Finite Group Databasebeta

← #28309769

Subgroups — #28309769

Order 13568.

312 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C2×#16811544C211yesmaximal, characteristic, complemented
2#20290277C211yesmaximal, characteristic, complemented
3#20289731C212yesmaximal, complemented
4#20289727C212yesmaximal, complemented
5#20285280C211yesmaximal, characteristic
6C22×#11719910—21no—
7#16811644—21no—
8#16811544—21no—
9#16811544C2212yes—
10#16811486—21no—
11#16811485C2212yes—
12#16811154—21no—
13#16811024C2211yescharacteristic
14#16811006C2211yescharacteristic
15#16810976—21no—
16#16810864C411yescharacteristic
17#16810850—21no—
18#16810717C411yescharacteristic
19#16810714—21no—
20D8×Dic53C411yescharacteristic
21C2×C8×Dic53C2211yescharacteristic
22C53×#466C411yescharacteristic
23C2×#11719910—22no—
24C2×#11719910—42no—
25C4⋊#88324—22no—
26C22⋊#88324—41no—
27#13133272—41no—
28C2×Dic212C2×C411yescharacteristic
29C2×#11719923—21no—
30C22×#88322D411yescharacteristic
31#13133210—41no—
32#13133208—21no—
33#13133204—21no—
34#13133198—41no—
35#13133196—21no—
36#13133182—21no—
37D4×Dic53—21no—
38D4×Dic53—41no—
39#13133173—41no—
40C2×C4×Dic53D411yescharacteristic
41#13133151C2×C411yescharacteristic
42#13133148—21no—
43#13133142—21no—
44#13133140—21no—
45#13133138—21no—
46C8×Dic53C2×C411yescharacteristic
47C8×Dic53D411yescharacteristic
48C8×Dic53D411yescharacteristic
49C8×Dic53C2311yescharacteristic
50C2×D8×C53C2×C411yescharacteristic
51C22×C8×C53C2×C411yescharacteristic
52#121×C53—21no—
53#107×C53C2×C411yescharacteristic
54#102×C53—21no—
55C2×D4×C53—21no—
56C2×D4×C53—41no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×Dic53—41no—
59C22×Dic53D811yescharacteristic
60C22×Dic53SD1611yescharacteristic
61C2×Dic106—21no—
62C2×#88324—42no—
63D8×C53—41no—
64C2×C8×C53—41no—
65C2×C8×C53C22⋊C411yescharacteristic
66C2×C8×C53C22×C411yescharacteristic
67C4⋊C4×C53—21no—
68C22⋊C4×C53—41no—
69C22⋊Dic53—41no—
70C22⋊Dic53—41no—
71C22⋊Dic53—41no—
72C22⋊Dic53—41no—
73#11719923—41no—
74#11719923—41no—
75C4⋊Dic53—21no—
76C4⋊Dic53—41no—
77#11719919—41no—
78#11719919—41no—
79#11719919—41no—
80C4×Dic53—41no—
81C4×Dic53C22⋊C411yescharacteristic
82C4×Dic53D811yescharacteristic
83C4×Dic53SD1611yescharacteristic
84C4×Dic53C2×D411yescharacteristic
85C2×#88322—21no—
86C2×#88322—21no—
87C2×#88322C22⋊C411yescharacteristic
88C2×#88322C2×D411yescharacteristic
89Dic212—41no—
90#11719910—42no—
91C23×C53—41no—
92C23×C53#10211yescharacteristic
93D4×C53—41no—
94D4×C53—41no—
95D4×C53—82no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—41no—
99C2×C4×C53#10211yescharacteristic
100C2×C4×C53C2×C22⋊C411yescharacteristic
101C2×Dic53—22no—
102C2×Dic53—41no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53—82no—
106C2×Dic53#10211yescharacteristic
107C2×Dic53#10211yescharacteristic
108C2×Dic53C2×D811yescharacteristic
109C2×Dic53C2×SD1611yescharacteristic
110Dic106—41no—
111Dic106—41no—
112#88324—82no—
113C8×C53—21no—
114C8×C53—21no—
115#88322—21no—
116#88322AGL(1,ℤ/8)11yescharacteristic
117#88322#13711yescharacteristic
118#11978—531nomaximal, Sylow
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—81no—
122C22×C53—81no—
123C22×C53—81no—
124C22×C53#35912yes—
125C22×C53C2×#10211yescharacteristic
126C4×C53—21no—
127C4×C53—81no—
128C4×C53#41311yescharacteristic
129C4×C53#41711yescharacteristic
130Dic53—21no—
131Dic53—21no—
132Dic53—21no—
133Dic53—21no—
134Dic53—81no—
135Dic53—81no—
136#3168—531no—
137C2×C4⋊C16—531no—
138#1568—531no—
139#1363—532no—
140#1360—532no—
141C2×C53—22no—
142C2×C53—161no—
143C2×C53#154711yescharacteristic
144C2×C53C2×#35911yescharacteristic
145C2×C53#214511yescharacteristic
146#529—1061no—
147C22×C16—1061no—
148#499—531no—
149#495—531no—
150#488—1061no—
151#467—531no—
152#466#615711yescharacteristic
153#465—1061no—
154C4×Q16—531no—
155C4×SD16—1061no—
156C4×D8—531no—
157#432—1061no—
158C2×C4×C8—531no—
159C4⋊C16—532no—
160C4⋊C16—1061no—
161#335—1061no—
162#334—532no—
163C53#1197811yescharacteristic, complemented, Sylow
164C2×Q16—531no—
165C2×SD16—1061no—
166C2×D8C2×#615711yescharacteristic
167C22×C8—531no—
168C22×C8C2×#615711yescharacteristic
169#126—2121no—
170#125—1061no—
171#124—1061no—
172#123—2121no—
173C22⋊Q8—1061no—
174#121—21no—
175C4×Q8—1061no—
176C4×D4—1061no—
177C4×D4—2121no—
178#117—2121no—
179C2×C42—531no—
180C2×C16—1062no—
181C2×C16—2122no—
182#107—531no—
183#107C2×#615711yescharacteristic
184#106—1061no—
185C4⋊C8—1062no—
186#103—1061no—
187#103—1061no—
188#102—21no—
189#102—1061no—
190C22⋊C8—2121no—
191#97—2121no—
192C4×C8—531no—
193C4×C8—531no—
194C4×C8—531no—
195C4×C8—531no—
196C2×Q8—1061no—
197C2×D4—21no—
198C2×D4—41no—
199C22×C4—531no—
200C22×C4—531no—
201C22×C4—2121no—
202C22×C4C22⋊#615711yescharacteristic
203Q16—2121no—
204SD16—2121no—
205SD16—2121no—
206D8—41no—
207C2×C8—41no—
208C2×C8—531no—
209C2×C8—531no—
210C2×C8—1061no—
211C2×C8—1061no—
212C2×C8—2122no—
213C2×C8C22⋊#615711yescharacteristic
214C2×C8C22×#615711yescharacteristic
215C4⋊C4—21no—
216C4⋊C4—1061no—
217C4⋊C4—2121no—
218C4⋊C4—2121no—
219C4⋊C4—2121no—
220C4⋊C4—2121no—
221C22⋊C4—41no—
222C22⋊C4—2121no—
223C22⋊C4—2121no—
224C22⋊C4—2121no—
225C22⋊C4—2121no—
226C42—531no—
227C42—531no—
228C42—531no—
229C42—531no—
230C42—2121no—
231C16—2122no—
232C23—41no—
233C23#1313327911yescharacteristic
234Q8—2121no—
235Q8—2121no—
236D4—41no—
237D4—41no—
238D4—82no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—531no—
243C2×C4—531no—
244C2×C4—531no—
245C2×C4—531no—
246C2×C4—1062no—
247C2×C4—4242no—
248C2×C4—2121no—
249C2×C4—2121no—
250C2×C4—2121no—
251C2×C4#1313327911yescharacteristic
252C2×C4C2×C22⋊#615711yescharacteristic
253C8—21no—
254C8—21no—
255C8—531no—
256C8—531no—
257C8—1061no—
258C8—4242no—
259C22—21no—
260C22—21no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22#1681153712yes—
265C22C2×#1313327911yescharacteristic
266C4—21no—
267C4—81no—
268C4—1061no—
269C4—1061no—
270C4—1061no—
271C4—1061no—
272C4—4241no—
273C4—4241no—
274C4#1681161011yescharacteristic
275C4#1681161411yescharacteristic
276C2—22no—
277C2—161no—
278C2#2029022711yescharacteristic
279C2C2×#1681153711yescharacteristic
280C2#2029103511yescharacteristic
subgroupG/H
centerC22C2×#13133279
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
Fitting, prime radicalC53×#466C4
socleC22×C53C2×#102
cosocleC8×Dic53C23
hypercenter#466#6157
hypocenterC53#11978
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#16811544C2semidirect
2. #20290277C2semidirect
3. #20289731C2semidirect
4. #20289727C2semidirect
163. C53#11978semidirect

Maximal subgroups

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

subgroupindex
C2×#168115442
#202902772
#202897312
#202897272
#202852802
#1197853