Finite Group Databasebeta

← #27271609

Subgroups — #27271609

Order 13568.

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

normal subgroups only

This group has 252 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#20275814C211yesmaximal, characteristic
2#20272842C211yesmaximal, characteristic
3C53×#1319C211yesmaximal, characteristic
4C22×#11719925—21no—
5C2×#13133217C411yescharacteristic
6C2×#13133140—21no—
7C2×#13133136—21no—
8#16810378C411yescharacteristic
9C53×#391C2211yescharacteristic
10C53×#324—21no—
11C22×Dic106—21no—
12C2×#11719925—42no—
13C2×#11719925—42no—
14C2×#11719925—41no—
15C2×#11719925—41no—
16C2×#11719917—41no—
17C22×#88322—21no—
18#13133217—42no—
19C2×C4⋊Dic53—41no—
20#13133140—42no—
21#13133136—22no—
22#13133136—41no—
23C22×Q8×C53D411yescharacteristic
24C2×C4⋊C4×C53D411yescharacteristic
25C2×C42×C53C2×C411yescharacteristic
26C22⋊C8×C53—41no—
27#95×C53—41no—
28C2×Q8×C53—21no—
29C2×Q8×C53—41no—
30C2×Q8×C53—41no—
31C2×Q8×C53D811yescharacteristic
32C2×Q8×C53SD1611yescharacteristic
33C22×C4×C53—21no—
34C22×C4×C53—21no—
35C22×C4×C53—21no—
36C22×C4×C53C22⋊C411yescharacteristic
37C22×Dic53—41no—
38C2×Dic106—22no—
39C2×Dic106—41no—
40C2×Dic106—41no—
41C2×Dic106—41no—
42C2×Dic106—41no—
43C2×Dic106—41no—
44C2×C8×C53—81no—
45C4⋊C4×C53—41no—
46C42×C53—41no—
47#11719925—44no—
48#11719925—44no—
49C4⋊Dic53—42no—
50C4⋊Dic53—42no—
51#11719917—42no—
52#11719917—81no—
53C2×#88322—22no—
54C2×#88322—41no—
55C2×#88322—41no—
56C2×#88322—81no—
57C23×C53#9911yescharacteristic
58Q8×C53—41no—
59Q8×C53—41no—
60Q8×C53—81no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—22no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—41no—
67C2×C4×C53—41no—
68C2×C4×C53—41no—
69C2×C4×C53—41no—
70C2×C4×C53—41no—
71C2×C4×C53—81no—
72C2×C4×C53#10211yescharacteristic
73C2×C4×C53#10411yescharacteristic
74C2×Dic53—42no—
75C2×Dic53—81no—
76C2×Dic53—81no—
77Dic106—42no—
78Dic106—42no—
79Dic106—81no—
80Dic106—81no—
81C8×C53—161no—
82#88322—42no—
83#88322—82no—
84#7405—531nomaximal, Sylow
85C22×C53—21no—
86C22×C53—21no—
87C22×C53#32711yescharacteristic
88C22×C53AΣL(1,𝔽16)11yescharacteristic
89C22×C53#35311yescharacteristic
90C4×C53—21no—
91C4×C53—21no—
92C4×C53—42no—
93C4×C53—41no—
94C4×C53—41no—
95C4×C53—81no—
96Dic53—82no—
97#2016—531no—
98#1394—531no—
99#1319D5311yescharacteristic
100C2×C53—21no—
101C2×C53—21no—
102C2×C53#134311yescharacteristic
103C2×C53#140711yescharacteristic
104C2×C53#141111yescharacteristic
105C22×Q16—1061no—
106C2×C4⋊Q8—531no—
107C2×#103—1061no—
108C2×#101—1061no—
109#391C2×D5311yescharacteristic
110#336—531no—
111#324—21no—
112C53#740511yescharacteristic, complemented, Sylow
113C22×Q8—1061no—
114C22×Q8D21211yescharacteristic
115C2×Q16—2122no—
116C2×Q16—2122no—
117C2×Q16—2121no—
118C2×Q16—2121no—
119C2×M16—2121no—
120C22×C8—1061no—
121C4⋊Q8—2122no—
122C2×C4⋊C4—2121no—
123C2×C4⋊C4C22⋊D5311yescharacteristic
124C2×C42C4×D5311yescharacteristic
125#103—2122no—
126#101—1062no—
127#101—2121no—
128C22⋊C8—41no—
129#95—41no—
130C2×Q8—21no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—1062no—
134C2×Q8—2121no—
135C2×Q8—2121no—
136C2×Q8—2121no—
137C2×Q8—2121no—
138C2×Q8—2121no—
139C2×Q8#1171991311yescharacteristic
140C2×Q8D42411yescharacteristic
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—2121no—
145C22×C4#1171992111yescharacteristic
146Q16—2124no—
147Q16—2124no—
148M16—4241no—
149M16—2122no—
150C2×C8—81no—
151C2×C8—1062no—
152C2×C8—4241no—
153C2×C8—2121no—
154C2×C8—2121no—
155C4⋊C4—41no—
156C4⋊C4—2122no—
157C4⋊C4—2122no—
158C42—41no—
159C23#1313313111yescharacteristic
160Q8—41no—
161Q8—41no—
162Q8—81no—
163Q8—4241no—
164Q8—4241no—
165Q8—2122no—
166Q8—2122no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—81no—
178C2×C4—4241no—
179C2×C4—4241no—
180C2×C4—2122no—
181C2×C4#1313313911yescharacteristic
182C2×C4#1313314311yescharacteristic
183C8—161no—
184C8—4242no—
185C8—2122no—
186C22—21no—
187C22—21no—
188C22#1681035811yescharacteristic
189C22#1681041111yescharacteristic
190C22#1681041511yescharacteristic
191C4—21no—
192C4—21no—
193C4—42no—
194C4—41no—
195C4—41no—
196C4—81no—
197C4—4242no—
198C2—21no—
199C2—21no—
200C2#2027271311yescharacteristic
201C2#2027287411yescharacteristic
202C2#2027288311yescharacteristic
subgroupG/H
centerC22#16810415
commutatorC2×C42×C53C2×C4
Frattini#391C2×D53
FittingC53×#1319C2
socleC22×C53#353
prime radicalC23×C53#99
cosocleC53×#391C22
hypercenter#1319D53
hypocenterC53#7405
LCSG ▷ C2×C42×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ #1319

Complements

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

NH ≅ G/Ntype
112. C53#7405semidirect

Maximal subgroups

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

subgroupindex
#202758142
#202728422
C53×#13192
#740553