Finite Group Databasebeta

← #27532709

Subgroups — #27532709

Order 13568.

688 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 135 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#20275892C218yesmaximal, complemented
2#20275882C214yesmaximal, complemented
3#20274633C212yesmaximal
4C53×#1782C211yesmaximal, characteristic
5#16811249C2214yes—
6C2×#13133132C2214yes—
7C2×#13133132C2214yes—
8#16810486C2214yes—
9#16810390C2214yes—
10#16810389C2218yes—
11C22×#41×C53C2211yescharacteristic
12C2×#99×C53C2214yes—
13C53×#381C2212yes—
14#13133196—44no—
15#13133194—48no—
16C2×C22⋊Dic53C2318yes—
17#13133132—44no—
18#13133132—44no—
19#13133124—44no—
20C2×#41×C53—42no—
21C2×#41×C53—44no—
22C22×Q8×C53Q811yescharacteristic, complemented
23C22×D4×C53Q811yescharacteristic, complemented
24C22×D4×C53C2312yes—
25C23×C4×C53Q812yescomplemented
26C23×C4×C53C2311yescharacteristic
27C2×C22⋊C4×C53C2314yes—
28#99×C53—44no—
29#95×C53—42no—
30C24×C53C2×Q812yes—
31C24×C53C2411yescharacteristic
32#41×C53—416no—
33C2×Q8×C53—41no—
34C2×Q8×C53—42no—
35C2×D4×C53—24no—
36C2×D4×C53—24no—
37C2×D4×C53—24no—
38C2×D4×C53—41no—
39C2×D4×C53—42no—
40C22×C4×C53—22no—
41C22×C4×C53—24no—
42C22×C4×C53—24no—
43C22×C4×C53—42no—
44C22×C4×C53—42no—
45C22×C4×C53—42no—
46C22×C4×C53C2×Q811yescharacteristic
47C22×C4×C53C2×Q812yes—
48C22×C4×C53C2×Q811yescharacteristic
49C22×Dic53—28no—
50C2×Dic106—44no—
51C22⋊C4×C53—28no—
52C22⋊Dic53—28no—
53C22⋊Dic53—28no—
54C4⋊Dic53—44no—
55#11719919—48no—
56C23×C53—22no—
57C23×C53—22no—
58C23×C53—22no—
59C23×C53—81no—
60C23×C53—82no—
61C23×C53C22×Q811yescharacteristic
62C23×C53H(5,𝔽2)14yes—
63C23×C53H(5,𝔽2)12yes—
64Q8×C53—44no—
65D4×C53—44no—
66D4×C53—48no—
67C2×C4×C53—21no—
68C2×C4×C53—22no—
69C2×C4×C53—21no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—48no—
75C2×C4×C53—44no—
76C2×C4×C53—44no—
77C2×C4×C53—84no—
78C2×Dic53—48no—
79C2×Dic53—88no—
80Dic106—164no—
81#32874—531nomaximal, Sylow
82C22×C53—44no—
83C22×C53—42no—
84C22×C53—41no—
85C22×C53—42no—
86C22×C53—82no—
87C22×C53—82no—
88C22×C53—82no—
89C22×C53#54312yes—
90C22×C53#54311yescharacteristic
91C22×C53H(3,𝔽4)14yes—
92C4×C53—41no—
93C4×C53—41no—
94C4×C53—42no—
95C4×C53—84no—
96Dic53—88no—
97#2032—538no—
98#2030—534no—
99#1782—532no—
100#1782D5311yescharacteristic
101C2×C53—24no—
102C2×C53—22no—
103C2×C53—21no—
104C2×C53—82no—
105C2×C53#284011yescharacteristic
106C22×#41C2×D5311yescharacteristic
107#543—534no—
108C2×#99—534no—
109C2×#99—534no—
110C2×#99C2×D5314yes—
111#381—534no—
112#381C2×D5312yes—
113#342—538no—
114#342—534no—
115C53#3287411yescharacteristic, complemented, Sylow
116C2×#41—42no—
117C2×#41—44no—
118C22×Q8Dic10611yescharacteristic, complemented
119C22×D4Dic10611yescharacteristic, complemented
120C22×D4C22×D5312yes—
121C23×C4Dic10612yescomplemented
122C23×C4C22×D5311yescharacteristic
123C22⋊Q8—2128no—
124C22⋊Q8—2124no—
125C2×C22⋊C4—538no—
126C2×C22⋊C4C22×D5314yes—
127#99—44no—
128#99—2124no—
129#99—2124no—
130#95—42no—
131#95—2124no—
132C24C2×Dic10612yes—
133C24C23×D5311yescharacteristic
134#41—416no—
135C2×Q8—41no—
136C2×Q8—42no—
137C2×Q8—2124no—
138C2×D4—24no—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—41no—
142C2×D4—42no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—1068no—
150C22×C4C2×Dic10611yescharacteristic
151C22×C4C2×Dic10612yes—
152C22×C4C2×Dic10611yescharacteristic
153C4⋊C4—2128no—
154C4⋊C4—2124no—
155C22⋊C4—28no—
156C22⋊C4—1068no—
157C22⋊C4—1068no—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—81no—
162C23—82no—
163C23C22×Dic10611yescharacteristic
164C23#1313326214yes—
165C23#1313326212yes—
166Q8—44no—
167Q8—8484no—
168D4—44no—
169D4—48no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—48no—
178C2×C4—44no—
179C2×C4—44no—
180C2×C4—84no—
181C2×C4—4248no—
182C2×C4—2128no—
183C22—42no—
184C22—44no—
185C22—41no—
186C22—42no—
187C22—82no—
188C22—82no—
189C22—82no—
190C22#1681124912yes—
191C22#1681124911yescharacteristic
192C22#1681136614yes—
193C4—41no—
194C4—41no—
195C4—42no—
196C4—84no—
197C4—4248no—
198C2—24no—
199C2—21no—
200C2—22no—
201C2—82no—
202C2#2028296811yescharacteristic
subgroupG/H
centerC2#20282968
commutator, cosocleC24×C53C24
FrattiniC24C23×D53
FittingC53×#1782C2
socleC2×C53#2840
prime radicalC22×#41×C53C22
hypercenter#1782D53
hypocenterC53#32874
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C24 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. #20275892C2semidirect
2. #20275882C2semidirect
22. C22×Q8×C53Q8semidirect
23. C22×D4×C53Q8semidirect
25. C23×C4×C53Q8semidirect
115. C53#32874semidirect
118. C22×Q8Dic106semidirect
119. C22×D4Dic106semidirect
121. C23×C4Dic106semidirect

Maximal subgroups

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

subgroupindex
#202758922
#202758822
#202746332
C53×#17822
#3287453