Finite Group Databasebeta

← #28308102

Subgroups — #28308102

Order 13568.

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

normal subgroups only

This group has 382 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#20283412C211yesmaximal, characteristic
5C2×#13133291—22no—
6#16811519—22no—
7#16811519C2214yes—
8C2×#13133224C2211yescharacteristic
9C22×C8×D53C411yescharacteristic
10#16811130C411yescharacteristic
11#16810697—22no—
12#16810686—22no—
13#16810649C2211yescharacteristic
14#16810586C2211yescharacteristic
15#16810581C411yescharacteristic
16C53×#408C411yescharacteristic
17#13133291—24no—
18#13133291—42no—
19C2×#11719910—42no—
20C22×C4×D53C2×C411yescharacteristic
21#13133224—41no—
22#13133224—41no—
23C2×#11719917D412yes—
24C2×#11719917C2311yescharacteristic
25C22×#88322C2×C411yescharacteristic
26C22×#88322D412yes—
27C2×C8×D53—21no—
28C2×C8×D53—21no—
29C2×C8×D53—41no—
30C2×C8×D53—41no—
31#13133198C812yes—
32#13133193—21no—
33#13133188—21no—
34#13133185C812yes—
35C4×C22⋊D53—22no—
36#13133176C2×C411yescharacteristic
37#13133146—22no—
38#13133129—22no—
39#13133127—21no—
40C8×Dic53—21no—
41C2×#41×C53C2×C411yescharacteristic
42C2×M16×C53C2×C411yescharacteristic
43C22×C8×C53C2×C411yescharacteristic
44C22⋊C8×C53—22no—
45C23×D53C2×C811yescharacteristic
46#41×C53—41no—
47#41×C53—41no—
48C2×Q8×C53M1611yescharacteristic
49C2×D4×C53C2×C812yes—
50C2×D4×C53M1611yescharacteristic
51C22×C4×C53C22⋊C412yes—
52C22×C4×C53C22×C411yescharacteristic
53C2×C22⋊D53—22no—
54C22×Dic53C2×C811yescharacteristic
55C2×C4×D53—21no—
56C2×C4×D53—21no—
57C2×C4×D53—41no—
58C2×C4×D53—41no—
59M16×C53—41no—
60C2×C8×C53—21no—
61C2×C8×C53—21no—
62C2×C8×C53—41no—
63C22⋊Dic53—22no—
64C22⋊Dic53C2×C812yes—
65#11719921—22no—
66C4⋊Dic53—21no—
67#11719919—22no—
68C4×Dic53—21no—
69#11719917—42no—
70#11719917—41no—
71C2×#88322—21no—
72C2×#88322—21no—
73C2×#88322—22no—
74C2×#88322—22no—
75C2×#88322C22⋊C411yescharacteristic
76C2×#88322C22⋊C411yescharacteristic
77C2×#88322C2×D412yes—
78C8×D53—42no—
79C8×D53—41no—
80C8×D53—41no—
81#11719910—44no—
82C23×C53C22⋊C812yes—
83C23×C53C22×C811yescharacteristic
84C22×D53—21no—
85C22×D53—42no—
86C22×D53—41no—
87Q8×C53—41no—
88D4×C53—42no—
89D4×C53—41no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53C22⋊C812yes—
95C2×C4×C53C2×C22⋊C411yescharacteristic
96C2×C4×C53C2×M1611yescharacteristic
97C22⋊D53—44no—
98C2×Dic53—21no—
99C2×Dic53—22no—
100C2×Dic53—41no—
101C4×D53—42no—
102C4×D53—41no—
103C4×D53—41no—
104C8×C53—41no—
105C8×C53—41no—
106#88322—22no—
107#88322—21no—
108#88322#9911yescharacteristic
109#88322#9911yescharacteristic
110#11483—531nomaximal, Sylow
111C22×C53—21no—
112C22×C53—22no—
113C22×C53—42no—
114C22×C53—41no—
115C22×C53C2×C22⋊C811yescharacteristic
116C2×D53—22no—
117C2×D53—41no—
118C2×D53—42no—
119C2×D53—41no—
120C2×D53—42no—
121C4×C53—21no—
122C4×C53—22no—
123C4×C53C2×#9911yescharacteristic
124C4×C53#41311yescharacteristic
125Dic53—42no—
126Dic53—41no—
127#2929—531no—
128#2115—532no—
129#1328—532no—
130#1320—532no—
131C2×C53—21no—
132C2×C53—41no—
133C2×C53—42no—
134C2×C53#145911yescharacteristic
135D53—42no—
136C2×#131—531no—
137C23×C8—531no—
138#525—531no—
139C2×M32—1062no—
140#435—1062no—
141C8×D4—1062no—
142#424—531no—
143#408#615711yescharacteristic
144#407—531no—
145#405—531no—
146#349—534no—
147#349—1062no—
148C53#1148311yescharacteristic, complemented, Sylow
149C2×#41C2×#615711yescharacteristic
150C23×C4—531no—
151#131—2121no—
152#131—2121no—
153C2×M16—532no—
154C2×M16—531no—
155C2×M16C2×#615711yescharacteristic
156C22×C8—532no—
157C22×C8—531no—
158C22×C8—1061no—
159C22×C8—1061no—
160C22×C8—2121no—
161C22×C8—2121no—
162C22×C8C2×#615711yescharacteristic
163#123—532no—
164C22⋊Q8—1061no—
165#121—1061no—
166#120—532no—
167C4×D4—1062no—
168#117—531no—
169M32—1064no—
170M32—2122no—
171C2×C16—2122no—
172C4⋊C8—1062no—
173C22⋊C8—22no—
174C22⋊C8—1062no—
175#97—1061no—
176C4×C8—1061no—
177C24—531no—
178#41—41no—
179#41—41no—
180C2×Q8#1171993611yescharacteristic
181C2×D4—1062no—
182C2×D4D53⋊C812yes—
183C2×D4#1171993611yescharacteristic
184C22×C4—531no—
185C22×C4—1061no—
186C22×C4—1061no—
187C22×C4—2121no—
188C22×C4—2121no—
189C22×C4C22⋊#615712yes—
190C22×C4C22×#615711yescharacteristic
191M16—41no—
192M16—2121no—
193M16—2122no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—41no—
197C2×C8—531no—
198C2×C8—531no—
199C2×C8—532no—
200C2×C8—1062no—
201C2×C8—1062no—
202C2×C8—1061no—
203C2×C8—1061no—
204C2×C8—2121no—
205C2×C8—2121no—
206C2×C8—2122no—
207C4⋊C4—1062no—
208C4⋊C4—1061no—
209C22⋊C4—532no—
210C22⋊C4—1062no—
211C22⋊C4—1062no—
212C42—1061no—
213C16—2124no—
214C23—1061no—
215C23—2121no—
216C23—2122no—
217C23#1313327412yes—
218C23C2×D53⋊C811yescharacteristic
219Q8—41no—
220D4—41no—
221D4—42no—
222D4—2124no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—1062no—
228C2×C4—1061no—
229C2×C4—2122no—
230C2×C4—2121no—
231C2×C4—2121no—
232C2×C4—2121no—
233C2×C4#1313327412yes—
234C2×C4C2×C22⋊#615711yescharacteristic
235C2×C4C2×#1171993611yescharacteristic
236C8—41no—
237C8—41no—
238C8—531no—
239C8—531no—
240C8—1062no—
241C8—1061no—
242C22—21no—
243C22—22no—
244C22—42no—
245C22—41no—
246C22—1062no—
247C22—2121no—
248C22—2122no—
249C22—2122no—
250C22—2121no—
251C22C2×#1313327411yescharacteristic
252C4—21no—
253C4—22no—
254C4—2122no—
255C4—2121no—
256C4D53⋊#9911yescharacteristic
257C4#1681161011yescharacteristic
258C2—21no—
259C2—41no—
260C2—42no—
261C2—2122no—
262C2#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#11483
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
148. C53#11483semidirect

Maximal subgroups

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

subgroupindex
#202909512
#202896642
#202896482
#202834122
#1148353