Finite Group Databasebeta

← #27288230

Subgroups — #27288230

Order 13568.

386 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 75 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#20286096C211yesmaximal, characteristic
2#20273435C212yesmaximal, complemented
3#20273325C212yesmaximal, complemented
4#20272626C211yesmaximal, characteristic, complemented
5C53×#1849C211yesmaximal, characteristic, complemented
6#16811173C2212yes—
7#16810914—22no—
8#16810876—22no—
9#16810819—22no—
10#16810680C2212yes—
11#16810368—22no—
12#16810366—22no—
13#16810362—24no—
14C2×#108×C53—22no—
15C53×#423C2211yescharacteristic
16C53×#423C2211yescharacteristic
17C2×#97×C53C2211yescharacteristic
18C2×#11719945—22no—
19C2×D424—22no—
20#13133217D412yes—
21#13133214D412yes—
22#13133205—22no—
23#13133191—22no—
24#13133191—22no—
25C4×D212—22no—
26C4×D212—22no—
27#13133149—22no—
28C4⋊#88322—22no—
29#13133139—24no—
30C22⋊#88322—22no—
31#13133128—22no—
32C2×M16×C53—21no—
33C2×M16×C53D412yes—
34C22×C8×C53—21no—
35C2×C42×C53C2311yescharacteristic
36#108×C53—44no—
37C4⋊C8×C53—22no—
38C4⋊C8×C53—22no—
39#97×C53—21no—
40#97×C53D412yes—
41C22×C4×C53C2×D411yescharacteristic
42C22×C4×C53#4112yes—
43C2×C22⋊D53—42no—
44#11719945—84no—
45C2×D212—22no—
46C2×D212—22no—
47C2×D212—22no—
48C2×C4×D53—42no—
49C2×Dic106—22no—
50M16×C53—42no—
51M16×C53—42no—
52C2×C8×C53—22no—
53C2×C8×C53—22no—
54C2×C8×C53—22no—
55C2×C8×C53—22no—
56C2×C8×C53—41no—
57C42×C53C2×D412yes—
58C42×C53C2×D411yescharacteristic
59C42×C53#4111yescharacteristic
60#11719921—42no—
61#11719921—42no—
62C4⋊Dic53—22no—
63C4⋊Dic53—22no—
64C2×#88322—42no—
65D424—82no—
66C23×C53#12411yescharacteristic
67C22×D53—42no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—42no—
71C2×C4×C53#12011yescharacteristic
72C2×C4×C53#12111yescharacteristic
73C2×C4×C53#12112yes—
74C2×C4×C53#12312yes—
75C22⋊D53—84no—
76C2×Dic53—42no—
77D212—42no—
78D212—42no—
79D212—82no—
80D212—82no—
81C4×D53—84no—
82Dic106—82no—
83C8×C53—24no—
84C8×C53—42no—
85C8×C53—81no—
86#88322—82no—
87#10524—531nomaximal, Sylow
88C22×C53—22no—
89C22×C53—21no—
90C22×C53#39411yescharacteristic
91C2×D53—82no—
92C2×D53—82no—
93C2×D53—82no—
94C4×C53—22no—
95C4×C53—22no—
96C4×C53—21no—
97C4×C53#45311yescharacteristic
98C4×C53#45611yescharacteristic
99Dic53—82no—
100#3253—531no—
101#1849D5311yescharacteristic, complemented
102#1557—532no—
103#1530—532no—
104#1307—531no—
105C2×C53—41no—
106C2×C53#203911yescharacteristic
107C2×C53#204812yes—
108D53—162no—
109#532—532no—
110#476—1062no—
111#469—1062no—
112#459—1062no—
113#432—532no—
114C2×#108—22no—
115#423C2×D5311yescharacteristic
116#423C2×D5311yescharacteristic
117C2×#97C2×D5311yescharacteristic
118#332—1062no—
119#331—1062no—
120#329—1064no—
121C53#1052411yescharacteristic, complemented, Sylow
122C2×#41—1062no—
123C2×D8—1062no—
124C2×M16—21no—
125C2×M16C22⋊D5312yes—
126C22×C8—21no—
127C4⋊Q8—532no—
128#127—532no—
129#124—1062no—
130#121—1062no—
131#121—1062no—
132C4×D4—1062no—
133C4×D4—1062no—
134C2×C42C22×D5311yescharacteristic
135#108—44no—
136#106—1062no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—1062no—
140#102—1064no—
141C22⋊C8—1062no—
142#97—21no—
143#97—1062no—
144#97D21212yes—
145#41—4244no—
146C2×Q8—1062no—
147C2×D4—1062no—
148C2×D4—1062no—
149C2×D4—1062no—
150C2×D4—2122no—
151C22×C4—2122no—
152C22×C4#1171994912yes—
153C22×C4C2×C22⋊D5311yescharacteristic
154D8—4242no—
155M16—42no—
156M16—42no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—41no—
162C2×C8—2122no—
163C4⋊C4—1062no—
164C4⋊C4—1062no—
165C22⋊C4—2122no—
166C22⋊C4—2122no—
167C42C2×D21211yescharacteristic
168C42#1171994511yescharacteristic
169C42D4×D5312yes—
170C23—2122no—
171C23#1313320611yescharacteristic
172Q8—4242no—
173D4—4242no—
174D4—4242no—
175D4—4244no—
176D4—2122no—
177D4—2122no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—4244no—
182C2×C4—2122no—
183C2×C4#1313318511yescharacteristic
184C2×C4#1313319111yescharacteristic
185C2×C4#1313319212yes—
186C2×C4#1313320212yes—
187C8—24no—
188C8—42no—
189C8—81no—
190C8—4242no—
191C22—22no—
192C22—21no—
193C22—4242no—
194C22—4242no—
195C22—4242no—
196C22#1681054011yescharacteristic
197C4—22no—
198C4—22no—
199C4—21no—
200C4—4242no—
201C4#1681079011yescharacteristic
202C4#1681080511yescharacteristic
203C2—41no—
204C2—8482no—
205C2#2027593911yescharacteristic
206C2#2027599012yes—
subgroupG/H
centerC22#16810540
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1849C2
socleC22×C53#394
prime radical#20286096C2
hypercenter#1849D53
hypocenterC53#10524
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1849

Complements

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

NH ≅ G/Ntype
2. #20273435C2semidirect
3. #20273325C2semidirect
4. #20272626C2semidirect
5. C53×#1849C2semidirect
101. #1849D53semidirect
121. C53#10524semidirect

Maximal subgroups

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

subgroupindex
#202860962
#202734352
#202733252
#202726262
C53×#18492
#1052453