Finite Group Databasebeta

← #27281443

Subgroups — #27281443

Order 13568.

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

normal subgroups only

This group has 446 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#20273317C212yesmaximal, complemented
3#20273049C211yesmaximal, characteristic
4#20272608C212yesmaximal, complemented
5C53×#2611C211yesmaximal, characteristic
6#16810684—24no—
7#16810680C2212yes—
8#16810647—21no—
9#16810587—21no—
10C4×#11719917C2212yes—
11#16810365—24no—
12#16810349—21no—
13#16810347—21no—
14C2×#125×C53C2211yescharacteristic
15C2×#117×C53C2211yescharacteristic
16C53×#400C411yescharacteristic
17C53×#399C411yescharacteristic
18C53×#398C411yescharacteristic
19C53×#397C411yescharacteristic
20C53×#389—21no—
21C53×#388—21no—
22C53×#387—21no—
23C53×#385—21no—
24C53×#376C2211yescharacteristic
25C2×#11719917—22no—
26C4⋊#88322—24no—
27C4⋊#88322—22no—
28C22⋊#88322—24no—
29C22⋊#88322—22no—
30#13133128—22no—
31#13133128D414yes—
32C4×#88322—22no—
33C23×C4×C53D411yescharacteristic
34#125×C53—22no—
35#125×C53—22no—
36#117×C53—22no—
37#117×C53—22no—
38C2×C4⋊C4×C53—21no—
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×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C42×C53—21no—
51C2×C42×C53D411yescharacteristic
52C2×C42×C53C2311yescharacteristic
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
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—41no—
68C22×C4×C53—41no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C4⋊C4×C53—22no—
76C4⋊C4×C53—22no—
77C4⋊C4×C53—22no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—42no—
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
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
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×C53C4×D412yes—
129C2×C4×C53#12011yescharacteristic
130C2×C4×C53#12112yes—
131C2×C4×C53#12311yescharacteristic
132#88322—44no—
133#88322—82no—
134#9390—531nomaximal, Sylow
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—42no—
138C22×C53—41no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53—81no—
142C22×C53#38611yescharacteristic
143C22×C53#42111yescharacteristic
144C22×C53#42111yescharacteristic
145C4×C53—22no—
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—22no—
149C4×C53—42no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53—41no—
153C4×C53—81no—
154C4×C53—81no—
155C4×C53—81no—
156#2921—531no—
157#2611D5311yescharacteristic
158#1528—532no—
159#1464—531no—
160#1300—532no—
161C2×C53—21no—
162C2×C53—21no—
163C2×C53—81no—
164C2×C53#188411yescharacteristic
165C2×C53#189711yescharacteristic
166C2×C53#191511yescharacteristic
167C2×#125C2×D5311yescharacteristic
168C2×#117C2×D5311yescharacteristic
169#433—1064no—
170#432—532no—
171#423—1061no—
172#407—1061no—
173C4×M16—532no—
174#400Dic5311yescharacteristic
175#399Dic5311yescharacteristic
176#398Dic5311yescharacteristic
177#397Dic5311yescharacteristic
178#389—21no—
179#388—21no—
180#387—21no—
181#385—21no—
182#376C2×D5311yescharacteristic
183#330—1064no—
184#324—1061no—
185#323—1061no—
186C53#939011yescharacteristic, complemented, Sylow
187C23×C4C22⋊D5311yescharacteristic
188C2×M16—1062no—
189#125—22no—
190#125—22no—
191#117—22no—
192#117—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×Dic5311yescharacteristic
198C2×C4⋊C4C2×Dic5311yescharacteristic
199C2×C4⋊C4C2×Dic5311yescharacteristic
200C2×C4⋊C4C22⋊D5311yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4C2×Dic5311yescharacteristic
204C2×C22⋊C4C22⋊D5311yescharacteristic
205C2×C42—21no—
206C2×C42C22⋊D5311yescharacteristic
207C2×C42C22×D5311yescharacteristic
208C4⋊C8—1064no—
209C4⋊C8—1062no—
210C22⋊C8—1064no—
211C22⋊C8—1062no—
212#97—534no—
213#97—1062no—
214C4×C8—1062no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95C2×Dic5311yescharacteristic
221#95C2×Dic5311yescharacteristic
222C24C22⋊Dic5311yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4C22⋊Dic5311yescharacteristic
232C22×C4C22⋊Dic5311yescharacteristic
233C22×C4C22⋊Dic5311yescharacteristic
234C22×C4C22×Dic5311yescharacteristic
235C22×C4C2×C22⋊D5311yescharacteristic
236C22×C4C2×C22⋊D5311yescharacteristic
237M16—2122no—
238M16—2122no—
239C2×C8—1064no—
240C2×C8—2122no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C42—21no—
255C42—21no—
256C42—42no—
257C42D4×D5312yes—
258C42#1171994712yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×C22⋊Dic5311yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
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×C4D4×Dic5312yes—
291C2×C4#1313318511yescharacteristic
292C2×C4#1313318912yes—
293C2×C4#1313319811yescharacteristic
294C8—4242no—
295C8—2124no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#1681050311yescharacteristic
304C22#1681064011yescharacteristic
305C22#1681064011yescharacteristic
306C4—22no—
307C4—22no—
308C4—21no—
309C4—21no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—81no—
315C4—81no—
316C4—81no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#2027509911yescharacteristic
321C2#2027516411yescharacteristic
322C2#2027526811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2611C2
socleC22×C53#386
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2611D53
hypocenterC53#9390
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2611

Complements

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

NH ≅ G/Ntype
2. #20273317C2semidirect
4. #20272608C2semidirect
186. C53#9390semidirect

Maximal subgroups

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

subgroupindex
#202833532
#202733172
#202730492
#202726082
C53×#26112
#939053