Finite Group Databasebeta

← #27300931

Subgroups — #27300931

Order 13568.

460 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#16810422C211yesmaximal, characteristic
2#20275109C211yesmaximal, characteristic
3#20274929C212yesmaximal
4#20272842C212yesmaximal
5C53×#2369C211yesmaximal, characteristic
6#16811101C2211yescharacteristic
7#16810663—22no—
8#16810632—21no—
9C2×#13133136C2211yescharacteristic
10C2×#13133136C2212yes—
11#16810422C414yescomplemented
12#16810383—21no—
13#16810383—22no—
14C2×C4⋊Q8×C53C2211yescharacteristic
15C53×#391—21no—
16C53×#391C2212yes—
17C53×#389—22no—
18C4×C4⋊C4×C53—22no—
19C4×C4⋊C4×C53—21no—
20C2×#11719917—22no—
21C2×#11719917—21no—
22Q8×Dic53—41no—
23Q8×Dic53—41no—
24#13133176—21no—
25#13133176—22no—
26#13133150—42no—
27#13133148—42no—
28#13133141—41no—
29#13133141—41no—
30#13133136—22no—
31#13133136—22no—
32#13133136C2×C414yes—
33C22×Q8×C53D411yescharacteristic
34C22×Q8×C53C2311yescharacteristic
35C4⋊Q8×C53—22no—
36C4⋊Q8×C53C2×C412yes—
37C4⋊Q8×C53Q812yes—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53D412yes—
43C2×C42×C53—22no—
44C2×C42×C53—21no—
45C2×C42×C53D411yescharacteristic
46C2×C42×C53D412yes—
47#95×C53—22no—
48#95×C53—22no—
49#95×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—22no—
61C22×C4×C53—21no—
62C22×C4×C53—22no—
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D412yes—
65C4⋊C4×C53—24no—
66C4⋊C4×C53—24no—
67C4⋊C4×C53—24no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—81no—
71C42×C53—21no—
72C42×C53—44no—
73C42×C53—42no—
74C42×C53—81no—
75C42×C53C4⋊C412yes—
76C22⋊Dic53—41no—
77C4⋊Dic53—21no—
78C4⋊Dic53—21no—
79C4⋊Dic53—22no—
80C4⋊Dic53—22no—
81C4×Dic53—41no—
82C4×Dic53—42no—
83#11719917—24no—
84#11719917—42no—
85#11719917—41no—
86#11719917—41no—
87C2×#88322—42no—
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—24no—
103C2×C4×C53—22no—
104C2×C4×C53—42no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53C2×C4⋊C411yescharacteristic
112C2×C4×C53C4×D412yes—
113C2×C4×C53C22⋊Q812yes—
114C2×C4×C53#12311yescharacteristic
115C2×Dic53—41no—
116C2×Dic53—41no—
117#88322—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—22no—
127C4×C53—21no—
128C4×C53—21no—
129C4×C53—22no—
130C4×C53—44no—
131C4×C53—42no—
132C4×C53—82no—
133Dic53—81no—
134Dic53—81no—
135#2369D5311yescharacteristic
136C2×#356—531no—
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—532no—
150C2×#101—531no—
151#391—21no—
152#391C2×D5312yes—
153#389—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—21no—
156#356—534no—
157#339—1062no—
158#339—1061no—
159C53#1297311yescharacteristic, complemented, Sylow
160C22×Q8C22⋊D5311yescharacteristic
161C22×Q8C22×D5311yescharacteristic
162C2×M16—1062no—
163C2×M16—1061no—
164C4⋊Q8—22no—
165C4⋊Q8Dic10612yes—
166C4⋊Q8C4×D5312yes—
167C4×Q8—2121no—
168C4×Q8—2121no—
169#117—1061no—
170#117—1062no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4C22⋊D5312yes—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42C22⋊D5311yescharacteristic
179C2×C42C22⋊D5312yes—
180#107—2122no—
181#106—2122no—
182#103—2121no—
183#103—2121no—
184#101—534no—
185#101—1062no—
186#101—1062no—
187#95—22no—
188#95—22no—
189#95—22no—
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#1171994512yes—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4C2×C22⋊D5311yescharacteristic
204C22×C4C2×C22⋊D5312yes—
205M16—1064no—
206M16—2121no—
207M16—2121no—
208M16—2122no—
209C2×C8—2121no—
210C2×C8—2122no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—81no—
217C4⋊C4—1061no—
218C4⋊C4—1061no—
219C4⋊C4—1062no—
220C4⋊C4—1062no—
221C22⋊C4—2121no—
222C42—21no—
223C42—42no—
224C42—44no—
225C42—81no—
226C42—2122no—
227C42—2121no—
228C42#1171991912yes—
229C23#1313318611yescharacteristic
230Q8—41no—
231Q8—41no—
232Q8—42no—
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—24no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—2121no—
252C2×C4—2121no—
253C2×C4C2×#1171991911yescharacteristic
254C2×C4C4×C22⋊D5312yes—
255C2×C4#1313319612yes—
256C2×C4#1313320011yescharacteristic
257C8—4241no—
258C8—2124no—
259C22—21no—
260C22—21no—
261C22#1681049611yescharacteristic
262C22#1681081011yescharacteristic
263C22#1681081311yescharacteristic
264C4—24no—
265C4—21no—
266C4—22no—
267C4—21no—
268C4—22no—
269C4—44no—
270C4—42no—
271C4—82no—
272C4—4241no—
273C4—4241no—
274C2—21no—
275C2—21no—
276C2#2027499611yescharacteristic
277C2#2027674111yescharacteristic
278C2#2027674511yescharacteristic
subgroupG/H
centerC22#16810810
commutatorC2×Q8×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#2369C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
cosocleC22×Q8×C53C23
hypercenter#2369D53
hypocenterC53#12973
LCSG ▷ C2×Q8×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2369

Complements

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

NH ≅ G/Ntype
11. #16810422C4semidirect
159. C53#12973semidirect

Maximal subgroups

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

subgroupindex
C2×#168104222
#202751092
#202749292
#202728422
C53×#23692
#1297353