Finite Group Databasebeta

← #27358997

Subgroups — #27358997

Order 13568.

870 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 191 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#20288525C211yesmaximal, characteristic
2#20285866C212yesmaximal
3#20273756C218yesmaximal
4#20273213C212yesmaximal
5#20273039C211yesmaximal, characteristic
6C53×#2568C211yesmaximal, characteristic
7Q8×Dic106—22no—
8#16811352—22no—
9#16811349—24no—
10#16811254—21no—
11#16811254C2214yes—
12#16811180—22no—
13#16811180C2212yes—
14#16811101—21no—
15#16811026—22no—
16#16810975—22no—
17#16810958C2218yes—
18#16810915—24no—
19#16810676C2212yes—
20#16810359C2218yes—
21#16810349C2214yes—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4×Q8×C53C2211yescharacteristic
24C53×#400C2214yes—
25C53×#391—21no—
26C53×#391—22no—
27C53×#382—21no—
28C4×C4⋊C4×C53C2211yescharacteristic
29#13133217—22no—
30#13133217—22no—
31#13133217D412yes—
32#13133217D412yes—
33#13133216—24no—
34#13133216—28no—
35#13133209—24no—
36#13133209—22no—
37#13133208—24no—
38#13133208—28no—
39#13133196—22no—
40#13133196—24no—
41#13133196C2314yes—
42Q8×Dic53—24no—
43Q8×Dic53—24no—
44C4×Dic106—24no—
45C4×Dic106—24no—
46C4×Dic106—22no—
47C4×Dic106D414yes—
48#13133176—24no—
49#13133176—22no—
50#13133150—28no—
51C4⋊#88322—22no—
52#13133140—28no—
53C22⋊#88322C2314yes—
54C4×#88322—22no—
55C22×Q8×C53—21no—
56C22×Q8×C53D411yescharacteristic
57C4⋊Q8×C53—22no—
58C4⋊Q8×C53—22no—
59C4×Q8×C53—22no—
60C4×Q8×C53—22no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53D411yescharacteristic
64C2×C4⋊C4×C53C2312yes—
65C2×C4⋊C4×C53C2312yes—
66C2×C42×C53—21no—
67C2×C42×C53D412yes—
68C2×C42×C53C2311yescharacteristic
69#95×C53—24no—
70#95×C53—24no—
71#95×C53C2312yes—
72C2×Q8×C53—22no—
73C2×Q8×C53—22no—
74C2×Q8×C53—44no—
75C2×Q8×C53—41no—
76C2×Q8×C53—41no—
77C2×Q8×C53Q1614yes—
78C22×C4×C53—22no—
79C22×C4×C53—22no—
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D412yes—
82C22×C4×C53C2×D412yes—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2411yescharacteristic
85C2×Dic106—22no—
86C2×Dic106—24no—
87C2×Dic106C2×D414yes—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—24no—
90C4⋊C4×C53—24no—
91C4⋊C4×C53—42no—
92C4⋊C4×C53—42no—
93C42×C53—22no—
94C42×C53—22no—
95C42×C53—42no—
96C42×C53C2×D412yes—
97C42×C53C2×D412yes—
98C22⋊Dic53—24no—
99C4⋊Dic53—24no—
100C4⋊Dic53—24no—
101C4⋊Dic53—24no—
102C4⋊Dic53—22no—
103C4⋊Dic53—22no—
104C4⋊Dic53—22no—
105C4⋊Dic53C2×D414yes—
106#11719919—28no—
107#11719919—24no—
108#11719919—24no—
109C4×Dic53—28no—
110C4×Dic53—24no—
111C2×#88322—24no—
112C23×C53C22×D411yescharacteristic
113Q8×C53—28no—
114Q8×C53—82no—
115Q8×C53—82no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—41no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—44no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53#12011yescharacteristic
128C2×C4×C53#12012yes—
129C2×C4×C53#12011yescharacteristic
130C2×C4×C53C2×Q1614yes—
131C2×C4×C53C2×Q1612yes—
132C2×C4×C53#13712yes—
133C2×C4×C53C22×D412yes—
134C2×Dic53—24no—
135C2×Dic53—24no—
136Dic106—44no—
137Dic106—44no—
138Dic106—84no—
139#88322—44no—
140#18411—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—21no—
143C22×C53C2×#12011yescharacteristic
144C22×C53C22×Q1611yescharacteristic
145C22×C53C2×#13711yescharacteristic
146C4×C53—24no—
147C4×C53—21no—
148C4×C53—22no—
149C4×C53—42no—
150C4×C53—82no—
151C4×C53—82no—
152C4×C53#45114yes—
153C4×C53#45412yes—
154Dic53—44no—
155Dic53—44no—
156#3502—531no—
157#3230—532no—
158#2568D5311yescharacteristic
159#1617—538no—
160#1503—532no—
161#1461—531no—
162C2×C53—21no—
163C2×C53—21no—
164C2×C53C2×#45111yescharacteristic
165C2×C53C2×#45411yescharacteristic
166C2×C53#302811yescharacteristic
167Q82—1062no—
168#557—1064no—
169#557—1062no—
170#544—534no—
171#544—1061no—
172#533—532no—
173#533—1062no—
174C2×C4⋊Q8C2×D5311yescharacteristic
175#519—1061no—
176C2×C4×Q8C2×D5311yescharacteristic
177#500—1062no—
178#487—1062no—
179#484—538no—
180#477—1064no—
181#431—532no—
182#400C2×D5314yes—
183#391—21no—
184#391—22no—
185#382—21no—
186C4×C4⋊C4C2×D5311yescharacteristic
187#328—538no—
188#324—534no—
189C53#1841111yescharacteristic, complemented, Sylow
190C22×Q8—21no—
191C22×Q8C22⋊D5311yescharacteristic
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—532no—
195C4⋊Q8—532no—
196C4⋊Q8—1062no—
197C4⋊Q8—1062no—
198C4⋊Q8—1068no—
199C4⋊Q8—1064no—
200#125—1064no—
201#125—1068no—
202#125—1064no—
203#125—1062no—
204C22⋊Q8—534no—
205C22⋊Q8—1062no—
206C22⋊Q8—1064no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—534no—
210C4×Q8—1064no—
211C4×Q8—1064no—
212C4×Q8—1064no—
213C4×Q8—1064no—
214C4×Q8—1062no—
215#117—1062no—
216#117—1064no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C22⋊D5311yescharacteristic
220C2×C4⋊C4C22×D5312yes—
221C2×C4⋊C4C22×D5312yes—
222C2×C42—21no—
223C2×C42C22⋊D5312yes—
224C2×C42C22×D5311yescharacteristic
225#107—1068no—
226C4⋊C8—1062no—
227#103—1068no—
228C22⋊C8—534no—
229C4×C8—1062no—
230#95—24no—
231#95—24no—
232#95C22×D5312yes—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—44no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—534no—
239C2×Q8—1064no—
240C2×Q8—1062no—
241C2×Q8#1171992514yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4C2×C22⋊D5311yescharacteristic
245C22×C4C2×C22⋊D5312yes—
246C22×C4C2×C22⋊D5312yes—
247C22×C4C2×C22⋊D5311yescharacteristic
248C22×C4C23×D5311yescharacteristic
249C2×C8—1064no—
250C4⋊C4—22no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—534no—
256C4⋊C4—1064no—
257C4⋊C4—1064no—
258C4⋊C4—1064no—
259C4⋊C4—1068no—
260C4⋊C4—1062no—
261C4⋊C4—1062no—
262C4⋊C4—1064no—
263C4⋊C4—1062no—
264C4⋊C4—1064no—
265C22⋊C4—1064no—
266C42—22no—
267C42—22no—
268C42—42no—
269C42—1068no—
270C42—1064no—
271C42D4×D5312yes—
272C42D4×D5312yes—
273C23C22×C22⋊D5311yescharacteristic
274Q8—28no—
275Q8—82no—
276Q8—82no—
277Q8—4244no—
278Q8—2124no—
279Q8—2124no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—1064no—
292C2×C4—1064no—
293C2×C4#1313318511yescharacteristic
294C2×C4#1313318512yes—
295C2×C4#1313318511yescharacteristic
296C2×C4C2×#1171992514yes—
297C2×C4C2×#1171992512yes—
298C2×C4#1313324712yes—
299C2×C4C2×D4×D5312yes—
300C8—2124no—
301C22—21no—
302C22—21no—
303C22C2×#1313318511yescharacteristic
304C22C22×#1171992511yescharacteristic
305C22C2×#1313324711yescharacteristic
306C4—24no—
307C4—21no—
308C4—22no—
309C4—42no—
310C4—82no—
311C4—82no—
312C4—2124no—
313C4—2124no—
314C4#1681077814yes—
315C4#1681079512yes—
316C2—21no—
317C2—21no—
318C2C2×#1681077811yescharacteristic
319C2C2×#1681079511yescharacteristic
320C2#2028412011yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2568C2
socleC22×C53C2×#120
prime radicalC23×C53C22×D4
hypercenter#2568D53
hypocenterC53#18411
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2568

Complements

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

NH ≅ G/Ntype
189. C53#18411semidirect

Maximal subgroups

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

subgroupindex
#202885252
#202858662
#202737562
#202732132
#202730392
C53×#25682
#1841153