Finite Group Databasebeta

← #28299729

Subgroups — #28299729

Order 13568.

384 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 94 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#20290643C212yesmaximal
2#20289533C214yesmaximal
3#125×Dic53C211yesmaximal, characteristic
4#16811489—22no—
5#16811489—22no—
6#16811489C2214yes—
7#16810522C2212yes—
8#16810509—22no—
9#16810489C411yescharacteristic
10C4⋊C4×Dic53—22no—
11C4⋊C4×Dic53C412yes—
12C42×Dic53C2211yescharacteristic
13C2×#125×C53C411yescharacteristic
14C22×#88324—24no—
15C2×C4⋊Dic53—22no—
16C2×C4⋊Dic53C2×C412yes—
17C2×C4×Dic53—22no—
18C2×C4×Dic53C2×C411yescharacteristic
19C2×C4×Dic53D412yes—
20C2×C4×Dic53D412yes—
21C2×C4×Dic53C2311yescharacteristic
22#13133124—22no—
23#13133124—22no—
24#13133124—22no—
25#125×C53—24no—
26C2×C4⋊C4×C53—22no—
27C2×C4⋊C4×C53C2×C412yes—
28C2×C42×C53C2×C411yescharacteristic
29C22×C4×C53—22no—
30C22×C4×C53C22⋊C412yes—
31C22×C4×C53C22×C411yescharacteristic
32C22×Dic53—22no—
33C22×Dic53C22⋊C411yescharacteristic
34C22×Dic53C22⋊C411yescharacteristic
35C22×Dic53C2×D412yes—
36C2×#88324—44no—
37C2×#88324—44no—
38C2×#88324—44no—
39C4⋊C4×C53—28no—
40C4⋊C4×C53—22no—
41C4⋊C4×C53M1614yes—
42C42×C53—22no—
43C4⋊Dic53—44no—
44C4⋊Dic53—42no—
45C4×Dic53—22no—
46C4×Dic53—24no—
47C4×Dic53—42no—
48C4×Dic53—41no—
49C4×Dic53—44no—
50C23×C53C2×C22⋊C411yescharacteristic
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—24no—
54C2×C4×C53—22no—
55C2×C4×C53—22no—
56C2×C4×C53—22no—
57C2×C4×C53—42no—
58C2×C4×C53—42no—
59C2×C4×C53C2×M1612yes—
60C2×Dic53—22no—
61C2×Dic53—22no—
62C2×Dic53—22no—
63C2×Dic53—21no—
64C2×Dic53—22no—
65C2×Dic53—42no—
66C2×Dic53—42no—
67C2×Dic53—42no—
68C2×Dic53#10112yes—
69C2×Dic53#13514yes—
70#88324—84no—
71#7650—531nomaximal, Sylow
72C22×C53#40711yescharacteristic
73C22×C53#41011yescharacteristic
74C22×C53C2×#10111yescharacteristic
75C22×C53#41612yes—
76C22×C53#42112yes—
77C4×C53—22no—
78C4×C53—42no—
79C4×C53—44no—
80Dic53—24no—
81Dic53—22no—
82Dic53—41no—
83Dic53—82no—
84C4×#125—531no—
85#1839—532no—
86#1271—534no—
87C2×C53#146711yescharacteristic
88C2×C53#148812yes—
89C2×C53#152112yes—
90C2×C53#153411yescharacteristic
91C2×C53#154411yescharacteristic
92C2×#125#615711yescharacteristic
93#389—532no—
94#387—1062no—
95#382—531no—
96C4×C4⋊C4—532no—
97C4×C4⋊C4—1062no—
98C43—531no—
99#336—534no—
100#336—1062no—
101#336—1062no—
102C53#765011yescharacteristic, complemented, Sylow
103C22×C8—1064no—
104#125—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—532no—
107C2×C4⋊C4—1062no—
108C2×C4⋊C4C2×#615712yes—
109C2×C42—532no—
110C2×C42—531no—
111C2×C42—532no—
112C2×C42—531no—
113C2×C42—1062no—
114C2×C42C2×#615711yescharacteristic
115#95—1062no—
116#95—1062no—
117#95—1062no—
118C22×C4—22no—
119C22×C4—531no—
120C22×C4—532no—
121C22×C4—531no—
122C22×C4—1062no—
123C22×C4C22⋊#615712yes—
124C22×C4C22×#615711yescharacteristic
125C2×C8—2124no—
126C2×C8—2124no—
127C2×C8—2124no—
128C4⋊C4—28no—
129C4⋊C4—22no—
130C4⋊C4—2122no—
131C4⋊C4—2124no—
132C4⋊C4#1171994014yes—
133C42—22no—
134C42—1062no—
135C42—1064no—
136C42—2121no—
137C42—2122no—
138C42—2124no—
139C23C2×C22⋊#615711yescharacteristic
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—22no—
143C2×C4—24no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—532no—
149C2×C4—534no—
150C2×C4—1062no—
151C2×C4—1062no—
152C2×C4—1062no—
153C2×C4—1061no—
154C2×C4—1062no—
155C2×C4—2122no—
156C2×C4—2122no—
157C2×C4—2122no—
158C2×C4C2×#1171994012yes—
159C8—4244no—
160C22#1681159311yescharacteristic
161C22#1681160411yescharacteristic
162C22D53⋊#10111yescharacteristic
163C22#1681161312yes—
164C22#1681162012yes—
165C4—22no—
166C4—42no—
167C4—44no—
168C4—1064no—
169C4—1062no—
170C4—4242no—
171C4—2121no—
172C2#2028997811yescharacteristic
173C2#2029004612yes—
174C2#2029015812yes—
175C2#2029019811yescharacteristic
176C2#2029022111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#125×C53C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#125#6157
hypocenterC53#7650
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
102. C53#7650semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202895332
#125×Dic532
#765053