Finite Group Databasebeta

← #27277117

Subgroups — #27277117

Order 13568.

404 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 101 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#20283426C211yesmaximal, characteristic
2#20273473C212yesmaximal, complemented
3#20273448C212yesmaximal
4#20272628C211yesmaximal, characteristic, complemented
5C53×#1309C211yesmaximal, characteristic, complemented
6C2×#13133225C412yes—
7#16811173C2211yescharacteristic
8#16810700—22no—
9#16810688—22no—
10#16810647C2212yes—
11#16810591C412yes—
12C2×C4×#88322C2211yescharacteristic
13#16810375—22no—
14#16810375—22no—
15#16810371—22no—
16#16810366—22no—
17C53×#423C2212yes—
18C53×#423C2211yescharacteristic
19C2×#11719945C2×C412yes—
20#13133225—44no—
21C2×#11719917—22no—
22C2×#11719912—22no—
23C22×#88322C2×C411yescharacteristic
24C22×#88322C2×C411yescharacteristic
25C2×C8×D53—22no—
26#13133217D411yescharacteristic
27#13133214D411yescharacteristic
28#13133205Q812yes—
29#13133191—22no—
30C4×D212—22no—
31C4⋊#88322—22no—
32C4⋊#88322—22no—
33#13133129—24no—
34C4×#88322D412yes—
35C4×#88322D412yes—
36C2×M16×C53—22no—
37C2×M16×C53C2×C412yes—
38C2×C42×C53C2311yescharacteristic
39C4⋊C8×C53—22no—
40C4⋊C8×C53—22no—
41C4⋊C8×C53—22no—
42C22×C4×C53C22×C411yescharacteristic
43C22×C4×C53#4112yes—
44C2×C22⋊D53—22no—
45#11719945—42no—
46#11719945—42no—
47C2×D212—22no—
48C2×D212C4⋊C412yes—
49C2×C4×D53—22no—
50C2×Dic106C4⋊C412yes—
51M16×C53—42no—
52M16×C53—82no—
53C2×C8×C53—22no—
54C2×C8×C53—42no—
55C42×C53C2×D412yes—
56C42×C53C2×D411yescharacteristic
57C42×C53C2×Q811yescharacteristic
58#11719921—24no—
59C4⋊Dic53—22no—
60#11719917—82no—
61C2×#88322—22no—
62C2×#88322—41no—
63C2×#88322—42no—
64C2×#88322C22⋊C412yes—
65C2×#88322C22⋊C411yescharacteristic
66C2×#88322C22⋊C411yescharacteristic
67#11719912—44no—
68C8×D53—44no—
69C23×C53#11711yescharacteristic
70C22×D53—22no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53C2×C22⋊C411yescharacteristic
76C2×C4×C53C2×C4⋊C411yescharacteristic
77C2×C4×C53#12112yes—
78C2×C4×C53C22⋊Q812yes—
79C22⋊D53—44no—
80C2×Dic53—22no—
81D212—42no—
82D212—42no—
83D212—42no—
84C4×D53—42no—
85C4×D53—42no—
86Dic106—42no—
87C8×C53—42no—
88C8×C53—82no—
89#88322—21no—
90#88322—22no—
91#88322—82no—
92#88322#10012yes—
93#8638—531nomaximal, Sylow
94C22×C53—21no—
95C22×C53—22no—
96C22×C53#38011yescharacteristic
97C2×D53—42no—
98C2×D53—44no—
99C4×C53—21no—
100C4×C53—22no—
101C4×C53C2×#10011yescharacteristic
102C4×C53#41311yescharacteristic
103C4×C53#47212yes—
104C4×C53#47312yes—
105Dic53—42no—
106#2931—531no—
107#1567—532no—
108#1560—532no—
109#1309—531no—
110#1309D5311yescharacteristic, complemented
111C2×C53—41no—
112C2×C53#180011yescharacteristic
113C2×C53#181612yes—
114D53—82no—
115C2×#131—532no—
116#532—531no—
117#436—1062no—
118C8×D4—1062no—
119#423—532no—
120#423C2×D5312yes—
121#423C2×D5311yescharacteristic
122#408—532no—
123C2×C4×C8—531no—
124#335—1062no—
125#335—1062no—
126#333—1062no—
127#331—1062no—
128C53#863811yescharacteristic, complemented, Sylow
129C2×#41—532no—
130#131—2124no—
131C2×M16—22no—
132C2×M16—1062no—
133C2×M16—1062no—
134C2×M16C4×D5312yes—
135C22×C8—531no—
136C22×C8—531no—
137C22×C8—1062no—
138C4⋊Q8—531no—
139#127—531no—
140#124—532no—
141#121—1062no—
142C4×D4—1062no—
143C2×C42C22×D5311yescharacteristic
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—1062no—
148C4⋊C8—1062no—
149C22⋊C8—1064no—
150C4×C8—532no—
151C4×C8—532no—
152#41—2122no—
153#41—2122no—
154C2×Q8—532no—
155C2×D4—532no—
156C2×D4—1062no—
157C2×D4—1062no—
158C22×C4—1062no—
159C22×C4C2×C4×D5311yescharacteristic
160C22×C4#1171994512yes—
161M16—42no—
162M16—82no—
163M16—4242no—
164M16—2124no—
165C2×C8—22no—
166C2×C8—42no—
167C2×C8—531no—
168C2×C8—532no—
169C2×C8—531no—
170C2×C8—1062no—
171C2×C8—2124no—
172C2×C8—2121no—
173C2×C8—2122no—
174C4⋊C4—1062no—
175C22⋊C4—1064no—
176C42D4×D5312yes—
177C42D4×D5311yescharacteristic
178C42Q8×D5311yescharacteristic
179C23—1062no—
180C23#1313317411yescharacteristic
181Q8—2122no—
182D4—2122no—
183D4—2124no—
184D4—2122no—
185D4—2122no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—1062no—
191C2×C4—2122no—
192C2×C4—2122no—
193C2×C4C22⋊C4×D5311yescharacteristic
194C2×C4C4⋊C4×D5311yescharacteristic
195C2×C4#1313319012yes—
196C2×C4#1313319512yes—
197C8—42no—
198C8—82no—
199C8—532no—
200C8—1061no—
201C8—1062no—
202C8—4242no—
203C22—21no—
204C22—22no—
205C22—2122no—
206C22—2124no—
207C22#1681048211yescharacteristic
208C4—21no—
209C4—22no—
210C4—2122no—
211C4#100×D5311yescharacteristic
212C4#1681060611yescharacteristic
213C4#1681089212yes—
214C4#1681089712yes—
215C2—41no—
216C2—4242no—
217C2#2027470611yescharacteristic
218C2#2027477812yes—
subgroupG/H
centerC22#16810482
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1309C2
socleC22×C53#380
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#1309D53
hypocenterC53#8638
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1309

Complements

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

NH ≅ G/Ntype
2. #20273473C2semidirect
4. #20272628C2semidirect
5. C53×#1309C2semidirect
110. #1309D53semidirect
128. C53#8638semidirect

Maximal subgroups

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

subgroupindex
#202834262
#202734732
#202734482
#202726282
C53×#13092
#863853