Finite Group Databasebeta

← #28308106

Subgroups — #28308106

Order 13568.

368 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 86 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#20290951C212yesmaximal, complemented
2#20289664C212yesmaximal, complemented
3#20289648C212yesmaximal, complemented
4#20283421C211yesmaximal, characteristic
5C2×#13133291—22no—
6#16811519—22no—
7#16811519C2214yes—
8C2×#13133224C2211yescharacteristic
9C22×#11719912C411yescharacteristic
10#16811130C411yescharacteristic
11#16810698—22no—
12#16810691—22no—
13#16810649C2211yescharacteristic
14#16810586C2211yescharacteristic
15#16810581C411yescharacteristic
16C53×#408C411yescharacteristic
17#13133291—24no—
18#13133291—42no—
19C2×#11719910—42no—
20C22×C4×D53C2×C411yescharacteristic
21#13133224—42no—
22C2×#11719917D412yes—
23C2×#11719917C2311yescharacteristic
24C2×#11719912—21no—
25C2×#11719912—21no—
26C2×#11719912—42no—
27C22×#88322C2×C411yescharacteristic
28C22×#88322D412yes—
29#13133198C812yes—
30#13133193—21no—
31#13133188—21no—
32#13133185C812yes—
33C4×C22⋊D53—22no—
34#13133176C2×C411yescharacteristic
35#13133146—22no—
36#13133129—22no—
37#13133127—21no—
38C8×Dic53—21no—
39C2×#41×C53C2×C411yescharacteristic
40C2×M16×C53C2×C411yescharacteristic
41C22×C8×C53C2×C411yescharacteristic
42C22⋊C8×C53—22no—
43C23×D53C2×C811yescharacteristic
44#41×C53—41no—
45#41×C53—41no—
46C2×Q8×C53M1611yescharacteristic
47C2×D4×C53C2×C812yes—
48C2×D4×C53M1611yescharacteristic
49C22×C4×C53C22⋊C412yes—
50C22×C4×C53C22×C411yescharacteristic
51C2×C22⋊D53—22no—
52C22×Dic53C2×C811yescharacteristic
53C2×C4×D53—21no—
54C2×C4×D53—21no—
55C2×C4×D53—41no—
56C2×C4×D53—41no—
57M16×C53—41no—
58C2×C8×C53—21no—
59C2×C8×C53—21no—
60C2×C8×C53—41no—
61C22⋊Dic53—22no—
62C22⋊Dic53C2×C812yes—
63#11719921—22no—
64C4⋊Dic53—21no—
65#11719919—22no—
66C4×Dic53—21no—
67#11719917—42no—
68#11719917—41no—
69C2×#88322—22no—
70C2×#88322—22no—
71C2×#88322—22no—
72C2×#88322C22⋊C411yescharacteristic
73C2×#88322C22⋊C411yescharacteristic
74C2×#88322C2×D412yes—
75#11719912—42no—
76#11719912—42no—
77#11719910—44no—
78C23×C53C22⋊C812yes—
79C23×C53C22×C811yescharacteristic
80C22×D53—21no—
81C22×D53—41no—
82C22×D53—81no—
83Q8×C53—41no—
84D4×C53—42no—
85D4×C53—41no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53C22⋊C812yes—
91C2×C4×C53C2×C22⋊C411yescharacteristic
92C2×C4×C53C2×M1611yescharacteristic
93C22⋊D53—82no—
94C2×Dic53—21no—
95C2×Dic53—22no—
96C2×Dic53—41no—
97C4×D53—42no—
98C4×D53—41no—
99C4×D53—41no—
100C8×C53—41no—
101C8×C53—41no—
102#88322—22no—
103#88322—21no—
104#88322#9912yes—
105#11484—531nomaximal, Sylow
106C22×C53—21no—
107C22×C53—22no—
108C22×C53—42no—
109C22×C53—41no—
110C22×C53C2×C22⋊C811yescharacteristic
111C2×D53—41no—
112C2×D53—41no—
113C2×D53—41no—
114C2×D53—82no—
115C4×C53—21no—
116C4×C53—22no—
117C4×C53C2×#9911yescharacteristic
118C4×C53#41311yescharacteristic
119Dic53—42no—
120Dic53—41no—
121#2930—531no—
122#2115—532no—
123#1328—532no—
124#1320—532no—
125C2×C53—21no—
126C2×C53—41no—
127C2×C53—42no—
128C2×C53#145911yescharacteristic
129D53—81no—
130C2×#131—531no—
131C22×M16—531no—
132#525—531no—
133C2×M32—1062no—
134#436—1062no—
135#435—1062no—
136#424—531no—
137#408#615711yescharacteristic
138#407—531no—
139#405—531no—
140#349—534no—
141#349—1062no—
142C53#1148411yescharacteristic, complemented, Sylow
143C2×#41C2×#615711yescharacteristic
144C23×C4—531no—
145#131—2122no—
146C2×M16—532no—
147C2×M16—531no—
148C2×M16—1061no—
149C2×M16—1061no—
150C2×M16—2122no—
151C2×M16C2×#615711yescharacteristic
152C22×C8—532no—
153C22×C8—531no—
154C22×C8C2×#615711yescharacteristic
155#123—532no—
156C22⋊Q8—1061no—
157#121—1061no—
158#120—532no—
159C4×D4—1062no—
160#117—531no—
161M32—1064no—
162M32—2122no—
163C2×C16—2122no—
164C4⋊C8—1062no—
165C22⋊C8—22no—
166C22⋊C8—1062no—
167#97—1061no—
168C4×C8—1061no—
169C24—531no—
170#41—41no—
171#41—41no—
172C2×Q8#1171993611yescharacteristic
173C2×D4—1062no—
174C2×D4D53⋊C812yes—
175C2×D4#1171993611yescharacteristic
176C22×C4—531no—
177C22×C4—1061no—
178C22×C4—1061no—
179C22×C4—2121no—
180C22×C4—2121no—
181C22×C4C22⋊#615712yes—
182C22×C4C22×#615711yescharacteristic
183M16—41no—
184M16—2122no—
185M16—2122no—
186M16—2121no—
187M16—2122no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—41no—
191C2×C8—531no—
192C2×C8—531no—
193C2×C8—532no—
194C2×C8—1062no—
195C2×C8—1062no—
196C2×C8—1062no—
197C4⋊C4—1062no—
198C4⋊C4—1061no—
199C22⋊C4—532no—
200C22⋊C4—1062no—
201C22⋊C4—1062no—
202C42—1061no—
203C16—2124no—
204C23—1061no—
205C23—4241no—
206C23—2121no—
207C23#1313327412yes—
208C23C2×D53⋊C811yescharacteristic
209Q8—41no—
210D4—41no—
211D4—42no—
212D4—4242no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—1062no—
218C2×C4—1061no—
219C2×C4—2122no—
220C2×C4—2121no—
221C2×C4—2121no—
222C2×C4—2121no—
223C2×C4#1313327412yes—
224C2×C4C2×C22⋊#615711yescharacteristic
225C2×C4C2×#1171993611yescharacteristic
226C8—41no—
227C8—41no—
228C8—532no—
229C8—1062no—
230C8—1061no—
231C22—21no—
232C22—22no—
233C22—42no—
234C22—41no—
235C22—4242no—
236C22—2121no—
237C22—2121no—
238C22—2121no—
239C22C2×#1313327411yescharacteristic
240C4—21no—
241C4—22no—
242C4—2122no—
243C4—2121no—
244C4D53⋊#9911yescharacteristic
245C4#1681161011yescharacteristic
246C2—21no—
247C2—41no—
248C2—42no—
249C2—4241no—
250C2#2028995311yescharacteristic
subgroupG/H
centerC4D53⋊#99
commutatorC23×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#408C4
socleC2×C53#1459
prime radical#16811130C4
cosocleC2×#11719917C23
hypercenter#408#6157
hypocenterC53#11484
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #20290951C2semidirect
2. #20289664C2semidirect
3. #20289648C2semidirect
142. C53#11484semidirect

Maximal subgroups

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

subgroupindex
#202909512
#202896642
#202896482
#202834212
#1148453