Finite Group Databasebeta

← #28299103

Subgroups — #28299103

Order 13568.

194 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 36 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#20289812C211yesmaximal, characteristic
2#20289646C211yesmaximal, characteristic
3#20276252C211yesmaximal, characteristic
4C2×#13133278—21no—
5#16811468—21no—
6C2×#13133151—21no—
7C2×#13133142—21no—
8#16810529C2211yescharacteristic
9#16810493C411yescharacteristic
10C53×#336C411yescharacteristic
11C2×#11719940—41no—
12#13133278—22no—
13#13133278—41no—
14C22⋊#88324—41no—
15C22×Dic106D411yescharacteristic
16C2×C4⋊Dic53—21no—
17C2×C4⋊Dic53—41no—
18C2×C4⋊Dic53D411yescharacteristic
19#13133151—42no—
20#13133142—42no—
21#13133124—41no—
22C22×C8×C53—21no—
23C2×C42×C53C2×C411yescharacteristic
24C22×C4×C53—21no—
25C22×C4×C53C22⋊C411yescharacteristic
26C22×Dic53—21no—
27C22×Dic53—21no—
28C22×Dic53—41no—
29C2×Dic106—21no—
30C2×Dic106—41no—
31C2×Dic106—41no—
32C2×Dic106SD1611yescharacteristic
33C2×Dic106Q1611yescharacteristic
34#11719940—42no—
35#11719940—81no—
36C2×#88324—81no—
37C2×#88324—81no—
38C2×C8×C53—22no—
39C2×C8×C53—41no—
40C2×C8×C53—41no—
41C42×C53—41no—
42C4⋊Dic53—41no—
43C4⋊Dic53—41no—
44C4⋊Dic53—41no—
45C4⋊Dic53—82no—
46C23×C53#9911yescharacteristic
47C2×C4×C53—21no—
48C2×C4×C53—21no—
49C2×C4×C53—22no—
50C2×C4×C53—41no—
51C2×C4×C53—41no—
52C2×C4×C53#10311yescharacteristic
53C2×C4×C53#10411yescharacteristic
54C2×Dic53—21no—
55C2×Dic53—21no—
56C2×Dic53—41no—
57C2×Dic53—41no—
58C2×Dic53—41no—
59C2×Dic53—81no—
60C2×Dic53—82no—
61C2×Dic53—81no—
62Dic106—41no—
63Dic106—41no—
64Dic106—81no—
65#88324—82no—
66#88324—161no—
67C8×C53—42no—
68#7436—531nomaximal, Sylow
69C22×C53—21no—
70C22×C53—21no—
71C22×C53#32811yescharacteristic
72C22×C53#35211yescharacteristic
73C22×C53#35311yescharacteristic
74C4×C53—21no—
75C4×C53—21no—
76C4×C53—42no—
77Dic53—41no—
78Dic53—41no—
79Dic53—81no—
80Dic53—161no—
81#2095—531no—
82#1394—531no—
83#1319—531no—
84C2×C53—21no—
85C2×C53—21no—
86C2×C53#135211yescharacteristic
87C2×C53#140611yescharacteristic
88C2×C53#141111yescharacteristic
89C2×#107—1061no—
90C2×#103—1061no—
91C2×#101—1061no—
92#391—531no—
93#384—531no—
94#336#615711yescharacteristic
95#324—1061no—
96C53#743611yescharacteristic, complemented, Sylow
97C22×Q8—531no—
98C2×M16—2121no—
99C22×C8—21no—
100C2×C4⋊C4—531no—
101C2×C4⋊C4—1061no—
102C2×C4⋊C4—2121no—
103C2×C42C2×#615711yescharacteristic
104#107—2122no—
105#103—2122no—
106#101—1062no—
107#101—2121no—
108C22⋊C8—2121no—
109#95—2121no—
110C2×Q8—531no—
111C2×Q8—531no—
112C2×Q8—1061no—
113C2×Q8—2121no—
114C2×Q8—2121no—
115C22×C4—21no—
116C22×C4—1061no—
117C22×C4—1061no—
118C22×C4—2121no—
119C22×C4C22⋊#615711yescharacteristic
120M16—4241no—
121M16—2122no—
122C2×C8—22no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—4241no—
126C2×C8—4241no—
127C4⋊C4—4242no—
128C4⋊C4—2121no—
129C4⋊C4—2121no—
130C4⋊C4—2121no—
131C42—41no—
132C23#1313327611yescharacteristic
133Q8—4241no—
134Q8—2121no—
135Q8—2121no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—1061no—
142C2×C4—1061no—
143C2×C4—4242no—
144C2×C4—4241no—
145C2×C4—4241no—
146C2×C4—2121no—
147C2×C4—2121no—
148C2×C4—2121no—
149C2×C4#1313328011yescharacteristic
150C2×C4#1313328111yescharacteristic
151C8—42no—
152C8—8481no—
153C8—4242no—
154C22—21no—
155C22—21no—
156C22#1681147611yescharacteristic
157C22#1681152611yescharacteristic
158C22#1681152811yescharacteristic
159C4—21no—
160C4—21no—
161C4—42no—
162C4—8481no—
163C4—4241no—
164C4—2121no—
165C4—2121no—
166C2—21no—
167C2—21no—
168C2#2028971111yescharacteristic
169C2#2028984311yescharacteristic
170C2#2028985311yescharacteristic
subgroupG/H
centerC22#16811528
commutatorC2×C42×C53C2×C4
FrattiniC2×C42C2×#6157
FittingC53×#336C4
socleC22×C53#353
prime radicalC23×C53#99
cosocle#16810529C22
hypercenter#336#6157
hypocenterC53#7436
LCSG ▷ C2×C42×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #336

Complements

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

NH ≅ G/Ntype
96. C53#7436semidirect

Maximal subgroups

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

subgroupindex
#202898122
#202896462
#202762522
#743653