Finite Group Databasebeta

← #27293289

Subgroups — #27293289

Order 13568.

386 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#16810675C211yesmaximal, characteristic
2#20276511C212yesmaximal, complemented
3#20272655C211yesmaximal, characteristic, complemented
4#20272638C212yesmaximal
5C53×#1322C211yesmaximal, characteristic, complemented
6C2×C42×D53C411yescharacteristic
7#16810675—22no—
8#16810675—22no—
9C2×C4⋊#88322C411yescharacteristic
10C2×#13133129C411yescharacteristic
11C2×#13133129C2211yescharacteristic
12C2×C8×Dic53C411yescharacteristic
13C2×C8×Dic53C2211yescharacteristic
14#16810441—22no—
15C22⋊#11719908C2212yes—
16C16×Dic53—22no—
17C2×C4⋊C8×C53C2211yescharacteristic
18C53×C22⋊C16C2212yes—
19C22×C4×D53C811yescharacteristic
20C22×C4×D53C811yescharacteristic
21C22×C4×D53C2×C411yescharacteristic
22C22×#88322C2×C411yescharacteristic
23C22×#88322C2×C411yescharacteristic
24C2×C4×Dic53C811yescharacteristic
25C2×C4×Dic53C811yescharacteristic
26C2×C4×Dic53C2×C411yescharacteristic
27C42×D53—21no—
28C42×D53—21no—
29C42×D53—21no—
30C42×D53—21no—
31C2×#11719908—22no—
32C4⋊#88322—22no—
33#13133129—22no—
34#13133129—22no—
35C8×Dic53—21no—
36C8×Dic53—21no—
37C8×Dic53D412yes—
38C8×Dic53D412yes—
39C22×C8×C53C2×C411yescharacteristic
40C22×C8×C53C2311yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42C2×C16×C53—22no—
43C4⋊C8×C53—22no—
44C23×D53C2×C811yescharacteristic
45C22×C4×C53C2×C811yescharacteristic
46C22×C4×C53C2×C811yescharacteristic
47C22×C4×C53C22×C411yescharacteristic
48C22×Dic53C2×C811yescharacteristic
49C22×Dic53C2×C811yescharacteristic
50C22×Dic53C2×C811yescharacteristic
51C2×C4×D53—21no—
52C2×C4×D53—21no—
53C2×C4×D53—22no—
54C2×C4×D53—21no—
55C2×C4×D53—21no—
56C2×C4×D53—21no—
57C2×C4×D53—21no—
58C2×C4×D53—42no—
59C2×C4×D53—41no—
60C2×C4×D53—41no—
61C2×C4×D53C1614yescomplemented
62C2×C8×C53—21no—
63C2×C8×C53—41no—
64C2×C8×C53—41no—
65C2×C8×C53C2×D412yes—
66C42×C53—21no—
67C42×C53—21no—
68C4×Dic53—21no—
69C4×Dic53—21no—
70C4×Dic53—21no—
71C4×Dic53—21no—
72C4×Dic53C22⋊C411yescharacteristic
73C4×Dic53C22⋊C411yescharacteristic
74C4×Dic53M1611yescharacteristic
75C4×Dic53M1611yescharacteristic
76C2×#88322—21no—
77C2×#88322—41no—
78C2×#88322—41no—
79C2×#88322C22⋊C412yes—
80C16×C53—42no—
81#11719908—42no—
82C23×C53C22×C811yescharacteristic
83C22×D53—21no—
84C22×D53—21no—
85C22×D53—41no—
86C22×D53—41no—
87C22×D53C2×C1612yes—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—41no—
94C2×C4×C53C2×C1612yes—
95C2×C4×C53C2×C22⋊C411yescharacteristic
96C2×C4×C53C2×M1611yescharacteristic
97C2×Dic53—22no—
98C2×Dic53—21no—
99C2×Dic53—41no—
100C2×Dic53C22⋊C811yescharacteristic
101C2×Dic53C22⋊C811yescharacteristic
102C2×Dic53C22⋊C811yescharacteristic
103C2×Dic53C22⋊C811yescharacteristic
104C2×Dic53C2×C1612yes—
105C2×Dic53M3212yes—
106C4×D53—24no—
107C4×D53—24no—
108C4×D53—42no—
109C4×D53—42no—
110C4×D53—41no—
111C4×D53—41no—
112C4×D53—42no—
113C8×C53—22no—
114C8×C53—41no—
115#88322—22no—
116#88322—41no—
117#11459—531nomaximal, Sylow
118C22×C53—21no—
119C22×C53—21no—
120C22×C53C2×C22⋊C811yescharacteristic
121C22×C53C22×C1611yescharacteristic
122C22×C53C2×M3211yescharacteristic
123C2×D53—22no—
124C2×D53—42no—
125C2×D53—41no—
126C2×D53—41no—
127C2×D53—42no—
128C4×C53—22no—
129C4×C53—21no—
130C4×C53—41no—
131C4×C53#41011yescharacteristic
132C4×C53#41311yescharacteristic
133Dic53—21no—
134Dic53—41no—
135Dic53C22⋊C1612yes—
136Dic53C22⋊C1612yes—
137Dic53#34912yes—
138C2×#431—531no—
139#2162—532no—
140#1322—531no—
141#1322D5311yescharacteristic, complemented
142#1314—532no—
143C2×C53—21no—
144C2×C53—21no—
145C2×C53#145811yescharacteristic
146C2×C53C2×C22⋊C1611yescharacteristic
147C2×C53C2×#34911yescharacteristic
148D53—42no—
149C22×C42—531no—
150#431—1062no—
151#431—1062no—
152C2×C4⋊C8—531no—
153C2×C4⋊C8C2×D5311yescharacteristic
154C2×C22⋊C8—531no—
155C2×C22⋊C8—531no—
156C2×C4×C8—531no—
157C2×C4×C8—531no—
158C4⋊C16—1062no—
159C22⋊C16—532no—
160C22⋊C16C2×D5312yes—
161C4×C16—1062no—
162C53#1145911yescharacteristic, complemented, Sylow
163C23×C4—531no—
164C23×C4—531no—
165C23×C4—531no—
166C22×C8—531no—
167C22×C8—531no—
168C22×C8C4×D5311yescharacteristic
169C22×C8C22×D5311yescharacteristic
170C2×C42—531no—
171C2×C42—531no—
172C2×C42—531no—
173C2×C42—1061no—
174C2×C42—1061no—
175C2×C42—1061no—
176C2×C42—1061no—
177C2×C42C4×D5311yescharacteristic
178C2×C16—22no—
179C2×C16—1062no—
180C4⋊C8—22no—
181C4⋊C8—1062no—
182C22⋊C8—1062no—
183C22⋊C8—1062no—
184C4×C8—532no—
185C4×C8—532no—
186C4×C8—1061no—
187C4×C8—1061no—
188C24—531no—
189C22×C4—531no—
190C22×C4—531no—
191C22×C4—534no—
192C22×C4—531no—
193C22×C4—1061no—
194C22×C4—1061no—
195C22×C4—1062no—
196C22×C4—1061no—
197C22×C4—1061no—
198C22×C4—1061no—
199C22×C4—1061no—
200C22×C4—2121no—
201C22×C4—2121no—
202C22×C4—2122no—
203C22×C4C8×D5311yescharacteristic
204C22×C4C8×D5311yescharacteristic
205C22×C4C2×C4×D5311yescharacteristic
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—532no—
210C2×C8—1061no—
211C2×C8—2121no—
212C2×C8—2121no—
213C2×C8D4×D5312yes—
214C42—21no—
215C42—21no—
216C42—531no—
217C42—531no—
218C42—531no—
219C42—531no—
220C42—1061no—
221C42—1061no—
222C42—1061no—
223C42—1061no—
224C16—42no—
225C16—2122no—
226C23—532no—
227C23—1061no—
228C23—1061no—
229C23—2121no—
230C23—2121no—
231C23C2×C8×D5311yescharacteristic
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—532no—
239C2×C4—532no—
240C2×C4—531no—
241C2×C4—531no—
242C2×C4—531no—
243C2×C4—531no—
244C2×C4—1064no—
245C2×C4—1064no—
246C2×C4—1062no—
247C2×C4—1061no—
248C2×C4—2122no—
249C2×C4—2122no—
250C2×C4—2122no—
251C2×C4—2121no—
252C2×C4—2121no—
253C2×C4—2121no—
254C2×C4C16×D5312yes—
255C2×C4C22⋊C4×D5311yescharacteristic
256C2×C4M16×D5311yescharacteristic
257C8—22no—
258C8—41no—
259C8—1062no—
260C8—2121no—
261C22—21no—
262C22—21no—
263C22—1062no—
264C22—2122no—
265C22—2122no—
266C22—2121no—
267C22—2121no—
268C22C22⋊C8×D5311yescharacteristic
269C22C2×C16×D5311yescharacteristic
270C22M32×D5311yescharacteristic
271C4—22no—
272C4—21no—
273C4—41no—
274C4—532no—
275C4—532no—
276C4—532no—
277C4—1061no—
278C4—2121no—
279C4#1681059511yescharacteristic
280C4#1681060611yescharacteristic
281C2—21no—
282C2—21no—
283C2—2122no—
284C2#2027302811yescharacteristic
285C2C22⋊C16×D5311yescharacteristic
286C2#349×D5311yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC22×C53C22×C16
FrattiniC22×C8C22×D53
FittingC53×#1322C2
socleC22×C53C2×C22⋊C8
prime radicalC23×D53C2×C8
cosocleC22×C8×C53C23
hypercenter#1322D53
hypocenterC53#11459
LCSG ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
2. #20276511C2semidirect
3. #20272655C2semidirect
5. C53×#1322C2semidirect
61. C2×C4×D53C16semidirect
141. #1322D53semidirect
162. C53#11459semidirect

Maximal subgroups

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

subgroupindex
C2×#168106752
#202765112
#202726552
#202726382
C53×#13222
#1145953