Finite Group Databasebeta

← C53×#7316 #27214828

Subgroups — C53×#7316 #27214828

Order 13568.

620 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C53×#1783C211yesmaximal, characteristic
2C53×#1781C212yesmaximal
3C23×D4×C53C2211yescharacteristic
4C2×#100×C53—22no—
5C2×#99×C53—22no—
6C2×#99×C53—22no—
7C53×#407C412yescomplemented
8C53×#379C414yescomplemented
9C25×C53C2×C412yes—
10C22×D4×C53—42no—
11C22×D4×C53—44no—
12C22×D4×C53D412yes—
13C22×D4×C53D411yescharacteristic
14C22×D4×C53Q811yescharacteristic
15C23×C4×C53C2×C411yescharacteristic
16C2×M16×C53—22no—
17C2×C22⋊C4×C53—24no—
18C2×C22⋊C4×C53—44no—
19#100×C53—44no—
20#99×C53—44no—
21#99×C53—44no—
22C22⋊C8×C53—42no—
23C24×C53—24no—
24C24×C53—42no—
25C24×C53—42no—
26C24×C53—84no—
27C24×C53C4211yescharacteristic
28C24×C53C22⋊C412yes—
29C24×C53C4⋊C412yes—
30C2×D4×C53—24no—
31C2×D4×C53—24no—
32C2×D4×C53—24no—
33C2×D4×C53—42no—
34C2×D4×C53—41no—
35C2×D4×C53—41no—
36C2×D4×C53—42no—
37C2×D4×C53—416no—
38C22×C4×C53—22no—
39C22×C4×C53—42no—
40C22×C4×C53—44no—
41C22×C4×C53C22⋊C411yescharacteristic
42C22×C4×C53C4⋊C411yescharacteristic
43M16×C53—44no—
44C2×C8×C53—42no—
45C22⋊C4×C53—44no—
46C22⋊C4×C53—44no—
47C22⋊C4×C53—88no—
48C23×C53—21no—
49C23×C53—22no—
50C23×C53—22no—
51C23×C53—22no—
52C23×C53—22no—
53C23×C53—22no—
54C23×C53—22no—
55C23×C53—42no—
56C23×C53—48no—
57C23×C53—48no—
58C23×C53—84no—
59C23×C53—84no—
60C23×C53—82no—
61C23×C53—82no—
62C23×C53—84no—
63C23×C53—84no—
64C23×C53—84no—
65C23×C53#9511yescharacteristic
66C23×C53#9914yes—
67D4×C53—48no—
68D4×C53—44no—
69D4×C53—44no—
70C2×C4×C53—22no—
71C2×C4×C53—21no—
72C2×C4×C53—22no—
73C2×C4×C53—44no—
74C2×C4×C53—84no—
75C2×C4×C53—88no—
76C2×C4×C53#9912yes—
77C8×C53—82no—
78#7316C5311yesmaximal, characteristic, direct product factor, Sylow
79C22×C53—24no—
80C22×C53—22no—
81C22×C53—24no—
82C22×C53—41no—
83C22×C53—44no—
84C22×C53—42no—
85C22×C53—41no—
86C22×C53—44no—
87C22×C53—41no—
88C22×C53—42no—
89C22×C53—82no—
90C22×C53—82no—
91C22×C53—82no—
92C22×C53—82no—
93C22×C53—816no—
94C22×C53#34212yes—
95C22×C53#34211yescharacteristic
96C4×C53—22no—
97C4×C53—41no—
98C4×C53—164no—
99#1783C2×C5311yescharacteristic
100#1781C2×C5312yes—
101C2×C53—22no—
102C2×C53—21no—
103C2×C53—44no—
104C2×C53—42no—
105C2×C53—82no—
106C2×C53#130411yescharacteristic
107C23×D4C22×C5311yescharacteristic
108C2×#100—22no—
109C2×#99—22no—
110C2×#99—22no—
111#407C4×C5312yescomplemented
112#379C4×C5314yescomplemented
113C53#731611yescharacteristic, direct product factor, Sylow
114C25C2×C4×C5312yes—
115C22×D4—42no—
116C22×D4—44no—
117C22×D4D4×C5312yes—
118C22×D4D4×C5311yescharacteristic
119C22×D4Q8×C5311yescharacteristic
120C23×C4C2×C4×C5311yescharacteristic
121C2×M16—22no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—44no—
124#100—44no—
125#99—44no—
126#99—44no—
127C22⋊C8—42no—
128C24—24no—
129C24—42no—
130C24—42no—
131C24—84no—
132C24C42×C5311yescharacteristic
133C24C22⋊C4×C5312yes—
134C24C4⋊C4×C5312yes—
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—24no—
138C2×D4—42no—
139C2×D4—41no—
140C2×D4—41no—
141C2×D4—42no—
142C2×D4—416no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—44no—
146C22×C4C22⋊C4×C5311yescharacteristic
147C22×C4C4⋊C4×C5311yescharacteristic
148M16—44no—
149C2×C8—42no—
150C22⋊C4—44no—
151C22⋊C4—44no—
152C22⋊C4—88no—
153C23—21no—
154C23—22no—
155C23—22no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—42no—
161C23—48no—
162C23—48no—
163C23—84no—
164C23—84no—
165C23—82no—
166C23—82no—
167C23—84no—
168C23—84no—
169C23—84no—
170C23#95×C5311yescharacteristic
171C23#99×C5314yes—
172D4—48no—
173D4—44no—
174D4—44no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—44no—
179C2×C4—84no—
180C2×C4—88no—
181C2×C4#99×C5312yes—
182C8—82no—
183C22—24no—
184C22—22no—
185C22—24no—
186C22—41no—
187C22—44no—
188C22—42no—
189C22—41no—
190C22—44no—
191C22—41no—
192C22—42no—
193C22—82no—
194C22—82no—
195C22—82no—
196C22—82no—
197C22—816no—
198C22C53×#34212yes—
199C22C53×#34211yescharacteristic
200C4—22no—
201C4—41no—
202C4—164no—
203C2—22no—
204C2—21no—
205C2—44no—
206C2—42no—
207C2—82no—
208C2C53×#130411yescharacteristic
subgroupG/H
center, socleC2×C53#1304
commutatorC24C42×C53
Frattini, cosocleC23×D4C22×C53
prime radicalC23×D4×C53C22
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C23×C53 ◁ C23×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
7. C53×#407C4semidirect
8. C53×#379C4semidirect
78. #7316C53direct (N × H)
111. #407C4×C53semidirect
112. #379C4×C53semidirect
113. C53#7316direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#17832
C53×#17812
#731653