Finite Group Databasebeta

← #28312481

Subgroups — #28312481

Order 13568.

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

normal subgroups only

This group has 318 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#20291008C212yesmaximal
2#20289858C212yesmaximal, complemented
3#20289857C212yesmaximal, complemented
4#20285063C211yesmaximal, characteristic
5#16811609—22no—
6#16811534C2214yes—
7#16811532—22no—
8#16811173C2211yescharacteristic
9#16810993C2211yescharacteristic
10#16810993C2211yescharacteristic
11#16810989C411yescharacteristic
12#16810989C411yescharacteristic
13#16810985—22no—
14#16810617—22no—
15C2×#13133128C411yescharacteristic
16C53×#533C411yescharacteristic
17D53⋊M16—42no—
18#13133278—24no—
19#13133277—42no—
20C2×#11719945D412yes—
21C22×#88322—21no—
22#13133217C2311yescharacteristic
23#13133214C2×C411yescharacteristic
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×D212C22⋊C412yes—
49C2×C4×D53—22no—
50C2×Dic106C2×D412yes—
51#11719940—44no—
52#11719936—82no—
53D53⋊C8—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—
72Q8×C53—82no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—41no—
78C2×C4×C53—41no—
79C2×C4×C53—81no—
80C2×C4×C53#9911yescharacteristic
81C2×C4×C53C2×C22⋊C411yescharacteristic
82C22⋊D53—82no—
83C2×Dic53—22no—
84D212—42no—
85D212—82no—
86C4×D53—42no—
87C4×D53—42no—
88Dic106—42no—
89#88324—84no—
90#88322—41no—
91#88322—41no—
92#13598—531nomaximal, Sylow
93C22×C53—21no—
94C22×C53—41no—
95C22×C53C2×#9911yescharacteristic
96C2×D53—42no—
97C2×D53—82no—
98C4×C53—21no—
99C4×C53—41no—
100C4×C53—41no—
101C4×C53—81no—
102C4×C53—81no—
103C4×C53#35411yescharacteristic
104C4×C53#35411yescharacteristic
105Dic53—42no—
106#3135—531no—
107#2132—532no—
108#1413—532no—
109#1413—532no—
110C2×C53—21no—
111C2×C53—41no—
112C2×C53C2×#35411yescharacteristic
113D53—82no—
114#533#615711yescharacteristic
115#532—531no—
116#491—531no—
117#491—531no—
118#490—531no—
119#490—531no—
120#489—1062no—
121#416—1062no—
122#413—1062no—
123C2×#97—531no—
124#356—534no—
125#355—1062no—
126C53#1359811yescharacteristic, complemented, Sylow
127C2×#41—532no—
128C2×M16—2122no—
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—
141C2×C42C2×#615711yescharacteristic
142#103—1062no—
143#103—1062no—
144#102—1062no—
145#102—1062no—
146#101—1064no—
147#100—2122no—
148#97—531no—
149#97—531no—
150#97—1061no—
151#41—2122no—
152#41—2122no—
153C2×Q8—41no—
154C2×Q8—532no—
155C2×D4—532no—
156C2×D4—1062no—
157C2×D4—2121no—
158C22×C4—21no—
159C22×C4—1062no—
160C22×C4C22⋊#615711yescharacteristic
161M16—4242no—
162M16—2124no—
163C2×C8—1061no—
164C2×C8—1061no—
165C2×C8—4242no—
166C2×C8—2121no—
167C2×C8—2121no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—2121no—
174C22⋊C4—41no—
175C22⋊C4—2122no—
176C42—21no—
177C42—41no—
178C42C22⋊#615711yescharacteristic
179C42C22×#615711yescharacteristic
180C23—1062no—
181C23#1313327511yescharacteristic
182Q8—82no—
183Q8—2122no—
184D4—4242no—
185D4—4242no—
186D4—2122no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—81no—
194C2×C4—1062no—
195C2×C4—2122no—
196C2×C4—2122no—
197C2×C4#1313327511yescharacteristic
198C2×C4C2×C22⋊#615711yescharacteristic
199C8—4244no—
200C8—2121no—
201C8—2121no—
202C22—21no—
203C22—41no—
204C22—4242no—
205C22—2122no—
206C22C2×#1313327511yescharacteristic
207C4—21no—
208C4—41no—
209C4—41no—
210C4—81no—
211C4—81no—
212C4—2122no—
213C4#1681152911yescharacteristic
214C4#1681152911yescharacteristic
215C2—21no—
216C2—41no—
217C2—4242no—
218C2C2×#1681152911yescharacteristic
subgroupG/H
centerC4#16811529
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC53×#533C4
socleC2×C53C2×#354
prime radical#16811173C22
cosocle#13133217C23
hypercenter#533#6157
hypocenterC53#13598
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. #20289858C2semidirect
3. #20289857C2semidirect
126. C53#13598semidirect

Maximal subgroups

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

subgroupindex
#202910082
#202898582
#202898572
#202850632
#1359853