Finite Group Databasebeta

← #27277548

Subgroups — #27277548

Order 13568.

404 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 115 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#20273029C212yesmaximal
2#20272548C212yesmaximal, complemented
3#20272544C212yesmaximal, complemented
4C53×#2298C211yesmaximal, characteristic
5#16810676C2214yes—
6#16810349—22no—
7#16810347—22no—
8C2×#117×C53C2211yescharacteristic
9C53×#388—22no—
10C53×#387—22no—
11C4×C4⋊C4×C53C412yes—
12C4×C22⋊C4×C53C412yes—
13C53×#376C2211yescharacteristic
14C43×C53C2211yescharacteristic
15C4⋊#88322—24no—
16C22⋊#88322—24no—
17C4×#88322—24no—
18C23×C4×C53C2×C411yescharacteristic
19#117×C53—22no—
20#117×C53—22no—
21C2×C4⋊C4×C53—22no—
22C2×C4⋊C4×C53D412yes—
23C2×C22⋊C4×C53—22no—
24C2×C22⋊C4×C53D412yes—
25C2×C42×C53—22no—
26C2×C42×C53C2×C412yes—
27C2×C42×C53C2×C412yes—
28C2×C42×C53C2×C411yescharacteristic
29C2×C42×C53C2311yescharacteristic
30#95×C53—22no—
31#95×C53—22no—
32#95×C53—22no—
33C24×C53C22⋊C411yescharacteristic
34C22×C4×C53—22no—
35C22×C4×C53—21no—
36C22×C4×C53—21no—
37C22×C4×C53—24no—
38C22×C4×C53—22no—
39C22×C4×C53C22⋊C411yescharacteristic
40C22×C4×C53C22⋊C412yes—
41C22×C4×C53C22×C411yescharacteristic
42C22×C4×C53C2×D412yes—
43C4⋊C4×C53—24no—
44C4⋊C4×C53—44no—
45C22⋊C4×C53—24no—
46C22⋊C4×C53—44no—
47C42×C53—21no—
48C42×C53—21no—
49C42×C53—24no—
50C42×C53—24no—
51C42×C53—44no—
52C42×C53#4112yes—
53C42×C53#4112yes—
54C2×#88322—44no—
55C23×C53—21no—
56C23×C53—22no—
57C23×C53—42no—
58C23×C53C2×C22⋊C411yescharacteristic
59C2×C4×C53—24no—
60C2×C4×C53—21no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—42no—
64C2×C4×C53—42no—
65C2×C4×C53—42no—
66C2×C4×C53—42no—
67C2×C4×C53—42no—
68C2×C4×C53—42no—
69C2×C4×C53—42no—
70C2×C4×C53—42no—
71C2×C4×C53—42no—
72C2×C4×C53#11711yescharacteristic
73C2×C4×C53#11711yescharacteristic
74C2×C4×C53#12314yes—
75C2×C4×C53#13114yes—
76#88322—84no—
77#8733—531nomaximal, Sylow
78C22×C53—22no—
79C22×C53—41no—
80C22×C53—42no—
81C22×C53—82no—
82C22×C53#38111yescharacteristic
83C22×C53#40812yes—
84C4×C53—24no—
85C4×C53—44no—
86C4×C53—41no—
87C4×C53—42no—
88C4×C53—82no—
89C4×C53#43312yes—
90C4×C53#43312yes—
91#2298D5311yescharacteristic
92#1458—532no—
93#1276—532no—
94#1274—532no—
95C2×C53—22no—
96C2×C53—81no—
97C2×C53#181811yescharacteristic
98C2×C53#182712yes—
99C2×#117C2×D5311yescharacteristic
100#431—534no—
101#388—22no—
102#387—22no—
103C4×C4⋊C4Dic5312yes—
104C4×C22⋊C4Dic5312yes—
105#376C2×D5311yescharacteristic
106C43C2×D5311yescharacteristic
107#324—1062no—
108#323—1062no—
109C53#873311yescharacteristic, complemented, Sylow
110C23×C4C2×Dic5311yescharacteristic
111#117—22no—
112#117—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4C22⋊D5312yes—
115C2×C22⋊C4—22no—
116C2×C22⋊C4C22⋊D5312yes—
117C2×C42—22no—
118C2×C42C2×Dic5312yes—
119C2×C42C2×Dic5312yes—
120C2×C42C2×Dic5311yescharacteristic
121C2×C42C22×D5311yescharacteristic
122C4⋊C8—1064no—
123C22⋊C8—1064no—
124C4×C8—1064no—
125#95—22no—
126#95—22no—
127#95—22no—
128C24C22⋊Dic5311yescharacteristic
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—24no—
133C22×C4—22no—
134C22×C4C22⋊Dic5311yescharacteristic
135C22×C4C22⋊Dic5312yes—
136C22×C4C22×Dic5311yescharacteristic
137C22×C4C2×C22⋊D5312yes—
138C2×C8—2124no—
139C4⋊C4—24no—
140C4⋊C4—44no—
141C22⋊C4—24no—
142C22⋊C4—44no—
143C42—21no—
144C42—21no—
145C42—24no—
146C42—24no—
147C42—44no—
148C42#1171994512yes—
149C42#1171994512yes—
150C23—21no—
151C23—22no—
152C23—42no—
153C23C2×C22⋊Dic5311yescharacteristic
154C2×C4—24no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4#1313317611yescharacteristic
168C2×C4#1313317611yescharacteristic
169C2×C4#1313320014yes—
170C2×C4#1313322414yes—
171C8—4244no—
172C22—22no—
173C22—41no—
174C22—42no—
175C22—82no—
176C22#1681048611yescharacteristic
177C22#1681059012yes—
178C4—24no—
179C4—44no—
180C4—41no—
181C4—42no—
182C4—82no—
183C4#1681068412yes—
184C4#1681068412yes—
185C2—22no—
186C2—81no—
187C2#2027478611yescharacteristic
188C2#2027481912yes—
subgroupG/H
centerC2×C4#13133176
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2298C2
socleC22×C53#381
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2298D53
hypocenterC53#8733
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ #376 ◁ #2298

Complements

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

NH ≅ G/Ntype
2. #20272548C2semidirect
3. #20272544C2semidirect
109. C53#8733semidirect

Maximal subgroups

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

subgroupindex
#202730292
#202725482
#202725442
C53×#22982
#873353