Finite Group Databasebeta

← #27271159

Subgroups — #27271159

Order 13568.

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

normal subgroups only

This group has 248 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#20274765C211yesmaximal, characteristic
2#20274761C211yesmaximal, characteristic
3C53×#2113C211yesmaximal, characteristic
4C2×#13133153—21no—
5#16810661—21no—
6#16810607—21no—
7#16810597—21no—
8#16810381C412yescomplemented
9#16810380C412yes—
10C2×#131×C53C2211yescharacteristic
11C2×M32×C53C411yescharacteristic, complemented
12C22×C16×C53C411yescharacteristic, complemented
13C53×C22⋊C16—21no—
14C53×C22⋊C16—21no—
15C2×#11719917—22no—
16#13133176—22no—
17#13133153—41no—
18#13133153—41no—
19#13133151—41no—
20#13133149—41no—
21#13133136—41no—
22#13133134—41no—
23#13133132—41no—
24#13133132—41no—
25C2×#41×C53D411yescharacteristic
26#131×C53—41no—
27#131×C53—41no—
28C2×M16×C53C2×C411yescharacteristic
29C2×M16×C53C2×C411yescharacteristic
30C2×M16×C53Q811yescharacteristic
31C22×C8×C53C2×C411yescharacteristic
32C22×C8×C53D411yescharacteristic
33C22×C8×C53D411yescharacteristic
34M32×C53—41no—
35C2×C16×C53—21no—
36C2×C16×C53—21no—
37C2×C16×C53—41no—
38#41×C53—41no—
39#41×C53—41no—
40C2×Q8×C53SD1611yescharacteristic
41C2×D4×C53D811yescharacteristic
42C2×D4×C53SD1611yescharacteristic
43C2×D4×C53Q1611yescharacteristic
44C22×C4×C53C22⋊C411yescharacteristic
45C22×C4×C53C22⋊C411yescharacteristic
46C22×C4×C53C4⋊C411yescharacteristic
47M16×C53—41no—
48M16×C53—41no—
49M16×C53—41no—
50C2×C8×C53—21no—
51C2×C8×C53—21no—
52C2×C8×C53—21no—
53C2×C8×C53—21no—
54C2×C8×C53—21no—
55C2×C8×C53—21no—
56C2×C8×C53C4211yescharacteristic
57C2×C8×C53C22⋊C411yescharacteristic
58C2×C8×C53C4⋊C411yescharacteristic
59C2×C8×C53C4⋊C411yescharacteristic
60C22⋊Dic53—42no—
61C4⋊Dic53—42no—
62C4×Dic53—42no—
63#11719917—42no—
64#11719917—42no—
65C2×#88322—42no—
66C16×C53—41no—
67C16×C53—41no—
68C23×C53#10211yescharacteristic
69C23×C53#10311yescharacteristic
70C23×C53#10711yescharacteristic
71Q8×C53—41no—
72D4×C53—41no—
73D4×C53—41no—
74D4×C53—41no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53#9511yescharacteristic
82C2×C4×C53#10211yescharacteristic
83C2×C4×C53#10311yescharacteristic
84C2×C4×C53#10611yescharacteristic
85C2×Dic53—82no—
86C8×C53—21no—
87C8×C53—21no—
88C8×C53—21no—
89C8×C53#9911yescharacteristic
90C8×C53#9911yescharacteristic
91#88322—82no—
92#7227—531nomaximal, Sylow
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—21no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53#34011yescharacteristic
100C4×C53—21no—
101C4×C53—21no—
102C4×C53—21no—
103C4×C53H(3,ℤ/4)11yescharacteristic
104C4×C53#34211yescharacteristic
105Dic53—162no—
106#2113D5311yescharacteristic
107#1813—531no—
108#1812—531no—
109C2×C53—21no—
110C2×C53—41no—
111C2×C53—41no—
112C2×C53—41no—
113C2×C53#128911yescharacteristic
114C2×#131C2×D5311yescharacteristic
115C2×M32Dic5311yescharacteristic, complemented
116C22×C16Dic5311yescharacteristic, complemented
117C2×#108—1061no—
118#427—1061no—
119#413—1061no—
120#410—1061no—
121C22⋊C16—21no—
122C22⋊C16—21no—
123#338—532no—
124H(3,ℤ/4)—532no—
125C53#722711yescharacteristic, complemented, Sylow
126C2×#41D21211yescharacteristic
127#131—41no—
128#131—41no—
129C2×M16—1062no—
130C2×M16Dic10611yescharacteristic
131C2×M16C4×D5311yescharacteristic
132C2×M16C4×D5311yescharacteristic
133C22×C8C2×Dic5311yescharacteristic
134C22×C8C22⋊D5311yescharacteristic
135C22×C8C22⋊D5311yescharacteristic
136#117—1062no—
137M32—41no—
138C2×C16—21no—
139C2×C16—21no—
140C2×C16—41no—
141#108—2121no—
142#108—2121no—
143#107—2121no—
144#106—2121no—
145#101—2121no—
146#100—2121no—
147#99—2121no—
148#99—2121no—
149#41—41no—
150#41—41no—
151C2×Q8#1171991311yescharacteristic
152C2×D4#1171991311yescharacteristic
153C2×D4D42411yescharacteristic
154C2×D4Dic21211yescharacteristic
155C22×C4C4⋊Dic5311yescharacteristic
156C22×C4#1171992111yescharacteristic
157C22×C4#1171992111yescharacteristic
158M16—41no—
159M16—41no—
160M16—41no—
161M16—2122no—
162M16—2122no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—2122no—
170C2×C8C4×Dic5311yescharacteristic
171C2×C8#1171991911yescharacteristic
172C2×C8#1171991911yescharacteristic
173C2×C8C22⋊Dic5311yescharacteristic
174C4⋊C4—2122no—
175C22⋊C4—2122no—
176C42—2122no—
177C16—41no—
178C16—41no—
179C23#1313313911yescharacteristic
180C23#1313314211yescharacteristic
181C23#1313315111yescharacteristic
182Q8—41no—
183D4—41no—
184D4—41no—
185D4—41no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—4242no—
193C2×C4#1313312411yescharacteristic
194C2×C4#1313313911yescharacteristic
195C2×C4#1313314211yescharacteristic
196C2×C4#1313314911yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—4242no—
201C8#1313313211yescharacteristic
202C8#1313313211yescharacteristic
203C22—21no—
204C22—21no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22#1681038611yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—8482no—
214C4#1681038011yescharacteristic
215C4#1681039011yescharacteristic
216C2—21no—
217C2—41no—
218C2—41no—
219C2—41no—
220C2#2027258311yescharacteristic
subgroupG/H
centerC4#16810380
commutatorC2×C8×C53C42
FrattiniC2×#131C2×D53
FittingC53×#2113C2
socleC2×C53#1289
prime radicalC2×#41×C53D4
cosocleC2×#131×C53C22
hypercenter#2113D53
hypocenterC53#7227
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C8 ◁ #2113

Complements

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

NH ≅ G/Ntype
8. #16810381C4semidirect
11. C2×M32×C53C4semidirect
12. C22×C16×C53C4semidirect
115. C2×M32Dic53semidirect
116. C22×C16Dic53semidirect
125. C53#7227semidirect

Maximal subgroups

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

subgroupindex
#202747652
#202747612
C53×#21132
#722753