Finite Group Databasebeta

← #28309679

Subgroups — #28309679

Order 13568.

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

normal subgroups only

This group has 306 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
1#20291074C211yesmaximal, characteristic, complemented
2#20290265C211yesmaximal, characteristic, complemented
3#20289731C212yesmaximal, complemented
4#20289727C212yesmaximal, complemented
5#20285283C211yesmaximal, characteristic
6#16811642—21no—
7#16811544C2212yes—
8#16811518—21no—
9C4×#11719910—21no—
10#16811486—21no—
11#16811485C2212yes—
12#16811155—21no—
13#16811024C2211yescharacteristic
14#16811006C2211yescharacteristic
15#16810976—21no—
16#16810865C411yescharacteristic
17#16810856C411yescharacteristic
18#16810851—21no—
19C4×Dic212C411yescharacteristic
20C4×#11719923—21no—
21C4×C8×D53C2211yescharacteristic
22C4×D8×C53C411yescharacteristic
23C2×#11719910—22no—
24C4⋊#88324—22no—
25#13133274—41no—
26C4×#88324—41no—
27C2×Dic212C2×C411yescharacteristic
28C2×#11719923—21no—
29C2×C8×D53C2×C411yescharacteristic
30C2×C8×D53D411yescharacteristic
31#13133210—41no—
32#13133208—21no—
33#13133204—21no—
34#13133199—41no—
35#13133197—21no—
36#13133188—21no—
37C4×Dic106—21no—
38C4×C22⋊D53—41no—
39#13133176—41no—
40C42×D53D411yescharacteristic
41#13133151C2×C411yescharacteristic
42#13133148—21no—
43#13133142—21no—
44#13133140—21no—
45#13133138—21no—
46C4×#88322D411yescharacteristic
47C8×Dic53D411yescharacteristic
48C8×Dic53C2311yescharacteristic
49C2×D8×C53C2×C411yescharacteristic
50C4×D4×C53—21no—
51#107×C53C2×C411yescharacteristic
52#102×C53—21no—
53C4×C8×C53C2×C411yescharacteristic
54C2×D4×C53—21no—
55C22×C4×C53—41no—
56C2×C22⋊D53—41no—
57C2×C4×D53C22⋊C411yescharacteristic
58C2×C4×D53D811yescharacteristic
59C2×C4×D53SD1611yescharacteristic
60C2×Dic106—21no—
61C2×#88324—42no—
62D8×C53—41no—
63C2×C8×C53C22⋊C411yescharacteristic
64C2×C8×C53C22×C411yescharacteristic
65C4⋊C4×C53—21no—
66C22⋊C4×C53—41no—
67C42×C53C22⋊C411yescharacteristic
68C22⋊Dic53—41no—
69C22⋊Dic53—41no—
70C22⋊Dic53—41no—
71#11719923—42no—
72#11719921—41no—
73C4⋊Dic53—21no—
74C4⋊Dic53—41no—
75C4⋊Dic53—41no—
76#11719919—41no—
77#11719919—41no—
78C4×Dic53—41no—
79C4×Dic53D811yescharacteristic
80C4×Dic53SD1611yescharacteristic
81C4×Dic53C2×D411yescharacteristic
82C2×#88322C22⋊C411yescharacteristic
83C2×#88322C2×D411yescharacteristic
84Dic212—41no—
85C8×D53—22no—
86C8×D53—41no—
87#11719910—42no—
88C23×C53—41no—
89C22×D53#10211yescharacteristic
90D4×C53—41no—
91D4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—82no—
94C2×C4×C53#10211yescharacteristic
95C2×C4×C53#10211yescharacteristic
96C2×C4×C53C2×C22⋊C411yescharacteristic
97C22⋊D53—82no—
98C2×Dic53—41no—
99C2×Dic53—41no—
100C2×Dic53#10211yescharacteristic
101C2×Dic53C2×D811yescharacteristic
102C2×Dic53C2×SD1611yescharacteristic
103C4×D53—22no—
104C4×D53—21no—
105C4×D53—21no—
106C4×D53—41no—
107Dic106—41no—
108Dic106—41no—
109#88324—82no—
110C8×C53—21no—
111C8×C53—21no—
112#88322—21no—
113#88322#13512yes—
114#11943—531nomaximal, Sylow
115C22×C53—81no—
116C22×C53—81no—
117C22×C53—81no—
118C22×C53C2×#10211yescharacteristic
119C2×D53—21no—
120C2×D53—21no—
121C2×D53—21no—
122C4×C53—21no—
123C4×C53—81no—
124C4×C53#35912yes—
125C4×C53#41311yescharacteristic
126C4×C53#41611yescharacteristic
127Dic53—21no—
128Dic53—21no—
129Dic53—21no—
130Dic53—81no—
131Dic53—81no—
132#3168—531no—
133#2162—531no—
134#1564—531no—
135#1363—532no—
136#1360—532no—
137C2×C53—161no—
138C2×C53#154011yescharacteristic
139C2×C53C2×#35911yescharacteristic
140C2×C53#214511yescharacteristic
141D53—41no—
142#529—1061no—
143#499—531no—
144#495—531no—
145#488—1061no—
146#467—531no—
147#466—531no—
148#465—1061no—
149C4×Q16—531no—
150C4×SD16—1061no—
151C4×D8#615711yescharacteristic
152#431—1061no—
153C2×C4×C8—531no—
154C4⋊C16—532no—
155C22⋊C16—1061no—
156C4×C16—1061no—
157#335—1061no—
158#334—532no—
159C53#1194311yescharacteristic, complemented, Sylow
160C2×Q16—531no—
161C2×SD16—1061no—
162C2×D8C2×#615711yescharacteristic
163C22×C8—531no—
164C22×C8—531no—
165#126—2121no—
166#125—1061no—
167#124—1061no—
168#123—2121no—
169C22⋊Q8—1061no—
170#121—1061no—
171C4×Q8—1061no—
172C4×D4—21no—
173C4×D4—2121no—
174#117—2121no—
175C2×C42—531no—
176C2×C16—1062no—
177#107—531no—
178#107C2×#615711yescharacteristic
179#106—1061no—
180C4⋊C8—1062no—
181#103—1061no—
182#103—1061no—
183#102—21no—
184#102—1061no—
185C22⋊C8—2121no—
186C4×C8—531no—
187C4×C8—531no—
188C4×C8—531no—
189C4×C8—2121no—
190C4×C8C2×#615711yescharacteristic
191C2×Q8—1061no—
192C2×D4—21no—
193C2×D4—2121no—
194C22×C4—41no—
195C22×C4—531no—
196C22×C4—531no—
197C22×C4—531no—
198Q16—2121no—
199SD16—2122no—
200D8—41no—
201C2×C8—531no—
202C2×C8—531no—
203C2×C8—1062no—
204C2×C8—2121no—
205C2×C8—2122no—
206C2×C8C22⋊#615711yescharacteristic
207C2×C8C22×#615711yescharacteristic
208C4⋊C4—21no—
209C4⋊C4—1061no—
210C4⋊C4—2121no—
211C4⋊C4—2121no—
212C4⋊C4—2121no—
213C4⋊C4—2121no—
214C22⋊C4—41no—
215C22⋊C4—2121no—
216C22⋊C4—2121no—
217C22⋊C4—2121no—
218C22⋊C4—2121no—
219C42—531no—
220C42—531no—
221C42—531no—
222C42—2121no—
223C42C22⋊#615711yescharacteristic
224C16—2122no—
225C23—41no—
226C23—531no—
227Q8—2121no—
228Q8—2121no—
229D4—41no—
230D4—41no—
231D4—4242no—
232C2×C4—41no—
233C2×C4—82no—
234C2×C4—531no—
235C2×C4—531no—
236C2×C4—531no—
237C2×C4—1062no—
238C2×C4—1061no—
239C2×C4—1061no—
240C2×C4—2121no—
241C2×C4—2121no—
242C2×C4—2121no—
243C2×C4#1313327911yescharacteristic
244C2×C4#1313327911yescharacteristic
245C2×C4C2×C22⋊#615711yescharacteristic
246C8—21no—
247C8—21no—
248C8—532no—
249C8—1061no—
250C8—4242no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22—1061no—
255C22—1061no—
256C22—1061no—
257C22C2×#1313327911yescharacteristic
258C4—21no—
259C4—81no—
260C4—1061no—
261C4—1061no—
262C4—1061no—
263C4—4241no—
264C4—4241no—
265C4#1681153712yes—
266C4#1681160911yescharacteristic
267C4#1681161311yescharacteristic
268C2—161no—
269C2—2121no—
270C2#2029021211yescharacteristic
271C2C2×#1681153711yescharacteristic
272C2#2029103511yescharacteristic
subgroupG/H
centerC2×C4#13133279
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC4×D8×C53C4
socleC22×C53C2×#102
prime radical#16810856C4
cosocleC8×Dic53C23
hypercenterC4×D8#6157
hypocenterC53#11943
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #20291074C2semidirect
2. #20290265C2semidirect
3. #20289731C2semidirect
4. #20289727C2semidirect
159. C53#11943semidirect

Maximal subgroups

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

subgroupindex
#202910742
#202902652
#202897312
#202897272
#202852832
#1194353