Finite Group Databasebeta

← #27271721

Subgroups — #27271721

Order 13568.

198 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 41 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#20276060C211yesmaximal, characteristic
2#20272661C211yesmaximal, characteristic
3C53×#1296C211yesmaximal, characteristic
4#16811254C411yescharacteristic
5#16810663—21no—
6#16810392C411yescharacteristic
7#16810383—21no—
8#16810349—21no—
9C2×C4⋊C8×C53—21no—
10C53×#384C2211yescharacteristic
11C2×#11719917—21no—
12#13133216—41no—
13#13133208—41no—
14#13133196—41no—
15#13133176—21no—
16#13133150—41no—
17#13133148—41no—
18C22⋊#88322—41no—
19C22×C8×C53—41no—
20C2×C4⋊C4×C53—41no—
21C2×C4⋊C4×C53C2×C411yescharacteristic
22C2×C4⋊C4×C53D411yescharacteristic
23C2×C42×C53D411yescharacteristic
24C4⋊C8×C53—22no—
25C4⋊C8×C53—41no—
26C22×C4×C53—21no—
27C22×C4×C53—21no—
28C22×C4×C53—21no—
29C22×C4×C53C22⋊C411yescharacteristic
30C2×Dic106—41no—
31C2×C8×C53—42no—
32C2×C8×C53—81no—
33C2×C8×C53—81no—
34C4⋊C4×C53—41no—
35C4⋊C4×C53—41no—
36C4⋊C4×C53—81no—
37C4⋊C4×C53—81no—
38C42×C53—21no—
39C42×C53SD1611yescharacteristic
40C42×C53Q1611yescharacteristic
41C22⋊Dic53—41no—
42C4⋊Dic53—21no—
43C4⋊Dic53—21no—
44C4⋊Dic53—41no—
45#11719919—42no—
46C4×Dic53—41no—
47#11719917—22no—
48#11719917—41no—
49C2×#88322—41no—
50C2×#88322—81no—
51C23×C53#9911yescharacteristic
52C2×C4×C53—21no—
53C2×C4×C53—21no—
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53—81no—
63C2×C4×C53—81no—
64C2×C4×C53#10311yescharacteristic
65C2×C4×C53#10411yescharacteristic
66C2×Dic53—41no—
67C2×Dic53—41no—
68Dic106—161no—
69C8×C53—82no—
70#88322—42no—
71#88322—161no—
72#7443—531nomaximal, Sylow
73C22×C53—21no—
74C22×C53—21no—
75C22×C53#32811yescharacteristic
76C22×C53AΣL(1,𝔽16)11yescharacteristic
77C22×C53#35611yescharacteristic
78C4×C53—21no—
79C4×C53—21no—
80C4×C53—21no—
81C4×C53—21no—
82C4×C53—21no—
83C4×C53—21no—
84C4×C53—81no—
85C4×C53—81no—
86Dic53—81no—
87Dic53—81no—
88#2060—531no—
89#1325—531no—
90#1296D5311yescharacteristic
91C2×C53—21no—
92C2×C53—21no—
93C2×C53#135411yescharacteristic
94C2×C53#140211yescharacteristic
95C2×C53#140511yescharacteristic
96#544—531no—
97#428—1061no—
98C2×C4⋊C8—21no—
99#384C2×D5311yescharacteristic
100#343—531no—
101#339—1061no—
102#324—1061no—
103C53#744311yescharacteristic, complemented, Sylow
104C2×M16—1061no—
105C22×C8—41no—
106C4⋊Q8—2121no—
107#125—2121no—
108C22⋊Q8—2121no—
109#117—1061no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4C4×D5311yescharacteristic
112C2×C4⋊C4D21211yescharacteristic
113C2×C42C22⋊D5311yescharacteristic
114#107—2121no—
115#106—2121no—
116C4⋊C8—22no—
117C4⋊C8—41no—
118C22⋊C8—2121no—
119C2×Q8—2121no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4#1171992111yescharacteristic
124M16—1062no—
125M16—2121no—
126C2×C8—42no—
127C2×C8—81no—
128C2×C8—81no—
129C2×C8—4241no—
130C2×C8—2121no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—81no—
134C4⋊C4—81no—
135C4⋊C4—1061no—
136C4⋊C4—1061no—
137C4⋊C4—2121no—
138C4⋊C4—2122no—
139C22⋊C4—2121no—
140C42—21no—
141C42—2121no—
142C42#1171992311yescharacteristic
143C42#1171992511yescharacteristic
144C23#1313313111yescharacteristic
145Q8—8481no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—81no—
157C2×C4—81no—
158C2×C4—2121no—
159C2×C4—2121no—
160C2×C4#1313314011yescharacteristic
161C2×C4#1313314511yescharacteristic
162C8—82no—
163C8—8481no—
164C8—2122no—
165C22—21no—
166C22—21no—
167C22#1681035911yescharacteristic
168C22#1681041111yescharacteristic
169C22#1681042111yescharacteristic
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—81no—
177C4—81no—
178C4—4241no—
179C4—4241no—
180C2—21no—
181C2—21no—
182C2#2027274411yescharacteristic
183C2#2027286111yescharacteristic
184C2#2027287011yescharacteristic
subgroupG/H
centerC22#16810411
commutatorC2×C4⋊C4×C53C2×C4
Frattini#384C2×D53
FittingC53×#1296C2
socleC22×C53AΣL(1,𝔽16)
prime radicalC23×C53#99
cosocleC53×#384C22
hypercenter#1296D53
hypocenterC53#7443
LCSG ▷ C2×C4⋊C4×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #384 ◁ #1296

Complements

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

NH ≅ G/Ntype
103. C53#7443semidirect

Maximal subgroups

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

subgroupindex
#202760602
#202726612
C53×#12962
#744353