Finite Group Databasebeta

← #28299691

Subgroups — #28299691

Order 13568.

468 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 98 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#20290432C212yesmaximal
2#20289536C214yesmaximal
3#20277282C211yesmaximal, characteristic
4C2×#13133272—22no—
5#16811489—22no—
6#16811489C2214yes—
7C2×#13133217C411yescharacteristic
8C2×#13133182C412yes—
9#16810528C2212yes—
10#16810519—22no—
11C4⋊C4×Dic53—22no—
12C42×Dic53C2211yescharacteristic
13C53×#384C411yescharacteristic
14C22×#88324—24no—
15#13133272—28no—
16C22×Dic106C2×C412yes—
17#13133217—24no—
18#13133182—44no—
19C2×C4⋊Dic53—22no—
20C2×#11719919—22no—
21C2×C4×Dic53—22no—
22C2×C4×Dic53C2×C411yescharacteristic
23C2×C4×Dic53D412yes—
24C2×C4×Dic53D412yes—
25C2×C4×Dic53C2311yescharacteristic
26#13133124—24no—
27C2×C4⋊C4×C53—22no—
28C2×C4⋊C4×C53C2×C412yes—
29C2×C42×C53C2×C411yescharacteristic
30C22×C4×C53—22no—
31C22×C4×C53C22⋊C412yes—
32C22×C4×C53C22×C411yescharacteristic
33C22×Dic53—22no—
34C22×Dic53C22⋊C411yescharacteristic
35C22×Dic53C22⋊C411yescharacteristic
36C22×Dic53C2×D412yes—
37C2×Dic106—22no—
38C2×Dic106—42no—
39C2×Dic106—42no—
40C2×Dic106M1614yes—
41C2×#88324—28no—
42C2×#88324—44no—
43C2×#88324—44no—
44C4⋊C4×C53—42no—
45C4⋊C4×C53—44no—
46C42×C53—22no—
47C4⋊Dic53—28no—
48#11719919—44no—
49C4×Dic53—24no—
50C4×Dic53—22no—
51C4×Dic53—24no—
52C4×Dic53—41no—
53C4×Dic53—44no—
54C23×C53C2×C22⋊C411yescharacteristic
55C2×C4×C53—22no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—22no—
59C2×C4×C53—42no—
60C2×C4×C53—44no—
61C2×C4×C53#10414yes—
62C2×C4×C53C2×M1612yes—
63C2×Dic53—22no—
64C2×Dic53—22no—
65C2×Dic53—22no—
66C2×Dic53—24no—
67C2×Dic53—21no—
68C2×Dic53—22no—
69C2×Dic53—42no—
70C2×Dic53—42no—
71C2×Dic53#10012yes—
72C2×Dic53#13714yes—
73Dic106—44no—
74Dic106—44no—
75#88324—48no—
76#7639—531nomaximal, Sylow
77C22×C53#40711yescharacteristic
78C22×C53#41011yescharacteristic
79C22×C53C2×#10011yescharacteristic
80C22×C53#41912yes—
81C22×C53C2×#10412yes—
82C4×C53—22no—
83C4×C53—24no—
84C4×C53—82no—
85Dic53—22no—
86Dic53—24no—
87Dic53—44no—
88Dic53—41no—
89C4×C4⋊Q8—531no—
90#1745—532no—
91#1272—534no—
92C2×C53#146611yescharacteristic
93C2×C53#149112yes—
94C2×C53#150612yes—
95C2×C53#152511yescharacteristic
96C2×C53#155811yescharacteristic
97C2×C4⋊Q8—531no—
98C2×C4×Q8—532no—
99C2×#97—1062no—
100#391—532no—
101#389—1062no—
102#384#615711yescharacteristic
103C4×C4⋊C4—1062no—
104C43—531no—
105#336—534no—
106#336—1062no—
107C53#763911yescharacteristic, complemented, Sylow
108C22×Q8—532no—
109C22×C8—1064no—
110C4⋊Q8—1064no—
111C4×Q8—2124no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—1062no—
114C2×C4⋊C4—1062no—
115C2×C4⋊C4C2×#615712yes—
116C2×C42—532no—
117C2×C42—532no—
118C2×C42—531no—
119C2×C42—531no—
120C2×C42—1062no—
121C2×C42C2×#615711yescharacteristic
122#97—1068no—
123#95—1064no—
124C2×Q8—534no—
125C2×Q8—1062no—
126C2×Q8—2122no—
127C2×Q8—2122no—
128C22×C4—22no—
129C22×C4—531no—
130C22×C4—532no—
131C22×C4—531no—
132C22×C4—1062no—
133C22×C4C22⋊#615712yes—
134C22×C4C22×#615711yescharacteristic
135C2×C8—1068no—
136C2×C8—2124no—
137C2×C8—2124no—
138C4⋊C4—42no—
139C4⋊C4—44no—
140C4⋊C4—1068no—
141C4⋊C4—2124no—
142C42—22no—
143C42—1062no—
144C42—1064no—
145C42—1064no—
146C42—2121no—
147C42—2124no—
148C23C2×C22⋊#615711yescharacteristic
149Q8—2124no—
150Q8—2124no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—42no—
156C2×C4—44no—
157C2×C4—532no—
158C2×C4—534no—
159C2×C4—1062no—
160C2×C4—1062no—
161C2×C4—1062no—
162C2×C4—1064no—
163C2×C4—1061no—
164C2×C4—1062no—
165C2×C4—2122no—
166C2×C4—2122no—
167C2×C4#1313328114yes—
168C2×C4D53⋊M1612yes—
169C8—2128no—
170C22#1681159511yescharacteristic
171C22#1681160411yescharacteristic
172C22D53⋊#10011yescharacteristic
173C22#1681161612yes—
174C22C2×#1313328112yes—
175C4—24no—
176C4—22no—
177C4—82no—
178C4—1064no—
179C4—1062no—
180C4—2121no—
181C4—2124no—
182C2#2028997611yescharacteristic
183C2#2029005912yes—
184C2#2029010712yes—
185C2#2029017011yescharacteristic
186C2#2029024911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#7639
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
107. C53#7639semidirect

Maximal subgroups

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

subgroupindex
#202904322
#202895362
#202772822
#763953