Finite Group Databasebeta

← #27300930

Subgroups — #27300930

Order 13568.

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

normal subgroups only

This group has 460 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#20277805C211yesmaximal, characteristic
2C2×#16810422C211yesmaximal, characteristic
3#20274930C212yesmaximal
4#20272841C212yesmaximal
5C53×#1886C211yesmaximal, characteristic
6#16810666—22no—
7C2×#13133136C2212yes—
8#16810530—21no—
9#16810530C2212yes—
10#16810522—22no—
11C4⋊C4×Dic53—22no—
12C4×C4⋊Dic53—21no—
13#16810422C414yescomplemented
14#16810382—22no—
15C2×C4⋊Q8×C53C2211yescharacteristic
16C53×#519C2211yescharacteristic
17C53×#419—21no—
18C2×#101×C53C2211yescharacteristic
19C53×#339—21no—
20C2×#11719917—22no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53D412yes—
25C2×C4×Dic53—22no—
26C2×C4×Dic53—21no—
27C2×C4×Dic53D412yes—
28#13133150—42no—
29#13133148—42no—
30#13133136—22no—
31#13133136C2×C414yes—
32#13133124—22no—
33#13133124—22no—
34#13133124—22no—
35C22×Q8×C53D411yescharacteristic
36C22×Q8×C53C2311yescharacteristic
37C2×M16×C53—21no—
38C4⋊Q8×C53—22no—
39C4⋊Q8×C53C2×C412yes—
40C4⋊Q8×C53Q812yes—
41C4×Q8×C53—41no—
42C4×Q8×C53—41no—
43#117×C53—21no—
44#117×C53—22no—
45C2×C4⋊C4×C53—22no—
46C2×C42×C53D411yescharacteristic
47#103×C53—41no—
48#103×C53—41no—
49#101×C53—22no—
50C2×Q8×C53—21no—
51C2×Q8×C53—41no—
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53—41no—
55C2×Q8×C53C4⋊C412yes—
56C2×Q8×C53C22×C411yescharacteristic
57C2×Q8×C53C2×Q811yescharacteristic
58C2×Q8×C53#4112yes—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D412yes—
63C22×Dic53—22no—
64C22×Dic53—22no—
65M16×C53—41no—
66M16×C53—41no—
67C2×C8×C53—41no—
68C4⋊C4×C53—24no—
69C4⋊C4×C53—24no—
70C4⋊C4×C53—21no—
71C4⋊C4×C53—21no—
72C4⋊C4×C53—22no—
73C4⋊C4×C53—22no—
74C22⋊C4×C53—41no—
75C42×C53—21no—
76C42×C53—41no—
77C42×C53—42no—
78C42×C53C4⋊C412yes—
79C4⋊Dic53—24no—
80C4⋊Dic53—42no—
81C4⋊Dic53—42no—
82C4⋊Dic53—81no—
83C4×Dic53—44no—
84C4×Dic53—42no—
85C4×Dic53—81no—
86#11719917—24no—
87#11719917—42no—
88C2×#88322—42no—
89C23×C53#12011yescharacteristic
90Q8×C53—41no—
91Q8×C53—41no—
92Q8×C53—42no—
93Q8×C53—42no—
94Q8×C53—81no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53C2×C4⋊C411yescharacteristic
108C2×C4×C53C4×D412yes—
109C2×C4×C53C22⋊Q812yes—
110C2×C4×C53#12311yescharacteristic
111C2×Dic53—24no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C8×C53—81no—
118#88322—44no—
119#12973—531nomaximal, Sylow
120C22×C53—21no—
121C22×C53—21no—
122C22×C53#38511yescharacteristic
123C22×C53B(4,𝔽2)11yescharacteristic
124C22×C53#45811yescharacteristic
125C4×C53—24no—
126C4×C53—22no—
127C4×C53—21no—
128C4×C53—21no—
129C4×C53—22no—
130C4×C53—42no—
131C4×C53—81no—
132C4×C53—81no—
133Dic53—44no—
134Dic53—82no—
135#2369—531no—
136C2×#356—531no—
137#1886D5311yescharacteristic
138#1857—532no—
139#1394—532no—
140C2×C53—21no—
141C2×C53—21no—
142C2×C53#186711yescharacteristic
143C2×C53#220411yescharacteristic
144C2×C53#220511yescharacteristic
145C2×C4⋊Q8C2×D5311yescharacteristic
146#519C2×D5311yescharacteristic
147#428—1062no—
148#419—21no—
149C2×#101—532no—
150C2×#101C2×D5311yescharacteristic
151#391—532no—
152#391—1061no—
153#389—1062no—
154C4×C4⋊C4—1062no—
155C4×C4⋊C4—1061no—
156#356—534no—
157#339—21no—
158#339—1062no—
159C53#1297311yescharacteristic, complemented, Sylow
160C22×Q8C22⋊D5311yescharacteristic
161C22×Q8C22×D5311yescharacteristic
162C2×M16—21no—
163C2×M16—1062no—
164C4⋊Q8—22no—
165C4⋊Q8Dic10612yes—
166C4⋊Q8C4×D5312yes—
167C4×Q8—41no—
168C4×Q8—41no—
169#117—22no—
170#117—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—532no—
173C2×C4⋊C4—1061no—
174C2×C4⋊C4—1061no—
175C2×C4⋊C4—1061no—
176C2×C42—532no—
177C2×C42—1061no—
178C2×C42—1062no—
179C2×C42C22⋊D5311yescharacteristic
180#107—2122no—
181#106—2122no—
182#103—41no—
183#103—41no—
184#101—22no—
185#101—534no—
186#101—1062no—
187#95—1062no—
188#95—1062no—
189#95—1062no—
190C2×Q8—21no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8#1171991912yes—
196C2×Q8C2×Dic10611yescharacteristic
197C2×Q8C2×C4×D5311yescharacteristic
198C2×Q8#1171994712yes—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—1062no—
202C22×C4—1062no—
203C22×C4D4×D5312yes—
204C22×C4C2×C22⋊D5311yescharacteristic
205M16—41no—
206M16—41no—
207M16—1064no—
208M16—2122no—
209C2×C8—41no—
210C2×C8—2122no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—1064no—
218C4⋊C4—4241no—
219C4⋊C4—2122no—
220C4⋊C4—2122no—
221C22⋊C4—41no—
222C42—21no—
223C42—42no—
224C42—41no—
225C42—4241no—
226C42—2122no—
227C42—2124no—
228C42#1171991912yes—
229C23#1313318511yescharacteristic
230Q8—41no—
231Q8—41no—
232Q8—42no—
233Q8—42no—
234Q8—81no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—1064no—
248C2×C4—2122no—
249C2×C4—2122no—
250C2×C4—2122no—
251C2×C4—2122no—
252C2×C4—2122no—
253C2×C4C2×#1171991911yescharacteristic
254C2×C4#1313317712yes—
255C2×C4#1313319412yes—
256C2×C4#1313319811yescharacteristic
257C8—81no—
258C8—2124no—
259C22—21no—
260C22—21no—
261C22#1681049511yescharacteristic
262C22#1681080911yescharacteristic
263C22#1681081211yescharacteristic
264C4—24no—
265C4—21no—
266C4—22no—
267C4—21no—
268C4—22no—
269C4—42no—
270C4—81no—
271C4—81no—
272C4—4242no—
273C4—2124no—
274C2—21no—
275C2—21no—
276C2#2027499511yescharacteristic
277C2#2027674011yescharacteristic
278C2#2027674411yescharacteristic
subgroupG/H
centerC22#16810809
commutatorC2×Q8×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#1886C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
cosocleC22×Q8×C53C23
hypercenter#1886D53
hypocenterC53#12973
LCSG ▷ C2×Q8×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1886

Complements

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

NH ≅ G/Ntype
13. #16810422C4semidirect
159. C53#12973semidirect

Maximal subgroups

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

subgroupindex
#202778052
C2×#168104222
#202749302
#202728412
C53×#18862
#1297353