Finite Group Databasebeta

← #27288186

Subgroups — #27288186

Order 13568.

364 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 79 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#20273413C212yesmaximal
3#20273226C212yesmaximal
4#20272585C211yesmaximal, characteristic
5C53×#1843C211yesmaximal, characteristic
6#16811180C2212yes—
7#16810919—22no—
8#16810864—22no—
9#16810834—22no—
10#16810680C2212yes—
11#16810370—22no—
12#16810368—22no—
13#16810359—24no—
14C2×C4⋊C8×C53—22no—
15C2×C4⋊C8×C53C2211yescharacteristic
16C2×C4⋊C8×C53C2211yescharacteristic
17C53×#376C2211yescharacteristic
18C2×Dic212—22no—
19#13133217D412yes—
20#13133217D412yes—
21#13133209—22no—
22#13133196—22no—
23#13133196—22no—
24C4×Dic106—22no—
25C4×Dic106—22no—
26#13133176—22no—
27#13133151—22no—
28C4⋊#88322—22no—
29#13133142—24no—
30C22⋊#88322—22no—
31#13133128—22no—
32C22×C8×C53—21no—
33C22×C8×C53D412yes—
34C2×C42×C53—21no—
35C2×C42×C53C2311yescharacteristic
36C4⋊C8×C53—22no—
37C4⋊C8×C53—22no—
38C4⋊C8×C53—44no—
39#95×C53—21no—
40#95×C53D412yes—
41C22×C4×C53—22no—
42C22×C4×C53C2×D411yescharacteristic
43C22×C4×C53C2×D411yescharacteristic
44C22×C4×C53#4111yescharacteristic
45C2×Dic106—22no—
46C2×Dic106—22no—
47C2×C8×C53—22no—
48C2×C8×C53—42no—
49C2×C8×C53—42no—
50C2×C8×C53—41no—
51C42×C53—22no—
52C42×C53—22no—
53C42×C53C2×D412yes—
54C42×C53#4112yes—
55C22⋊Dic53—42no—
56C4⋊Dic53—22no—
57C4⋊Dic53—22no—
58C4⋊Dic53—22no—
59C4⋊Dic53—22no—
60#11719919—42no—
61#11719919—42no—
62C4×Dic53—42no—
63C2×#88322—42no—
64Dic212—82no—
65C23×C53#12111yescharacteristic
66C2×C4×C53—22no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—24no—
70C2×C4×C53—41no—
71C2×C4×C53—42no—
72C2×C4×C53—42no—
73C2×C4×C53#12011yescharacteristic
74C2×C4×C53#12112yes—
75C2×C4×C53#12312yes—
76C2×C4×C53#12411yescharacteristic
77C2×Dic53—42no—
78C2×Dic53—42no—
79Dic106—42no—
80Dic106—42no—
81Dic106—82no—
82C8×C53—42no—
83C8×C53—81no—
84#88322—82no—
85#10516—531nomaximal, Sylow
86C22×C53—21no—
87C22×C53—21no—
88C22×C53#39411yescharacteristic
89C22×C53#47312yes—
90C4×C53—22no—
91C4×C53—21no—
92C4×C53—22no—
93C4×C53—44no—
94C4×C53#45311yescharacteristic
95C4×C53#45611yescharacteristic
96Dic53—82no—
97Dic53—82no—
98#3249—531no—
99#1843D5311yescharacteristic
100#1552—532no—
101#1506—532no—
102#1290—531no—
103C2×C53—22no—
104C2×C53#203911yescharacteristic
105C2×C53#204711yescharacteristic
106C2×C53#205311yescharacteristic
107#533—532no—
108#477—1062no—
109#467—1062no—
110#462—1062no—
111#432—532no—
112C2×C4⋊C8—22no—
113C2×C4⋊C8C2×D5311yescharacteristic
114C2×C4⋊C8C2×D5311yescharacteristic
115#376C2×D5311yescharacteristic
116#332—1062no—
117#332—1062no—
118#328—1064no—
119C53#1051611yescharacteristic, complemented, Sylow
120C2×Q16—1062no—
121C22×C8—21no—
122C22×C8C22⋊D5312yes—
123C4⋊Q8—532no—
124C4⋊Q8—532no—
125#125—1062no—
126C22⋊Q8—1062no—
127C22⋊Q8—1062no—
128C4×Q8—1062no—
129C4×Q8—1062no—
130#117—1062no—
131C2×C42—21no—
132C2×C42C22×D5311yescharacteristic
133#107—1062no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—44no—
137C4⋊C8—1062no—
138#103—1064no—
139C22⋊C8—1062no—
140#97—1062no—
141#95—21no—
142#95D21212yes—
143C2×Q8—1062no—
144C2×Q8—1062no—
145C22×C4—22no—
146C22×C4C2×D21211yescharacteristic
147C22×C4#1171994511yescharacteristic
148C22×C4C2×C22⋊D5311yescharacteristic
149Q16—4242no—
150C2×C8—22no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—41no—
154C2×C8—2122no—
155C4⋊C4—1062no—
156C4⋊C4—1062no—
157C4⋊C4—1062no—
158C4⋊C4—1062no—
159C4⋊C4—2122no—
160C4⋊C4—2122no—
161C22⋊C4—2122no—
162C42—22no—
163C42—22no—
164C42—2122no—
165C42D4×D5312yes—
166C42#1171994912yes—
167C23#1313319111yescharacteristic
168Q8—4242no—
169Q8—2122no—
170Q8—2122no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—2122no—
179C2×C4—2122no—
180C2×C4#1313318511yescharacteristic
181C2×C4#1313319212yes—
182C2×C4#1313320212yes—
183C2×C4#1313320611yescharacteristic
184C8—42no—
185C8—81no—
186C8—4242no—
187C22—21no—
188C22—21no—
189C22#1681054011yescharacteristic
190C22#1681089812yes—
191C4—22no—
192C4—22no—
193C4—21no—
194C4—44no—
195C4—4242no—
196C4—4242no—
197C4#1681079011yescharacteristic
198C4#1681080511yescharacteristic
199C2—22no—
200C2#2027593911yescharacteristic
201C2#2027598311yescharacteristic
202C2#2027602111yescharacteristic
subgroupG/H
centerC22#16810540
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1843C2
socleC22×C53#394
prime radicalC23×C53#121
hypercenter#1843D53
hypocenterC53#10516
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1843

Complements

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

NH ≅ G/Ntype
119. C53#10516semidirect

Maximal subgroups

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

subgroupindex
#202860602
#202734132
#202732262
#202725852
C53×#18432
#1051653