Finite Group Databasebeta

← #27271586

Subgroups — #27271586

Order 13568.

258 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 37 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#20275824C211yesmaximal, characteristic
2#20272653C211yesmaximal, characteristic, complemented
3C53×#1391C211yesmaximal, characteristic, complemented
4C2×#13133242—21no—
5C22×D424C411yescharacteristic, complemented
6C2×#13133133—21no—
7#16810405—21no—
8#16810388C411yescharacteristic, complemented
9C53×#410—21no—
10C53×#408C2211yescharacteristic
11C22×D212—21no—
12#13133242—42no—
13#13133242—42no—
14C2×#11719924—41no—
15C2×D424—42no—
16C2×D424—41no—
17C2×#11719922—41no—
18C2×#11719917—21no—
19#13133157—41no—
20#13133133—42no—
21C2×#41×C53D411yescharacteristic
22C2×M16×C53D411yescharacteristic
23C22×C8×C53C2×C411yescharacteristic
24#117×C53—41no—
25#99×C53—22no—
26#99×C53—41no—
27C22⋊C8×C53—41no—
28C23×D53—41no—
29#41×C53—41no—
30#41×C53—41no—
31C2×Q8×C53SD1611yescharacteristic
32C2×D4×C53—21no—
33C2×D4×C53D811yescharacteristic
34C22×C4×C53—21no—
35C22×C4×C53C22⋊C411yescharacteristic
36C2×D212—22no—
37C2×D212—41no—
38C2×D212—41no—
39C2×D212—41no—
40C2×D212—41no—
41C2×D212—41no—
42M16×C53—41no—
43C2×C8×C53—21no—
44C2×C8×C53—21no—
45C2×C8×C53—41no—
46C4⋊C4×C53—81no—
47C22⋊C4×C53—42no—
48C42×C53—81no—
49#11719924—44no—
50#11719922—44no—
51#11719917—22no—
52#11719917—41no—
53C2×#88322—41no—
54D424—42no—
55D424—42no—
56#11719908—81no—
57C23×C53—21no—
58C23×C53#10411yescharacteristic
59C22×D53—42no—
60C22×D53—81no—
61C22×D53—81no—
62C22×D53—162no—
63Q8×C53—41no—
64D4×C53—41no—
65D4×C53—81no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—41no—
70C2×C4×C53—41no—
71C2×C4×C53—82no—
72C2×C4×C53#9911yescharacteristic
73C2×C4×C53#10211yescharacteristic
74D212—42no—
75D212—42no—
76D212—81no—
77D212—81no—
78C8×C53—41no—
79C8×C53—41no—
80#88322—42no—
81#7398—531nomaximal, Sylow
82C22×C53—21no—
83C22×C53—41no—
84C22×C53—41no—
85C22×C53—81no—
86C22×C53#32711yescharacteristic
87C2×D53—82no—
88C2×D53—162no—
89C2×D53—162no—
90C2×D53—162no—
91C4×C53—21no—
92C4×C53—41no—
93C4×C53—162no—
94C4×C53AΣL(1,𝔽16)11yescharacteristic
95C4×C53#35311yescharacteristic
96#2018—531no—
97#1391D5311yescharacteristic, complemented
98#1321—531no—
99C2×C53—21no—
100C2×C53—41no—
101C2×C53—81no—
102C2×C53#134311yescharacteristic
103D53—162no—
104C2×AGL(1,ℤ/8)—1061no—
105C22×D8—531no—
106C2×#100—1061no—
107#410—21no—
108#408C2×D5311yescharacteristic
109#349—1061no—
110#341—531no—
111C53#739811yescharacteristic, complemented, Sylow
112C2×#41C22⋊D5311yescharacteristic
113C22×D4—1061no—
114AGL(1,ℤ/8)—2122no—
115AGL(1,ℤ/8)—2122no—
116C2×SD16—2121no—
117C2×D8—2121no—
118C2×D8—2122no—
119C2×D8—2121no—
120C2×M16—1061no—
121C2×M16D21211yescharacteristic
122C22×C8C4×D5311yescharacteristic
123#117—41no—
124M32—2121no—
125#100—2122no—
126#99—22no—
127#99—41no—
128C22⋊C8—41no—
129C24—2121no—
130#41—41no—
131#41—41no—
132C2×Q8#1171992411yescharacteristic
133C2×D4—21no—
134C2×D4—1062no—
135C2×D4—2121no—
136C2×D4—2121no—
137C2×D4—2121no—
138C2×D4—2121no—
139C2×D4—2121no—
140C2×D4#1171992211yescharacteristic
141C22×C4—21no—
142C22×C4#1171992111yescharacteristic
143SD16—2124no—
144D8—2122no—
145D8—2122no—
146D8—2124no—
147M16—41no—
148M16—1062no—
149M16—2121no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—41no—
153C2×C8—2121no—
154C4⋊C4—81no—
155C22⋊C4—42no—
156C42—81no—
157C16—4241no—
158C23—21no—
159C23—8482no—
160C23—4241no—
161C23—4241no—
162C23—2122no—
163C23#1313314511yescharacteristic
164Q8—41no—
165D4—41no—
166D4—81no—
167D4—4241no—
168D4—4241no—
169D4—2122no—
170D4—2122no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—82no—
177C2×C4#1313313111yescharacteristic
178C2×C4#1313313711yescharacteristic
179C8—41no—
180C8—41no—
181C8—2122no—
182C22—21no—
183C22—41no—
184C22—41no—
185C22—81no—
186C22—8482no—
187C22—8482no—
188C22—8482no—
189C22—4242no—
190C22#1681035611yescharacteristic
191C4—21no—
192C4—41no—
193C4—162no—
194C4#1681040911yescharacteristic
195C4#1681041511yescharacteristic
196C2—21no—
197C2—41no—
198C2—81no—
199C2—8482no—
200C2#2027271111yescharacteristic
subgroupG/H
centerC2#20272711
commutatorC22×C8×C53C2×C4
Frattini#408C2×D53
FittingC53×#1391C2
socleC2×C53#1343
prime radical#20275824C2
cosocleC53×#408C22
hypercenter#1391D53
hypocenterC53#7398
LCSG ▷ C22×C8×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #408 ◁ #1391

Complements

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

NH ≅ G/Ntype
2. #20272653C2semidirect
3. C53×#1391C2semidirect
5. C22×D424C4semidirect
8. #16810388C4semidirect
97. #1391D53semidirect
111. C53#7398semidirect

Maximal subgroups

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

subgroupindex
#202758242
#202726532
C53×#13912
#739853