Finite Group Databasebeta

← #27282017

Subgroups — #27282017

Order 13568.

660 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 107 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#20277313C211yesmaximal, characteristic
2#20274789C212yesmaximal
3#20274789C211yesmaximal, characteristic
4#20272619C212yesmaximal
5C53×#2483C211yesmaximal, characteristic
6#16810516—24no—
7#16810483—21no—
8#16810479C2212yes—
9#16810479C2212yes—
10C4×C22⋊Dic53—21no—
11#16810390—24no—
12#16810390—22no—
13C22×C22⋊C4×C53C2211yescharacteristic
14C22×C22⋊C4×C53C2211yescharacteristic
15C53×#397—22no—
16C53×#385—21no—
17C53×#385C412yescomplemented
18C53×#381—21no—
19C53×#381C412yescomplemented
20C53×#379C2211yescharacteristic
21C2×C4⋊Dic53—22no—
22C2×C22⋊Dic53—22no—
23C2×C22⋊Dic53—22no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26C2×C22⋊Dic53—22no—
27C2×C22⋊Dic53D414yes—
28C2×C4×Dic53—22no—
29#13133124—22no—
30#13133124—22no—
31C25×C53C2311yescharacteristic
32C23×C4×C53C2×C411yescharacteristic
33C23×C4×C53C2×C411yescharacteristic
34C23×C4×C53D411yescharacteristic
35C23×C4×C53D411yescharacteristic
36C2×C4⋊C4×C53—22no—
37C2×C22⋊C4×C53—22no—
38C2×C22⋊C4×C53—22no—
39C2×C22⋊C4×C53—21no—
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53C2×C412yes—
46C2×C22⋊C4×C53C2×C412yes—
47C2×C22⋊C4×C53D412yes—
48#95×C53—22no—
49#95×C53—22no—
50#95×C53—22no—
51C24×C53—42no—
52C24×C53—42no—
53C24×C53—81no—
54C24×C53C22×C411yescharacteristic
55C24×C53C2×D411yescharacteristic
56C24×C53C2×D411yescharacteristic
57C24×C53C2×D412yes—
58C24×C53#4112yes—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53—81no—
71C22×C4×C53—81no—
72C22×C4×C53C22⋊C412yes—
73C22×C4×C53C22⋊C412yes—
74C22×Dic53—24no—
75C22×Dic53—42no—
76C4⋊C4×C53—82no—
77C22⋊C4×C53—24no—
78C22⋊C4×C53—24no—
79C22⋊C4×C53—42no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—42no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—81no—
85C22⋊Dic53—44no—
86C22⋊Dic53—44no—
87C22⋊Dic53—44no—
88C22⋊Dic53—44no—
89C22⋊Dic53—82no—
90C22⋊Dic53—82no—
91C4⋊Dic53—82no—
92C4×Dic53—82no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53—22no—
96C23×C53—22no—
97C23×C53—24no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—42no—
102C23×C53—41no—
103C23×C53—42no—
104C23×C53—42no—
105C23×C53—42no—
106C23×C53—41no—
107C23×C53—42no—
108C23×C53—88no—
109C23×C53C2×C22⋊C411yescharacteristic
110C23×C53C4×D412yes—
111C23×C53#12011yescharacteristic
112C23×C53#12112yes—
113C23×C53#12311yescharacteristic
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—81no—
128C2×C4×C53—81no—
129C2×C4×C53—82no—
130C2×C4×C53—82no—
131C2×Dic53—44no—
132C2×Dic53—44no—
133C2×Dic53—44no—
134C2×Dic53—82no—
135C2×Dic53—82no—
136C2×Dic53—82no—
137#9500—531nomaximal, Sylow
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—22no—
143C22×C53—21no—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—21no—
148C22×C53—22no—
149C22×C53—22no—
150C22×C53—21no—
151C22×C53—44no—
152C22×C53—44no—
153C22×C53—84no—
154C22×C53—82no—
155C22×C53—84no—
156C22×C53#38611yescharacteristic
157C22×C53#41012yes—
158C22×C53#45812yes—
159C4×C53—42no—
160C4×C53—42no—
161C4×C53—82no—
162C4×C53—81no—
163C4×C53—81no—
164Dic53—84no—
165Dic53—162no—
166#2483D5311yescharacteristic
167#2312—531no—
168#1819—532no—
169#1819—531no—
170#1304—532no—
171C2×C53—22no—
172C2×C53—22no—
173C2×C53—42no—
174C2×C53—41no—
175C2×C53—42no—
176C2×C53#189011yescharacteristic
177C2×C53#189911yescharacteristic
178C2×C53#191311yescharacteristic
179C22×C22⋊C4C2×D5311yescharacteristic
180C22×C22⋊C4C2×D5311yescharacteristic
181#397—22no—
182#388—1064no—
183#385—21no—
184#385Dic5312yescomplemented
185#381—21no—
186#381Dic5312yescomplemented
187#380—1061no—
188#379—532no—
189#379—532no—
190#379C2×D5311yescharacteristic
191C4×C22⋊C4—1061no—
192#342—1064no—
193#342—1062no—
194C53#950011yescharacteristic, complemented, Sylow
195C25C22×D5311yescharacteristic
196C23×C4C2×Dic5311yescharacteristic
197C23×C4C2×Dic5311yescharacteristic
198C23×C4C22⋊D5311yescharacteristic
199C23×C4C22⋊D5311yescharacteristic
200C2×C4⋊C4—22no—
201C2×C4⋊C4—1062no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—534no—
211C2×C22⋊C4—1062no—
212C2×C22⋊C4—1062no—
213C2×C22⋊C4—1062no—
214C2×C22⋊C4—1062no—
215C2×C22⋊C4—1062no—
216C2×C22⋊C4C2×Dic5312yes—
217C2×C22⋊C4C2×Dic5312yes—
218C2×C22⋊C4C22⋊D5312yes—
219C2×C42—1062no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—1062no—
224#95—1062no—
225C24—42no—
226C24—42no—
227C24—81no—
228C24D4×D5312yes—
229C24#1171994712yes—
230C24C22×Dic5311yescharacteristic
231C24C2×C22⋊D5311yescharacteristic
232C24C2×C22⋊D5311yescharacteristic
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—1064no—
247C22×C4—2122no—
248C22×C4C22⋊Dic5312yes—
249C22×C4C22⋊Dic5312yes—
250C4⋊C4—82no—
251C4⋊C4—4242no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—42no—
255C22⋊C4—41no—
256C22⋊C4—42no—
257C22⋊C4—41no—
258C22⋊C4—42no—
259C22⋊C4—81no—
260C22⋊C4—4242no—
261C22⋊C4—4242no—
262C22⋊C4—2124no—
263C22⋊C4—2124no—
264C22⋊C4—2124no—
265C22⋊C4—2124no—
266C42—4242no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—24no—
271C23—22no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—41no—
282C23—88no—
283C23C2×C22⋊Dic5311yescharacteristic
284C23D4×Dic5312yes—
285C23#1313318511yescharacteristic
286C23#1313318912yes—
287C23#1313319811yescharacteristic
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—4242no—
306C2×C4—4242no—
307C2×C4—4242no—
308C2×C4—2124no—
309C2×C4—2124no—
310C2×C4—2124no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—21no—
317C22—22no—
318C22—21no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—21no—
324C22—44no—
325C22—44no—
326C22—82no—
327C22—84no—
328C22—84no—
329C22#1681050311yescharacteristic
330C22#1681059712yes—
331C22#1681081212yes—
332C4—42no—
333C4—42no—
334C4—82no—
335C4—81no—
336C4—81no—
337C4—8482no—
338C4—4244no—
339C2—22no—
340C2—22no—
341C2—42no—
342C2—42no—
343C2—41no—
344C2#2027512911yescharacteristic
345C2#2027517411yescharacteristic
346C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC24×C53C22×C4
FrattiniC25C22×D53
FittingC53×#2483C2
socleC22×C53#386
prime radical, cosocleC25×C53C23
hypercenter#2483D53
hypocenterC53#9500
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ #2483

Complements

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

NH ≅ G/Ntype
17. C53×#385C4semidirect
19. C53×#381C4semidirect
184. #385Dic53semidirect
186. #381Dic53semidirect
194. C53#9500semidirect

Maximal subgroups

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

subgroupindex
#202773132
#202747892
#202726192
C53×#24832
#950053