Finite Group Databasebeta

← #28309687

Subgroups — #28309687

Order 13568.

312 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 76 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#20291078C211yesmaximal, characteristic, complemented
2D53⋊#334C211yesmaximal, characteristic, complemented
3#20289731C212yesmaximal, complemented
4#20289727C212yesmaximal, complemented
5#20285283C211yesmaximal, characteristic
6D53⋊(C4⋊C8)—21no—
7#16811544C2212yes—
8#16811518—21no—
9#16811511—21no—
10#16811485C2212yes—
11#16811484—21no—
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×D53⋊C8—41no—
24C2×#11719910—22no—
25#13133284—41no—
26C4⋊#88324—22no—
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—
62D53⋊C8—82no—
63D8×C53—41no—
64C2×C8×C53C22⋊C411yescharacteristic
65C2×C8×C53C22×C411yescharacteristic
66C4⋊C4×C53—21no—
67C22⋊C4×C53—41no—
68C42×C53C22⋊C411yescharacteristic
69C22⋊Dic53—41no—
70C22⋊Dic53—41no—
71C22⋊Dic53—41no—
72#11719923—42no—
73#11719921—41no—
74C4⋊Dic53—21no—
75C4⋊Dic53—41no—
76C4⋊Dic53—41no—
77#11719919—41no—
78#11719919—41no—
79C4×Dic53—41no—
80C4×Dic53D811yescharacteristic
81C4×Dic53SD1611yescharacteristic
82C4×Dic53C2×D411yescharacteristic
83C2×#88322C22⋊C411yescharacteristic
84C2×#88322C2×D411yescharacteristic
85Dic212—41no—
86C8×D53—22no—
87C8×D53—41no—
88#11719910—42no—
89C23×C53—41no—
90C22×D53#10211yescharacteristic
91D4×C53—41no—
92D4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—82no—
95C2×C4×C53#10211yescharacteristic
96C2×C4×C53#10211yescharacteristic
97C2×C4×C53C2×C22⋊C411yescharacteristic
98C22⋊D53—82no—
99C2×Dic53—41no—
100C2×Dic53—41no—
101C2×Dic53#10211yescharacteristic
102C2×Dic53C2×D811yescharacteristic
103C2×Dic53C2×SD1611yescharacteristic
104C4×D53—22no—
105C4×D53—21no—
106C4×D53—21no—
107C4×D53—41no—
108Dic106—41no—
109Dic106—41no—
110#88324—82no—
111C8×C53—21no—
112C8×C53—21no—
113#88322—21no—
114#88322#13512yes—
115#11945—531nomaximal, Sylow
116C22×C53—81no—
117C22×C53—81no—
118C22×C53—81no—
119C22×C53C2×#10211yescharacteristic
120C2×D53—21no—
121C2×D53—21no—
122C2×D53#36012yes—
123C4×C53—21no—
124C4×C53—21no—
125C4×C53—81no—
126C4×C53#41311yescharacteristic
127C4×C53#41611yescharacteristic
128Dic53—21no—
129Dic53—21no—
130Dic53—21no—
131Dic53—81no—
132Dic53—81no—
133#3168—531no—
134#2163—531no—
135C2×#334—531no—
136#1363—532no—
137#1360—532no—
138C2×C53—161no—
139C2×C53#154011yescharacteristic
140C2×C53#214311yescharacteristic
141C2×C53C2×#36011yescharacteristic
142D53—22no—
143#529—1061no—
144#499—531no—
145#495—531no—
146#488—1061no—
147#467—531no—
148#466—531no—
149#465—1061no—
150C4×Q16—531no—
151C4×SD16—1061no—
152C4×D8#615711yescharacteristic
153C2×C4⋊C8—1061no—
154C2×C4×C8—531no—
155C4⋊C16—532no—
156C22⋊C16—1061no—
157#346—1061no—
158#334—532no—
159#334—1061no—
160C53#1194511yescharacteristic, complemented, Sylow
161C2×Q16—531no—
162C2×SD16—1061no—
163C2×D8C2×#615711yescharacteristic
164C22×C8—531no—
165C22×C8—531no—
166C22×C8—2121no—
167#126—2121no—
168#125—1061no—
169#124—1061no—
170#123—2121no—
171C22⋊Q8—1061no—
172#121—1061no—
173C4×Q8—1061no—
174C4×D4—21no—
175C4×D4—2121no—
176#117—2121no—
177C2×C42—531no—
178C2×C16—1062no—
179#107—531no—
180#107C2×#615711yescharacteristic
181#106—1061no—
182C4⋊C8—1062no—
183C4⋊C8—2121no—
184#103—1061no—
185#103—1061no—
186#102—21no—
187#102—1061no—
188C4×C8—531no—
189C4×C8—531no—
190C4×C8—531no—
191C4×C8C2×#615711yescharacteristic
192C2×Q8—1061no—
193C2×D4—21no—
194C2×D4—2121no—
195C22×C4—41no—
196C22×C4—531no—
197C22×C4—531no—
198C22×C4—531no—
199Q16—2121no—
200SD16—2122no—
201D8—41no—
202C2×C8—531no—
203C2×C8—531no—
204C2×C8—1062no—
205C2×C8—4242no—
206C2×C8—2121no—
207C2×C8—2122no—
208C2×C8C22⋊#615711yescharacteristic
209C2×C8C22×#615711yescharacteristic
210C4⋊C4—21no—
211C4⋊C4—1061no—
212C4⋊C4—2121no—
213C4⋊C4—2121no—
214C4⋊C4—2121no—
215C4⋊C4—2121no—
216C22⋊C4—41no—
217C22⋊C4—2121no—
218C22⋊C4—2121no—
219C22⋊C4—2121no—
220C22⋊C4—2121no—
221C42—531no—
222C42—531no—
223C42—531no—
224C42—2121no—
225C42C22⋊#615711yescharacteristic
226C16—2122no—
227C23—41no—
228C23—531no—
229Q8—2121no—
230Q8—2121no—
231D4—41no—
232D4—41no—
233D4—4242no—
234C2×C4—41no—
235C2×C4—82no—
236C2×C4—531no—
237C2×C4—531no—
238C2×C4—531no—
239C2×C4—1062no—
240C2×C4—1061no—
241C2×C4—1061no—
242C2×C4—2121no—
243C2×C4—2121no—
244C2×C4—2121no—
245C2×C4#1313327911yescharacteristic
246C2×C4#1313327911yescharacteristic
247C2×C4C2×C22⋊#615711yescharacteristic
248C8—21no—
249C8—21no—
250C8—532no—
251C8—1061no—
252C8—4242no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22—532no—
257C22—1061no—
258C22—1061no—
259C22C2×#1313327911yescharacteristic
260C4—21no—
261C4—21no—
262C4—81no—
263C4—1061no—
264C4—1061no—
265C4—1061no—
266C4—4241no—
267C4—4241no—
268C4#1681160911yescharacteristic
269C4#1681161311yescharacteristic
270C2—161no—
271C2—1062no—
272C2#2029021211yescharacteristic
273C2#2029103011yescharacteristic
274C2D53⋊#36011yescharacteristic
subgroupG/H
centerC22C2×#13133279
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC4×D8×C53C4
socleC22×C53C2×#102
prime radical#16810856C4
cosocleC8×Dic53C23
hypercenterC4×D8#6157
hypocenterC53#11945
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #20291078C2semidirect
2. D53⋊#334C2semidirect
3. #20289731C2semidirect
4. #20289727C2semidirect
160. C53#11945semidirect

Maximal subgroups

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

subgroupindex
#202910782
D53⋊#3342
#202897312
#202897272
#202852832
#1194553