Finite Group Databasebeta

← #28299725

Subgroups — #28299725

Order 13568.

468 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 94 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#20290643C212yesmaximal, complemented
2#20289535C212yesmaximal, complemented
3#20289531C212yesmaximal
4#124×Dic53C211yesmaximal, characteristic
5#16811489—22no—
6#16811489—22no—
7#16811489C2214yes—
8C2×Q8×Dic53C411yescharacteristic
9C2×D4×Dic53C411yescharacteristic
10#16810530C2211yescharacteristic
11#16810526C2211yescharacteristic
12#16810516—22no—
13#16810489C411yescharacteristic
14C22⋊C4×Dic53—22no—
15C42×Dic53C2211yescharacteristic
16C2×#124×C53C411yescharacteristic
17C22×#88324—24no—
18C23×Dic53—21no—
19Q8×Dic53—42no—
20D4×Dic53—42no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53C2×C411yescharacteristic
23C2×C4⋊Dic53C2×C411yescharacteristic
24C2×C22⋊Dic53—21no—
25C2×C22⋊Dic53—22no—
26C2×C4×Dic53—21no—
27C2×C4×Dic53C2×C411yescharacteristic
28C2×C4×Dic53D412yes—
29C2×C4×Dic53D412yes—
30C2×C4×Dic53C2311yescharacteristic
31#13133124—22no—
32#13133124—22no—
33C22×Q8×C53C2×C411yescharacteristic
34C22×D4×C53C2×C411yescharacteristic
35#124×C53—22no—
36#124×C53—22no—
37C2×C22⋊C4×C53—22no—
38C2×C42×C53C2×C411yescharacteristic
39C24×C53—21no—
40C2×Q8×C53—21no—
41C2×Q8×C53—41no—
42C2×Q8×C53—41no—
43C2×Q8×C53M1612yes—
44C2×D4×C53—21no—
45C2×D4×C53—41no—
46C2×D4×C53—41no—
47C2×D4×C53M1612yes—
48C22×C4×C53—21no—
49C22×C4×C53C22⋊C412yes—
50C22×C4×C53C22×C411yescharacteristic
51C22×Dic53—21no—
52C22×Dic53—21no—
53C22×Dic53—44no—
54C22×Dic53—42no—
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C22⋊C411yescharacteristic
57C22×Dic53C2×D412yes—
58C2×#88324—44no—
59C2×#88324—44no—
60C2×#88324—44no—
61C22⋊C4×C53—28no—
62C42×C53—21no—
63C42×C53—21no—
64C22⋊Dic53—42no—
65C22⋊Dic53—44no—
66C4⋊Dic53—42no—
67C4⋊Dic53—41no—
68C4⋊Dic53—41no—
69C4×Dic53—22no—
70C4×Dic53—24no—
71C4×Dic53—42no—
72C4×Dic53—41no—
73C4×Dic53—42no—
74C23×C53—22no—
75C23×C53—41no—
76C23×C53—41no—
77C23×C53—42no—
78C23×C53—42no—
79C23×C53C2×C22⋊C411yescharacteristic
80Q8×C53—42no—
81Q8×C53—42no—
82D4×C53—42no—
83D4×C53—42no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53C2×M1611yescharacteristic
93C2×C4×C53C2×M1611yescharacteristic
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—21no—
98C2×Dic53—22no—
99C2×Dic53—41no—
100C2×Dic53—41no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—84no—
104C2×Dic53#10012yes—
105C2×Dic53#13514yes—
106#88324—84no—
107#7649—531nomaximal, Sylow
108C22×C53—42no—
109C22×C53—44no—
110C22×C53—81no—
111C22×C53—82no—
112C22×C53—81no—
113C22×C53#40711yescharacteristic
114C22×C53#41011yescharacteristic
115C22×C53C2×#10011yescharacteristic
116C22×C53#41612yes—
117C22×C53#42112yes—
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—42no—
121C4×C53—42no—
122Dic53—24no—
123Dic53—22no—
124Dic53—41no—
125Dic53—81no—
126Dic53—81no—
127C4×#124—531no—
128#1839—532no—
129#1272—532no—
130#1270—532no—
131C2×C53—82no—
132C2×C53#146611yescharacteristic
133C2×C53#148812yes—
134C2×C53#152112yes—
135C2×C53#153311yescharacteristic
136C2×C53#154311yescharacteristic
137C2×#124#615711yescharacteristic
138C2×C4×Q8—531no—
139C2×C4×D4—531no—
140#391—531no—
141#390—531no—
142#388—1062no—
143#382—531no—
144C4×C22⋊C4—1062no—
145C43—531no—
146#336—534no—
147#336—1062no—
148#336—1062no—
149C53#764911yescharacteristic, complemented, Sylow
150C22×Q8C2×#615711yescharacteristic
151C22×D4C2×#615711yescharacteristic
152C23×C4—1061no—
153C22×C8—1064no—
154#124—22no—
155#124—22no—
156C4×Q8—2122no—
157C4×D4—2122no—
158C2×C4⋊C4—531no—
159C2×C4⋊C4—531no—
160C2×C4⋊C4—1061no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—1061no—
163C2×C22⋊C4—1062no—
164C2×C42—532no—
165C2×C42—531no—
166C2×C42—532no—
167C2×C42—531no—
168C2×C42—1061no—
169C2×C42C2×#615711yescharacteristic
170#95—1062no—
171#95—1062no—
172C24—21no—
173C2×Q8—21no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8#1171994012yes—
177C2×D4—21no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4#1171994012yes—
181C22×C4—21no—
182C22×C4—531no—
183C22×C4—532no—
184C22×C4—531no—
185C22×C4—1061no—
186C22×C4—1061no—
187C22×C4—2122no—
188C22×C4—2124no—
189C22×C4C22⋊#615712yes—
190C22×C4C22×#615711yescharacteristic
191C2×C8—2124no—
192C2×C8—2124no—
193C2×C8—2124no—
194C4⋊C4—2121no—
195C4⋊C4—2121no—
196C4⋊C4—2122no—
197C22⋊C4—28no—
198C22⋊C4—2124no—
199C22⋊C4—2122no—
200C42—21no—
201C42—21no—
202C42—1062no—
203C42—1064no—
204C42—2121no—
205C42—2122no—
206C42—2122no—
207C23—22no—
208C23—41no—
209C23—41no—
210C23—42no—
211C23—42no—
212C23C2×C22⋊#615711yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—42no—
216D4—42no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—532no—
226C2×C4—534no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1061no—
231C2×C4—1062no—
232C2×C4—4244no—
233C2×C4—2121no—
234C2×C4—2121no—
235C2×C4—2122no—
236C2×C4—2122no—
237C2×C4C2×#1171994011yescharacteristic
238C2×C4C2×#1171994011yescharacteristic
239C8—4244no—
240C22—42no—
241C22—44no—
242C22—82no—
243C22—81no—
244C22—81no—
245C22#1681159311yescharacteristic
246C22#1681160311yescharacteristic
247C22D53⋊#10011yescharacteristic
248C22#1681161312yes—
249C22#1681162012yes—
250C4—21no—
251C4—21no—
252C4—42no—
253C4—42no—
254C4—1064no—
255C4—1062no—
256C4—4241no—
257C4—4241no—
258C4—2121no—
259C2—82no—
260C2#2028997511yescharacteristic
261C2#2029004612yes—
262C2#2029015612yes—
263C2#2029019511yescharacteristic
264C2#2029021911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#124×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C4×Dic53C23
hypercenterC2×#124#6157
hypocenterC53#7649
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #20290643C2semidirect
2. #20289535C2semidirect
149. C53#7649semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202895352
#202895312
#124×Dic532
#764953