Finite Group Databasebeta

← #27300925

Subgroups — #27300925

Order 13568.

492 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 95 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#20277827C211yesmaximal, characteristic
2C2×#16810420C211yesmaximal, characteristic
3#20274930C212yesmaximal, complemented
4#20272841C212yesmaximal, complemented
5C53×#1888C211yesmaximal, characteristic
6#16810666—22no—
7C2×#13133136C2212yes—
8#16810530C2212yes—
9#16810516—22no—
10#16810503—21no—
11C22⋊C4×Dic53—22no—
12#16810469—21no—
13#16810420C414yescomplemented
14#16810382—22no—
15C2×#124×C53C2211yescharacteristic
16C53×#519C2211yescharacteristic
17C53×#418—21no—
18C2×#101×C53C2211yescharacteristic
19C53×#339—21no—
20C23×Dic53—21no—
21C2×#11719917—22no—
22C2×C4⋊Dic53D412yes—
23C2×C22⋊Dic53—22no—
24C2×C22⋊Dic53—21no—
25C2×C4×Dic53—22no—
26C2×C4×Dic53D412yes—
27#13133150—42no—
28#13133148—42no—
29#13133136—22no—
30#13133136C2×C414yescomplemented
31#13133124—21no—
32#13133124—21no—
33#13133124—22no—
34#13133124—22no—
35C22×Q8×C53C2311yescharacteristic
36C22×D4×C53D411yescharacteristic
37C2×M16×C53—21no—
38#124×C53—22no—
39#124×C53C2×C412yes—
40#124×C53Q812yes—
41C4×Q8×C53—42no—
42#117×C53—21no—
43#117×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C42×C53D411yescharacteristic
46#102×C53—42no—
47#101×C53—22no—
48C24×C53—21no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53C22×C411yescharacteristic
52C2×Q8×C53C2×Q811yescharacteristic
53C2×Q8×C53#4112yes—
54C2×D4×C53—21no—
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53C4⋊C412yes—
58C22×C4×C53—21no—
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D412yes—
61C22×Dic53—22no—
62C22×Dic53—22no—
63C22×Dic53—44no—
64C22×Dic53—81no—
65M16×C53—41no—
66M16×C53—41no—
67C2×C8×C53—41no—
68C4⋊C4×C53—22no—
69C4⋊C4×C53—22no—
70C4⋊C4×C53—22no—
71C22⋊C4×C53—24no—
72C22⋊C4×C53—24no—
73C22⋊C4×C53—41no—
74C42×C53—21no—
75C42×C53—41no—
76C42×C53—42no—
77C42×C53C4⋊C412yes—
78C22⋊Dic53—44no—
79C22⋊Dic53—81no—
80C4⋊Dic53—42no—
81C4×Dic53—44no—
82C4×Dic53—42no—
83#11719917—24no—
84#11719917—42no—
85C2×#88322—42no—
86C23×C53—22no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53—41no—
90C23×C53—41no—
91C23×C53#12011yescharacteristic
92Q8×C53—41no—
93Q8×C53—41no—
94Q8×C53—42no—
95D4×C53—42no—
96D4×C53—81no—
97C2×C4×C53—21no—
98C2×C4×C53—22no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53C2×C4⋊C411yescharacteristic
105C2×C4×C53C4×D412yes—
106C2×C4×C53C22⋊Q812yes—
107C2×C4×C53#12311yescharacteristic
108C2×Dic53—24no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—84no—
115C8×C53—81no—
116#88322—44no—
117#12972—531nomaximal, Sylow
118C22×C53—22no—
119C22×C53—44no—
120C22×C53—42no—
121C22×C53—81no—
122C22×C53—81no—
123C22×C53—82no—
124C22×C53#38511yescharacteristic
125C22×C53B(4,𝔽2)11yescharacteristic
126C22×C53#45811yescharacteristic
127C4×C53—22no—
128C4×C53—21no—
129C4×C53—21no—
130C4×C53—22no—
131C4×C53—42no—
132C4×C53—82no—
133Dic53—44no—
134Dic53—82no—
135#2372—531no—
136C2×#355—531no—
137#1888D5311yescharacteristic
138#1857—532no—
139#1394—532no—
140C2×C53—21no—
141C2×C53—21no—
142C2×C53—82no—
143C2×C53#186711yescharacteristic
144C2×C53#220412yes—
145C2×#124C2×D5311yescharacteristic
146#519C2×D5311yescharacteristic
147#428—1062no—
148#418—21no—
149C2×#101—532no—
150C2×#101C2×D5311yescharacteristic
151#391—532no—
152#388—1062no—
153#386—1061no—
154C4×C22⋊C4—1062no—
155#376—1061no—
156#355—534no—
157#339—21no—
158#339—1062no—
159C53#1297211yescharacteristic, complemented, Sylow
160C22×Q8C22×D5311yescharacteristic
161C22×D4C22⋊D5311yescharacteristic
162C23×C4—1061no—
163C2×M16—21no—
164C2×M16—1062no—
165#124—22no—
166#124Dic10612yes—
167#124C4×D5312yes—
168C4×Q8—42no—
169#117—22no—
170#117—21no—
171C2×C4⋊C4—532no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—1061no—
174C2×C22⋊C4—1062no—
175C2×C42—532no—
176C2×C42—1062no—
177C2×C42C22⋊D5311yescharacteristic
178#107—2122no—
179#106—2122no—
180#102—42no—
181#101—22no—
182#101—534no—
183#101—1062no—
184#95—1061no—
185#95—1061no—
186#95—1062no—
187#95—1062no—
188C24—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C2×Dic10611yescharacteristic
192C2×Q8C2×C4×D5311yescharacteristic
193C2×Q8#1171994712yes—
194C2×D4—21no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4#1171991912yes—
198C22×C4—21no—
199C22×C4—1062no—
200C22×C4—1062no—
201C22×C4—4241no—
202C22×C4—2124no—
203C22×C4D4×D5312yes—
204C22×C4C2×C22⋊D5311yescharacteristic
205M16—41no—
206M16—41no—
207M16—1064no—
208M16—2122no—
209C2×C8—41no—
210C2×C8—2122no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—2122no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—41no—
218C22⋊C4—4241no—
219C22⋊C4—2124no—
220C42—21no—
221C42—42no—
222C42—41no—
223C42—2122no—
224C42—2124no—
225C42#1171991912yes—
226C23—22no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—41no—
231C23#1313318511yescharacteristic
232Q8—41no—
233Q8—41no—
234Q8—42no—
235D4—42no—
236D4—81no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—1064no—
245C2×C4—4244no—
246C2×C4—2122no—
247C2×C4—2122no—
248C2×C4—2122no—
249C2×C4—2122no—
250C2×C4—2122no—
251C2×C4C2×#1171991911yescharacteristic
252C2×C4#1313317712yes—
253C2×C4#1313319412yes—
254C2×C4#1313319811yescharacteristic
255C8—81no—
256C8—2124no—
257C22—22no—
258C22—44no—
259C22—42no—
260C22—81no—
261C22—81no—
262C22—82no—
263C22#1681049511yescharacteristic
264C22#1681080911yescharacteristic
265C22#1681081211yescharacteristic
266C4—21no—
267C4—22no—
268C4—21no—
269C4—22no—
270C4—42no—
271C4—82no—
272C4—4242no—
273C4—2124no—
274C2—21no—
275C2—21no—
276C2—82no—
277C2#2027499511yescharacteristic
278C2#2027673912yes—
subgroupG/H
centerC22#16810809
commutatorC2×Q8×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#1888C2
socleC22×C53B(4,𝔽2)
prime radicalC22×D4×C53D4
cosocleC22×Q8×C53C23
hypercenter#1888D53
hypocenterC53#12972
LCSG ▷ C2×Q8×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1888

Complements

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

NH ≅ G/Ntype
3. #20274930C2semidirect
4. #20272841C2semidirect
13. #16810420C4semidirect
30. #13133136C2×C4semidirect
159. C53#12972semidirect

Maximal subgroups

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

subgroupindex
#202778272
C2×#168104202
#202749302
#202728412
C53×#18882
#1297253