Finite Group Databasebeta

← #27288192

Subgroups — #27288192

Order 13568.

344 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 77 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#20286060C211yesmaximal, characteristic
2#20274834C211yesmaximal, characteristic
3#20273416C212yesmaximal
4#20273228C212yesmaximal
5C53×#1290C211yesmaximal, characteristic
6#16811180C2212yes—
7#16810919—22no—
8#16810864—22no—
9#16810834—22no—
10C2×C4⋊#88322C2211yescharacteristic
11#16810469C2211yescharacteristic
12#16810377—22no—
13#16810370—22no—
14#16810368—22no—
15#16810361—24no—
16C53×#432C2212yes—
17C2×C4⋊C8×C53C2211yescharacteristic
18C2×Dic212—22no—
19C22×#88322—21no—
20#13133217D412yes—
21#13133217D412yes—
22#13133209—22no—
23#13133196—22no—
24#13133196—22no—
25C4×Dic106—22no—
26C4×Dic106—22no—
27#13133176—22no—
28C2×C4×Dic53—21no—
29#13133151—22no—
30C4⋊#88322—22no—
31#13133142—24no—
32#13133124—21no—
33#13133124D412yes—
34C22×C8×C53D412yes—
35C2×C42×C53C2311yescharacteristic
36C4⋊C8×C53—22no—
37C4⋊C8×C53—22no—
38C22⋊C8×C53—22no—
39#97×C53—22no—
40C22×C4×C53C2×D411yescharacteristic
41C22×C4×C53C2×D411yescharacteristic
42C22×C4×C53#4111yescharacteristic
43C22×Dic53—22no—
44C2×Dic106—22no—
45C2×Dic106—22no—
46C2×C8×C53—22no—
47C2×C8×C53—42no—
48C2×C8×C53—42no—
49C42×C53C2×D412yes—
50C42×C53#4112yes—
51C22⋊Dic53—42no—
52C4⋊Dic53—22no—
53C4⋊Dic53—22no—
54C4⋊Dic53—22no—
55C4⋊Dic53—22no—
56#11719919—42no—
57#11719919—42no—
58C4×Dic53—41no—
59C4×Dic53—41no—
60C4×Dic53—42no—
61C2×#88322—42no—
62C2×#88322—41no—
63Dic212—82no—
64C23×C53#12111yescharacteristic
65C2×C4×C53—22no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—41no—
69C2×C4×C53#12011yescharacteristic
70C2×C4×C53#12112yes—
71C2×C4×C53#12312yes—
72C2×C4×C53#12411yescharacteristic
73C2×Dic53—42no—
74C2×Dic53—42no—
75C2×Dic53—42no—
76C2×Dic53—42no—
77C2×Dic53—42no—
78Dic106—42no—
79Dic106—42no—
80Dic106—82no—
81C8×C53—42no—
82C8×C53—82no—
83#88322—81no—
84#10517—531nomaximal, Sylow
85C22×C53—21no—
86C22×C53—21no—
87C22×C53#39411yescharacteristic
88C22×C53#47312yes—
89C4×C53—22no—
90C4×C53—21no—
91C4×C53—22no—
92C4×C53#45411yescharacteristic
93C4×C53#45511yescharacteristic
94Dic53—82no—
95Dic53—82no—
96Dic53—82no—
97#3249—531no—
98#1832—531no—
99#1552—532no—
100#1506—532no—
101#1290D5311yescharacteristic
102C2×C53—22no—
103C2×C53#204011yescharacteristic
104C2×C53#204711yescharacteristic
105C2×C53#205311yescharacteristic
106#533—532no—
107#477—1062no—
108#467—1062no—
109#462—1062no—
110#432C2×D5312yes—
111C2×C4⋊C8—531no—
112C2×C4⋊C8C2×D5311yescharacteristic
113#376—531no—
114#336—1062no—
115#332—1062no—
116#332—1062no—
117#328—1064no—
118C53#1051711yescharacteristic, complemented, Sylow
119C2×Q16—1062no—
120C22×C8—1061no—
121C22×C8D21212yes—
122C4⋊Q8—532no—
123C4⋊Q8—532no—
124#125—1062no—
125C22⋊Q8—1062no—
126C22⋊Q8—1062no—
127C4×Q8—1062no—
128C4×Q8—1062no—
129#117—1062no—
130C2×C42—1061no—
131C2×C42C22×D5311yescharacteristic
132#107—1062no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—1062no—
136#103—1064no—
137C22⋊C8—22no—
138#97—22no—
139#95—532no—
140#95—1061no—
141C2×Q8—1062no—
142C2×Q8—1062no—
143C22×C4—1062no—
144C22×C4C2×D21211yescharacteristic
145C22×C4D4×D5311yescharacteristic
146C22×C4#1171994911yescharacteristic
147Q16—4242no—
148C2×C8—22no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—2122no—
152C2×C8—2121no—
153C4⋊C4—1062no—
154C4⋊C4—1062no—
155C4⋊C4—1062no—
156C4⋊C4—1062no—
157C4⋊C4—2122no—
158C4⋊C4—2122no—
159C22⋊C4—2122no—
160C42—2121no—
161C42—2121no—
162C42—2122no—
163C42#1171994512yes—
164C42D4×D5312yes—
165C23#1313319211yescharacteristic
166Q8—4242no—
167Q8—2122no—
168Q8—2122no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—2122no—
174C2×C4—2122no—
175C2×C4—2122no—
176C2×C4—2122no—
177C2×C4—2122no—
178C2×C4#1313318711yescharacteristic
179C2×C4#1313319012yes—
180C2×C4#1313320212yes—
181C2×C4#1313320511yescharacteristic
182C8—42no—
183C8—82no—
184C8—4241no—
185C22—21no—
186C22—21no—
187C22#1681054111yescharacteristic
188C22#1681089912yes—
189C4—22no—
190C4—22no—
191C4—21no—
192C4—4242no—
193C4—4242no—
194C4—4242no—
195C4#1681079811yescharacteristic
196C4#1681080311yescharacteristic
197C2—22no—
198C2#2027594511yescharacteristic
199C2#2027598411yescharacteristic
200C2#2027602211yescharacteristic
subgroupG/H
centerC22#16810541
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1290C2
socleC22×C53#394
prime radicalC23×C53#121
hypercenter#1290D53
hypocenterC53#10517
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1290

Complements

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

NH ≅ G/Ntype
118. C53#10517semidirect

Maximal subgroups

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

subgroupindex
#202860602
#202748342
#202734162
#202732282
C53×#12902
#1051753