Finite Group Databasebeta

← #27277615

Subgroups — #27277615

Order 13568.

566 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 99 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#20277301C211yesmaximal, characteristic
2#20274639C211yesmaximal, characteristic
3#20272624C214yesmaximal
4C53×#1784C211yesmaximal, characteristic
5C2×#13133176C412yes—
6C2×#13133136—22no—
7#16810587C2212yes—
8#16810530—22no—
9#16810530—22no—
10#16810486C412yes—
11#16810486C2212yes—
12C23×Q8×C53C2211yescharacteristic
13C2×#101×C53—22no—
14C53×#407C2212yes—
15C2×#11719917—22no—
16#13133176—44no—
17C2×C4⋊Dic53—22no—
18C2×C22⋊Dic53C2×C414yes—
19C2×C4×Dic53—22no—
20#13133136—44no—
21C22⋊#88322—24no—
22#13133124—24no—
23#13133124—22no—
24#13133124—22no—
25C22×Q8×C53—24no—
26C22×Q8×C53—44no—
27C22×Q8×C53D412yes—
28C22×Q8×C53D412yes—
29C23×C4×C53C2×C411yescharacteristic
30C23×C4×C53C2×C411yescharacteristic
31C23×C4×C53C2311yescharacteristic
32C2×M16×C53—22no—
33#101×C53—44no—
34C22⋊C8×C53—24no—
35C24×C53C22×C411yescharacteristic
36C2×Q8×C53—24no—
37C2×Q8×C53—24no—
38C2×Q8×C53—24no—
39C2×Q8×C53—24no—
40C2×Q8×C53—42no—
41C2×Q8×C53—42no—
42C2×Q8×C53—416no—
43C22×C4×C53—22no—
44C22×C4×C53—24no—
45C22×C4×C53—22no—
46C22×C4×C53—42no—
47C22×C4×C53—42no—
48C22×C4×C53C22⋊C412yes—
49C22×C4×C53C22⋊C412yes—
50C22×C4×C53C2×D411yescharacteristic
51C22×C4×C53C2×D411yescharacteristic
52C22×C4×C53#4114yes—
53C22×Dic53—24no—
54M16×C53—44no—
55C2×C8×C53—42no—
56C22⋊Dic53—44no—
57C4⋊Dic53—44no—
58C4×Dic53—44no—
59#11719917—44no—
60C2×#88322—42no—
61C23×C53—22no—
62C23×C53—42no—
63C23×C53C2×C22⋊C411yescharacteristic
64C23×C53#11712yes—
65Q8×C53—48no—
66Q8×C53—48no—
67C2×C4×C53—28no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—44no—
76C2×C4×C53—44no—
77C2×C4×C53#9914yes—
78C2×C4×C53#12312yes—
79C2×C4×C53#12312yes—
80C2×Dic53—48no—
81C2×Dic53—44no—
82C8×C53—82no—
83#88322—82no—
84#8754—531nomaximal, Sylow
85C22×C53—22no—
86C22×C53—22no—
87C22×C53—24no—
88C22×C53—44no—
89C22×C53#38111yescharacteristic
90C22×C53C2×#9912yes—
91C4×C53—22no—
92C4×C53—22no—
93C4×C53—24no—
94C4×C53—42no—
95Dic53—84no—
96#2310—531no—
97#1784—531no—
98#1784D5311yescharacteristic
99#1306—534no—
100C2×C53—22no—
101C2×C53—42no—
102C2×C53#181911yescharacteristic
103C2×C53#183012yes—
104C23×Q8C2×D5311yescharacteristic
105C2×#117—532no—
106C2×#101—22no—
107C2×#101—1062no—
108#407—532no—
109#407C2×D5312yes—
110#391—1062no—
111#391—1062no—
112#381—532no—
113#381—532no—
114C53#875411yescharacteristic, complemented, Sylow
115C22×Q8—24no—
116C22×Q8—44no—
117C22×Q8D21212yes—
118C22×Q8C22⋊D5312yes—
119C23×C4C4×D5311yescharacteristic
120C23×C4C4×D5311yescharacteristic
121C23×C4C22×D5311yescharacteristic
122C2×M16—22no—
123C2×M16—1062no—
124#117—2124no—
125C2×C4⋊C4—1062no—
126C2×C22⋊C4—534no—
127C2×C42—1062no—
128#101—44no—
129#101—2124no—
130C22⋊C8—24no—
131C22⋊C8—1064no—
132#95—1062no—
133#95—1062no—
134#95—1064no—
135C24C2×C4×D5311yescharacteristic
136C2×Q8—24no—
137C2×Q8—24no—
138C2×Q8—24no—
139C2×Q8—24no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—416no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—22no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—1064no—
149C22×C4#1171992112yes—
150C22×C4#1171992112yes—
151C22×C4C2×D21211yescharacteristic
152C22×C4#1171994714yes—
153C22×C4C2×C22⋊D5311yescharacteristic
154M16—44no—
155M16—2124no—
156C2×C8—42no—
157C2×C8—2122no—
158C4⋊C4—2124no—
159C22⋊C4—2124no—
160C42—2124no—
161C23—22no—
162C23—42no—
163C23C2×#1171992111yescharacteristic
164C23#1313317312yes—
165Q8—48no—
166Q8—48no—
167C2×C4—28no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—44no—
176C2×C4—44no—
177C2×C4—2128no—
178C2×C4—2124no—
179C2×C4#1313313114yes—
180C2×C4#1313319812yes—
181C2×C4#1313320112yes—
182C8—82no—
183C8—4242no—
184C22—22no—
185C22—22no—
186C22—24no—
187C22—44no—
188C22#1681048411yescharacteristic
189C22C2×#1313313112yes—
190C4—22no—
191C4—22no—
192C4—24no—
193C4—42no—
194C4—4244no—
195C2—22no—
196C2—42no—
197C2#2027478711yescharacteristic
198C2#2027482712yes—
subgroupG/H
centerC22#16810484
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1784C2
socleC22×C53#381
cosocleC23×C4×C53C23
hypercenter#1784D53
hypocenterC53#8754
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
114. C53#8754semidirect

Maximal subgroups

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

subgroupindex
#202773012
#202746392
#202726242
C53×#17842
#875453