Finite Group Databasebeta

← #27293589

Subgroups — #27293589

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#20283405C211yesmaximal, characteristic
2#20276528C212yesmaximal, complemented
3#20272652C212yesmaximal, complemented
4#20272651C211yesmaximal, characteristic, complemented
5C53×#1320C211yesmaximal, characteristic, complemented
6C2×#13133225C411yescharacteristic
7C23×#88322C411yescharacteristic
8#16811134C411yescharacteristic
9#16810691—22no—
10#16810686—22no—
11#16810651C411yescharacteristic
12#16810591C2211yescharacteristic
13#16810581C2211yescharacteristic
14#16810443—22no—
15#16810405C2212yes—
16#16810400—22no—
17C53×#407C2211yescharacteristic
18C53×#349C2212yes—
19C2×#11719945C2×C411yescharacteristic
20#13133225—42no—
21C2×#11719912—21no—
22C22×#88322—21no—
23C22×#88322—21no—
24C22×#88322—42no—
25C22×#88322C2×C411yescharacteristic
26C22×#88322C2×C411yescharacteristic
27C2×C8×D53—21no—
28#13133200—21no—
29#13133196—21no—
30#13133191—21no—
31#13133186—21no—
32C4×C22⋊D53—22no—
33#13133176C2×C411yescharacteristic
34#13133157—22no—
35#13133146—22no—
36#13133129—22no—
37#13133127D412yes—
38C8×Dic53D412yes—
39C23×C4×C53C2×C411yescharacteristic
40C2×M16×C53C2×C411yescharacteristic
41C2×M16×C53C2311yescharacteristic
42M32×C53—22no—
43C22⋊C8×C53—22no—
44C24×C53M1611yescharacteristic
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53—41no—
48C22×C4×C53—41no—
49C22×C4×C53M1611yescharacteristic
50C22×C4×C53C22×C411yescharacteristic
51C2×C22⋊D53—22no—
52C2×C22⋊D53—21no—
53#11719945—41no—
54#11719945—41no—
55C2×D212M1611yescharacteristic
56C2×C4×D53—21no—
57C2×Dic106M1611yescharacteristic
58M16×C53—41no—
59M16×C53—41no—
60C2×C8×C53—21no—
61C2×C8×C53C2×D412yes—
62C22⋊Dic53—22no—
63C22⋊Dic53—21no—
64#11719921—22no—
65C4⋊Dic53—21no—
66#11719919—22no—
67C4×Dic53C22⋊C412yes—
68C2×#88322—22no—
69C2×#88322—21no—
70C2×#88322—42no—
71C2×#88322—42no—
72C2×#88322—41no—
73C2×#88322C22⋊C411yescharacteristic
74C2×#88322C22⋊C411yescharacteristic
75#11719912—42no—
76C8×D53—42no—
77C16×C53—42no—
78#11719908—42no—
79C23×C53—21no—
80C23×C53—41no—
81C23×C53—42no—
82C23×C53C2×M1611yescharacteristic
83C22×D53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—42no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53C2×C22⋊C411yescharacteristic
92C2×C4×C53C2×M1611yescharacteristic
93C22⋊D53—42no—
94C22⋊D53—44no—
95C2×Dic53—21no—
96C2×Dic53—22no—
97D212—41no—
98C4×D53—41no—
99C4×D53—41no—
100Dic106—41no—
101C8×C53—22no—
102C8×C53—41no—
103#88322—21no—
104#88322—41no—
105#88322#10012yes—
106#11537—531nomaximal, Sylow
107C22×C53—22no—
108C22×C53—21no—
109C22×C53—41no—
110C22×C53—41no—
111C22×C53—42no—
112C22×C53—42no—
113C22×C53—41no—
114C22×C53#40711yescharacteristic
115C2×D53—42no—
116C2×D53—41no—
117C4×C53—21no—
118C4×C53—41no—
119C4×C53#41011yescharacteristic
120C4×C53C2×#10011yescharacteristic
121Dic53—42no—
122Dic53—41no—
123#2929—531no—
124#2166—532no—
125#1321—532no—
126#1320—531no—
127#1320D5311yescharacteristic, complemented
128C2×C53—21no—
129C2×C53—42no—
130C2×C53—41no—
131C2×C53#146611yescharacteristic
132D53—81no—
133C2×#131—531no—
134C23×C8—531no—
135#525—531no—
136#435—1062no—
137C8×D4—1062no—
138#424—531no—
139#408—531no—
140#407C2×D5311yescharacteristic
141#405—531no—
142#364—1062no—
143#349—532no—
144#349C2×D5312yes—
145#347—1062no—
146C53#1153711yescharacteristic, complemented, Sylow
147C2×#41—531no—
148C23×C4C4×D5311yescharacteristic
149#131—2122no—
150C2×M16—1061no—
151C2×M16C4×D5311yescharacteristic
152C2×M16C22×D5311yescharacteristic
153C22×C8—531no—
154C22×C8—531no—
155C22×C8—1061no—
156C22×C8—1061no—
157C22×C8—1061no—
158C22×C8—2122no—
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#1171991211yescharacteristic
173#41—2121no—
174#41—2121no—
175C2×Q8—531no—
176C2×D4—531no—
177C2×D4—1062no—
178C2×D4—1061no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—1061no—
184C22×C4#1171991211yescharacteristic
185C22×C4C2×C4×D5311yescharacteristic
186M16—41no—
187M16—41no—
188M16—2122no—
189C2×C8—21no—
190C2×C8—531no—
191C2×C8—531no—
192C2×C8—1062no—
193C2×C8—1061no—
194C2×C8—2122no—
195C2×C8—2122no—
196C2×C8—2122no—
197C2×C8—2121no—
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—41no—
209C23—42no—
210C23—1061no—
211C23C2×#1171991211yescharacteristic
212Q8—2121no—
213D4—2121no—
214D4—2124no—
215D4—2122no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—1062no—
224C2×C4—1061no—
225C2×C4—2121no—
226C2×C4—2121no—
227C2×C4C22⋊C4×D5311yescharacteristic
228C2×C4M16×D5311yescharacteristic
229C8—22no—
230C8—41no—
231C8—532no—
232C8—1061no—
233C8—2121no—
234C22—22no—
235C22—21no—
236C22—41no—
237C22—41no—
238C22—42no—
239C22—42no—
240C22—41no—
241C22—2122no—
242C22—2121no—
243C22#1681058511yescharacteristic
244C4—21no—
245C4—41no—
246C4—2122no—
247C4—2121no—
248C4#1681059511yescharacteristic
249C4#100×D5311yescharacteristic
250C2—21no—
251C2—42no—
252C2—41no—
253C2—4241no—
254C2#2027305411yescharacteristic
subgroupG/H
centerC4#16810595
commutatorC22×C4×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#1320C2
socleC2×C53#1466
prime radical#16811134C4
cosocleC2×M16×C53C23
hypercenter#1320D53
hypocenterC53#11537
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1320

Complements

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

NH ≅ G/Ntype
2. #20276528C2semidirect
3. #20272652C2semidirect
4. #20272651C2semidirect
5. C53×#1320C2semidirect
127. #1320D53semidirect
146. C53#11537semidirect

Maximal subgroups

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

subgroupindex
#202834052
#202765282
#202726522
#202726512
C53×#13202
#1153753