Finite Group Databasebeta

← #27293417

Subgroups — #27293417

Order 13568.

364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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#20283423C211yesmaximal, characteristic
2#20276511C212yesmaximal, complemented
3#20272641C211yesmaximal, characteristic, complemented
4#20272640C212yesmaximal, complemented
5C53×#1315C211yesmaximal, characteristic, complemented
6C2×#13133225C411yescharacteristic
7C2×#13133224C411yescharacteristic
8#16811169C411yescharacteristic
9#16810698—22no—
10#16810686—22no—
11#16810646C411yescharacteristic
12#16810589C2211yescharacteristic
13C2×C8×Dic53C2211yescharacteristic
14#16810441—22no—
15C22⋊#11719908C2212yes—
16C16×Dic53—22no—
17C53×#408C2211yescharacteristic
18C53×C22⋊C16C2212yes—
19C2×#11719945C2×C411yescharacteristic
20#13133225—42no—
21#13133224—42no—
22C2×#11719917—21no—
23C2×#11719917C2×C411yescharacteristic
24C2×#11719912—21no—
25C22×#88322—21no—
26C22×#88322C2×C411yescharacteristic
27C2×C8×D53—21no—
28#13133215C811yescharacteristic
29#13133213C811yescharacteristic
30#13133206C811yescharacteristic
31#13133204C811yescharacteristic
32#13133189—22no—
33C4×C22⋊D53—22no—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×#11719908—22no—
36#13133146—22no—
37#13133129—22no—
38C8×Dic53D412yes—
39C8×Dic53D412yes—
40C2×#41×C53C2×C411yescharacteristic
41C2×M16×C53C2×C411yescharacteristic
42C22×C8×C53C2311yescharacteristic
43C2×C16×C53—22no—
44C22⋊C8×C53—22no—
45#41×C53—41no—
46#41×C53—41no—
47C2×Q8×C53C2×C811yescharacteristic
48C2×D4×C53—21no—
49C2×D4×C53C2×C811yescharacteristic
50C22×C4×C53—21no—
51C22×C4×C53C22×C411yescharacteristic
52C2×C22⋊D53—21no—
53C2×C22⋊D53—22no—
54C22×Dic53M1612yes—
55#11719945—41no—
56#11719945—41no—
57C2×D212C2×C811yescharacteristic
58C2×C4×D53—21no—
59C2×Dic106C2×C811yescharacteristic
60M16×C53—41no—
61C2×C8×C53—21no—
62C2×C8×C53—41no—
63C2×C8×C53C2×D412yes—
64C22⋊Dic53—22no—
65#11719921—22no—
66#11719919—22no—
67C4×Dic53C22⋊C412yes—
68C4×Dic53C2×C811yescharacteristic
69C4×Dic53C2×C811yescharacteristic
70#11719917—42no—
71#11719917—41no—
72C2×#88322—22no—
73C2×#88322—21no—
74C2×#88322—42no—
75C2×#88322C22⋊C411yescharacteristic
76C2×#88322C22⋊C411yescharacteristic
77#11719912—42no—
78C8×D53—42no—
79C16×C53—42no—
80#11719908—42no—
81C23×C53—21no—
82C23×C53C2×M1611yescharacteristic
83C22×D53—21no—
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×C53C22×C811yescharacteristic
94C22⋊D53—42no—
95C22⋊D53—44no—
96C2×Dic53—24no—
97C2×Dic53—21no—
98C2×Dic53C22⋊C812yes—
99C2×Dic53C22⋊C812yes—
100D212—41no—
101C4×D53—41no—
102C4×D53—41no—
103Dic106—41no—
104C8×C53—22no—
105C8×C53—41no—
106#88322—21no—
107#88322—41no—
108#88322#10012yes—
109#11490—531nomaximal, Sylow
110C22×C53—22no—
111C22×C53—21no—
112C22×C53—42no—
113C22×C53—41no—
114C22×C53C2×C22⋊C811yescharacteristic
115C2×D53—42no—
116C2×D53—41no—
117C4×C53—21no—
118C4×C53—41no—
119C4×C53#41011yescharacteristic
120C4×C53C2×#10011yescharacteristic
121Dic53—22no—
122Dic53—41no—
123Dic53#35014yes—
124#2931—531no—
125#2162—532no—
126#1315—531no—
127#1315—532no—
128#1315D5311yescharacteristic, complemented
129C2×C53—41no—
130C2×C53—81no—
131C2×C53#146011yescharacteristic
132C2×C53C2×#35012yes—
133D53—81no—
134C2×#131—531no—
135C2×#131—531no—
136#532—531no—
137#436—1062no—
138C8×D4—1062no—
139#423—531no—
140#408—531no—
141#408C2×D5311yescharacteristic
142C2×C4×C8—531no—
143C4⋊C16—1062no—
144C22⋊C16—532no—
145C22⋊C16C2×D5312yes—
146C4×C16—1062no—
147C53#1149011yescharacteristic, complemented, Sylow
148C2×#41—531no—
149C2×#41C4×D5311yescharacteristic
150#131—2122no—
151#131—2122no—
152C2×M16—531no—
153C2×M16—1061no—
154C2×M16—1061no—
155C2×M16C4×D5311yescharacteristic
156C22×C8—531no—
157C22×C8—1061no—
158C22×C8—1061no—
159C22×C8C22×D5311yescharacteristic
160C4⋊Q8—531no—
161#127—531no—
162#124—531no—
163#124—531no—
164#121—1062no—
165C4×D4—1062no—
166C2×C42—531no—
167C2×C16—22no—
168C2×C16—1062no—
169C4⋊C8—1062no—
170C22⋊C8—22no—
171C22⋊C8—1062no—
172C4×C8—532no—
173C4×C8—532no—
174#41—41no—
175#41—41no—
176#41—2121no—
177#41—2121no—
178C2×Q8—531no—
179C2×Q8C8×D5311yescharacteristic
180C2×D4—21no—
181C2×D4—531no—
182C2×D4—1062no—
183C2×D4—1061no—
184C2×D4C8×D5311yescharacteristic
185C22×C4—21no—
186C22×C4—532no—
187C22×C4—1061no—
188C22×C4C2×C4×D5311yescharacteristic
189M16—41no—
190M16—2122no—
191M16—2122no—
192M16—2121no—
193C2×C8—21no—
194C2×C8—41no—
195C2×C8—531no—
196C2×C8—531no—
197C2×C8—1062no—
198C2×C8—1061no—
199C2×C8—2122no—
200C2×C8—2122no—
201C2×C8D4×D5312yes—
202C4⋊C4—1062no—
203C22⋊C4—1062no—
204C22⋊C4—1062no—
205C42—532no—
206C42—531no—
207C42—531no—
208C16—42no—
209C16—2122no—
210C23—21no—
211C23—1061no—
212C23M16×D5311yescharacteristic
213Q8—41no—
214Q8—2121no—
215D4—41no—
216D4—42no—
217D4—2121no—
218D4—2124no—
219D4—2122no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—532no—
226C2×C4—532no—
227C2×C4—1064no—
228C2×C4—1061no—
229C2×C4—2121no—
230C2×C4—2121no—
231C2×C4C22⋊C4×D5311yescharacteristic
232C2×C4C2×C8×D5311yescharacteristic
233C8—22no—
234C8—41no—
235C8—532no—
236C8—1061no—
237C8—2121no—
238C22—22no—
239C22—21no—
240C22—42no—
241C22—41no—
242C22—2122no—
243C22—2121no—
244C22C22⋊C8×D5311yescharacteristic
245C4—21no—
246C4—41no—
247C4—534no—
248C4—1062no—
249C4—2121no—
250C4#1681059511yescharacteristic
251C4#100×D5311yescharacteristic
252C2—41no—
253C2—81no—
254C2—4241no—
255C2#2027303411yescharacteristic
256C2#350×D5312yes—
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#1315C2
socleC22×C53C2×C22⋊C8
prime radical#16811169C4
cosocleC22×C8×C53C23
hypercenter#1315D53
hypocenterC53#11490
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #20276511C2semidirect
3. #20272641C2semidirect
4. #20272640C2semidirect
5. C53×#1315C2semidirect
128. #1315D53semidirect
147. C53#11490semidirect

Maximal subgroups

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

subgroupindex
#202834232
#202765112
#202726412
#202726402
C53×#13152
#1149053