Finite Group Databasebeta

← #27281961

Subgroups — #27281961

Order 13568.

602 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2975C211yesmaximal, 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
18C53×#436—24no—
19C53×#431C2212yes—
20C53×#407—22no—
21C22×D212C2×C411yescharacteristic
22C22×C22⋊D53—22no—
23C22×C4×D53—21no—
24C2×#11719917—22no—
25#13133188—44no—
26C4×D212—22no—
27C4×D212—22no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C22⋊Dic53—22no—
31C2×#11719921—22no—
32C2×#11719921—21no—
33C4⋊#88322—22no—
34C4⋊#88322—22no—
35#13133134—42no—
36#13133124—22no—
37#13133124—21no—
38#13133124C2×C412yes—
39C22×D4×C53D411yescharacteristic
40C23×C4×C53D412yes—
41C2×M16×C53—24no—
42C4×D4×C53—22no—
43C4×D4×C53—22no—
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C22⋊C4×C53C2×C412yes—
46C2×C42×C53C2311yescharacteristic
47C4⋊C8×C53—22no—
48C22⋊C8×C53—24no—
49C22⋊C8×C53—22no—
50C4×C8×C53D414yescomplemented
51C24×C53C22⋊C412yes—
52C23×D53—21no—
53C2×D4×C53—22no—
54C2×D4×C53—21no—
55C2×D4×C53—22no—
56C2×D4×C53—21no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—82no—
60C22×C4×C53C22⋊C412yes—
61C22×C4×C53C22×C411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C2×C22⋊D53—44no—
65C2×C22⋊D53—44no—
66C2×C22⋊D53—82no—
67C22×Dic53—22no—
68C22×Dic53—21no—
69C2×D212—22no—
70C2×D212—42no—
71C2×C4×D53—42no—
72C2×C4×D53—42no—
73C2×C4×D53—41no—
74C2×C4×D53—41no—
75M16×C53—84no—
76C2×C8×C53—24no—
77C4⋊C4×C53—22no—
78C22⋊C4×C53—24no—
79C42×C53C2×D412yes—
80C42×C53#4112yes—
81C22⋊Dic53—44no—
82#11719921—41no—
83#11719921—41no—
84#11719921—82no—
85C4⋊Dic53—22no—
86C4⋊Dic53—42no—
87#11719917—42no—
88#11719917—42no—
89C2×#88322—42no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53C2×C22⋊C411yescharacteristic
96C22×D53—22no—
97C22×D53—41no—
98C22×D53—41no—
99C22×D53—41no—
100C22×D53—41no—
101C22×D53—41no—
102C22×D53—41no—
103D4×C53—42no—
104D4×C53—42no—
105C2×C4×C53—22no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—44no—
109C2×C4×C53—42no—
110C2×C4×C53—82no—
111C2×C4×C53C4×D412yes—
112C2×C4×C53#12011yescharacteristic
113C2×C4×C53#12112yes—
114C2×C4×C53#12311yescharacteristic
115C2×C4×C53AGL(1,ℤ/8)12yes—
116C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
117C2×C4×C53#13711yescharacteristic
118C22⋊D53—88no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—42no—
125D212—24no—
126D212—42no—
127C4×D53—82no—
128C4×D53—82no—
129C8×C53—44no—
130#88322—82no—
131#9489—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—44no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—42no—
138C22×C53—82no—
139C22×C53#38611yescharacteristic
140C22×C53#41711yescharacteristic
141C22×C53#41811yescharacteristic
142C2×D53—42no—
143C2×D53—42no—
144C2×D53—42no—
145C2×D53—81no—
146C2×D53—81no—
147C2×D53—81no—
148C2×D53—81no—
149C4×C53—22no—
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—42no—
153C4×C53#46812yes—
154C4×C53#46912yes—
155Dic53—82no—
156Dic53—81no—
157#2975D5311yescharacteristic, complemented
158#2535—531no—
159#1800—531no—
160#1486—532no—
161#1286—532no—
162C2×C53—21no—
163C2×C53—21no—
164C2×C53—82no—
165C2×C53#189111yescharacteristic
166C2×C53#189511yescharacteristic
167C2×C53#191111yescharacteristic
168D53—82no—
169C2×#121—1061no—
170C2×C4×D4—531no—
171C2×C4×D4C2×D5311yescharacteristic
172#436—24no—
173#431C2×D5312yes—
174#423—532no—
175C2×#100—1061no—
176#407—22no—
177#397—532no—
178#392—532no—
179#386—1062no—
180#382—531no—
181#380—1061no—
182#343—1061no—
183D4⋊C8—1064no—
184C53#948911yescharacteristic, complemented, Sylow
185C22×D4—531no—
186C22×D4—1062no—
187C22×D4D21211yescharacteristic
188C23×C4—1061no—
189C23×C4C22⋊D5312yes—
190C2×M16—24no—
191C2×M16—1062no—
192#121—2124no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—1062no—
196C4×D4—1062no—
197C2×C4⋊C4—531no—
198C2×C4⋊C4—1061no—
199C2×C4⋊C4D21211yescharacteristic
200C2×C22⋊C4—1062no—
201C2×C22⋊C4—1062no—
202C2×C22⋊C4—1061no—
203C2×C22⋊C4C4×D5312yes—
204C2×C42C22×D5311yescharacteristic
205C4⋊C8—22no—
206C4⋊C8—1062no—
207C4⋊C8—1062no—
208#100—2122no—
209C22⋊C8—24no—
210C22⋊C8—22no—
211C4×C8C22⋊D5314yescomplemented
212#95—532no—
213#95—1061no—
214#95—1062no—
215C24—1061no—
216C24#1171992112yes—
217C2×D4—22no—
218C2×D4—21no—
219C2×D4—22no—
220C2×D4—21no—
221C2×D4—1062no—
222C2×D4—4242no—
223C2×D4—2122no—
224C2×D4—2124no—
225C2×D4—2124no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—82no—
229C22×C4—1062no—
230C22×C4—1061no—
231C22×C4—2122no—
232C22×C4—2122no—
233C22×C4—2121no—
234C22×C4—2121no—
235C22×C4#1171992112yes—
236C22×C4C2×C4×D5311yescharacteristic
237C22×C4C2×D21211yescharacteristic
238C22×C4C2×C22⋊D5311yescharacteristic
239M16—84no—
240M16—2122no—
241M16—2122no—
242C2×C8—24no—
243C2×C8—2122no—
244C4⋊C4—22no—
245C4⋊C4—1062no—
246C4⋊C4—2122no—
247C22⋊C4—24no—
248C22⋊C4—4242no—
249C22⋊C4—2124no—
250C22⋊C4—2121no—
251C22⋊C4—2121no—
252C42#1171994512yes—
253C42C2×C22⋊D5312yes—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—42no—
258C23—42no—
259C23—1062no—
260C23—2121no—
261C23—2121no—
262C23—2121no—
263C23—2121no—
264C23—2121no—
265C23—2121no—
266C23C2×#1171992111yescharacteristic
267D4—42no—
268D4—42no—
269D4—1064no—
270D4—4248no—
271D4—2122no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—4242no—
279C2×C4—4242no—
280C2×C4—2122no—
281C2×C4—2122no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2122no—
286C2×C4C4×C22⋊D5312yes—
287C2×C4#1313318611yescharacteristic
288C2×C4#1313319112yes—
289C2×C4#1313320011yescharacteristic
290C2×C4#1313324512yes—
291C2×C4#1313324511yescharacteristic
292C2×C4#1313325011yescharacteristic
293C8—44no—
294C8—4242no—
295C22—21no—
296C22—21no—
297C22—44no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—4241no—
303C22—4241no—
304C22—4241no—
305C22—4241no—
306C22—2122no—
307C22—2122no—
308C22—2122no—
309C22#1681050211yescharacteristic
310C22#1681062511yescharacteristic
311C22#1681062911yescharacteristic
312C4—21no—
313C4—21no—
314C4—22no—
315C4—42no—
316C4—4242no—
317C4—4241no—
318C4#1681086912yes—
319C4#1681087612yes—
320C2—21no—
321C2—21no—
322C2—82no—
323C2—4242no—
324C2#2027513311yescharacteristic
325C2#2027515311yescharacteristic
326C2#2027524711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2975C2
socleC22×C53#386
prime radical#20279612C2
cosocleC2×C42×C53C23
hypercenter#2975D53
hypocenterC53#9489
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
2. #20274707C2semidirect
3. #20273141C2semidirect
4. #20272574C2semidirect
5. C53×#2975C2semidirect
50. C4×C8×C53D4semidirect
157. #2975D53semidirect
184. C53#9489semidirect
211. 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×#29752
#948953