Finite Group Databasebeta

← #28298584

Subgroups — #28298584

Order 13568.

372 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 66 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#20290579C211yesmaximal, characteristic
2#20290575C211yesmaximal, characteristic
3#20275007C211yesmaximal, characteristic
4#16811626—21no—
5C2×#13133277—21no—
6C2×#13133275—21no—
7#16811565—21no—
8#16811503C412yescomplemented
9#16811501C412yes—
10C2×C4×C22⋊D53C2211yescharacteristic
11C2×C22⋊#88322C411yescharacteristic, complemented
12C2×#13133129—21no—
13#16810377—21no—
14C23×C8×C53C411yescharacteristic, complemented
15C2×#11719940—22no—
16C2×C4⋊#6157—41no—
17C2×C22⋊#6157—22no—
18#13133284—41no—
19C4⋊#88324—41no—
20#13133277—41no—
21#13133277—41no—
22#13133275—41no—
23#13133275—41no—
24(C2×#6157)⋊C4—41no—
25C22×C22⋊D53D411yescharacteristic
26C22×C4×D53D411yescharacteristic
27C22×#88322—21no—
28C4×C22⋊D53—41no—
29C4×C22⋊D53—41no—
30C2×#11719919C2×C411yescharacteristic
31C2×C22⋊Dic53Q811yescharacteristic
32C2×#11719921C2×C411yescharacteristic
33C2×C4×Dic53D411yescharacteristic
34C22⋊#88322—41no—
35#13133129—42no—
36C23×C4×C53C2×C411yescharacteristic
37C22×C8×C53—21no—
38C22×C8×C53—41no—
39C22×C8×C53—42no—
40C24×C53C4⋊C411yescharacteristic
41C23×D53C22⋊C411yescharacteristic
42C22×#6157—42no—
43C22×C4×C53—21no—
44C22×C4×C53—21no—
45C22×C4×C53—41no—
46C22×C4×C53—41no—
47C22×C4×C53C4211yescharacteristic
48C22×C4×C53C22⋊C411yescharacteristic
49C2×C22⋊D53—41no—
50C2×C22⋊D53—41no—
51C2×C22⋊D53D811yescharacteristic
52C2×C22⋊D53SD1611yescharacteristic
53C2×C22⋊D53SD1611yescharacteristic
54C2×C22⋊D53Q1611yescharacteristic
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C4⋊C411yescharacteristic
57C22×Dic53C4⋊C411yescharacteristic
58C2×C4×D53—21no—
59C2×C4×D53—21no—
60C2×C4×D53—41no—
61C2×C4×D53—41no—
62C22⋊#6157—42no—
63C22⋊#6157—42no—
64#11719940—42no—
65#11719940—42no—
66C2×#88324—42no—
67C4⋊#6157—81no—
68C4⋊#6157—81no—
69C2×C8×C53—22no—
70C2×C8×C53—41no—
71C2×C8×C53—42no—
72C2×C8×C53—41no—
73C2×C8×C53—42no—
74C22⋊Dic53—41no—
75#11719921—41no—
76#11719919—41no—
77C4×Dic53—21no—
78C4×Dic53—21no—
79C2×#88322—41no—
80C2×#88322—41no—
81C2×#88322—41no—
82C23×C53—21no—
83C23×C53—21no—
84C23×C53—41no—
85C23×C53—41no—
86C23×C53#9511yescharacteristic
87C23×C53#10611yescharacteristic
88C23×C53#10711yescharacteristic
89C22×D53—21no—
90C22×D53—21no—
91C22×D53—41no—
92C22×D53—41no—
93C22×D53#10211yescharacteristic
94C22×D53#10311yescharacteristic
95C2×#6157—82no—
96C2×#6157—82no—
97C2×#6157—82no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—42no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53#9911yescharacteristic
106C2×C4×C53#9911yescharacteristic
107C22⋊D53—41no—
108C22⋊D53—41no—
109C22⋊D53—41no—
110C22⋊D53—41no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53#10211yescharacteristic
118C2×Dic53#10311yescharacteristic
119C4×D53—41no—
120C4×D53—41no—
121C4×D53—81no—
122#88324—82no—
123C8×C53—42no—
124#88322—81no—
125#7236—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—41no—
131C22×C53—41no—
132C22×C53—42no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53H(3,ℤ/4)11yescharacteristic
136C22×C53#34011yescharacteristic
137C22×C53#34211yescharacteristic
138C2×D53—21no—
139C2×D53—21no—
140C2×D53—41no—
141C2×D53—41no—
142C2×D53—41no—
143C2×D53—41no—
144C2×D53—81no—
145#6157—162no—
146C4×C53—21no—
147C4×C53—41no—
148C4×C53AΣL(1,𝔽16)11yescharacteristic
149C4×C53#35211yescharacteristic
150Dic53—21no—
151Dic53—21no—
152Dic53—41no—
153Dic53—41no—
154#1869—531no—
155#1800—531no—
156#1799—531no—
157C2×C53—21no—
158C2×C53—21no—
159C2×C53—41no—
160C2×C53—41no—
161C2×C53#128911yescharacteristic
162C2×C53#139011yescharacteristic
163C2×C53#139511yescharacteristic
164D53—41no—
165D53—41no—
166C23×C8#615711yescharacteristic, complemented
167C2×C4×D4—531no—
168#423—1061no—
169C2×#100—1061no—
170C2×#99—1061no—
171C2×C22⋊C8—531no—
172C2×C22⋊C8—1061no—
173#380—1061no—
174#343—532no—
175#342—532no—
176#336—1061no—
177C53#723611yescharacteristic, complemented, Sylow
178C22×D4—531no—
179C23×C4—531no—
180C23×C4C2×#615711yescharacteristic
181C2×M16—1062no—
182C22×C8—21no—
183C22×C8—42no—
184C22×C8—41no—
185C22×C8—1061no—
186C4×D4—2121no—
187C4×D4—2121no—
188C2×C4⋊C4—531no—
189C2×C4⋊C4—2121no—
190C2×C22⋊C4—531no—
191C2×C22⋊C4—531no—
192C2×C22⋊C4—1062no—
193C2×C42—531no—
194C4⋊C8—2121no—
195C4⋊C8—2121no—
196#100—2121no—
197#100—2121no—
198#99—2121no—
199#99—2121no—
200C22⋊C8—2122no—
201C22⋊C8—2121no—
202#95—2121no—
203C24—531no—
204C24C4⋊#615711yescharacteristic
205C2×D4—531no—
206C2×D4—531no—
207C2×D4—531no—
208C2×D4—531no—
209C2×D4—2121no—
210C2×D4—2121no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—1061no—
219C22×C4—1061no—
220C22×C4—2121no—
221C22×C4—2121no—
222C22×C4—2122no—
223C22×C4C4×#615711yescharacteristic
224C22×C4C22⋊#615711yescharacteristic
225M16—2122no—
226M16—2122no—
227C2×C8—22no—
228C2×C8—41no—
229C2×C8—42no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—2121no—
233C2×C8—2121no—
234C2×C8—2122no—
235C2×C8—2121no—
236C4⋊C4—4241no—
237C4⋊C4—4241no—
238C4⋊C4—2121no—
239C22⋊C4—2121no—
240C22⋊C4—2121no—
241C22⋊C4—2122no—
242C22⋊C4—2122no—
243C42—1061no—
244C42—1061no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—531no—
250C23—531no—
251C23—1061no—
252C23—1061no—
253C23—2121no—
254C23—2121no—
255C23(C2×#6157)⋊C411yescharacteristic
256C23#1313328511yescharacteristic
257C23#1313328611yescharacteristic
258D4—2121no—
259D4—2121no—
260D4—2121no—
261D4—2121no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—531no—
270C2×C4—531no—
271C2×C4—1061no—
272C2×C4—1061no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—4241no—
276C2×C4—4242no—
277C2×C4—4242no—
278C2×C4—4242no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4#1313327511yescharacteristic
284C2×C4#1313327611yescharacteristic
285C8—42no—
286C8—4241no—
287C8—4242no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—41no—
293C22—41no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—1061no—
298C22—1061no—
299C22—4241no—
300C22—2121no—
301C22—2121no—
302C22—2121no—
303C22—2121no—
304C22#1681149111yescharacteristic
305C22#1681149711yescharacteristic
306C22#1681150111yescharacteristic
307C4—21no—
308C4—41no—
309C4—8482no—
310C4—1061no—
311C4—1061no—
312C4—2121no—
313C4—2121no—
314C4#1681152311yescharacteristic
315C4#1681152511yescharacteristic
316C2—21no—
317C2—21no—
318C2—41no—
319C2—41no—
320C2—2121no—
321C2—2121no—
322C2#2028957811yescharacteristic
323C2#2028979911yescharacteristic
324C2#2028981511yescharacteristic
subgroupG/H
centerC22#16811501
commutatorC22×C4×C53C42
FrattiniC23×C4C2×#6157
FittingC23×C8×C53C4
socleC22×C53#342
prime radicalC22×C22⋊D53D4
cosocleC2×C4×C22⋊D53C22
hypercenterC23×C8#6157
hypocenterC53#7236
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
8. #16811503C4semidirect
11. C2×C22⋊#88322C4semidirect
14. C23×C8×C53C4semidirect
166. C23×C8#6157semidirect
177. C53#7236semidirect

Maximal subgroups

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

subgroupindex
#202905792
#202905752
#202750072
#723653