Finite Group Databasebeta

← #27300926

Subgroups — #27300926

Order 13568.

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

normal subgroups only

This group has 492 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×#16810420C211yesmaximal, characteristic
2#20275119C211yesmaximal, characteristic
3#20274929C212yesmaximal, complemented
4#20272842C212yesmaximal, complemented
5C53×#2372C211yesmaximal, characteristic
6#16811101C2211yescharacteristic
7#16810663—22no—
8#16810628—21no—
9C2×#13133136C2211yescharacteristic
10C2×#13133136C2212yes—
11#16810420C414yescomplemented
12#16810383—21no—
13#16810383—22no—
14C2×#124×C53C2211yescharacteristic
15C53×#391C2212yes—
16C53×#388—22no—
17C53×#386—21no—
18C4×C22⋊C4×C53—22no—
19C53×#376—21no—
20C2×#11719917—22no—
21C2×#11719917—21no—
22Q8×Dic53—42no—
23#13133176—21no—
24#13133176—22no—
25#13133150—42no—
26#13133148—42no—
27#13133138—42no—
28#13133136—22no—
29#13133136—22no—
30#13133136C2×C414yescomplemented
31C22×Q8×C53C2311yescharacteristic
32C22×D4×C53D411yescharacteristic
33C23×C4×C53—21no—
34#124×C53—22no—
35#124×C53C2×C412yes—
36#124×C53Q812yes—
37C2×C4⋊C4×C53D412yes—
38C2×C22⋊C4×C53—22no—
39C2×C22⋊C4×C53—21no—
40C2×C22⋊C4×C53—22no—
41C2×C42×C53—22no—
42C2×C42×C53D411yescharacteristic
43C2×C42×C53D412yes—
44#95×C53—21no—
45#95×C53—21no—
46#95×C53—22no—
47#95×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—22no—
59C22×C4×C53—21no—
60C22×C4×C53—22no—
61C22×C4×C53—44no—
62C22×C4×C53—81no—
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D412yes—
65C4⋊C4×C53—42no—
66C22⋊C4×C53—24no—
67C22⋊C4×C53—24no—
68C22⋊C4×C53—44no—
69C22⋊C4×C53—81no—
70C42×C53—21no—
71C42×C53—44no—
72C42×C53—42no—
73C42×C53C4⋊C412yes—
74C22⋊Dic53—41no—
75C4⋊Dic53—22no—
76C4⋊Dic53—22no—
77C4⋊Dic53—22no—
78C4×Dic53—41no—
79C4×Dic53—42no—
80#11719917—24no—
81#11719917—42no—
82#11719917—41no—
83#11719917—41no—
84C2×#88322—42no—
85C2×#88322—41no—
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—24no—
102C2×C4×C53—22no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—84no—
110C2×C4×C53C2×C4⋊C411yescharacteristic
111C2×C4×C53C4×D412yes—
112C2×C4×C53C22⋊Q812yes—
113C2×C4×C53#12311yescharacteristic
114C2×Dic53—42no—
115#88322—44no—
116#88322—81no—
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—44no—
133C4×C53—82no—
134Dic53—82no—
135#2372D5311yescharacteristic
136C2×#355—531no—
137#1888—531no—
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#519—531no—
147#428—1062no—
148#418—1061no—
149C2×#101—532no—
150C2×#101—531no—
151#391C2×D5312yes—
152#388—22no—
153#386—21no—
154C4×C22⋊C4—22no—
155#376—21no—
156#355—534no—
157#339—1062no—
158#339—1061no—
159C53#1297211yescharacteristic, complemented, Sylow
160C22×Q8C22×D5311yescharacteristic
161C22×D4C22⋊D5311yescharacteristic
162C23×C4—21no—
163C2×M16—1062no—
164C2×M16—1061no—
165#124—22no—
166#124Dic10612yes—
167#124C4×D5312yes—
168C4×Q8—2122no—
169#117—1061no—
170#117—1062no—
171C2×C4⋊C4C22⋊D5312yes—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C42—22no—
176C2×C42C22⋊D5311yescharacteristic
177C2×C42C22⋊D5312yes—
178#107—2122no—
179#106—2122no—
180#102—2122no—
181#101—534no—
182#101—1062no—
183#101—1062no—
184#95—21no—
185#95—21no—
186#95—22no—
187#95—22no—
188C24—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C2×Dic10611yescharacteristic
192C2×Q8C2×C4×D5311yescharacteristic
193C2×Q8#1171994512yes—
194C2×D4—21no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4#1171991912yes—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—44no—
202C22×C4—81no—
203C22×C4C2×C22⋊D5311yescharacteristic
204C22×C4C2×C22⋊D5312yes—
205M16—1064no—
206M16—2121no—
207M16—2121no—
208M16—2122no—
209C2×C8—2121no—
210C2×C8—2122no—
211C4⋊C4—42no—
212C4⋊C4—1062no—
213C4⋊C4—1062no—
214C4⋊C4—1062no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—44no—
218C22⋊C4—81no—
219C22⋊C4—2121no—
220C42—21no—
221C42—42no—
222C42—44no—
223C42—2122no—
224C42—2121no—
225C42#1171991912yes—
226C23—22no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—41no—
231C23#1313318611yescharacteristic
232Q8—41no—
233Q8—41no—
234Q8—42no—
235D4—42no—
236D4—81no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—84no—
250C2×C4—2122no—
251C2×C4C2×#1171991911yescharacteristic
252C2×C4C4×C22⋊D5312yes—
253C2×C4#1313319612yes—
254C2×C4#1313320011yescharacteristic
255C8—4241no—
256C8—2124no—
257C22—22no—
258C22—44no—
259C22—42no—
260C22—81no—
261C22—81no—
262C22—82no—
263C22#1681049611yescharacteristic
264C22#1681081011yescharacteristic
265C22#1681081311yescharacteristic
266C4—21no—
267C4—22no—
268C4—21no—
269C4—22no—
270C4—44no—
271C4—42no—
272C4—82no—
273C4—4242no—
274C2—21no—
275C2—21no—
276C2—82no—
277C2#2027499611yescharacteristic
278C2#2027674112yes—
subgroupG/H
centerC22#16810810
commutatorC2×Q8×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#2372C2
socleC22×C53B(4,𝔽2)
prime radicalC22×D4×C53D4
cosocleC22×Q8×C53C23
hypercenter#2372D53
hypocenterC53#12972
LCSG ▷ C2×Q8×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2372

Complements

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

NH ≅ G/Ntype
3. #20274929C2semidirect
4. #20272842C2semidirect
11. #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
C2×#168104202
#202751192
#202749292
#202728422
C53×#23722
#1297253