Finite Group Databasebeta

← #28301907

Subgroups — #28301907

Order 13568.

534 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C8×C22⋊#6157C212yesmaximal, complemented
2#20290092C212yesmaximal
3#20289597C211yesmaximal, characteristic, complemented
4#20289595C211yesmaximal, characteristic, complemented
5#20283408C211yesmaximal, characteristic
6D53⋊(C22⋊C8)C2212yes—
7C2×C8×#6157—22no—
8C4×C22⋊#6157C2212yes—
9(C2×#6157)⋊C8—22no—
10#16811476C414yescomplemented
11#16811475C414yescomplemented
12C2×#13133225C411yescharacteristic, complemented
13C22×C8×D53C2211yescharacteristic
14#16811135C2211yescharacteristic
15#16810693—22no—
16#16810688—22no—
17#16810589C411yescharacteristic, complemented
18#16810586C2211yescharacteristic
19#16810579C411yescharacteristic, complemented
20C53×#424C411yescharacteristic, complemented
21C2×D53⋊C8—22no—
22C2×C22⋊#6157—22no—
23C2×C4×#6157—22no—
24#13133274C2×C414yes—
25C22⋊#88324—22no—
26C8×#6157—44no—
27(C2×#6157)⋊C4C2×C414yes—
28C2×#11719945C2×C411yescharacteristic
29C22×C4×D53C2311yescharacteristic
30#13133225—42no—
31C2×#11719917C2×C411yescharacteristic
32C2×#11719912—21no—
33C22×#88322C2×C411yescharacteristic
34C2×C8×D53—21no—
35C2×C8×D53—42no—
36C2×C8×D53D412yes—
37C2×C8×D53D412yes—
38#13133201—21no—
39#13133197C2×C412yes—
40#13133192C2×C412yes—
41#13133187—21no—
42C4×D212—22no—
43#13133129—22no—
44#13133129—22no—
45#13133128—21no—
46C4×#88322—21no—
47C2×M16×C53C2×C411yescharacteristic
48C22×C8×C53C2×C411yescharacteristic
49#117×C53C2×C411yescharacteristic
50C4⋊C8×C53—22no—
51C23×D53#4111yescharacteristic
52C22×#6157—24no—
53C22×C4×C53C22×C411yescharacteristic
54C2×C22⋊D53—21no—
55C22×Dic53#4111yescharacteristic
56#11719945—41no—
57#11719945—41no—
58C2×D212—22no—
59C2×D212C4211yescharacteristic
60C2×C4×D53—21no—
61C2×C4×D53—41no—
62C2×C4×D53—41no—
63C2×C4×D53C22×C412yes—
64C2×C4×D53C2×D412yes—
65C2×Dic106C4211yescharacteristic
66C22⋊#6157—28no—
67C2×#88324—24no—
68C4×#6157—42no—
69C4×#6157—42no—
70D53⋊C8—42no—
71D53⋊C8—42no—
72M16×C53—41no—
73C2×C8×C53—21no—
74C2×C8×C53—41no—
75C2×C8×C53C22⋊C412yes—
76C4⋊C4×C53C4212yes—
77C22⋊C4×C53—21no—
78C42×C53—21no—
79#11719921—22no—
80#11719921—22no—
81C4⋊Dic53—22no—
82#11719917—41no—
83C2×#88322—22no—
84C2×#88322—21no—
85C2×#88322C22⋊C411yescharacteristic
86C2×#88322C22⋊C411yescharacteristic
87#11719912—42no—
88C8×D53—24no—
89C8×D53—42no—
90C8×D53—42no—
91C23×C53#11711yescharacteristic
92C22×D53—24no—
93C22×D53—21no—
94C22×D53—41no—
95C22×D53C4×D412yes—
96C2×#6157—28no—
97C2×#6157—44no—
98C2×#6157—44no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53C2×C4211yescharacteristic
103C2×C4×C53C2×C22⋊C411yescharacteristic
104C22⋊D53—42no—
105C2×Dic53—21no—
106C2×Dic53—41no—
107C2×Dic53C4×D412yes—
108D212—41no—
109D212—44no—
110C4×D53—22no—
111C4×D53—22no—
112C4×D53—42no—
113C4×D53—41no—
114C4×D53—41no—
115Dic106—41no—
116#88324—44no—
117C8×C53—22no—
118C8×C53—41no—
119#88322—21no—
120#88322—41no—
121#88322#9912yes—
122#8238—531nomaximal, Sylow
123C22×C53—21no—
124C22×C53—22no—
125C22×C53C4×C22⋊C411yescharacteristic
126C2×D53—22no—
127C2×D53—22no—
128C2×D53—44no—
129C2×D53—41no—
130C2×D53—42no—
131C2×D53#44314yes—
132#6157—48no—
133C4×C53—21no—
134C4×C53—42no—
135C4×C53C2×#9911yescharacteristic
136C4×C53#41011yescharacteristic
137Dic53—22no—
138Dic53—41no—
139#2929—531no—
140C8×C22⋊C4—532no—
141#1501—532no—
142#1298—531no—
143#1297—531no—
144C2×C53—41no—
145C2×C53C4×#9911yescharacteristic
146C2×C53#176412yes—
147D53—24no—
148D53—81no—
149C2×#131—531no—
150C23×C8—531no—
151#525—531no—
152#435—1062no—
153C8×D4—1062no—
154#424#615711yescharacteristic, complemented
155#408—531no—
156#407—531no—
157C2×C22⋊C8—532no—
158#405—531no—
159C2×C4×C8—1062no—
160C4×C22⋊C4—532no—
161#336—1062no—
162#328—534no—
163#327—534no—
164C53#823811yescharacteristic, complemented, Sylow
165C2×#41—531no—
166C23×C4—531no—
167#131—2122no—
168C2×M16—531no—
169C2×M16—1061no—
170C2×M16C2×#615711yescharacteristic
171C22×C8—531no—
172C22×C8—532no—
173C22×C8—532no—
174C22×C8—1061no—
175C22×C8—1062no—
176C22×C8—2122no—
177C22×C8C2×#615711yescharacteristic
178#123—1061no—
179C22⋊Q8—532no—
180#121—532no—
181#120—1061no—
182C4×D4—1062no—
183#117C2×#615711yescharacteristic
184C2×C22⋊C4—1062no—
185C2×C42—1062no—
186C4⋊C8—22no—
187C22⋊C8—534no—
188C22⋊C8—1062no—
189C22⋊C8—1062no—
190C22⋊C8—1062no—
191#97—1061no—
192C4×C8—1061no—
193C4×C8—2124no—
194#95—534no—
195C24—531no—
196#41—2121no—
197#41—2121no—
198C2×Q8—531no—
199C2×D4—531no—
200C2×D4—1062no—
201C2×D4—1061no—
202C22×C4—531no—
203C22×C4—532no—
204C22×C4—532no—
205C22×C4—1064no—
206C22×C4—1061no—
207C22×C4—2121no—
208C22×C4—2121no—
209C22×C4C22×#615711yescharacteristic
210M16—41no—
211M16—2122no—
212M16—2121no—
213C2×C8—21no—
214C2×C8—41no—
215C2×C8—531no—
216C2×C8—531no—
217C2×C8—1064no—
218C2×C8—1062no—
219C2×C8—1064no—
220C2×C8—1061no—
221C2×C8—2122no—
222C2×C8—2122no—
223C2×C8—2122no—
224C2×C8—2122no—
225C2×C8C22⋊#615712yes—
226C4⋊C4—1062no—
227C4⋊C4C4×#615712yes—
228C22⋊C4—21no—
229C22⋊C4—1062no—
230C22⋊C4—1062no—
231C22⋊C4—1068no—
232C42—21no—
233C42—2122no—
234C42—2122no—
235C23—532no—
236C23—1064no—
237C23—1061no—
238C23—2121no—
239C23#1313329611yescharacteristic
240Q8—2121no—
241D4—2121no—
242D4—2124no—
243D4—2122no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—532no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—1068no—
251C2×C4—1061no—
252C2×C4—2122no—
253C2×C4—2121no—
254C2×C4—2124no—
255C2×C4—2124no—
256C2×C4—2121no—
257C2×C4—2121no—
258C2×C4C2×C4×#615711yescharacteristic
259C2×C4C2×C22⋊#615711yescharacteristic
260C8—22no—
261C8—41no—
262C8—532no—
263C8—1061no—
264C8—2124no—
265C8—2121no—
266C22—21no—
267C22—22no—
268C22—534no—
269C22—1062no—
270C22—1062no—
271C22—2124no—
272C22—2121no—
273C22—2122no—
274C22C4×C22⋊#615711yescharacteristic
275C4—21no—
276C4—42no—
277C4—1062no—
278C4—2128no—
279C4—2121no—
280C4D53⋊#9911yescharacteristic
281C4#1681160411yescharacteristic
282C2—41no—
283C2—1064no—
284C2—4241no—
285C2#2029046011yescharacteristic
286C2#2029048912yes—
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC53×#424C4
socleC22×C53C4×C22⋊C4
prime radical#16811135C22
cosocleC22×C4×D53C23
hypercenter#424#6157
hypocenterC53#8238
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #424

Complements

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

NH ≅ G/Ntype
1. C8×C22⋊#6157C2semidirect
3. #20289597C2semidirect
4. #20289595C2semidirect
10. #16811476C4semidirect
11. #16811475C4semidirect
12. C2×#13133225C4semidirect
17. #16810589C4semidirect
19. #16810579C4semidirect
20. C53×#424C4semidirect
154. #424#6157semidirect
164. C53#8238semidirect

Maximal subgroups

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

subgroupindex
C8×C22⋊#61572
#202900922
#202895972
#202895952
#202834082
#823853