Finite Group Databasebeta

← #27280100

Subgroups — #27280100

Order 13568.

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

normal subgroups only

This group has 458 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#20280536C211yesmaximal, characteristic
2#20273397C212yesmaximal, complemented
3#20273029C211yesmaximal, characteristic
4#20272575C212yesmaximal, complemented
5C53×#2921C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7#16810676C2212yes—
8#16810562—21no—
9#16810559—21no—
10#16810543C412yes—
11#16810535C412yes—
12#16810521—21no—
13#16810489C2211yescharacteristic
14#16810483—21no—
15#16810371—24no—
16#16810349—21no—
17#16810347—21no—
18C2×#117×C53C2211yescharacteristic
19C53×#433—24no—
20C53×#423C2212yes—
21C53×#407—21no—
22C4×M16×C53—21no—
23C22×Dic106C2×C412yes—
24#13133217—22no—
25#13133217—22no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29C2×#11719919—22no—
30C2×C22⋊Dic53—22no—
31C4⋊#88322—22no—
32C22⋊#88322—22no—
33C4×#88322—22no—
34#13133124—22no—
35#13133124—22no—
36#13133124—21no—
37#13133124C2×C412yes—
38C23×C4×C53D411yescharacteristic
39C2×M16×C53—22no—
40#117×C53—22no—
41#117×C53—22no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53C2×C411yescharacteristic
45C2×C22⋊C4×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47C2×C42×C53C2311yescharacteristic
48C4⋊C8×C53—22no—
49C4⋊C8×C53D414yes—
50C22⋊C8×C53—24no—
51#97×C53—22no—
52C4×C8×C53—22no—
53C24×C53C22⋊C411yescharacteristic
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×Dic53—22no—
65C22×Dic53—22no—
66C2×Dic106—24no—
67C2×Dic106—82no—
68M16×C53—42no—
69M16×C53—42no—
70C2×C8×C53—24no—
71C4⋊C4×C53—22no—
72C4⋊C4×C53—22no—
73C22⋊C4×C53—22no—
74C22⋊C4×C53—22no—
75C42×C53—21no—
76C42×C53—21no—
77C42×C53C2×D412yes—
78C42×C53#4112yes—
79C22⋊Dic53—82no—
80C4⋊Dic53—41no—
81C4⋊Dic53—42no—
82C4⋊Dic53—41no—
83C4⋊Dic53—42no—
84#11719919—82no—
85C2×#88322—42no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—41no—
90C23×C53—41no—
91C23×C53C2×C22⋊C411yescharacteristic
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53C4×D412yes—
104C2×C4×C53#12011yescharacteristic
105C2×C4×C53#12112yes—
106C2×C4×C53#12311yescharacteristic
107C2×C4×C53#13512yes—
108C2×C4×C53#13712yes—
109C2×Dic53—42no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115Dic106—84no—
116C8×C53—44no—
117#88322—82no—
118#9192—531nomaximal, Sylow
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—42no—
122C22×C53—41no—
123C22×C53—41no—
124C22×C53—41no—
125C22×C53—81no—
126C22×C53#38611yescharacteristic
127C22×C53#41611yescharacteristic
128C22×C53#41911yescharacteristic
129C4×C53—22no—
130C4×C53—22no—
131C4×C53—41no—
132C4×C53—41no—
133C4×C53—41no—
134C4×C53#45212yes—
135C4×C53#48512yes—
136Dic53—82no—
137Dic53—82no—
138#2921D5311yescharacteristic
139#2628—531no—
140#1548—532no—
141#1458—531no—
142#1286—532no—
143C2×C53—21no—
144C2×C53—21no—
145C2×C53—81no—
146C2×C53#187311yescharacteristic
147C2×C53#190911yescharacteristic
148C2×C53#191411yescharacteristic
149C2×C4⋊Q8—531no—
150C2×#117C2×D5311yescharacteristic
151#433—24no—
152#431—532no—
153#423C2×D5312yes—
154#407—21no—
155C4×M16—21no—
156#400—1061no—
157#399—1061no—
158#395—532no—
159#393—532no—
160#389—1061no—
161#382—531no—
162#380—1061no—
163#333—1064no—
164#324—1061no—
165#323—1061no—
166C53#919211yescharacteristic, complemented, Sylow
167C22×Q8—532no—
168C23×C4C22⋊D5311yescharacteristic
169C2×M16—22no—
170C4⋊Q8—1062no—
171C4⋊Q8—1062no—
172#117—22no—
173#117—22no—
174C2×C4⋊C4—1062no—
175C2×C4⋊C4—1061no—
176C2×C4⋊C4—1061no—
177C2×C4⋊C4—1061no—
178C2×C4⋊C4C4×D5311yescharacteristic
179C2×C4⋊C4D21211yescharacteristic
180C2×C22⋊C4—1062no—
181C2×C22⋊C4C4×D5311yescharacteristic
182C2×C22⋊C4D21211yescharacteristic
183C2×C42C22⋊D5311yescharacteristic
184C2×C42C22×D5311yescharacteristic
185C4⋊C8—22no—
186C4⋊C8—1062no—
187C4⋊C8C22⋊D5314yes—
188C22⋊C8—24no—
189C22⋊C8—1062no—
190#97—22no—
191C4×C8—22no—
192C4×C8—1062no—
193#95—532no—
194#95—1061no—
195#95—1062no—
196#95—1062no—
197C24#1171992111yescharacteristic
198C2×Q8—1064no—
199C2×Q8—4242no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—1062no—
205C22×C4—1062no—
206C22×C4#1171992111yescharacteristic
207C22×C4#1171992111yescharacteristic
208C22×C4#1171992111yescharacteristic
209C22×C4C2×C4×D5311yescharacteristic
210C22×C4C2×D21211yescharacteristic
211C22×C4C2×C22⋊D5311yescharacteristic
212M16—42no—
213M16—42no—
214C2×C8—24no—
215C2×C8—2122no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—4242no—
219C4⋊C4—2121no—
220C4⋊C4—2122no—
221C4⋊C4—2121no—
222C4⋊C4—2122no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—4242no—
226C42—21no—
227C42—21no—
228C42#1171994512yes—
229C42C2×C22⋊D5312yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23C2×#1171992111yescharacteristic
236Q8—4244no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—2122no—
249C2×C4—2122no—
250C2×C4—2122no—
251C2×C4—2122no—
252C2×C4—2122no—
253C2×C4—2122no—
254C2×C4C4×C22⋊D5312yes—
255C2×C4#1313318611yescharacteristic
256C2×C4#1313319112yes—
257C2×C4#1313320011yescharacteristic
258C2×C4#1313323712yes—
259C2×C4#1313324712yes—
260C8—44no—
261C8—4242no—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22#1681050211yescharacteristic
270C22#1681061711yescharacteristic
271C22#1681063111yescharacteristic
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—4242no—
278C4—4242no—
279C4#1681078512yes—
280C4#1681096412yes—
281C2—21no—
282C2—21no—
283C2—81no—
284C2#2027502711yescharacteristic
285C2#2027523711yescharacteristic
286C2#2027526211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2921C2
socleC22×C53#386
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2921D53
hypocenterC53#9192
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #20273397C2semidirect
4. #20272575C2semidirect
166. C53#9192semidirect

Maximal subgroups

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

subgroupindex
#202805362
#202733972
#202730292
#202725752
C53×#29212
#919253