Finite Group Databasebeta

← #27278470

Subgroups — #27278470

Order 13568.

468 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 111 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#20274850C212yesmaximal, complemented
3C2×#16810363C211yesmaximal, characteristic
4#20272597C212yesmaximal, complemented
5C53×#1959C211yesmaximal, characteristic
6C2×#13133128C2212yes—
7#16810530—21no—
8#16810516—22no—
9#16810503—21no—
10#16810498—22no—
11#16810489C2212yes—
12#16810385—22no—
13#16810385—22no—
14#16810363C414yescomplemented
15C2×#124×C53C2211yescharacteristic
16C2×#103×C53—21no—
17C2×#102×C53—21no—
18C2×#97×C53C2211yescharacteristic
19C53×#384C2211yescharacteristic
20C23×Dic53—21no—
21C22×#88322—22no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53D412yes—
24C2×C4⋊Dic53D412yes—
25C2×C22⋊Dic53—22no—
26C2×C22⋊Dic53—21no—
27C2×C4×Dic53—21no—
28#13133128—22no—
29#13133128C2×C414yescomplemented
30#13133124—22no—
31#13133124—22no—
32#13133124—22no—
33C22×Q8×C53D411yescharacteristic
34C22×D4×C53D411yescharacteristic
35C22×C8×C53—21no—
36#124×C53—22no—
37#124×C53C2×C412yes—
38#124×C53Q812yes—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53—21no—
41C2×C22⋊C4×C53—22no—
42C2×C42×C53C2311yescharacteristic
43#103×C53—42no—
44#102×C53—42no—
45#97×C53—22no—
46C24×C53—21no—
47C2×Q8×C53—21no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53C4⋊C412yes—
51C2×D4×C53—21no—
52C2×D4×C53—41no—
53C2×D4×C53—41no—
54C2×D4×C53C4⋊C412yes—
55C22×C4×C53—21no—
56C22×C4×C53—41no—
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53C2×D412yes—
59C22×Dic53—22no—
60C22×Dic53—22no—
61C22×Dic53—81no—
62C22×Dic53—82no—
63C2×C8×C53—22no—
64C2×C8×C53—41no—
65C2×C8×C53—41no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—44no—
69C22⋊C4×C53—24no—
70C22⋊C4×C53—24no—
71C42×C53C22×C411yescharacteristic
72C42×C53C2×Q811yescharacteristic
73C42×C53#4112yes—
74C22⋊Dic53—82no—
75C22⋊Dic53—81no—
76C4⋊Dic53—42no—
77C4⋊Dic53—42no—
78C4⋊Dic53—81no—
79C4×Dic53—81no—
80C2×#88322—24no—
81C2×#88322—42no—
82C2×#88322—42no—
83C23×C53—22no—
84C23×C53—41no—
85C23×C53—42no—
86C23×C53—42no—
87C23×C53—41no—
88C23×C53#12011yescharacteristic
89Q8×C53—42no—
90Q8×C53—81no—
91D4×C53—42no—
92D4×C53—81no—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—81no—
101C2×C4×C53—82no—
102C2×C4×C53C2×C4⋊C411yescharacteristic
103C2×C4×C53C4×D412yes—
104C2×C4×C53C22⋊Q812yes—
105C2×C4×C53#12311yescharacteristic
106C2×Dic53—42no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—84no—
113C8×C53—42no—
114#88322—44no—
115#8914—531nomaximal, Sylow
116C22×C53—42no—
117C22×C53—44no—
118C22×C53—81no—
119C22×C53—81no—
120C22×C53—82no—
121C22×C53#38511yescharacteristic
122C22×C53#45311yescharacteristic
123C22×C53#45412yes—
124C22×C53#45512yes—
125C22×C53#45511yescharacteristic
126C4×C53—21no—
127C4×C53—21no—
128C4×C53—22no—
129C4×C53—22no—
130C4×C53—42no—
131C4×C53—161no—
132Dic53—82no—
133Dic53—82no—
134#2379—531no—
135#1959D5311yescharacteristic
136#1836—532no—
137C2×#329—531no—
138#1295—532no—
139C2×C53—82no—
140C2×C53#165711yescharacteristic
141C2×C53#165911yescharacteristic
142C2×C53#166212yes—
143C2×C53#185611yescharacteristic
144C2×C53#186012yes—
145C2×#124C2×D5311yescharacteristic
146C2×#103—21no—
147C2×#102—21no—
148C2×#97—532no—
149C2×#97C2×D5311yescharacteristic
150#391—1061no—
151#388—1062no—
152#386—1061no—
153#385—1062no—
154#384C2×D5311yescharacteristic
155#382—532no—
156#340—1062no—
157#340—1062no—
158#329—534no—
159C53#891411yescharacteristic, complemented, Sylow
160C22×Q8C22⋊D5311yescharacteristic
161C22×D4C22⋊D5311yescharacteristic
162C23×C4—1061no—
163C22×C8—21no—
164C22×C8—1062no—
165#124—22no—
166#124Dic10612yes—
167#124C4×D5312yes—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4—532no—
172C2×C4⋊C4—1061no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—1062no—
175C2×C22⋊C4—1061no—
176C2×C42—1061no—
177C2×C42C22×D5311yescharacteristic
178#103—42no—
179#102—42no—
180#97—22no—
181#97—534no—
182#97—1062no—
183#95—1062no—
184#95—1062no—
185#95—1062no—
186C24—21no—
187C2×Q8—21no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8#1171991912yes—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4#1171991912yes—
195C22×C4—21no—
196C22×C4—41no—
197C22×C4—1062no—
198C22×C4—1062no—
199C22×C4—4241no—
200C22×C4—4242no—
201C22×C4D4×D5312yes—
202C22×C4C2×C22⋊D5311yescharacteristic
203C2×C8—22no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—1064no—
207C2×C8—2122no—
208C2×C8—2122no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—44no—
212C4⋊C4—4241no—
213C4⋊C4—2122no—
214C4⋊C4—2122no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—4242no—
218C22⋊C4—4241no—
219C42—4241no—
220C42C2×Dic10611yescharacteristic
221C42C2×C4×D5311yescharacteristic
222C42#1171994712yes—
223C23—22no—
224C23—41no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23#1313318511yescharacteristic
229Q8—42no—
230Q8—81no—
231D4—42no—
232D4—81no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—81no—
241C2×C4—82no—
242C2×C4—4244no—
243C2×C4—2122no—
244C2×C4—2122no—
245C2×C4—2122no—
246C2×C4—2122no—
247C2×C4—2122no—
248C2×C4—2122no—
249C2×C4C2×#1171991911yescharacteristic
250C2×C4#1313317712yes—
251C2×C4#1313319412yes—
252C2×C4#1313319811yescharacteristic
253C8—42no—
254C8—2124no—
255C22—44no—
256C22—42no—
257C22—81no—
258C22—81no—
259C22—82no—
260C22#1681049511yescharacteristic
261C22#1681079011yescharacteristic
262C22#1681079412yes—
263C22#1681079912yes—
264C22#1681080011yescharacteristic
265C4—22no—
266C4—22no—
267C4—21no—
268C4—21no—
269C4—42no—
270C4—161no—
271C4—4242no—
272C4—4242no—
273C2—82no—
274C2#2027401111yescharacteristic
275C2#2027401911yescharacteristic
276C2#2027403712yes—
277C2#2027492411yescharacteristic
278C2#2027494612yes—
subgroupG/H
centerC23#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1959C2
socleC23×C53#120
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#1959D53
hypocenterC53#8914
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1959

Complements

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

NH ≅ G/Ntype
2. #20274850C2semidirect
4. #20272597C2semidirect
14. #16810363C4semidirect
29. #13133128C2×C4semidirect
159. C53#8914semidirect

Maximal subgroups

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

subgroupindex
#202778892
#202748502
C2×#168103632
#202725972
C53×#19592
#891453