Finite Group Databasebeta

← #27278449

Subgroups — #27278449

Order 13568.

500 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 119 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#20277889C211yesmaximal, characteristic
2#20275532C211yesmaximal, characteristic
3#20274883C212yesmaximal, complemented
4#20272597C212yesmaximal, complemented
5C2×C53×#329C211yesmaximal, characteristic
6C2×#13133151—22no—
7C2×#13133149—22no—
8C2×#13133141—21no—
9C2×#13133138—21no—
10C2×#13133128C2211yescharacteristic
11#16810530—21no—
12#16810516—22no—
13#16810503—21no—
14#16810498—22no—
15#16810493C2211yescharacteristic
16#16810489C2212yes—
17C2×#124×C53C2211yescharacteristic
18C2×#97×C53C2212yes—
19C53×#329C414yescomplemented
20C23×Dic53—21no—
21C22×#88322—21no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53D412yes—
26C2×C4⋊Dic53D412yes—
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53—21no—
29C2×C4×Dic53—21no—
30#13133151—44no—
31#13133149—44no—
32#13133141—42no—
33#13133138—42no—
34#13133128—22no—
35#13133124—22no—
36#13133124—22no—
37#13133124—22no—
38C22×Q8×C53D411yescharacteristic
39C22×D4×C53D411yescharacteristic
40C22×C8×C53—22no—
41#124×C53—22no—
42#124×C53C2×C412yes—
43#124×C53Q812yes—
44C2×C22⋊C4×C53—22no—
45C2×C42×C53C2311yescharacteristic
46#97×C53—22no—
47#97×C53C2×C414yescomplemented
48C24×C53—21no—
49C2×Q8×C53—21no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C2×Q8×C53C4⋊C412yes—
53C2×D4×C53—21no—
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C2×D4×C53C4⋊C412yes—
57C22×C4×C53—21no—
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53C2×D412yes—
60C22×Dic53—22no—
61C22×Dic53—22no—
62C22×Dic53—41no—
63C22×Dic53—82no—
64C22×Dic53—81no—
65C2×C8×C53—24no—
66C2×C8×C53—24no—
67C2×C8×C53—24no—
68C22⋊C4×C53—24no—
69C22⋊C4×C53—24no—
70C42×C53C22×C411yescharacteristic
71C42×C53C2×Q811yescharacteristic
72C42×C53#4112yes—
73C22⋊Dic53—82no—
74C22⋊Dic53—81no—
75C4⋊Dic53—42no—
76C4⋊Dic53—42no—
77C4⋊Dic53—41no—
78C4⋊Dic53—41no—
79C4⋊Dic53—42no—
80C4⋊Dic53—42no—
81C4⋊Dic53—81no—
82C4×Dic53—81no—
83C2×#88322—22no—
84C2×#88322—41no—
85C2×#88322—41no—
86C23×C53—22no—
87C23×C53—41no—
88C23×C53—42no—
89C23×C53—41no—
90C23×C53—42no—
91C23×C53#12011yescharacteristic
92Q8×C53—42no—
93Q8×C53—81no—
94D4×C53—42no—
95D4×C53—81no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53C2×C4⋊C411yescharacteristic
104C2×C4×C53C4×D412yes—
105C2×C4×C53C22⋊Q812yes—
106C2×C4×C53#12311yescharacteristic
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—84no—
114C2×Dic53—82no—
115C2×Dic53—81no—
116C8×C53—28no—
117#88322—42no—
118#8910—531nomaximal, Sylow
119C22×C53—42no—
120C22×C53—44no—
121C22×C53—81no—
122C22×C53—81no—
123C22×C53—82no—
124C22×C53#38511yescharacteristic
125C22×C53#45312yes—
126C22×C53#45311yescharacteristic
127C22×C53#45511yescharacteristic
128C22×C53#45612yes—
129C4×C53—21no—
130C4×C53—21no—
131C4×C53—22no—
132C4×C53—22no—
133C4×C53—42no—
134Dic53—82no—
135Dic53—82no—
136Dic53—161no—
137#2379—531no—
138#1959—531no—
139#1847—532no—
140C2×#329D5311yescharacteristic
141#1295—532no—
142C2×C53—82no—
143C2×C53#165511yescharacteristic
144C2×C53#166011yescharacteristic
145C2×C53#166112yes—
146C2×C53#185611yescharacteristic
147C2×C53#185912yes—
148C2×#124C2×D5311yescharacteristic
149C2×#107—1062no—
150C2×#106—1062no—
151C2×#103—1061no—
152C2×#102—1061no—
153C2×#97—531no—
154C2×#97C2×D5312yes—
155#391—1061no—
156#388—1062no—
157#386—1061no—
158#385—1062no—
159#384—531no—
160#382—532no—
161#329Dic5314yescomplemented
162C53#891011yescharacteristic, complemented, Sylow
163C22×Q8D21211yescharacteristic
164C22×D4D21211yescharacteristic
165C23×C4—1061no—
166C22×C8—22no—
167C22×C8—1061no—
168#124—22no—
169#124Dic10612yes—
170#124C2×Dic5312yes—
171C2×C4⋊C4—532no—
172C2×C4⋊C4—532no—
173C2×C4⋊C4—1061no—
174C2×C4⋊C4—1061no—
175C2×C4⋊C4—1062no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—1062no—
178C2×C22⋊C4—1061no—
179C2×C42—1061no—
180C2×C42C22×D5311yescharacteristic
181#107—2124no—
182#106—2124no—
183#103—2122no—
184#102—2122no—
185#97—22no—
186#97—1062no—
187#97C2×Dic5314yescomplemented
188#95—1062no—
189#95—1062no—
190#95—1062no—
191C24—21no—
192C2×Q8—21no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8C4⋊Dic5312yes—
196C2×D4—21no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4C4⋊Dic5312yes—
200C22×C4—21no—
201C22×C4—1062no—
202C22×C4—1062no—
203C22×C4—4242no—
204C22×C4—4241no—
205C22×C4—2121no—
206C22×C4C2×D21211yescharacteristic
207C22×C4D4×D5312yes—
208C2×C8—24no—
209C2×C8—24no—
210C2×C8—24no—
211C2×C8—1062no—
212C2×C8—2121no—
213C2×C8—2121no—
214C4⋊C4—4241no—
215C4⋊C4—2121no—
216C4⋊C4—2121no—
217C4⋊C4—2122no—
218C4⋊C4—2122no—
219C4⋊C4—2122no—
220C4⋊C4—2122no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—4242no—
224C22⋊C4—4241no—
225C42—4241no—
226C42C2×Dic10611yescharacteristic
227C42#1171994712yes—
228C42C22×Dic5311yescharacteristic
229C23—22no—
230C23—42no—
231C23—41no—
232C23—41no—
233C23—42no—
234C23#1313318711yescharacteristic
235Q8—42no—
236Q8—81no—
237D4—42no—
238D4—81no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—4242no—
247C2×C4—4241no—
248C2×C4—4244no—
249C2×C4—2122no—
250C2×C4—2122no—
251C2×C4—2122no—
252C2×C4—2122no—
253C2×C4—2122no—
254C2×C4—2122no—
255C2×C4C2×C4⋊Dic5311yescharacteristic
256C2×C4D4×Dic5312yes—
257C2×C4#1313319412yes—
258C2×C4#1313320111yescharacteristic
259C8—28no—
260C8—2122no—
261C22—42no—
262C22—44no—
263C22—81no—
264C22—81no—
265C22—82no—
266C22#1681049811yescharacteristic
267C22#1681079212yes—
268C22#1681079311yescharacteristic
269C22#1681080311yescharacteristic
270C22#1681080712yes—
271C4—21no—
272C4—21no—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—8481no—
277C4—4242no—
278C4—4242no—
279C2—82no—
280C2#2027400511yescharacteristic
281C2#2027402711yescharacteristic
282C2#2027403512yes—
283C2#2027492711yescharacteristic
284C2#2027494312yes—
subgroupG/H
centerC23#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#329C2
socleC23×C53#120
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenterC2×#329D53
hypocenterC53#8910
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
3. #20274883C2semidirect
4. #20272597C2semidirect
19. C53×#329C4semidirect
47. #97×C53C2×C4semidirect
161. #329Dic53semidirect
162. C53#8910semidirect
187. #97C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202778892
#202755322
#202748832
#202725972
C2×C53×#3292
#891053