Finite Group Databasebeta

← #27281982

Subgroups — #27281982

Order 13568.

458 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 103 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#20283353C211yesmaximal, characteristic
2#20273162C212yesmaximal, complemented
3#20273049C211yesmaximal, characteristic
4#20272579C212yesmaximal, complemented
5C53×#2599C211yesmaximal, characteristic
6#16810684—24no—
7#16810680C2212yes—
8#16810647—21no—
9#16810587—21no—
10C4×#11719917C2212yes—
11Q8⋊#88322—24no—
12#16810349—21no—
13#16810347—21no—
14C2×C4×Q8×C53C2211yescharacteristic
15C2×#117×C53C2211yescharacteristic
16C53×#398C411yescharacteristic
17C53×#397C411yescharacteristic
18C53×#395C411yescharacteristic
19C53×#393C411yescharacteristic
20C53×#388—21no—
21C53×#387—21no—
22C53×#387—21no—
23C53×#385—21no—
24C53×#383C2211yescharacteristic
25C2×#11719917—22no—
26C4⋊#88322—24no—
27C4⋊#88322—22no—
28C22⋊#88322—24no—
29C22⋊#88322—22no—
30#13133128—22no—
31#13133128—22no—
32C4×#88322D414yes—
33C22×Q8×C53C2×C411yescharacteristic
34C23×C4×C53D411yescharacteristic
35C4×Q8×C53—22no—
36C4×Q8×C53—22no—
37#117×C53—22no—
38#117×C53—22no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C42×C53—21no—
50C2×C42×C53D411yescharacteristic
51C2×C42×C53C2311yescharacteristic
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53C2×C411yescharacteristic
59#95×C53C2×C411yescharacteristic
60C24×C53C22⋊C411yescharacteristic
61C2×Q8×C53—22no—
62C2×Q8×C53—42no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C4⋊C4×C53—22no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—42no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C22⋊C4×C53—22no—
85C22⋊C4×C53—22no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—42no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53—42no—
91C42×C53C2×D412yes—
92C42×C53#4112yes—
93#11719917—42no—
94#11719917—42no—
95C2×#88322—24no—
96C2×#88322—42no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53C2×C22⋊C411yescharacteristic
103Q8×C53—24no—
104Q8×C53—42no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—22no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53C4×D412yes—
131C2×C4×C53#12011yescharacteristic
132C2×C4×C53#12112yes—
133C2×C4×C53#12311yescharacteristic
134#88322—44no—
135#88322—82no—
136#9493—531nomaximal, Sylow
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—42no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53—81no—
144C22×C53#38611yescharacteristic
145C22×C53#42111yescharacteristic
146C22×C53#42111yescharacteristic
147C4×C53—22no—
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—22no—
151C4×C53—41no—
152C4×C53—42no—
153C4×C53—41no—
154C4×C53—41no—
155C4×C53—81no—
156C4×C53—81no—
157C4×C53—81no—
158#2921—531no—
159#2599D5311yescharacteristic
160#1491—532no—
161#1464—531no—
162#1288—532no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53—81no—
166C2×C53#189111yescharacteristic
167C2×C53#189711yescharacteristic
168C2×C53#190911yescharacteristic
169C2×C4×Q8C2×D5311yescharacteristic
170C2×#117C2×D5311yescharacteristic
171#433—1064no—
172#432—532no—
173#423—1061no—
174#407—1061no—
175C4×M16—532no—
176#398Dic5311yescharacteristic
177#397Dic5311yescharacteristic
178#395Dic5311yescharacteristic
179#393Dic5311yescharacteristic
180#388—21no—
181#387—21no—
182#387—21no—
183#385—21no—
184#383C2×D5311yescharacteristic
185Q8⋊C8—1064no—
186#324—1061no—
187#323—1061no—
188C53#949311yescharacteristic, complemented, Sylow
189C22×Q8C2×Dic5311yescharacteristic
190C23×C4C22⋊D5311yescharacteristic
191C2×M16—1062no—
192C4×Q8—22no—
193C4×Q8—22no—
194#117—22no—
195#117—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C2×Dic5311yescharacteristic
200C2×C4⋊C4C2×Dic5311yescharacteristic
201C2×C4⋊C4C22⋊D5311yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4C2×Dic5311yescharacteristic
205C2×C22⋊C4C22⋊D5311yescharacteristic
206C2×C42—21no—
207C2×C42C22⋊D5311yescharacteristic
208C2×C42C22×D5311yescharacteristic
209C4⋊C8—1064no—
210C4⋊C8—1062no—
211C22⋊C8—1064no—
212C22⋊C8—1062no—
213#97—1062no—
214#97—1062no—
215C4×C8—534no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95C2×Dic5311yescharacteristic
223#95C2×Dic5311yescharacteristic
224C24C22⋊Dic5311yescharacteristic
225C2×Q8—22no—
226C2×Q8—42no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4C22⋊Dic5311yescharacteristic
236C22×C4C22⋊Dic5311yescharacteristic
237C22×C4C22⋊Dic5311yescharacteristic
238C22×C4C22×Dic5311yescharacteristic
239C22×C4C2×C22⋊D5311yescharacteristic
240C22×C4C2×C22⋊D5311yescharacteristic
241M16—2122no—
242M16—2122no—
243C2×C8—1064no—
244C2×C8—2122no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C42—21no—
257C42—21no—
258C42—42no—
259C42D4×D5312yes—
260C42#1171994712yes—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊Dic5311yescharacteristic
267Q8—24no—
268Q8—42no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4D4×Dic5312yes—
295C2×C4#1313318511yescharacteristic
296C2×C4#1313318912yes—
297C2×C4#1313319811yescharacteristic
298C8—4242no—
299C8—2124no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22#1681050311yescharacteristic
308C22#1681064011yescharacteristic
309C22#1681064011yescharacteristic
310C4—22no—
311C4—22no—
312C4—21no—
313C4—21no—
314C4—41no—
315C4—42no—
316C4—41no—
317C4—41no—
318C4—81no—
319C4—81no—
320C4—81no—
321C2—21no—
322C2—21no—
323C2—81no—
324C2#2027513411yescharacteristic
325C2#2027516411yescharacteristic
326C2#2027523811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2599C2
socleC22×C53#386
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2599D53
hypocenterC53#9493
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2599

Complements

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

NH ≅ G/Ntype
2. #20273162C2semidirect
4. #20272579C2semidirect
188. C53#9493semidirect

Maximal subgroups

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

subgroupindex
#202833532
#202731622
#202730492
#202725792
C53×#25992
#949353