Finite Group Databasebeta

← #28312467

Subgroups — #28312467

Order 13568.

326 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 60 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#20290988C212yesmaximal
2#20289855C212yesmaximal, complemented
3#20289854C212yesmaximal, complemented
4#20285063C211yesmaximal, characteristic
5#16811602—22no—
6#16811530—22no—
7#16811528C2214yes—
8#16811173C2211yescharacteristic
9#16810993C411yescharacteristic, complemented
10#16810993C411yescharacteristic, complemented
11#16810989C2211yescharacteristic
12#16810989C2211yescharacteristic
13#16810985—22no—
14#16810617—22no—
15C2×#13133128C411yescharacteristic, complemented
16C53×#533C411yescharacteristic, complemented
17#13133296—42no—
18#13133276—24no—
19#13133275—42no—
20C2×#11719945D412yes—
21C22×#88322—21no—
22#13133217C2×C411yescharacteristic
23#13133214C2311yescharacteristic
24#13133205D412yes—
25#13133191—41no—
26C4×D212—41no—
27#13133140—22no—
28#13133140—22no—
29#13133137—22no—
30#13133137—22no—
31#13133128—21no—
32#13133128C2×C411yescharacteristic
33#13133128C2×C411yescharacteristic
34C4⋊Q8×C53—21no—
35#125×C53C2×C411yescharacteristic
36#125×C53C2×C411yescharacteristic
37C22⋊Q8×C53—41no—
38C4×Q8×C53—41no—
39#117×C53—21no—
40C2×C42×C53C2×C411yescharacteristic
41C2×Q8×C53—41no—
42C22×C4×C53—21no—
43C22×C4×C53C22⋊C411yescharacteristic
44C2×C22⋊D53—22no—
45#11719945—42no—
46#11719945—42no—
47C2×D212—41no—
48C2×D212C2×D412yes—
49C2×C4×D53—22no—
50C2×Dic106C22⋊C412yes—
51C22⋊#6157—44no—
52C4⋊#6157—82no—
53C4×#6157—82no—
54C4⋊C4×C53—21no—
55C4⋊C4×C53—21no—
56C4⋊C4×C53—41no—
57C4⋊C4×C53—41no—
58C4⋊C4×C53—41no—
59C22⋊C4×C53—41no—
60C42×C53—21no—
61C42×C53—41no—
62C42×C53C22⋊C411yescharacteristic
63C42×C53C22×C411yescharacteristic
64#11719921—42no—
65C4⋊Dic53—41no—
66C2×#88322—21no—
67C2×#88322—21no—
68C2×#88322—41no—
69C2×#88322—41no—
70C23×C53#9911yescharacteristic
71C22×D53—22no—
72C2×#6157—84no—
73Q8×C53—82no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—41no—
79C2×C4×C53—41no—
80C2×C4×C53—81no—
81C2×C4×C53#9911yescharacteristic
82C2×C4×C53C2×C22⋊C411yescharacteristic
83C22⋊D53—82no—
84C2×Dic53—22no—
85D212—42no—
86D212—82no—
87C4×D53—42no—
88C4×D53—42no—
89Dic106—42no—
90#88322—41no—
91#88322—41no—
92#13594—531nomaximal, Sylow
93C22×C53—21no—
94C22×C53—41no—
95C22×C53C2×#9911yescharacteristic
96C2×D53—42no—
97C2×D53—82no—
98#6157—164no—
99C4×C53—21no—
100C4×C53—41no—
101C4×C53—41no—
102C4×C53—81no—
103C4×C53—81no—
104C4×C53#35411yescharacteristic
105C4×C53#35411yescharacteristic
106Dic53—42no—
107#3135—531no—
108#2126—532no—
109#1412—532no—
110#1412—532no—
111C2×C53—21no—
112C2×C53—41no—
113C2×C53C2×#35411yescharacteristic
114D53—82no—
115#533#615711yescharacteristic, complemented
116#532—531no—
117#491—531no—
118#491—531no—
119#490—531no—
120#490—531no—
121#489—1062no—
122#416—1062no—
123#410—1062no—
124C2×#97—531no—
125#354—1062no—
126#353—534no—
127C53#1359411yescharacteristic, complemented, Sylow
128C2×#41—532no—
129C22×C8—1061no—
130C4⋊Q8—21no—
131C4⋊Q8—531no—
132#127—531no—
133#125C2×#615711yescharacteristic
134#125C2×#615711yescharacteristic
135#124—532no—
136C22⋊Q8—41no—
137#121—2121no—
138C4×Q8—41no—
139C4×D4—2121no—
140#117—21no—
141#117—2122no—
142C2×C42C2×#615711yescharacteristic
143#103—1062no—
144#103—1062no—
145#102—1062no—
146#102—1062no—
147#99—1064no—
148#99—2122no—
149#97—531no—
150#97—531no—
151#97—1061no—
152#41—2122no—
153#41—2122no—
154C2×Q8—41no—
155C2×Q8—532no—
156C2×D4—532no—
157C2×D4—1062no—
158C2×D4—2121no—
159C22×C4—21no—
160C22×C4—1062no—
161C22×C4C22⋊#615711yescharacteristic
162C2×C8—1061no—
163C2×C8—1061no—
164C2×C8—2121no—
165C2×C8—2121no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—4242no—
172C4⋊C4—2121no—
173C22⋊C4—41no—
174C22⋊C4—2122no—
175C22⋊C4—2124no—
176C42—21no—
177C42—41no—
178C42—4242no—
179C42C22⋊#615711yescharacteristic
180C42C22×#615711yescharacteristic
181C23—1062no—
182C23#1313327511yescharacteristic
183Q8—82no—
184Q8—2122no—
185D4—4242no—
186D4—4242no—
187D4—2122no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—81no—
195C2×C4—1062no—
196C2×C4—4244no—
197C2×C4—2122no—
198C2×C4—2122no—
199C2×C4#1313327511yescharacteristic
200C2×C4C2×C22⋊#615711yescharacteristic
201C8—2121no—
202C8—2121no—
203C22—21no—
204C22—41no—
205C22—4242no—
206C22—2122no—
207C22C2×#1313327511yescharacteristic
208C4—21no—
209C4—41no—
210C4—41no—
211C4—81no—
212C4—81no—
213C4—8484no—
214C4—2122no—
215C4#1681152911yescharacteristic
216C4#1681152911yescharacteristic
217C2—21no—
218C2—41no—
219C2—4242no—
220C2C2×#1681152911yescharacteristic
subgroupG/H
centerC4#16811529
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC53×#533C4
socleC2×C53C2×#354
prime radical#16811173C22
cosocle#13133214C23
hypercenter#533#6157
hypocenterC53#13594
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #20289855C2semidirect
3. #20289854C2semidirect
9. #16810993C4semidirect
10. #16810993C4semidirect
15. C2×#13133128C4semidirect
16. C53×#533C4semidirect
115. #533#6157semidirect
127. C53#13594semidirect

Maximal subgroups

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

subgroupindex
#202909882
#202898552
#202898542
#202850632
#1359453