Finite Group Databasebeta

← #28299571

Subgroups — #28299571

Order 13568.

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

normal subgroups only

This group has 440 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#20290642C212yesmaximal
2#20289536C214yesmaximal
3#20277282C211yesmaximal, characteristic
4#16811489—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—
15C22×Dic106C2×C412yes—
16#13133217—22no—
17#13133217C814yescomplemented
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×Dic106C2×C814yes—
41C2×#88324—44no—
42C2×#88324—44no—
43C2×#88324—44no—
44C4⋊C4×C53—44no—
45C4⋊C4×C53—42no—
46C42×C53C2×C812yes—
47C42×C53M1612yes—
48C4⋊Dic53—28no—
49#11719919—44no—
50C4×Dic53—22no—
51C4×Dic53—24no—
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—44no—
62C2×C4×C53C22⋊C814yes—
63C2×C4×C53C22×C811yescharacteristic
64C2×C4×C53C2×M1611yescharacteristic
65C2×Dic53—22no—
66C2×Dic53—22no—
67C2×Dic53—22no—
68C2×Dic53—24no—
69C2×Dic53—21no—
70C2×Dic53—22no—
71C2×Dic53—42no—
72C2×Dic53—42no—
73C2×Dic53#10012yes—
74C2×Dic53#13714yes—
75Dic106—44no—
76Dic106—44no—
77#88324—84no—
78#7605—531nomaximal, Sylow
79C22×C53C2×C22⋊C811yescharacteristic
80C22×C53#41011yescharacteristic
81C22×C53C2×#10011yescharacteristic
82C22×C53#41912yes—
83C22×C53#42112yes—
84C4×C53—21no—
85C4×C53—21no—
86C4×C53—24no—
87C4×C53—82no—
88Dic53—22no—
89Dic53—24no—
90Dic53—41no—
91Dic53—44no—
92C4×C4⋊Q8—531no—
93#1839—532no—
94#1272—534no—
95C2×C53#146011yescharacteristic
96C2×C53#147912yes—
97C2×C53#151512yes—
98C2×C53#153611yescharacteristic
99C2×C53#155811yescharacteristic
100C2×C4⋊Q8—531no—
101C2×C4×Q8—532no—
102#391—532no—
103#389—1062no—
104#384#615711yescharacteristic
105C4×C4⋊C4—1062no—
106C43—531no—
107#336—534no—
108#336—1062no—
109#336—1062no—
110C53#760511yescharacteristic, complemented, Sylow
111C22×Q8—532no—
112C22×C8—1064no—
113C4⋊Q8—534no—
114C4⋊Q8—1062no—
115C4×Q8—2124no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—1062no—
118C2×C4⋊C4—1062no—
119C2×C4⋊C4C2×#615712yes—
120C2×C42—532no—
121C2×C42—531no—
122C2×C42—532no—
123C2×C42—531no—
124C2×C42—1062no—
125C2×C42C2×#615711yescharacteristic
126#95—1064no—
127C2×Q8—534no—
128C2×Q8—1062no—
129C2×Q8—2122no—
130C2×Q8—2122no—
131C22×C4—22no—
132C22×C4—531no—
133C22×C4—532no—
134C22×C4—531no—
135C22×C4—1062no—
136C22×C4C22⋊#615712yes—
137C22×C4C22×#615711yescharacteristic
138C2×C8—2124no—
139C2×C8—2124no—
140C2×C8—2124no—
141C4⋊C4—42no—
142C4⋊C4—44no—
143C4⋊C4—1068no—
144C4⋊C4—2124no—
145C42—1062no—
146C42—1064no—
147C42—2121no—
148C42—2122no—
149C42—2124no—
150C42D53⋊C812yes—
151C42#1171993612yes—
152C23C2×C22⋊#615711yescharacteristic
153Q8—2124no—
154Q8—2124no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—42no—
160C2×C4—44no—
161C2×C4—532no—
162C2×C4—534no—
163C2×C4—1062no—
164C2×C4—1062no—
165C2×C4—1062no—
166C2×C4—1064no—
167C2×C4—1061no—
168C2×C4—1062no—
169C2×C4—2122no—
170C2×C4—2122no—
171C2×C4#1313327414yes—
172C2×C4C2×D53⋊C811yescharacteristic
173C2×C4C2×#1171993611yescharacteristic
174C8—4244no—
175C22C2×#1313327411yescharacteristic
176C22#1681160411yescharacteristic
177C22D53⋊#10011yescharacteristic
178C22#1681161612yes—
179C22#1681162112yes—
180C4—21no—
181C4—21no—
182C4—24no—
183C4—82no—
184C4—1064no—
185C4—1062no—
186C4—2121no—
187C4—2124no—
188C2#2028995611yescharacteristic
189C2#2029001912yes—
190C2#2029013812yes—
191C2#2029020411yescharacteristic
192C2#2029024911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#7605
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
17. #13133217C8semidirect
110. C53#7605semidirect

Maximal subgroups

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

subgroupindex
#202906422
#202895362
#202772822
#760553