Finite Group Databasebeta

← #27277083

Subgroups — #27277083

Order 13568.

388 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 93 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#20273476C212yesmaximal, complemented
3#20273349C212yesmaximal
4#20272626C211yesmaximal, characteristic, complemented
5C53×#1307C211yesmaximal, characteristic, complemented
6C2×#13133225C412yes—
7#16811173C2211yescharacteristic
8#16810700—22no—
9#16810688—22no—
10#16810680C2212yes—
11#16810591C412yes—
12C2×C4×#88322C2211yescharacteristic
13#16810375—22no—
14#16810373—22no—
15#16810362—24no—
16C53×#432C2212yes—
17C53×#423C2211yescharacteristic
18C2×#11719945C2×C412yes—
19#13133225—44no—
20C2×#11719912—22no—
21C22×#88322C2×C411yescharacteristic
22C22×#88322C2×C411yescharacteristic
23C2×C8×D53—22no—
24#13133217Q811yescharacteristic
25#13133214Q811yescharacteristic
26#13133205D412yes—
27#13133191—22no—
28C4×D212—22no—
29C4⋊#88322—22no—
30C22⋊#88322—22no—
31#13133129—24no—
32#13133128—22no—
33C4×#88322D412yes—
34C4×#88322D412yes—
35C2×M16×C53C2×C412yes—
36C2×C42×C53C2311yescharacteristic
37C4⋊C8×C53—22no—
38C4⋊C8×C53—22no—
39C22⋊C8×C53—22no—
40#97×C53—22no—
41C22×C4×C53C22×C411yescharacteristic
42C22×C4×C53#4112yes—
43C2×C22⋊D53—22no—
44#11719945—42no—
45#11719945—42no—
46C2×D212—22no—
47C2×D212C4⋊C412yes—
48C2×C4×D53—22no—
49C2×Dic106C4⋊C412yes—
50M16×C53—42no—
51C2×C8×C53—22no—
52C2×C8×C53—42no—
53C42×C53C2×D412yes—
54C42×C53C2×D411yescharacteristic
55C42×C53C2×Q811yescharacteristic
56#11719921—24no—
57C4⋊Dic53—22no—
58C2×#88322—22no—
59C2×#88322—41no—
60C2×#88322—42no—
61C2×#88322C22⋊C412yes—
62C2×#88322C22⋊C411yescharacteristic
63C2×#88322C22⋊C411yescharacteristic
64#11719912—44no—
65C8×D53—44no—
66C23×C53#11711yescharacteristic
67C22×D53—22no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—42no—
71C2×C4×C53C2×C22⋊C411yescharacteristic
72C2×C4×C53C2×C4⋊C411yescharacteristic
73C2×C4×C53#12112yes—
74C2×C4×C53C22⋊Q812yes—
75C22⋊D53—44no—
76C2×Dic53—22no—
77D212—42no—
78D212—44no—
79C4×D53—42no—
80C4×D53—42no—
81Dic106—42no—
82C8×C53—42no—
83C8×C53—82no—
84#88322—21no—
85#88322—22no—
86#88322—82no—
87#88322#10012yes—
88#8630—531nomaximal, Sylow
89C22×C53—21no—
90C22×C53—22no—
91C22×C53#38011yescharacteristic
92C2×D53—42no—
93C2×D53—44no—
94C4×C53—21no—
95C4×C53—22no—
96C4×C53—22no—
97C4×C53C2×#10011yescharacteristic
98C4×C53#41311yescharacteristic
99Dic53—42no—
100#2931—531no—
101#1568—532no—
102#1536—532no—
103#1307—531no—
104#1307D5311yescharacteristic, complemented
105C2×C53—41no—
106C2×C53#180011yescharacteristic
107C2×C53#181412yes—
108D53—82no—
109C2×#131—532no—
110#532—531no—
111#436—1062no—
112C8×D4—1062no—
113#432—532no—
114#432C2×D5312yes—
115#423C2×D5311yescharacteristic
116#408—532no—
117C2×C4×C8—531no—
118#335—1062no—
119#334—1062no—
120#329—1064no—
121C53#863011yescharacteristic, complemented, Sylow
122C2×#41—532no—
123#131—2124no—
124C2×M16—1062no—
125C2×M16C4×D5312yes—
126C22×C8—531no—
127C22×C8—531no—
128C22×C8—1062no—
129C4⋊Q8—531no—
130#127—531no—
131#124—532no—
132#121—1062no—
133C4×D4—1062no—
134C2×C42C22×D5311yescharacteristic
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—1062no—
138C22⋊C8—22no—
139C22⋊C8—1064no—
140C22⋊C8—1062no—
141#97—22no—
142#97—1062no—
143C4×C8—532no—
144C4×C8—532no—
145#41—2122no—
146#41—2122no—
147C2×Q8—532no—
148C2×D4—532no—
149C2×D4—1062no—
150C2×D4—1062no—
151C22×C4—1062no—
152C22×C4C2×C4×D5311yescharacteristic
153C22×C4#1171994512yes—
154M16—42no—
155M16—2124no—
156C2×C8—22no—
157C2×C8—42no—
158C2×C8—531no—
159C2×C8—532no—
160C2×C8—531no—
161C2×C8—1062no—
162C2×C8—2124no—
163C2×C8—2121no—
164C2×C8—2122no—
165C4⋊C4—1062no—
166C22⋊C4—1064no—
167C42D4×D5312yes—
168C42D4×D5311yescharacteristic
169C42Q8×D5311yescharacteristic
170C23—1062no—
171C23#1313317411yescharacteristic
172Q8—2122no—
173D4—2122no—
174D4—2124no—
175D4—2124no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—1062no—
180C2×C4—2122no—
181C2×C4—2122no—
182C2×C4C22⋊C4×D5311yescharacteristic
183C2×C4C4⋊C4×D5311yescharacteristic
184C2×C4#1313319012yes—
185C2×C4#1313319512yes—
186C8—42no—
187C8—82no—
188C8—532no—
189C8—1061no—
190C8—1062no—
191C8—4242no—
192C22—21no—
193C22—22no—
194C22—2122no—
195C22—2124no—
196C22#1681048211yescharacteristic
197C4—21no—
198C4—22no—
199C4—22no—
200C4—2122no—
201C4#100×D5311yescharacteristic
202C4#1681060611yescharacteristic
203C2—41no—
204C2—4242no—
205C2#2027470611yescharacteristic
206C2#2027476812yes—
subgroupG/H
centerC22#16810482
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1307C2
socleC22×C53#380
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#1307D53
hypocenterC53#8630
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1307

Complements

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

NH ≅ G/Ntype
2. #20273476C2semidirect
4. #20272626C2semidirect
5. C53×#1307C2semidirect
104. #1307D53semidirect
121. C53#8630semidirect

Maximal subgroups

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

subgroupindex
#202834262
#202734762
#202733492
#202726262
C53×#13072
#863053