Finite Group Databasebeta

← #27293591

Subgroups — #27293591

Order 13568.

348 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 71 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#20283412C211yesmaximal, characteristic
2#20276528C212yesmaximal, complemented
3#20272653C211yesmaximal, characteristic, complemented
4#20272650C212yesmaximal, complemented
5C53×#1321C211yesmaximal, characteristic, complemented
6C2×#13133224C411yescharacteristic
7C22×C8×D53C411yescharacteristic
8#16811130C411yescharacteristic
9#16810697—22no—
10#16810686—22no—
11#16810649C411yescharacteristic
12#16810586C2211yescharacteristic
13#16810581C2211yescharacteristic
14#16810443—22no—
15#16810405C2212yes—
16#16810400—22no—
17C53×#408C2211yescharacteristic
18C53×#349C2212yes—
19C22×C4×D53C2×C411yescharacteristic
20#13133224—42no—
21C2×#11719917—21no—
22C2×#11719917C2×C411yescharacteristic
23C22×#88322—21no—
24C22×#88322C2×C411yescharacteristic
25C2×C8×D53—21no—
26C2×C8×D53—21no—
27C2×C8×D53—42no—
28#13133198—21no—
29#13133193—21no—
30#13133188—21no—
31#13133185—21no—
32C4×C22⋊D53—22no—
33#13133176C2×C411yescharacteristic
34#13133157—22no—
35#13133146—22no—
36#13133129—22no—
37#13133127D412yes—
38C8×Dic53D412yes—
39C2×#41×C53C2×C411yescharacteristic
40C2×M16×C53C2311yescharacteristic
41C22×C8×C53C2×C411yescharacteristic
42M32×C53—22no—
43C22⋊C8×C53—22no—
44C23×D53M1611yescharacteristic
45#41×C53—41no—
46#41×C53—41no—
47C2×Q8×C53M1611yescharacteristic
48C2×D4×C53—21no—
49C2×D4×C53M1611yescharacteristic
50C22×C4×C53—21no—
51C22×C4×C53C22×C411yescharacteristic
52C2×C22⋊D53—22no—
53C22×Dic53M1611yescharacteristic
54C2×C4×D53—21no—
55C2×C4×D53—21no—
56C2×C4×D53—41no—
57C2×C4×D53—41no—
58M16×C53—41no—
59C2×C8×C53—21no—
60C2×C8×C53—41no—
61C2×C8×C53C2×D412yes—
62C22⋊Dic53—22no—
63C22⋊Dic53—21no—
64#11719921—22no—
65C4⋊Dic53—21no—
66#11719919—22no—
67C4×Dic53C22⋊C412yes—
68#11719917—42no—
69#11719917—41no—
70C2×#88322—22no—
71C2×#88322—21no—
72C2×#88322—42no—
73C2×#88322C22⋊C411yescharacteristic
74C2×#88322C22⋊C411yescharacteristic
75C8×D53—42no—
76C8×D53—42no—
77C16×C53—42no—
78#11719908—42no—
79C23×C53—21no—
80C23×C53C2×M1611yescharacteristic
81C22×D53—21no—
82C22×D53—42no—
83C22×D53—41no—
84Q8×C53—41no—
85D4×C53—41no—
86D4×C53—42no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53C2×C22⋊C411yescharacteristic
93C2×C4×C53C2×M1611yescharacteristic
94C22⋊D53—44no—
95C2×Dic53—21no—
96C2×Dic53—22no—
97C2×Dic53—41no—
98C4×D53—42no—
99C4×D53—41no—
100C4×D53—41no—
101C8×C53—22no—
102C8×C53—41no—
103#88322—21no—
104#88322—41no—
105#88322#10012yes—
106#11537—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—41no—
109C22×C53—42no—
110C22×C53—41no—
111C22×C53#40711yescharacteristic
112C2×D53—22no—
113C2×D53—41no—
114C2×D53—42no—
115C2×D53—41no—
116C2×D53—42no—
117C4×C53—21no—
118C4×C53—41no—
119C4×C53#41011yescharacteristic
120C4×C53C2×#10011yescharacteristic
121Dic53—42no—
122Dic53—41no—
123#2929—531no—
124#2166—532no—
125#1321—531no—
126#1321D5311yescharacteristic, complemented
127#1320—532no—
128C2×C53—21no—
129C2×C53—41no—
130C2×C53—81no—
131C2×C53#146611yescharacteristic
132D53—42no—
133C2×#131—531no—
134C23×C8—531no—
135#525—531no—
136#435—1062no—
137C8×D4—1062no—
138#424—531no—
139#408C2×D5311yescharacteristic
140#407—531no—
141#405—531no—
142#364—1062no—
143#349—532no—
144#349C2×D5312yes—
145#347—1062no—
146C53#1153711yescharacteristic, complemented, Sylow
147C2×#41C4×D5311yescharacteristic
148C23×C4—531no—
149#131—2122no—
150C2×M16—531no—
151C2×M16—1061no—
152C2×M16C22×D5311yescharacteristic
153C22×C8—531no—
154C22×C8—1061no—
155C22×C8—1061no—
156C22×C8—1061no—
157C22×C8—2122no—
158C22×C8C4×D5311yescharacteristic
159#123—1061no—
160C22⋊Q8—1061no—
161#121—1061no—
162#120—1061no—
163C4×D4—1062no—
164#117—531no—
165M32—22no—
166M32—1062no—
167C4⋊C8—1062no—
168C22⋊C8—22no—
169C22⋊C8—1062no—
170#97—532no—
171C4×C8—532no—
172C24—531no—
173#41—41no—
174#41—41no—
175C2×Q8#1171991211yescharacteristic
176C2×D4—21no—
177C2×D4—1062no—
178C2×D4#1171991211yescharacteristic
179C22×C4—21no—
180C22×C4—531no—
181C22×C4—1061no—
182C22×C4—1061no—
183C22×C4—2121no—
184C22×C4—2121no—
185C22×C4C2×C4×D5311yescharacteristic
186M16—41no—
187M16—2122no—
188M16—2121no—
189C2×C8—21no—
190C2×C8—41no—
191C2×C8—531no—
192C2×C8—531no—
193C2×C8—1062no—
194C2×C8—1061no—
195C2×C8—2122no—
196C2×C8—2122no—
197C2×C8—2122no—
198C2×C8D4×D5312yes—
199C4⋊C4—1062no—
200C4⋊C4—1061no—
201C22⋊C4—1062no—
202C22⋊C4—1062no—
203C22⋊C4—1061no—
204C42—532no—
205C16—42no—
206C16—2122no—
207C23—21no—
208C23—1061no—
209C23—2121no—
210C23—2122no—
211C23M16×D5311yescharacteristic
212Q8—41no—
213D4—41no—
214D4—42no—
215D4—2124no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—1062no—
222C2×C4—1061no—
223C2×C4—2122no—
224C2×C4—2121no—
225C2×C4—2121no—
226C2×C4—2121no—
227C2×C4C22⋊C4×D5311yescharacteristic
228C2×C4C2×#1171991211yescharacteristic
229C8—22no—
230C8—41no—
231C8—532no—
232C8—1061no—
233C8—2121no—
234C22—21no—
235C22—41no—
236C22—42no—
237C22—41no—
238C22—1062no—
239C22—2121no—
240C22—2122no—
241C22—2122no—
242C22—2121no—
243C22#1681058511yescharacteristic
244C4—21no—
245C4—41no—
246C4—2122no—
247C4—2121no—
248C4#1681059511yescharacteristic
249C4#100×D5311yescharacteristic
250C2—21no—
251C2—41no—
252C2—81no—
253C2—2122no—
254C2#2027305411yescharacteristic
subgroupG/H
centerC4#16810595
commutatorC22×C4×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#1321C2
socleC2×C53#1466
prime radical#16811130C4
cosocleC2×M16×C53C23
hypercenter#1321D53
hypocenterC53#11537
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1321

Complements

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

NH ≅ G/Ntype
2. #20276528C2semidirect
3. #20272653C2semidirect
4. #20272650C2semidirect
5. C53×#1321C2semidirect
126. #1321D53semidirect
146. C53#11537semidirect

Maximal subgroups

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

subgroupindex
#202834122
#202765282
#202726532
#202726502
C53×#13212
#1153753