22×C4×C53—22no—
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53#4112yes—
61C2×D212—22no—
62C2×D212—24no—
63C2×D212—42no—
64C2×D212—42no—
65C2×D212—41no—
66C2×D212—42no—
67C2×D212—82no—
68C2×D212—82no—
69M16×C53—82no—
70C2×C8×C53—42no—
71C4⋊C4×C53—22no—
72C22⋊C4×C53—24no—
73C42×C53C22×C411yescharacteristic
74C42×C53C2×D411yescharacteristic
75C42×C53C2×D412yes—
76#11719924—48no—
77#11719922—48no—
78#11719917—22no—
79#11719917—22no—
80#11719917—82no—
81C2×#88322—22no—
82C2×#88322—22no—
83C2×#88322—22no—
84C2×#88322—82no—
85C23×C53—22no—
86C23×C53#12311yescharacteristic
87C22×D53—42no—
88C22×D53—82no—
89C22×D53—84no—
90Q8×C53—42no—
91D4×C53—42no—
92D4×C53—42no—
93D4×C53—42no—
94D4×C53—82no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53C2×C22⋊C411yescharacteristic
103C2×C4×C53C4×D412yes—
104C2×C4×C53#12011yescharacteristic
105C2×C4×C53#12112yes—
106D212—44no—
107D212—84no—
108D212—81no—
109D212—81no—
110D212—82no—
111D212—82no—
112C8×C53—82no—
113#88322—24no—
114#88322—42no—
115#88322—42no—
116#9437—531nomaximal, Sylow
117C22×C53—21no—
118C22×C53—22no—
119C22×C53—42no—
120C22×C53—44no—
121C22×C53#38611yescharacteristic
122C2×D53—82no—
123C2×D53—84no—
124C2×D53—164no—
125C4×C53—21no—
126C4×C53—22no—
127C4×C53—42no—
128C4×C53#45311yescharacteristic
129C4×C53#45311yescharacteristic
130C4×C53#47112yes—
131C4×C53#47212yes—
132#3151—531no—
133#2979—531no—
134#1560D5311yescharacteristic, complemented
135#1545—532no—
136#1299—532no—
137C2×C53—41no—
138C2×C53—82no—
139C2×C53#188711yescharacteristic
140C2×C53#190712yes—
141D53—162no—
142C2×AGL(1,ℤ/8)—1062no—
143#532C2×D5311yescharacteristic
144C2×#127—531no—
145#496—534no—
146#493—1061no—
147#492—1061no—
148#436—1062no—
149C8×D4—1062no—
150#431—532no—
151#423C2×D5312yes—
152#408—1062no—
153C4×M16—531no—
154#333—21no—
155#331—21no—
156#331—1062no—
157#331—1062no—
158C53#943711yescharacteristic, complemented, Sylow
159C2×#41D21212yes—
160C22×D4—1061no—
161C22×D4—1062no—
162AGL(1,ℤ/8)—2124no—
163AGL(1,ℤ/8)—2124no—
164C2×SD16—1064no—
165C2×D8—1064no—
166C2×M16—22no—
167C2×M16—1061no—
168C2×M16—1062no—
169C22×C8—1062no—
170C4⋊Q8C22⋊D5311yescharacteristic
171#127—532no—
172#127—1062no—
173#127—1061no—
174#127C22⋊D5311yescharacteristic
175#124C4×D5312yes—
176#121—22no—
177C4×D4—22no—
178C2×C42C22×D5311yescharacteristic
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—532no—
182C4⋊C8—532no—
183C22⋊C8—1064no—
184C22⋊C8—1062no—
185#97—532no—
186C4×C8—1062no—
187C4×C8—1061no—
188C24—2121no—
189#41—42no—
190#41—42no—
191C2×Q8#1171992112yes—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—1062no—
195C2×D4—1064no—
196C2×D4—4242no—
197C2×D4—4242no—
198C2×D4—2122no—
199C2×D4—2122no—
200C2×D4—2121no—
201C2×D4—2122no—
202C2×D4#1171992112yes—
203C22×C4—22no—
204C22×C4C2×D21211yescharacteristic
205C22×C4#1171994712yes—
206SD16—2128no—
207D8—2128no—
208M16—82no—
209M16—1062no—
210M16—1062no—
211M16—4242no—
212C2×C8—42no—
213C2×C8—1062no—
214C2×C8—1062no—
215C2×C8—1062no—
216C2×C8—4242no—
217C4⋊C4—22no—
218C22⋊C4—24no—
219C42C2×C4×D5311yescharacteristic
220C42D4×D5312yes—
221C42C2×C22⋊D5311yescharacteristic
222C23—22no—
223C23—4244no—
224C23—4242no—
225C23—2122no—
226C23#1313320111yescharacteristic
227Q8—42no—
228D4—42no—
229D4—42no—
230D4—42no—
231D4—82no—
232D4—4244no—
233D4—4241no—
234D4—4241no—
235D4—4242no—
236D4—4242no—
237D4—2124no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4C2×#1171992111yescharacteristic
246C2×C4#1313317712yes—
247C2×C4#1313318711yescharacteristic
248C2×C4#1313318912yes—
249C8—82no—
250C8—1064no—
251C8—2122no—
252C8—2122no—
253C22—22no—
254C22—21no—
255C22—44no—
256C22—42no—
257C22—8484no—
258C22—4244no—
259C22—4242no—
260C22#1681050011yescharacteristic
261C4—22no—
262C4—21no—
263C4—42no—
264C4#1681079311yescharacteristic
265C4#1681079311yescharacteristic
266C4#1681088812yes—
267C4#1681089112yes—
268C2—41no—
269C2—82no—
270C2—8482no—
271C2#2027511111yescharacteristic
272C2#2027522112yes—
subgroupG/H
centerC22#16810500
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1560C2
socleC22×C53#386
prime radical#20285168C2
cosocleC2×C42×C53C23
hypercenter#1560D53
hypocenterC53#9437
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1560

Complements

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

NH ≅ G/Ntype
2. #20283782C2semidirect
3. #20273384C2semidirect
4. #20272604C2semidirect
5. C53×#1560C2semidirect
134. #1560D53semidirect
158. C53#9437semidirect

Maximal subgroups

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

subgroupindex
#202851682
#202837822
#202733842
#202726042
C53×#15602
#943753