Finite Group Databasebeta

← #27360352

Subgroups — #27360352

Order 13568.

1004 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 171 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×#16811353C211yesmaximal, characteristic, complemented
2#20284833C212yesmaximal
3#20273853C214yesmaximal, complemented
4#20273791C214yesmaximal, complemented
5#20273441C211yesmaximal, characteristic, complemented
6#20273163C212yesmaximal, complemented
7C53×#3152C211yesmaximal, characteristic
8#16811353—24no—
9#16811353—22no—
10#16811353C2214yes—
11C2×#13133216—22no—
12C2×#13133215—22no—
13C2×C4×Dic106C2212yes—
14C2×#13133182—22no—
15#16810934—22no—
16#16810915C2214yes—
17#16810820C2214yes—
18#16810761—22no—
19#16810723—24no—
20#16810680C2212yes—
21#16810372C2214yes—
22#16810355C2218yes—
23C2×#137×C53—22no—
24C53×#532C2211yescharacteristic
25C2×C4⋊Q8×C53C2211yescharacteristic
26C53×#497—22no—
27C53×#494C2212yes—
28C53×#492C2212yes—
29C4×M16×C53C2211yescharacteristic
30C22×Dic106—22no—
31C22×Dic106D412yes—
32C2×#11719923—24no—
33#13133216—28no—
34#13133216—28no—
35#13133215—28no—
36#13133215—28no—
37C4×Dic106—22no—
38C4×Dic106—22no—
39C4×Dic106D414yes—
40C4×Dic106C2314yes—
41#13133182—28no—
42#13133182—28no—
43C2×C4⋊Dic53D412yes—
44C2×#11719919—24no—
45C2×#11719919—22no—
46C2×C4×Dic53—22no—
47C2×C4×Dic53—22no—
48#13133150—24no—
49C4⋊#88322C2314yes—
50#13133140—24no—
51#13133138—24no—
52C22⋊#88322—22no—
53#13133128—22no—
54C2×#41×C53—21no—
55C22×Q8×C53D412yes—
56#137×C53—48no—
57C2×Q16×C53—24no—
58C2×SD16×C53—24no—
59C2×M16×C53—21no—
60C4⋊Q8×C53—22no—
61C4⋊Q8×C53D412yes—
62C4⋊Q8×C53C2311yescharacteristic
63C4⋊Q8×C53C2312yes—
64#127×C53C2311yescharacteristic
65#124×C53—21no—
66#121×C53—22no—
67C4×D4×C53—22no—
68C2×C4⋊C4×C53—22no—
69C2×C42×C53C2311yescharacteristic
70#97×C53—21no—
71C4×C8×C53C2312yes—
72#41×C53—44no—
73C2×Q8×C53—22no—
74C2×Q8×C53—21no—
75C2×Q8×C53—44no—
76C2×Q8×C53C2×D414yes—
77C2×D4×C53—22no—
78C2×D4×C53—21no—
79C2×D4×C53—41no—
80C22×C4×C53—22no—
81C22×C4×C53—41no—
82C22×C4×C53C2×D412yes—
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—22no—
85C22×Dic53—22no—
86C2×Dic106—22no—
87C2×Dic106—28no—
88C2×Dic106—28no—
89C2×Dic106—84no—
90C2×Dic106C2×D414yes—
91Q16×C53—48no—
92SD16×C53—48no—
93M16×C53—24no—
94C2×C8×C53—22no—
95C4⋊C4×C53—24no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53—22no—
98C22⋊C4×C53—42no—
99C42×C53C2×D412yes—
100C42×C53C2×D411yescharacteristic
101C42×C53C2411yescharacteristic
102#11719923—48no—
103C4⋊Dic53—22no—
104C4⋊Dic53C2×D414yes—
105#11719919—216no—
106#11719919—24no—
107#11719919—24no—
108C4×Dic53—28no—
109C4×Dic53—24no—
110C4×Dic53—24no—
111C2×#88322—24no—
112C23×C53—41no—
113C23×C53#12011yescharacteristic
114Q8×C53—44no—
115Q8×C53—42no—
116Q8×C53—82no—
117D4×C53—42no—
118D4×C53—44no—
119D4×C53—82no—
120C2×C4×C53—22no—
121C2×C4×C53—24no—
122C2×C4×C53—24no—
123C2×C4×C53—44no—
124C2×C4×C53—41no—
125C2×C4×C53—82no—
126C2×C4×C53#12012yes—
127C2×C4×C53#12011yescharacteristic
128C2×C4×C53C22×D412yes—
129C2×C4×C53C22×D411yescharacteristic
130C2×Dic53—28no—
131C2×Dic53—24no—
132C2×Dic53—24no—
133C2×Dic53—44no—
134Dic106—28no—
135Dic106—24no—
136Dic106—88no—
137Dic106#13718yescomplemented
138C8×C53—24no—
139#88322—44no—
140#18546—531nomaximal, Sylow
141C22×C53—22no—
142C22×C53—82no—
143C22×C53—81no—
144C22×C53#45312yes—
145C22×C53C2×#12011yescharacteristic
146C4×C53—22no—
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—44no—
150C4×C53—81no—
151C4×C53C2×#13714yes—
152Dic53—28no—
153Dic53—44no—
154C2×Q82—531no—
155#3152D5311yescharacteristic
156#3112—532no—
157#1633—534no—
158#1623—534no—
159#1558—531no—
160#1491—532no—
161C2×C53—22no—
162C2×C53—161no—
163C2×C53#300412yes—
164C2×C53C2×#45311yescharacteristic
165C2×#137—22no—
166Q82—534no—
167Q82—1064no—
168Q82—1062no—
169#532C2×D5311yescharacteristic
170C2×C4⋊Q8—1062no—
171C2×C4⋊Q8—1062no—
172C2×C4⋊Q8C2×D5311yescharacteristic
173C2×C4×Q8—532no—
174C2×C4×Q8—1062no—
175#497—22no—
176#494C2×D5312yes—
177#492C2×D5312yes—
178#480—1062no—
179#477—534no—
180#460—534no—
181#448—1062no—
182#440—1064no—
183#432—532no—
184C4×M16C2×D5311yescharacteristic
185#333—534no—
186Q8⋊C8—538no—
187C53#1854611yescharacteristic, complemented, Sylow
188C2×#41—21no—
189C22×Q8—532no—
190C22×Q8—1062no—
191C22×Q8C22⋊D5312yes—
192#137—48no—
193C2×Q16—24no—
194C2×SD16—24no—
195C2×SD16—1064no—
196C2×M16—21no—
197C4⋊Q8—22no—
198C4⋊Q8—1068no—
199C4⋊Q8—1068no—
200C4⋊Q8—1068no—
201C4⋊Q8—1068no—
202C4⋊Q8C22⋊D5312yes—
203C4⋊Q8C22×D5311yescharacteristic
204C4⋊Q8C22×D5312yes—
205#127C22×D5311yescharacteristic
206#124—21no—
207#121—22no—
208C4×Q8—534no—
209C4×Q8—534no—
210C4×Q8—1068no—
211C4×Q8—1068no—
212C4×Q8—1062no—
213C4×Q8—1062no—
214C4×D4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—532no—
217C2×C4⋊C4—1064no—
218C2×C4⋊C4—1062no—
219C2×C42—1062no—
220C2×C42—1062no—
221C2×C42C22×D5311yescharacteristic
222#107—1064no—
223C4⋊C8—534no—
224#103—1064no—
225#102—1064no—
226C22⋊C8—1062no—
227#97—21no—
228#97—1062no—
229C4×C8C22×D5312yes—
230#41—44no—
231C2×Q8—22no—
232C2×Q8—21no—
233C2×Q8—44no—
234C2×Q8—534no—
235C2×Q8—1068no—
236C2×Q8—1068no—
237C2×Q8—1062no—
238C2×Q8—4244no—
239C2×Q8C2×C22⋊D5314yes—
240C2×D4—22no—
241C2×D4—21no—
242C2×D4—41no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—1062no—
246C22×C4—1062no—
247C22×C4D4×D5312yes—
248C22×C4C2×C22⋊D5311yescharacteristic
249Q16—48no—
250SD16—48no—
251SD16—2128no—
252M16—24no—
253C2×C8—22no—
254C2×C8—1064no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—534no—
259C4⋊C4—10616no—
260C4⋊C4—1062no—
261C4⋊C4—1064no—
262C4⋊C4—1064no—
263C22⋊C4—42no—
264C42—1068no—
265C42—1064no—
266C42—1064no—
267C42D4×D5312yes—
268C42C2×C22⋊D5311yescharacteristic
269C42C23×D5311yescharacteristic
270C23—41no—
271C23#1313318511yescharacteristic
272Q8—44no—
273Q8—42no—
274Q8—82no—
275Q8—538no—
276Q8—1068no—
277Q8—1064no—
278Q8—4248no—
279D4—42no—
280D4—44no—
281D4—82no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—44no—
286C2×C4—41no—
287C2×C4—82no—
288C2×C4—1068no—
289C2×C4—1064no—
290C2×C4—1064no—
291C2×C4—2124no—
292C2×C4#1313318512yes—
293C2×C4#1313318511yescharacteristic
294C2×C4C2×D4×D5312yes—
295C2×C4C22×C22⋊D5311yescharacteristic
296C8—24no—
297C8—2124no—
298C22—22no—
299C22—82no—
300C22—81no—
301C22#1681079012yes—
302C22C2×#1313318511yescharacteristic
303C4—22no—
304C4—21no—
305C4—21no—
306C4—44no—
307C4—81no—
308C4—1068no—
309C4—2124no—
310C4#137×D5314yes—
311C2—22no—
312C2—161no—
313C2#2028394712yes—
314C2C2×#1681079011yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3152C2
socleC22×C53C2×#120
prime radicalC53×#532C22
hypercenter#3152D53
hypocenterC53#18546
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. C2×#16811353C2semidirect
3. #20273853C2semidirect
4. #20273791C2semidirect
5. #20273441C2semidirect
6. #20273163C2semidirect
137. Dic106#137semidirect
187. C53#18546semidirect

Maximal subgroups

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

subgroupindex
C2×#168113532
#202848332
#202738532
#202737912
#202734412
#202731632
C53×#31522
#1854653