Finite Group Databasebeta

← #27300933

Subgroups — #27300933

Order 13568.

460 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 103 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
2#20275109C211yesmaximal, characteristic
3#20274932C212yesmaximal
4#20272841C212yesmaximal
5C2×C53×#356C211yesmaximal, characteristic
6#16811101C2211yescharacteristic
7#16810665—22no—
8#16810632—21no—
9C2×#13133136C2211yescharacteristic
10#16810530—21no—
11#16810530C2212yes—
12#16810522—22no—
13C4⋊C4×Dic53—22no—
14C4×C4⋊Dic53—21no—
15#16810384—22no—
16#16810383—21no—
17C2×C4⋊Q8×C53C2211yescharacteristic
18C2×#101×C53C2212yes—
19C53×#356C414yescomplemented
20C2×#11719917—21no—
21Q8×Dic53—41no—
22Q8×Dic53—41no—
23#13133176—21no—
24#13133176—22no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53D412yes—
29C2×C4×Dic53—22no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53D412yes—
32#13133151—42no—
33#13133149—42no—
34#13133141—41no—
35#13133141—41no—
36#13133136—22no—
37#13133124—22no—
38#13133124—22no—
39#13133124—22no—
40C22×Q8×C53D411yescharacteristic
41C22×Q8×C53C2311yescharacteristic
42C2×M16×C53—22no—
43C4⋊Q8×C53—22no—
44C4⋊Q8×C53C2×C412yes—
45C4⋊Q8×C53Q812yes—
46C2×C4⋊C4×C53—22no—
47C2×C42×C53D411yescharacteristic
48#101×C53—22no—
49#101×C53C2×C414yes—
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—24no—
66M16×C53—42no—
67C2×C8×C53—42no—
68C4⋊C4×C53—24no—
69C4⋊C4×C53—24no—
70C42×C53—21no—
71C42×C53C4⋊C412yes—
72C22⋊Dic53—41no—
73C4⋊Dic53—24no—
74C4⋊Dic53—21no—
75C4⋊Dic53—21no—
76C4⋊Dic53—22no—
77C4⋊Dic53—22no—
78C4⋊Dic53—42no—
79C4⋊Dic53—42no—
80C4⋊Dic53—81no—
81C4×Dic53—44no—
82C4×Dic53—41no—
83C4×Dic53—42no—
84C4×Dic53—42no—
85C4×Dic53—81no—
86#11719917—41no—
87#11719917—41no—
88C2×#88322—41no—
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×C53C2×C4⋊C411yescharacteristic
106C2×C4×C53C4×D412yes—
107C2×C4×C53C22⋊Q812yes—
108C2×C4×C53#12311yescharacteristic
109C2×Dic53—24no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C8×C53—44no—
118#88322—81no—
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—21no—
127C4×C53—22no—
128C4×C53—21no—
129C4×C53—22no—
130C4×C53—42no—
131Dic53—44no—
132Dic53—81no—
133Dic53—82no—
134Dic53—81no—
135#2369—531no—
136C2×#356D5311yescharacteristic
137#1886—531no—
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#519—531no—
147#428—1062no—
148#419—1061no—
149C2×#101—531no—
150C2×#101C2×D5312yes—
151#391—532no—
152#391—1061no—
153#389—1062no—
154C4×C4⋊C4—1062no—
155C4×C4⋊C4—1061no—
156#356Dic5314yescomplemented
157#339—1062no—
158#339—1061no—
159C53#1297311yescharacteristic, complemented, Sylow
160C22×Q8D21211yescharacteristic
161C22×Q8C22×D5311yescharacteristic
162C2×M16—22no—
163C2×M16—1061no—
164C4⋊Q8—22no—
165C4⋊Q8Dic10612yes—
166C4⋊Q8C2×Dic5312yes—
167C4×Q8—2121no—
168C4×Q8—2121no—
169#117—1061no—
170#117—1062no—
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×C42D21211yescharacteristic
180#107—2122no—
181#106—2122no—
182#103—2121no—
183#103—2121no—
184#101—22no—
185#101—1062no—
186#101C2×Dic5314yes—
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×Q8C4⋊Dic5312yes—
196C2×Q8C2×Dic10611yescharacteristic
197C2×Q8#1171994712yes—
198C2×Q8C22×Dic5311yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—1062no—
202C22×C4—1062no—
203C22×C4C2×D21211yescharacteristic
204C22×C4D4×D5312yes—
205M16—24no—
206M16—42no—
207M16—2121no—
208M16—2121no—
209C2×C8—42no—
210C2×C8—2121no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—1064no—
214C4⋊C4—1061no—
215C4⋊C4—1061no—
216C4⋊C4—1062no—
217C4⋊C4—1062no—
218C4⋊C4—4241no—
219C4⋊C4—2122no—
220C4⋊C4—2122no—
221C22⋊C4—2121no—
222C42—21no—
223C42—4241no—
224C42—2122no—
225C42—2124no—
226C42—2122no—
227C42—2121no—
228C42C4⋊Dic5312yes—
229C23#1313318711yescharacteristic
230Q8—41no—
231Q8—42no—
232Q8—41no—
233Q8—42no—
234Q8—81no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—1064no—
246C2×C4—2122no—
247C2×C4—2122no—
248C2×C4—2121no—
249C2×C4—2121no—
250C2×C4—2122no—
251C2×C4—2122no—
252C2×C4—2122no—
253C2×C4C2×C4⋊Dic5311yescharacteristic
254C2×C4D4×Dic5312yes—
255C2×C4#1313319412yes—
256C2×C4#1313320111yescharacteristic
257C8—44no—
258C8—4241no—
259C22—21no—
260C22—21no—
261C22#1681049811yescharacteristic
262C22#1681081111yescharacteristic
263C22#1681081411yescharacteristic
264C4—24no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—22no—
269C4—42no—
270C4—4242no—
271C4—4241no—
272C4—4241no—
273C4—2124no—
274C2—21no—
275C2—21no—
276C2#2027499811yescharacteristic
277C2#2027674311yescharacteristic
278C2#2027674611yescharacteristic
subgroupG/H
centerC22#16810811
commutatorC2×Q8×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC2×C53×#356C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
cosocleC22×Q8×C53C23
hypercenterC2×#356D53
hypocenterC53#12973
LCSG ▷ C2×Q8×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
19. C53×#356C4semidirect
156. #356Dic53semidirect
159. C53#12973semidirect

Maximal subgroups

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

subgroupindex
#202778052
#202751092
#202749322
#202728412
C2×C53×#3562
#1297353