Finite Group Databasebeta

← #27281941

Subgroups — #27281941

Order 13568.

626 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 131 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#20279612C211yesmaximal, characteristic
2#20274707C211yesmaximal, characteristic, complemented
3#20273141C212yesmaximal, complemented
4#20272574C212yesmaximal, complemented
5C53×#2972C211yesmaximal, characteristic, complemented
6C2×#13133188—21no—
7C2×C4×D212C2211yescharacteristic
8#16810647C2212yes—
9C2×#13133134—21no—
10#16810548C412yes—
11#16810532C412yes—
12#16810502—22no—
13#16810489C2211yescharacteristic
14#16810480—21no—
15#16810392—21no—
16#16810352—24no—
17C2×C4×D4×C53C2211yescharacteristic
18C8×D4×C53—24no—
19C53×#431C2212yes—
20C2×C22⋊C8×C53—22no—
21C22×D212C2×C411yescharacteristic
22C22×C22⋊D53—22no—
23C22×C4×D53—21no—
24C2×#11719917—22no—
25#13133188—42no—
26#13133188—42no—
27C4×D212—22no—
28C4×D212—22no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C22⋊Dic53—22no—
32C2×#11719921—22no—
33C2×#11719921—21no—
34C4⋊#88322—22no—
35C4⋊#88322—22no—
36#13133134—42no—
37#13133124—22no—
38#13133124—21no—
39#13133124C2×C412yes—
40C22×D4×C53D411yescharacteristic
41C23×C4×C53D412yes—
42C22×C8×C53—24no—
43C4×D4×C53—22no—
44C4×D4×C53—22no—
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C22⋊C4×C53C2×C412yes—
47C2×C42×C53C2311yescharacteristic
48C4⋊C8×C53—22no—
49C22⋊C8×C53—24no—
50C22⋊C8×C53—22no—
51C4×C8×C53D414yescomplemented
52C24×C53C22⋊C412yes—
53C23×D53—21no—
54C2×D4×C53—22no—
55C2×D4×C53—22no—
56C2×D4×C53D812yes—
57C2×D4×C53SD1612yes—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—82no—
61C22×C4×C53C22⋊C412yes—
62C22×C4×C53C22×C411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C2×C22⋊D53—44no—
66C2×C22⋊D53—44no—
67C2×C22⋊D53—82no—
68C22×Dic53—22no—
69C22×Dic53—21no—
70C2×D212—22no—
71C2×D212—42no—
72C2×C4×D53—42no—
73C2×C4×D53—41no—
74C2×C4×D53—41no—
75C2×C4×D53—42no—
76C2×C8×C53—24no—
77C2×C8×C53—84no—
78C4⋊C4×C53—22no—
79C22⋊C4×C53—24no—
80C42×C53C2×D412yes—
81C42×C53#4112yes—
82C22⋊Dic53—44no—
83#11719921—41no—
84#11719921—41no—
85#11719921—82no—
86C4⋊Dic53—22no—
87C4⋊Dic53—41no—
88C4⋊Dic53—41no—
89#11719917—42no—
90#11719917—42no—
91C2×#88322—42no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—42no—
95C23×C53—42no—
96C23×C53#10214yes—
97C23×C53C2×C22⋊C411yescharacteristic
98C22×D53—22no—
99C22×D53—41no—
100C22×D53—41no—
101C22×D53—41no—
102C22×D53—41no—
103C22×D53—41no—
104C22×D53—41no—
105D4×C53—24no—
106D4×C53—24no—
107C2×C4×C53—22no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—44no—
111C2×C4×C53—42no—
112C2×C4×C53—82no—
113C2×C4×C53C4×D412yes—
114C2×C4×C53#12011yescharacteristic
115C2×C4×C53#12112yes—
116C2×C4×C53#12311yescharacteristic
117C2×C4×C53C2×D811yescharacteristic
118C2×C4×C53C2×SD1611yescharacteristic
119C2×C4×C53#13512yes—
120C22⋊D53—88no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—42no—
126C2×Dic53—41no—
127D212—24no—
128D212—42no—
129C4×D53—82no—
130C4×D53—82no—
131C8×C53—44no—
132#88322—82no—
133#9485—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—24no—
137C22×C53—44no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—82no—
141C22×C53#38611yescharacteristic
142C22×C53C2×#10211yescharacteristic
143C22×C53#41611yescharacteristic
144C2×D53—42no—
145C2×D53—42no—
146C2×D53—42no—
147C2×D53—81no—
148C2×D53—81no—
149C2×D53—81no—
150C2×D53—81no—
151C4×C53—22no—
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—42no—
155C4×C53#46512yes—
156C4×C53#46612yes—
157Dic53—82no—
158Dic53—81no—
159#2972D5311yescharacteristic, complemented
160#2535—531no—
161#1800—531no—
162#1486—532no—
163#1286—532no—
164C2×C53—21no—
165C2×C53—21no—
166C2×C53—44no—
167C2×C53#189111yescharacteristic
168C2×C53#189311yescharacteristic
169C2×C53#191111yescharacteristic
170D53—82no—
171C2×#121—1061no—
172C2×C4×D4—531no—
173C2×C4×D4C2×D5311yescharacteristic
174C8×D4—24no—
175#431C2×D5312yes—
176#423—532no—
177C2×#100—1061no—
178C2×C22⋊C8—22no—
179#397—532no—
180#392—532no—
181#386—1062no—
182#382—531no—
183#380—1061no—
184#343—1061no—
185D4⋊C8—1064no—
186C53#948511yescharacteristic, complemented, Sylow
187C22×D4—531no—
188C22×D4—1062no—
189C22×D4D21211yescharacteristic
190C23×C4—1061no—
191C23×C4C22⋊D5312yes—
192C2×M16—1062no—
193C22×C8—24no—
194#121—2122no—
195#121—2122no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—1062no—
199C4×D4—1062no—
200C2×C4⋊C4—531no—
201C2×C4⋊C4—1061no—
202C2×C4⋊C4D21211yescharacteristic
203C2×C22⋊C4—1062no—
204C2×C22⋊C4—1062no—
205C2×C22⋊C4—1061no—
206C2×C22⋊C4C4×D5312yes—
207C2×C42C22×D5311yescharacteristic
208C4⋊C8—22no—
209C4⋊C8—1062no—
210C4⋊C8—1062no—
211#100—2122no—
212C22⋊C8—24no—
213C22⋊C8—22no—
214C4×C8C22⋊D5314yescomplemented
215#95—532no—
216#95—1061no—
217#95—1062no—
218C24—1061no—
219C24#1171992112yes—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—1062no—
223C2×D4—4242no—
224C2×D4—2122no—
225C2×D4—2124no—
226C2×D4—2124no—
227C2×D4#1171991312yes—
228C2×D4D42412yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—82no—
232C22×C4—1062no—
233C22×C4—1061no—
234C22×C4—2122no—
235C22×C4—2121no—
236C22×C4—2121no—
237C22×C4—2122no—
238C22×C4#1171992112yes—
239C22×C4C2×C4×D5311yescharacteristic
240C22×C4C2×D21211yescharacteristic
241C22×C4C2×C22⋊D5311yescharacteristic
242M16—2122no—
243M16—2122no—
244C2×C8—24no—
245C2×C8—84no—
246C2×C8—2122no—
247C4⋊C4—22no—
248C4⋊C4—1062no—
249C4⋊C4—2121no—
250C4⋊C4—2121no—
251C22⋊C4—24no—
252C22⋊C4—4242no—
253C22⋊C4—2124no—
254C22⋊C4—2121no—
255C22⋊C4—2121no—
256C42#1171994512yes—
257C42C2×C22⋊D5312yes—
258C23—22no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—1062no—
263C23—2121no—
264C23—2121no—
265C23—2121no—
266C23—2121no—
267C23—2121no—
268C23—2121no—
269C23#1313313914yes—
270C23C2×#1171992111yescharacteristic
271D4—24no—
272D4—24no—
273D4—1064no—
274D4—4248no—
275D4—2122no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—82no—
282C2×C4—4242no—
283C2×C4—4242no—
284C2×C4—2122no—
285C2×C4—2122no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4—2122no—
289C2×C4—2121no—
290C2×C4C4×C22⋊D5312yes—
291C2×C4#1313318611yescharacteristic
292C2×C4#1313319112yes—
293C2×C4#1313320011yescharacteristic
294C2×C4C2×D42411yescharacteristic
295C2×C4C2×#1171991311yescharacteristic
296C2×C4#1313323912yes—
297C8—44no—
298C8—4242no—
299C22—21no—
300C22—21no—
301C22—24no—
302C22—44no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22—4241no—
307C22—4241no—
308C22—4241no—
309C22—4241no—
310C22—2122no—
311C22—2122no—
312C22—2122no—
313C22#1681050211yescharacteristic
314C22C2×#1313313911yescharacteristic
315C22#1681062011yescharacteristic
316C4—21no—
317C4—21no—
318C4—22no—
319C4—42no—
320C4—4242no—
321C4—4241no—
322C4#1681085212yes—
323C4#1681085912yes—
324C2—21no—
325C2—21no—
326C2—44no—
327C2—4242no—
328C2#2027513311yescharacteristic
329C2#2027514311yescharacteristic
330C2#2027524711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2972C2
socleC22×C53#386
prime radical#20279612C2
cosocleC2×C42×C53C23
hypercenter#2972D53
hypocenterC53#9485
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #20274707C2semidirect
3. #20273141C2semidirect
4. #20272574C2semidirect
5. C53×#2972C2semidirect
51. C4×C8×C53D4semidirect
159. #2972D53semidirect
186. C53#9485semidirect
214. C4×C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202796122
#202747072
#202731412
#202725742
C53×#29722
#948553