Finite Group Databasebeta

← #28299562

Subgroups — #28299562

Order 13568.

408 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 110 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#20290630C212yesmaximal
2#20289534C214yesmaximal
3#20277270C211yesmaximal, characteristic
4C2×C4⋊#88324—22no—
5#16811489—22no—
6#16811489C2214yes—
7C2×#13133209C411yescharacteristic
8#16810519C2212yes—
9#16810505—22no—
10C4⋊C4×Dic53C412yes—
11#16810474—22no—
12C42×Dic53C2211yescharacteristic
13C53×#382C411yescharacteristic
14C22×#88324—24no—
15C4⋊#88324—44no—
16#13133209—22no—
17#13133209C814yescomplemented
18C2×C4⋊Dic53C2×C412yes—
19C2×#11719919—22no—
20C2×#11719919—22no—
21C2×C4×Dic53—22no—
22C2×C4×Dic53C2×C411yescharacteristic
23C2×C4×Dic53D412yes—
24C2×C4×Dic53D412yes—
25C2×C4×Dic53C2311yescharacteristic
26#13133124—22no—
27#13133124—22no—
28C2×C4⋊C4×C53C2×C412yes—
29C2×C42×C53C2×C411yescharacteristic
30#95×C53—22no—
31C22×C4×C53—22no—
32C22×C4×C53C22⋊C412yes—
33C22×C4×C53C22×C411yescharacteristic
34C22×Dic53—22no—
35C22×Dic53C22⋊C411yescharacteristic
36C22×Dic53C22⋊C411yescharacteristic
37C22×Dic53C2×D412yes—
38C2×#88324—44no—
39C2×#88324—44no—
40C2×#88324—44no—
41C4⋊C4×C53—42no—
42C42×C53C2×C812yes—
43C42×C53M1612yes—
44C4⋊Dic53—22no—
45C4⋊Dic53C2×C814yes—
46#11719919—28no—
47#11719919—42no—
48#11719919—42no—
49C4×Dic53—22no—
50C4×Dic53—22no—
51C4×Dic53—22no—
52C4×Dic53—42no—
53C4×Dic53—41no—
54C4×Dic53—44no—
55C23×C53C2×C22⋊C411yescharacteristic
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—22no—
59C2×C4×C53—22no—
60C2×C4×C53—42no—
61C2×C4×C53—42no—
62C2×C4×C53—42no—
63C2×C4×C53C22⋊C814yes—
64C2×C4×C53C22×C811yescharacteristic
65C2×C4×C53C2×M1611yescharacteristic
66C2×Dic53—22no—
67C2×Dic53—22no—
68C2×Dic53—22no—
69C2×Dic53—24no—
70C2×Dic53—21no—
71C2×Dic53—22no—
72C2×Dic53—42no—
73C2×Dic53—42no—
74C2×Dic53#10112yes—
75C2×Dic53AGL(1,ℤ/8)12yes—
76C2×Dic53#13712yes—
77#88324—84no—
78#7602—531nomaximal, Sylow
79C22×C53C2×C22⋊C811yescharacteristic
80C22×C53#41011yescharacteristic
81C22×C53C2×#10111yescharacteristic
82C22×C53#41712yes—
83C22×C53#42112yes—
84C4×C53—21no—
85C4×C53—21no—
86C4×C53—24no—
87C4×C53—82no—
88Dic53—22no—
89Dic53—22no—
90Dic53—22no—
91Dic53—41no—
92Dic53—44no—
93C4×#125—531no—
94#1831—532no—
95#1271—534no—
96C2×C53#146111yescharacteristic
97C2×C53#147712yes—
98C2×C53#151212yes—
99C2×C53#153411yescharacteristic
100C2×C53#156111yescharacteristic
101C2×#125—531no—
102C2×C4⋊C8—1062no—
103#389—532no—
104#387—1062no—
105#382#615711yescharacteristic
106C4×C4⋊C4—532no—
107C4×C4⋊C4—1062no—
108C43—531no—
109#336—534no—
110#336—1062no—
111C53#760211yescharacteristic, complemented, Sylow
112C22×C8—1064no—
113#125—534no—
114#125—1062no—
115C2×C4⋊C4—532no—
116C2×C4⋊C4—1062no—
117C2×C4⋊C4—1062no—
118C2×C4⋊C4C2×#615712yes—
119C2×C42—532no—
120C2×C42—531no—
121C2×C42—532no—
122C2×C42—531no—
123C2×C42—1062no—
124C2×C42C2×#615711yescharacteristic
125C4⋊C8—2124no—
126#95—22no—
127#95—1062no—
128#95—1062no—
129C22×C4—22no—
130C22×C4—531no—
131C22×C4—532no—
132C22×C4—531no—
133C22×C4—1062no—
134C22×C4C22⋊#615712yes—
135C22×C4C22×#615711yescharacteristic
136C2×C8—2124no—
137C2×C8—2124no—
138C2×C8—2124no—
139C4⋊C4—42no—
140C4⋊C4—534no—
141C4⋊C4—1068no—
142C4⋊C4—1062no—
143C4⋊C4—2122no—
144C4⋊C4—2122no—
145C42—1062no—
146C42—1062no—
147C42—1062no—
148C42—2121no—
149C42—2122no—
150C42—2124no—
151C42D53⋊C812yes—
152C42#1171993612yes—
153C23C2×C22⋊#615711yescharacteristic
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—532no—
162C2×C4—532no—
163C2×C4—532no—
164C2×C4—1062no—
165C2×C4—1062no—
166C2×C4—1062no—
167C2×C4—1064no—
168C2×C4—1061no—
169C2×C4—1062no—
170C2×C4—2122no—
171C2×C4—2122no—
172C2×C4#1313327414yes—
173C2×C4C2×D53⋊C811yescharacteristic
174C2×C4C2×#1171993611yescharacteristic
175C8—4244no—
176C22C2×#1313327411yescharacteristic
177C22#1681160311yescharacteristic
178C22D53⋊#10111yescharacteristic
179C22#1681161412yes—
180C22#1681162012yes—
181C4—21no—
182C4—21no—
183C4—24no—
184C4—82no—
185C4—1062no—
186C4—1062no—
187C4—1062no—
188C4—2121no—
189C4—2124no—
190C2#2028995911yescharacteristic
191C2#2029001212yes—
192C2#2029012712yes—
193C2#2029019811yescharacteristic
194C2#2029025511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#382C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#382#6157
hypocenterC53#7602
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
17. #13133209C8semidirect
111. C53#7602semidirect

Maximal subgroups

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

subgroupindex
#202906302
#202895342
#202772702
#760253