Finite Group Databasebeta

← #28299568

Subgroups — #28299568

Order 13568.

440 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 110 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#20290642C211yesmaximal, characteristic
2#20290642C211yesmaximal, characteristic
3#20289534C214yesmaximal
4#20277282C211yesmaximal, characteristic
5#16811489—21no—
6#16811489—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489C2212yes—
10#16811489C2212yes—
11C2×#13133217C411yescharacteristic
12C2×#13133182—21no—
13#16810528—21no—
14#16810519—21no—
15#16810519C2212yes—
16C4⋊C4×Dic53—21no—
17C4⋊C4×Dic53C412yes—
18C42×Dic53C2211yescharacteristic
19C53×#384C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22C22×Dic106—21no—
23#13133217—22no—
24#13133217C814yescomplemented
25#13133182—44no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53C2×C412yes—
28C2×#11719919—22no—
29C2×C4×Dic53—22no—
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53D411yescharacteristic
32C2×C4×Dic53D411yescharacteristic
33C2×C4×Dic53D411yescharacteristic
34C2×C4×Dic53D411yescharacteristic
35C2×C4×Dic53C2311yescharacteristic
36#13133124—22no—
37#13133124—22no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2×C412yes—
41C2×C42×C53C2×C411yescharacteristic
42C22×C4×C53—22no—
43C22×C4×C53C22⋊C411yescharacteristic
44C22×C4×C53C22⋊C411yescharacteristic
45C22×C4×C53C22×C411yescharacteristic
46C22×Dic53—22no—
47C22×Dic53C22⋊C411yescharacteristic
48C22×Dic53C22⋊C411yescharacteristic
49C22×Dic53C2×D411yescharacteristic
50C22×Dic53C2×D411yescharacteristic
51C2×Dic106—24no—
52C2×Dic106—42no—
53C2×Dic106—42no—
54C2×#88324—42no—
55C2×#88324—42no—
56C2×#88324—42no—
57C2×#88324—42no—
58C2×#88324—42no—
59C2×#88324—42no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—42no—
63C42×C53C2×C812yes—
64C42×C53M1612yes—
65C4⋊Dic53—24no—
66C4⋊Dic53—22no—
67C4⋊Dic53C2×C814yes—
68#11719919—44no—
69C4×Dic53—22no—
70C4×Dic53—22no—
71C4×Dic53—22no—
72C4×Dic53—41no—
73C4×Dic53—41no—
74C4×Dic53—41no—
75C4×Dic53—44no—
76C23×C53C2×C22⋊C411yescharacteristic
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53C22⋊C812yes—
87C2×C4×C53C22⋊C812yes—
88C2×C4×C53C22×C811yescharacteristic
89C2×C4×C53C2×M1611yescharacteristic
90C2×Dic53—21no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—24no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53#10112yes—
103C2×Dic53AGL(1,ℤ/8)12yes—
104C2×Dic53#13712yes—
105Dic106—48no—
106#88324—82no—
107#88324—82no—
108#7604—531nomaximal, Sylow
109C22×C53C2×C22⋊C811yescharacteristic
110C22×C53#41011yescharacteristic
111C22×C53C2×#10111yescharacteristic
112C22×C53#41811yescharacteristic
113C22×C53#41911yescharacteristic
114C22×C53#42111yescharacteristic
115C22×C53#42111yescharacteristic
116C4×C53—21no—
117C4×C53—21no—
118C4×C53—22no—
119C4×C53—22no—
120C4×C53—82no—
121Dic53—22no—
122Dic53—22no—
123Dic53—22no—
124Dic53—41no—
125Dic53—44no—
126C4×C4⋊Q8—531no—
127#1839—531no—
128#1839—531no—
129#1271—534no—
130C2×C53#146111yescharacteristic
131C2×C53#147811yescharacteristic
132C2×C53#147911yescharacteristic
133C2×C53#151411yescharacteristic
134C2×C53#151511yescharacteristic
135C2×C53#153711yescharacteristic
136C2×C53#155611yescharacteristic
137C2×C4⋊Q8—531no—
138C2×C4×Q8—1061no—
139#391—1061no—
140#389—532no—
141#389—1061no—
142#384#615711yescharacteristic
143C4×C4⋊C4—532no—
144C4×C4⋊C4—1061no—
145C43—531no—
146#336—532no—
147#336—532no—
148#336—1061no—
149#336—1061no—
150#336—1061no—
151#336—1061no—
152C53#760411yescharacteristic, complemented, Sylow
153C22×Q8—1061no—
154C22×C8—1062no—
155C22×C8—1062no—
156C4⋊Q8—534no—
157C4⋊Q8—1062no—
158C4×Q8—2124no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—532no—
162C2×C4⋊C4—1061no—
163C2×C4⋊C4—1062no—
164C2×C4⋊C4C2×#615712yes—
165C2×C42—531no—
166C2×C42—531no—
167C2×C42—531no—
168C2×C42—531no—
169C2×C42—531no—
170C2×C42—531no—
171C2×C42—1062no—
172C2×C42C2×#615711yescharacteristic
173#95—1062no—
174#95—1062no—
175C2×Q8—1064no—
176C2×Q8—2122no—
177C2×Q8—2122no—
178C22×C4—22no—
179C22×C4—531no—
180C22×C4—531no—
181C22×C4—531no—
182C22×C4—531no—
183C22×C4—1062no—
184C22×C4C22⋊#615711yescharacteristic
185C22×C4C22⋊#615711yescharacteristic
186C22×C4C22×#615711yescharacteristic
187C2×C8—2122no—
188C2×C8—2122no—
189C2×C8—2122no—
190C2×C8—2122no—
191C2×C8—2122no—
192C2×C8—2122no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—534no—
197C4⋊C4—1064no—
198C4⋊C4—1062no—
199C4⋊C4—2124no—
200C42—1062no—
201C42—1062no—
202C42—1062no—
203C42—2121no—
204C42—2121no—
205C42—2121no—
206C42—2124no—
207C42D53⋊C812yes—
208C42#1171993612yes—
209C23C2×C22⋊#615711yescharacteristic
210Q8—2128no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—532no—
221C2×C4—532no—
222C2×C4—532no—
223C2×C4—1061no—
224C2×C4—1061no—
225C2×C4—1061no—
226C2×C4—1061no—
227C2×C4—1061no—
228C2×C4—1061no—
229C2×C4—1064no—
230C2×C4—1061no—
231C2×C4—1061no—
232C2×C4—1061no—
233C2×C4—2122no—
234C2×C4—2122no—
235C2×C4#1313327412yes—
236C2×C4#1313327412yes—
237C2×C4C2×D53⋊C811yescharacteristic
238C2×C4C2×#1171993611yescharacteristic
239C8—4242no—
240C8—4242no—
241C22C2×#1313327411yescharacteristic
242C22#1681160311yescharacteristic
243C22D53⋊#10111yescharacteristic
244C22#1681161511yescharacteristic
245C22#1681161611yescharacteristic
246C22#1681162111yescharacteristic
247C22#1681162211yescharacteristic
248C4—21no—
249C4—21no—
250C4—22no—
251C4—22no—
252C4—82no—
253C4—1062no—
254C4—1062no—
255C4—1062no—
256C4—2121no—
257C4—2124no—
258C2#2028995911yescharacteristic
259C2#2029001611yescharacteristic
260C2#2029001911yescharacteristic
261C2#2029013511yescharacteristic
262C2#2029013911yescharacteristic
263C2#2029020711yescharacteristic
264C2#2029024511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#7604
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
24. #13133217C8semidirect
152. C53#7604semidirect

Maximal subgroups

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

subgroupindex
#202906422
#202895342
#202772822
#760453