Finite Group Databasebeta

← #27295277

Subgroups — #27295277

Order 13568.

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

normal subgroups only

This group has 296 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#20285365C211yesmaximal, characteristic
2#20276531C211yesmaximal, characteristic, complemented
3#20272747C214yesmaximal, complemented
4C53×#2110C211yesmaximal, characteristic, complemented
5#16811228C412yes—
6#16811025C411yescharacteristic
7#16811000C411yescharacteristic
8#16810988C2212yes—
9#16810876—21no—
10#16810859—21no—
11#16810736—21no—
12C4×D424—21no—
13#16810444—21no—
14#16810405—21no—
15#16810401C2212yes—
16C2×M32×C53—21no—
17C53×#405C2211yescharacteristic
18C53×#347—21no—
19C53×#347C2212yes—
20C2×D424—21no—
21C2×D424—21no—
22#13133214C2×C412yes—
23#13133209C2×C412yes—
24#13133202—42no—
25#13133191—22no—
26#13133190—42no—
27C4×D212—22no—
28#13133157—22no—
29#13133151—21no—
30#13133149—21no—
31#13133139—22no—
32#13133139—22no—
33C2×M16×C53D411yescharacteristic
34C22×C8×C53D411yescharacteristic
35#117×C53C2×C411yescharacteristic
36M32×C53—22no—
37M32×C53—22no—
38C2×C16×C53—22no—
39#97×C53D411yescharacteristic
40#97×C53C2311yescharacteristic
41C4×C8×C53C2×C411yescharacteristic
42C4×C8×C53D411yescharacteristic
43C22×C4×C53C22⋊C411yescharacteristic
44C2×C22⋊D53—42no—
45C2×D212—22no—
46C2×D212—42no—
47C2×C4×D53—42no—
48M16×C53—41no—
49C2×C8×C53—21no—
50C2×C8×C53—21no—
51C2×C8×C53C22⋊C411yescharacteristic
52C2×C8×C53C22⋊C411yescharacteristic
53C2×C8×C53C2×D411yescharacteristic
54C2×C8×C53C2×D411yescharacteristic
55C4⋊C4×C53—21no—
56C22⋊C4×C53—21no—
57C42×C53C22⋊C411yescharacteristic
58C42×C53C22×C411yescharacteristic
59#11719921—42no—
60#11719921—42no—
61C4⋊Dic53—22no—
62#11719919—42no—
63D424—42no—
64D424—41no—
65D424—41no—
66C16×C53—24no—
67#11719908—42no—
68C23×C53#10011yescharacteristic
69C22×D53—42no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53#10011yescharacteristic
75C2×C4×C53C2×C22⋊C411yescharacteristic
76C22⋊D53—162no—
77C2×Dic53—42no—
78D212—42no—
79D212—42no—
80D212—162no—
81C4×D53—162no—
82C8×C53—21no—
83C8×C53—21no—
84C8×C53—21no—
85C8×C53#10411yescharacteristic
86C8×C53#10411yescharacteristic
87#11913—531nomaximal, Sylow
88C22×C53—21no—
89C22×C53—41no—
90C22×C53C2×#10011yescharacteristic
91C2×D53—82no—
92C2×D53—82no—
93C2×D53—82no—
94C4×C53—21no—
95C4×C53—41no—
96C4×C53—41no—
97C4×C53C2×#10411yescharacteristic
98C4×C53#42111yescharacteristic
99Dic53—82no—
100#3176—531no—
101#2166—531no—
102#2110D5311yescharacteristic, complemented
103#1355—534no—
104C2×C53—21no—
105C2×C53—41no—
106C2×C53#153011yescharacteristic
107D53—162no—
108#540—532no—
109C2×M32—21no—
110#499—531no—
111#493—531no—
112#490—532no—
113#469—1061no—
114#466—1061no—
115#442—1061no—
116C4×D8—1061no—
117#405C2×D5311yescharacteristic
118#364—1061no—
119#349—1061no—
120#347—21no—
121#347—532no—
122#347C2×D5312yes—
123C53#1191311yescharacteristic, complemented, Sylow
124C2×D8—1061no—
125C2×D8—1061no—
126C2×M16D21211yescharacteristic
127C22×C8C22⋊D5311yescharacteristic
128#127—532no—
129#125—532no—
130#123—2122no—
131#121—1062no—
132#121—2122no—
133C4×D4—1062no—
134#117C4×D5311yescharacteristic
135M32—22no—
136M32—22no—
137M32—1062no—
138C2×C16—22no—
139#107—1061no—
140#106—1061no—
141#102—1062no—
142#102—1062no—
143#97C22⋊D5311yescharacteristic
144#97C22×D5311yescharacteristic
145C4×C8C4×D5311yescharacteristic
146C4×C8D21211yescharacteristic
147C2×D4—1062no—
148C2×D4—2122no—
149C2×D4—2122no—
150C22×C4—2122no—
151C22×C4#1171992111yescharacteristic
152D8—2122no—
153D8—2121no—
154D8—2121no—
155M16—41no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8#1171992111yescharacteristic
159C2×C8#1171992111yescharacteristic
160C2×C8C2×D21211yescharacteristic
161C2×C8C2×C22⋊D5311yescharacteristic
162C4⋊C4—21no—
163C4⋊C4—1062no—
164C4⋊C4—2122no—
165C22⋊C4—21no—
166C22⋊C4—2122no—
167C22⋊C4—2122no—
168C42#1171992111yescharacteristic
169C42C2×C4×D5311yescharacteristic
170C16—24no—
171C16—2122no—
172C23—2122no—
173C23#1313313311yescharacteristic
174D4—8482no—
175D4—8482no—
176D4—2122no—
177D4—2122no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—8482no—
183C2×C4—2122no—
184C2×C4#1313313311yescharacteristic
185C2×C4C2×#1171992111yescharacteristic
186C8—21no—
187C8—21no—
188C8—21no—
189C8#1313314511yescharacteristic
190C8#1313314511yescharacteristic
191C22—21no—
192C22—41no—
193C22—4242no—
194C22—4242no—
195C22—4242no—
196C22C2×#1313313311yescharacteristic
197C4—21no—
198C4—41no—
199C4—41no—
200C4—4242no—
201C4C2×#1313314511yescharacteristic
202C4#1681063911yescharacteristic
203C2—21no—
204C2—41no—
205C2—8482no—
206C2#2027332511yescharacteristic
subgroupG/H
centerC2#20273325
commutatorC42×C53C22×C4
Frattini#97C22×D53
FittingC53×#2110C2
socleC2×C53#1530
prime radical#20285365C2
cosocle#97×C53C23
hypercenter#2110D53
hypocenterC53#11913
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2110

Complements

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

NH ≅ G/Ntype
2. #20276531C2semidirect
3. #20272747C2semidirect
4. C53×#2110C2semidirect
102. #2110D53semidirect
123. C53#11913semidirect

Maximal subgroups

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

subgroupindex
#202853652
#202765312
#202727472
C53×#21102
#1191353