Finite Group Databasebeta

← #27279853

Subgroups — #27279853

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810489C211yesmaximal, characteristic
2#20274783C211yesmaximal, characteristic
3C2×#16810360C212yesmaximal
4#20272593C212yesmaximal
5C53×#2855C211yesmaximal, characteristic
6C22×C4⋊Dic53—21no—
7C2×C22⋊#88322C2212yes—
8#16810489—24no—
9C2×#13133124—21no—
10C2×#13133124C2212yes—
11#16810378—22no—
12#16810360—24no—
13C2×C22⋊Q8×C53C2211yescharacteristic
14C22×C42×C53C2211yescharacteristic
15C53×#391C412yescomplemented
16C53×#389C412yescomplemented
17C53×#387—22no—
18C53×#385—21no—
19C53×#381—21no—
20C53×#380C2211yescharacteristic
21C23×Dic53—22no—
22C22×#88322—22no—
23C2×C4⋊Dic53—24no—
24C2×C4⋊Dic53—42no—
25C2×C4⋊Dic53—42no—
26C22⋊#88322—22no—
27C22⋊#88322—22no—
28#13133124—24no—
29#13133124—22no—
30#13133124D414yes—
31C22×Q8×C53C2×C411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C23×C4×C53D411yescharacteristic
34C23×C4×C53C2311yescharacteristic
35C22⋊Q8×C53—22no—
36C22⋊Q8×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C412yes—
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C42×C53—22no—
44C2×C42×C53—22no—
45C2×C42×C53C2×C412yes—
46C2×C42×C53D412yes—
47#95×C53—22no—
48#95×C53—22no—
49#95×C53—22no—
50#95×C53—21no—
51C24×C53C2×D411yescharacteristic
52C2×Q8×C53—22no—
53C2×Q8×C53—42no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22⋊C412yes—
67C22×C4×C53SD1612yes—
68C22×C4×C53Q1612yes—
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D412yes—
72C22×C4×C53#4112yes—
73C22×Dic53—24no—
74C22×Dic53—44no—
75C22×Dic53—44no—
76C22×Dic53—42no—
77C22×Dic53—42no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—44no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—42no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—41no—
86C42×C53—24no—
87C42×C53—24no—
88C4⋊Dic53—48no—
89C2×#88322—44no—
90C2×#88322—42no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53C2×C22⋊C411yescharacteristic
94C23×C53#12011yescharacteristic
95C23×C53#12311yescharacteristic
96C23×C53C2×SD1611yescharacteristic
97C23×C53C2×Q1611yescharacteristic
98C23×C53AGL(1,ℤ/8)11yescharacteristic
99C23×C53#13711yescharacteristic
100Q8×C53—82no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53#10314yes—
124C2×C4×C53C4×D412yes—
125C2×C4×C53#12112yes—
126C2×Dic53—44no—
127C2×Dic53—44no—
128C2×Dic53—44no—
129C2×Dic53—84no—
130C2×Dic53—84no—
131#88322—82no—
132#9152—531nomaximal, Sylow
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—22no—
138C22×C53—22no—
139C22×C53—22no—
140C22×C53—22no—
141C22×C53#38611yescharacteristic
142C22×C53#41012yes—
143C22×C53C2×#10311yescharacteristic
144C22×C53#41711yescharacteristic
145C22×C53#42112yes—
146C22×C53#45011yescharacteristic
147C22×C53#45111yescharacteristic
148C22×C53#48311yescharacteristic
149C22×C53#48411yescharacteristic
150C4×C53—24no—
151C4×C53—22no—
152C4×C53—22no—
153C4×C53—42no—
154C4×C53—81no—
155C4×C53—81no—
156C4×C53—82no—
157Dic53—84no—
158#2855D5311yescharacteristic
159C2×#382—531no—
160#1817—531no—
161C2×#328—532no—
162#1294—532no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53—21no—
166C2×C53—21no—
167C2×C53#151212yes—
168C2×C53#151812yes—
169C2×C53#187211yescharacteristic
170C2×C53#189711yescharacteristic
171C2×C53#189911yescharacteristic
172C2×C22⋊Q8C2×D5311yescharacteristic
173C22×C4⋊C4—1061no—
174C22×C42C2×D5311yescharacteristic
175C2×C22⋊C8—532no—
176#391Dic5312yescomplemented
177#389Dic5312yescomplemented
178#387—22no—
179#385—21no—
180#382—1064no—
181#381—21no—
182#380C2×D5311yescharacteristic
183C2×#95—532no—
184C2×#95—1061no—
185#336—1062no—
186#328—1064no—
187C53#915211yescharacteristic, complemented, Sylow
188C22×Q8C2×Dic5311yescharacteristic
189C23×C4—1062no—
190C23×C4C22⋊D5311yescharacteristic
191C23×C4C22⋊D5311yescharacteristic
192C23×C4C22×D5311yescharacteristic
193C22×C8—1062no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—1064no—
199C2×C4⋊C4—2122no—
200C2×C4⋊C4—2122no—
201C2×C4⋊C4C2×Dic5311yescharacteristic
202C2×C4⋊C4C2×Dic5312yes—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C2×Dic5312yes—
208C2×C42C22⋊D5312yes—
209C22⋊C8—1062no—
210C22⋊C8—1062no—
211#95—21no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—534no—
216#95—1064no—
217#95—1062no—
218C24C2×C22⋊D5311yescharacteristic
219C2×Q8—22no—
220C2×Q8—42no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—1064no—
233C22×C4—2124no—
234C22×C4—2124no—
235C22×C4—2122no—
236C22×C4—2122no—
237C22×C4#1171992412yes—
238C22×C4#1171992512yes—
239C22×C4C22⋊Dic5312yes—
240C22×C4C22⋊Dic5312yes—
241C22×C4D4×D5312yes—
242C22×C4#1171994712yes—
243C22×C4C22×Dic5311yescharacteristic
244C22×C4C2×C22⋊D5311yescharacteristic
245C2×C8—2124no—
246C2×C8—2122no—
247C4⋊C4—22no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—2128no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C42—24no—
257C42—24no—
258C23—22no—
259C23—22no—
260C23C2×C22⋊Dic5311yescharacteristic
261C23#1313318511yescharacteristic
262C23#1313319811yescharacteristic
263C23C2×#1171992411yescharacteristic
264C23C2×#1171992511yescharacteristic
265C23#1313324211yescharacteristic
266C23#1313324711yescharacteristic
267Q8—82no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—4244no—
291C2×C4—4244no—
292C2×C4—2124no—
293C2×C4—2124no—
294C2×C4—2124no—
295C2×C4#1313314114yes—
296C2×C4D4×Dic5312yes—
297C2×C4#1313318912yes—
298C8—4242no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22#1681050311yescharacteristic
308C22#1681059712yes—
309C22C2×#1313314111yescharacteristic
310C22#1681062411yescharacteristic
311C22#1681064012yes—
312C22#1681077311yescharacteristic
313C22#1681077811yescharacteristic
314C22#1681095111yescharacteristic
315C22#1681095611yescharacteristic
316C4—24no—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—82no—
321C4—81no—
322C4—81no—
323C4—4244no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2#2027325312yes—
329C2#2027327912yes—
330C2#2027502311yescharacteristic
331C2#2027516411yescharacteristic
332C2#2027517411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2855C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2855D53
hypocenterC53#9152
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
15. C53×#391C4semidirect
16. C53×#389C4semidirect
176. #391Dic53semidirect
177. #389Dic53semidirect
187. C53#9152semidirect

Maximal subgroups

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

subgroupindex
C2×#168104892
#202747832
C2×#168103602
#202725932
C53×#28552
#915253