Finite Group Databasebeta

← #28298725

Subgroups — #28298725

Order 13568.

388 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 58 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
1C4×#13133277C211yesmaximal, characteristic
2C4×#13133275C211yesmaximal, characteristic
3#20275012C211yesmaximal, characteristic
4C2×#13133277—21no—
5C2×#13133275—21no—
6C4×#11719940—21no—
7C4×C22⋊#6157—21no—
8#16811503C412yescomplemented
9#16811501C412yes—
10C2×C4×C22⋊D53C2211yescharacteristic
11C2×C22⋊#88322C411yescharacteristic, complemented
12C2×#13133129—21no—
13#16810377—21no—
14C22×M16×C53C411yescharacteristic, complemented
15C2×#11719940—22no—
16C2×C22⋊#6157—22no—
17C2×C4×#6157—41no—
18#13133277—24no—
19#13133275—24no—
20#13133272—41no—
21C4×#88324—41no—
22(C2×#6157)⋊C4—41no—
23C22×C22⋊D53D411yescharacteristic
24C22×C4×D53D411yescharacteristic
25C22×#88322—21no—
26C4×C22⋊D53—41no—
27C4×C22⋊D53—41no—
28C2×#11719919C2×C411yescharacteristic
29C2×C22⋊Dic53Q811yescharacteristic
30C2×#11719921C2×C411yescharacteristic
31C2×C4×Dic53D411yescharacteristic
32C22⋊#88322—41no—
33#13133129—42no—
34C23×C4×C53C2×C411yescharacteristic
35C2×M16×C53—41no—
36C2×M16×C53—42no—
37C22×C8×C53—21no—
38C24×C53C4⋊C411yescharacteristic
39C23×D53C22⋊C411yescharacteristic
40C22×#6157—42no—
41C22×C4×C53—21no—
42C22×C4×C53—21no—
43C22×C4×C53—41no—
44C22×C4×C53—41no—
45C22×C4×C53C4211yescharacteristic
46C22×C4×C53C22⋊C411yescharacteristic
47C2×C22⋊D53—22no—
48C2×C22⋊D53—22no—
49C2×C22⋊D53—41no—
50C22×Dic53C22⋊C411yescharacteristic
51C22×Dic53C4⋊C411yescharacteristic
52C22×Dic53C4⋊C411yescharacteristic
53C2×C4×D53—21no—
54C2×C4×D53—21no—
55C2×C4×D53—41no—
56C2×C4×D53—41no—
57C22⋊#6157—24no—
58C22⋊#6157—42no—
59#11719940—24no—
60#11719940—42no—
61C2×#88324—42no—
62C4×#6157—81no—
63C4×#6157—81no—
64M16×C53—42no—
65M16×C53—42no—
66C2×C8×C53—22no—
67C2×C8×C53—41no—
68C2×C8×C53—41no—
69C22⋊Dic53—41no—
70#11719921—41no—
71#11719919—41no—
72C4×Dic53—21no—
73C4×Dic53—21no—
74C2×#88322—41no—
75C2×#88322—41no—
76C2×#88322—41no—
77C23×C53—21no—
78C23×C53—21no—
79C23×C53—21no—
80C23×C53—41no—
81C23×C53—41no—
82C23×C53#9511yescharacteristic
83C22×D53—21no—
84C22×D53—21no—
85C22×D53—41no—
86C22×D53—41no—
87C22×D53#10412yes—
88C2×#6157—44no—
89C2×#6157—82no—
90C2×#6157—82no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—42no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53#9911yescharacteristic
99C2×C4×C53#9911yescharacteristic
100C22⋊D53—44no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53—41no—
106C2×Dic53—41no—
107C2×Dic53#10412yes—
108C4×D53—41no—
109C4×D53—41no—
110C4×D53—81no—
111#88324—44no—
112C8×C53—42no—
113#88322—81no—
114#7282—531nomaximal, Sylow
115C22×C53—21no—
116C22×C53—21no—
117C22×C53—41no—
118C22×C53—41no—
119C22×C53—41no—
120C22×C53—81no—
121C22×C53—81no—
122C22×C53#33911yescharacteristic
123C22×C53#33911yescharacteristic
124C22×C53#34211yescharacteristic
125C2×D53—22no—
126C2×D53—41no—
127C2×D53—41no—
128C2×D53—42no—
129C2×D53—81no—
130#6157—84no—
131C4×C53—21no—
132C4×C53—41no—
133C4×C53AΣL(1,𝔽16)11yescharacteristic
134C4×C53#35211yescharacteristic
135Dic53—22no—
136Dic53—41no—
137Dic53—41no—
138#1870—531no—
139C4×#100—531no—
140C4×#99—531no—
141C2×C53—21no—
142C2×C53—21no—
143C2×C53—81no—
144C2×C53#129711yescharacteristic
145C2×C53#139011yescharacteristic
146C2×C53#139511yescharacteristic
147D53—42no—
148C22×M16#615711yescharacteristic, complemented
149C2×C4×D4—531no—
150C2×#100—1061no—
151C2×#99—1061no—
152C2×C22⋊C8—531no—
153C2×C22⋊C8—1061no—
154C4×M16—1061no—
155C4×C22⋊C4—1061no—
156#343—532no—
157#342—532no—
158#336—1061no—
159C53#728211yescharacteristic, complemented, Sylow
160C22×D4—531no—
161C23×C4—531no—
162C23×C4C2×#615711yescharacteristic
163C2×M16—42no—
164C2×M16—41no—
165C2×M16—1062no—
166C22×C8—21no—
167C22×C8—1061no—
168C4×D4—2121no—
169C4×D4—2121no—
170C2×C4⋊C4—531no—
171C2×C22⋊C4—531no—
172C2×C22⋊C4—531no—
173C2×C22⋊C4—1062no—
174C2×C42—531no—
175C2×C42—2121no—
176#100—1064no—
177#99—1064no—
178C22⋊C8—2122no—
179C22⋊C8—2121no—
180#97—2121no—
181C4×C8—2121no—
182#95—2121no—
183C24—531no—
184C24C4⋊#615711yescharacteristic
185C2×D4—1062no—
186C2×D4—1062no—
187C2×D4—2121no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—531no—
193C22×C4—531no—
194C22×C4—531no—
195C22×C4—1061no—
196C22×C4—1061no—
197C22×C4—2121no—
198C22×C4—2121no—
199C22×C4—2122no—
200C22×C4C4×#615711yescharacteristic
201C22×C4C22⋊#615711yescharacteristic
202M16—42no—
203M16—42no—
204M16—1064no—
205M16—2122no—
206C2×C8—22no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—2121no—
210C2×C8—2121no—
211C2×C8—2121no—
212C2×C8—2122no—
213C4⋊C4—2121no—
214C22⋊C4—1064no—
215C22⋊C4—2121no—
216C22⋊C4—2121no—
217C22⋊C4—2122no—
218C42—1061no—
219C42—1061no—
220C42—4241no—
221C42—4241no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—41no—
226C23—41no—
227C23—532no—
228C23—1061no—
229C23—1061no—
230C23—2121no—
231C23—2121no—
232C23(C2×#6157)⋊C411yescharacteristic
233D4—2124no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—532no—
242C2×C4—1061no—
243C2×C4—1061no—
244C2×C4—1061no—
245C2×C4—1061no—
246C2×C4—4241no—
247C2×C4—4242no—
248C2×C4—4242no—
249C2×C4—2124no—
250C2×C4—2121no—
251C2×C4—2121no—
252C2×C4—2121no—
253C2×C4—2121no—
254C2×C4#1313327511yescharacteristic
255C2×C4#1313327611yescharacteristic
256C8—42no—
257C8—4241no—
258C8—2124no—
259C22—21no—
260C22—21no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—81no—
265C22—81no—
266C22—1062no—
267C22—4241no—
268C22—2122no—
269C22—2121no—
270C22—2121no—
271C22#1681149511yescharacteristic
272C22#1681149511yescharacteristic
273C22#1681150111yescharacteristic
274C4—21no—
275C4—41no—
276C4—1062no—
277C4—4244no—
278C4—2121no—
279C4—2121no—
280C4#1681152311yescharacteristic
281C4#1681152511yescharacteristic
282C2—21no—
283C2—21no—
284C2—81no—
285C2—2122no—
286C2#2028959511yescharacteristic
287C2#2028979911yescharacteristic
288C2#2028981611yescharacteristic
subgroupG/H
centerC2×C4#13133275
commutatorC22×C4×C53C42
FrattiniC23×C4C2×#6157
FittingC22×M16×C53C4
socleC22×C53#342
prime radicalC22×C22⋊D53D4
cosocleC2×C4×C22⋊D53C22
hypercenterC22×M16#6157
hypocenterC53#7282
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
8. #16811503C4semidirect
11. C2×C22⋊#88322C4semidirect
14. C22×M16×C53C4semidirect
148. C22×M16#6157semidirect
159. C53#7282semidirect

Maximal subgroups

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

subgroupindex
C4×#131332772
C4×#131332752
#202750122
#728253