Finite Group Databasebeta

← #28308280

Subgroups — #28308280

Order 13568.

440 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 74 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#20289971C212yesmaximal
2#20289657C212yesmaximal, complemented
3#20289645C212yesmaximal, complemented
4#20277428C211yesmaximal, characteristic
5#16811593—22no—
6#16811468C2214yes—
7#16811466—22no—
8C2×Q8×Dic53C411yescharacteristic
9C2×D4×Dic53C2211yescharacteristic
10#16810530C2211yescharacteristic
11#16810526C411yescharacteristic
12#16810516—22no—
13#16810498—22no—
14#16810493C2211yescharacteristic
15#16810469C411yescharacteristic
16C2×#124×C53C411yescharacteristic
17C2×#11719940—42no—
18C22⋊#88324—24no—
19C22⋊#88324—42no—
20C23×Dic53D412yes—
21Q8×Dic53—42no—
22D4×Dic53—42no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53C2311yescharacteristic
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53D412yes—
29C2×C4×Dic53—21no—
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53C2×C411yescharacteristic
32#13133124—21no—
33#13133124—21no—
34#13133124—22no—
35C22×Q8×C53C2×C411yescharacteristic
36C22×D4×C53C2×C411yescharacteristic
37#124×C53—22no—
38#124×C53—22no—
39C2×C22⋊C4×C53—22no—
40C2×C42×C53C2×C411yescharacteristic
41C24×C53C22⋊C412yes—
42C2×Q8×C53—21no—
43C2×Q8×C53—41no—
44C2×Q8×C53—41no—
45C2×Q8×C53M1612yescomplemented
46C2×D4×C53—21no—
47C2×D4×C53—21no—
48C2×D4×C53—41no—
49C2×D4×C53—41no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53C22×C411yescharacteristic
53C22×Dic53—22no—
54C22×Dic53—22no—
55C22×Dic53—21no—
56C22×Dic53—21no—
57C22×Dic53—82no—
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C22⋊C411yescharacteristic
60C22×Dic53C2×D412yes—
61#11719940—84no—
62C2×#88324—44no—
63C22⋊C4×C53—28no—
64C42×C53—21no—
65C42×C53M1612yescomplemented
66C22⋊Dic53—42no—
67C22⋊Dic53—82no—
68C4⋊Dic53—41no—
69C4⋊Dic53—42no—
70C4⋊Dic53—41no—
71C4⋊Dic53—82no—
72C4×Dic53—22no—
73C4×Dic53—42no—
74C4×Dic53—41no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53—42no—
79C23×C53—42no—
80C23×C53C2×C22⋊C411yescharacteristic
81Q8×C53—42no—
82Q8×C53—42no—
83D4×C53—42no—
84D4×C53—42no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53C2×M1611yescharacteristic
94C2×C4×C53C2×M1611yescharacteristic
95C2×Dic53—22no—
96C2×Dic53—21no—
97C2×Dic53—41no—
98C2×Dic53—44no—
99C2×Dic53—42no—
100C2×Dic53—41no—
101C2×Dic53—41no—
102C2×Dic53—42no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53—41no—
106C2×Dic53—82no—
107C2×Dic53#9912yes—
108#88324—84no—
109#11535—531nomaximal, Sylow
110C22×C53—21no—
111C22×C53—21no—
112C22×C53—44no—
113C22×C53—42no—
114C22×C53—42no—
115C22×C53—42no—
116C22×C53—82no—
117C22×C53#40711yescharacteristic
118C22×C53C2×#9911yescharacteristic
119C22×C53#41311yescharacteristic
120C4×C53—21no—
121C4×C53—21no—
122C4×C53—42no—
123C4×C53—42no—
124Dic53—22no—
125Dic53—42no—
126Dic53—41no—
127Dic53—81no—
128Dic53—81no—
129#2326—531no—
130#1465—532no—
131#1325—532no—
132#1319—532no—
133C2×C53—21no—
134C2×C53—21no—
135C2×C53—82no—
136C2×C53#146511yescharacteristic
137C2×C53#212711yescharacteristic
138C2×C53#213511yescharacteristic
139C2×#124#615711yescharacteristic
140C2×C4×Q8—531no—
141C2×C4×D4—531no—
142#407—1062no—
143#391—531no—
144#390—531no—
145#388—1062no—
146#385—1062no—
147#384—531no—
148#376—531no—
149#324—534no—
150#323—1062no—
151C53#1153511yescharacteristic, complemented, Sylow
152C22×Q8C2×#615711yescharacteristic
153C22×D4C2×#615711yescharacteristic
154C23×C4—532no—
155C2×M16—2122no—
156#124—22no—
157#124—22no—
158C4×Q8—2122no—
159C4×D4—2122no—
160C2×C4⋊C4—531no—
161C2×C4⋊C4—531no—
162C2×C4⋊C4—1062no—
163C2×C4⋊C4—1061no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—532no—
166C2×C22⋊C4—1062no—
167C2×C42—531no—
168C2×C42—531no—
169C2×C42—1061no—
170C2×C42C2×#615711yescharacteristic
171C22⋊C8—1064no—
172C22⋊C8—2122no—
173#95—1061no—
174#95—1061no—
175#95—1062no—
176C24C22⋊#615712yes—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8#1171994012yescomplemented
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—41no—
184C2×D4—41no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—531no—
188C22×C4—531no—
189C22×C4—532no—
190C22×C4—1062no—
191C22×C4—1062no—
192C22×C4—1061no—
193C22×C4—1061no—
194C22×C4—4242no—
195C22×C4C22×#615711yescharacteristic
196M16—4244no—
197C2×C8—2124no—
198C4⋊C4—4242no—
199C4⋊C4—2121no—
200C4⋊C4—2121no—
201C4⋊C4—2122no—
202C22⋊C4—28no—
203C22⋊C4—4242no—
204C22⋊C4—2122no—
205C42—21no—
206C42—1062no—
207C42—2121no—
208C42—2122no—
209C42#1171994012yescomplemented
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×C22⋊#615711yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—42no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—532no—
229C2×C4—1061no—
230C2×C4—1062no—
231C2×C4—4242no—
232C2×C4—2121no—
233C2×C4—2124no—
234C2×C4—2121no—
235C2×C4—2122no—
236C2×C4—2121no—
237C2×C4—2121no—
238C2×C4—2121no—
239C2×C4—2122no—
240C2×C4—2121no—
241C2×C4C2×#1171994011yescharacteristic
242C2×C4C2×#1171994011yescharacteristic
243C8—4244no—
244C22—21no—
245C22—21no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22#1681159311yescharacteristic
252C22D53⋊#9911yescharacteristic
253C22#1681161011yescharacteristic
254C4—21no—
255C4—21no—
256C4—42no—
257C4—42no—
258C4—1062no—
259C4—4241no—
260C4—4241no—
261C4—2122no—
262C4—2121no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2028997111yescharacteristic
267C2#2029099311yescharacteristic
268C2#2029101811yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#124×C53C4
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×#124#6157
hypocenterC53#11535
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #20289657C2semidirect
3. #20289645C2semidirect
45. C2×Q8×C53M16semidirect
65. C42×C53M16semidirect
151. C53#11535semidirect
180. C2×Q8#11719940semidirect
209. C42#11719940semidirect

Maximal subgroups

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

subgroupindex
#202899712
#202896572
#202896452
#202774282
#1153553