Finite Group Databasebeta

← #27279046

Subgroups — #27279046

Order 13568.

376 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 99 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#20285237C211yesmaximal, characteristic
2#20273380C211yesmaximal, characteristic
3#20273187C212yesmaximal
4#20272613C212yesmaximal
5C53×#2987C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16811033C414yes—
8#16811029—22no—
9#16810676C2212yes—
10#16810659—22no—
11C2×#13133128C2211yescharacteristic
12#16810369—22no—
13#16810353—24no—
14C53×#533C2211yescharacteristic
15C53×#446—24no—
16C53×#433—22no—
17C53×#432C2212yes—
18C22×#88322—21no—
19#13133217C2×C412yes—
20#13133209—21no—
21#13133196—22no—
22C4×Dic106—22no—
23#13133176—21no—
24#13133150—24no—
25#13133148—24no—
26C4⋊#88322—22no—
27C22⋊#88322—22no—
28#13133128—21no—
29#13133128C2×C412yes—
30C4×#88322—22no—
31C4⋊Q8×C53D412yes—
32#125×C53C2×C412yes—
33C22⋊Q8×C53—22no—
34C4×Q8×C53—22no—
35#117×C53D412yes—
36C2×C42×C53C2311yescharacteristic
37C4⋊C8×C53—24no—
38C4⋊C8×C53D414yes—
39C22⋊C8×C53—22no—
40#97×C53—22no—
41#97×C53—22no—
42C4×C8×C53—22no—
43C2×Q8×C53—22no—
44C22×C4×C53C2×D411yescharacteristic
45C22×C4×C53#4112yes—
46C2×Dic106—22no—
47C2×C8×C53—24no—
48C4⋊C4×C53—24no—
49C4⋊C4×C53—22no—
50C4⋊C4×C53C22⋊C414yes—
51C22⋊C4×C53—22no—
52C42×C53—22no—
53C42×C53C22×C411yescharacteristic
54C42×C53C2×D411yescharacteristic
55C42×C53C2×D412yes—
56C22⋊Dic53—41no—
57C4⋊Dic53—22no—
58C4⋊Dic53—22no—
59#11719919—42no—
60C4×Dic53—41no—
61C2×#88322—24no—
62C2×#88322—22no—
63C2×#88322—42no—
64C23×C53#12311yescharacteristic
65Q8×C53—82no—
66C2×C4×C53—22no—
67C2×C4×C53—24no—
68C2×C4×C53—42no—
69C2×C4×C53C2×C22⋊C411yescharacteristic
70C2×C4×C53C4×D412yes—
71C2×C4×C53#12011yescharacteristic
72C2×C4×C53#12112yes—
73C2×Dic53—42no—
74Dic106—44no—
75C8×C53—44no—
76#88322—24no—
77#88322—82no—
78#9006—531nomaximal, Sylow
79C22×C53—22no—
80C22×C53—21no—
81C22×C53#38611yescharacteristic
82C4×C53—22no—
83C4×C53—22no—
84C4×C53—21no—
85C4×C53—44no—
86C4×C53#45612yes—
87Dic53—82no—
88#3162—531no—
89#2987D5311yescharacteristic
90#1544—531no—
91#1497—532no—
92#1302—532no—
93C2×C53—41no—
94C2×C53#188112yes—
95C2×C53#188611yescharacteristic
96#533—531no—
97#533C2×D5311yescharacteristic
98#501—534no—
99#501—1062no—
100#446—24no—
101#433—22no—
102#432C2×D5312yes—
103#431—532no—
104#427—1062no—
105C2×#97—531no—
106#332—1062no—
107Q8⋊C8—1064no—
108C53#900611yescharacteristic, complemented, Sylow
109C22×C8—1061no—
110C4⋊Q8—532no—
111C4⋊Q8D21212yes—
112#125—1061no—
113#125C4×D5312yes—
114C22⋊Q8—22no—
115C22⋊Q8—1062no—
116C4×Q8—22no—
117C4×Q8—1062no—
118#117—1061no—
119#117C22⋊D5312yes—
120C2×C42C22×D5311yescharacteristic
121#107—1064no—
122#106—1064no—
123C4⋊C8—24no—
124C4⋊C8—1062no—
125C4⋊C8C22⋊D5314yes—
126C22⋊C8—22no—
127C22⋊C8—1062no—
128#97—22no—
129#97—22no—
130#97—532no—
131#97—1061no—
132C4×C8—22no—
133C4×C8—1062no—
134C2×Q8—22no—
135C2×Q8—1062no—
136C22×C4#1171994512yes—
137C22×C4C2×C22⋊D5311yescharacteristic
138C2×C8—24no—
139C2×C8—1064no—
140C2×C8—1062no—
141C2×C8—2122no—
142C4⋊C4—24no—
143C4⋊C4—22no—
144C4⋊C4—1062no—
145C4⋊C4—1062no—
146C4⋊C4—2122no—
147C4⋊C4#1171992114yes—
148C22⋊C4—22no—
149C22⋊C4—2121no—
150C42—22no—
151C42—2121no—
152C42C2×C4×D5311yescharacteristic
153C42C2×D21211yescharacteristic
154C42C2×C22⋊D5312yes—
155C23#1313320011yescharacteristic
156Q8—82no—
157Q8—2124no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—42no—
161C2×C4—2122no—
162C2×C4C2×#1171992111yescharacteristic
163C2×C4C4×C22⋊D5312yes—
164C2×C4#1313318611yescharacteristic
165C2×C4#1313319112yes—
166C8—44no—
167C8—1064no—
168C8—4242no—
169C22—22no—
170C22—21no—
171C22#1681050211yescharacteristic
172C4—22no—
173C4—22no—
174C4—21no—
175C4—44no—
176C4—4242no—
177C4#1681080612yes—
178C2—41no—
179C2#2027507712yes—
180C2#2027510811yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2987C2
socleC22×C53#386
prime radicalC23×C53#123
cosocleC2×C42×C53C23
hypercenter#2987D53
hypocenterC53#9006
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2987

Complements

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

NH ≅ G/Ntype
108. C53#9006semidirect

Maximal subgroups

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

subgroupindex
#202852372
#202733802
#202731872
#202726132
C53×#29872
#900653