Finite Group Databasebeta

← C53×#13595 #27221107

Subgroups — C53×#13595 #27221107

Order 13568.

388 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 90 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×#3138C211yesmaximal, characteristic
2C2×C53×#353C212yesmaximal
3C53×#1412C214yesmaximal, complemented
4C53×#533C411yescharacteristic, complemented
5C2×#127×C53C2211yescharacteristic
6C53×#490C412yescomplemented
7C53×#490C2212yes—
8C53×#486—22no—
9C53×#418—22no—
10C2×#99×C53—22no—
11C4×M16×C53C411yescharacteristic, complemented
12C53×#353—22no—
13C53×#353C2214yes—
14C22×D4×C53—41no—
15C22×D4×C53D412yes—
16C2×M16×C53—21no—
17C4⋊Q8×C53—21no—
18#127×C53—41no—
19#127×C53C2×C411yescharacteristic
20#127×C53D412yes—
21#127×C53C2311yescharacteristic
22#125×C53C2×C412yescomplemented
23C22⋊Q8×C53—41no—
24C4×Q8×C53—41no—
25#117×C53—21no—
26C2×C22⋊C4×C53—42no—
27C2×C42×C53C2×C411yescharacteristic
28#102×C53—24no—
29#102×C53—24no—
30#99×C53—24no—
31#99×C53—42no—
32#97×C53C2×C412yescomplemented
33C4×C8×C53—21no—
34C24×C53—22no—
35C2×Q8×C53—41no—
36C2×D4×C53—22no—
37C2×D4×C53—42no—
38C2×D4×C53—42no—
39C2×D4×C53—41no—
40C2×D4×C53—42no—
41C2×D4×C53—41no—
42C2×D4×C53—84no—
43C2×D4×C53C22⋊C412yes—
44C2×D4×C53C2×D412yes—
45C22×C4×C53—21no—
46C22×C4×C53—82no—
47C22×C4×C53C22⋊C411yescharacteristic
48M16×C53—42no—
49C2×C8×C53—22no—
50C4⋊C4×C53—22no—
51C4⋊C4×C53—41no—
52C4⋊C4×C53—42no—
53C22⋊C4×C53—41no—
54C22⋊C4×C53—44no—
55C22⋊C4×C53—82no—
56C42×C53—21no—
57C42×C53—41no—
58C42×C53C22⋊C411yescharacteristic
59C42×C53C22×C411yescharacteristic
60C23×C53—22no—
61C23×C53—22no—
62C23×C53—42no—
63C23×C53—42no—
64C23×C53—84no—
65C23×C53#9911yescharacteristic
66Q8×C53—82no—
67D4×C53—42no—
68D4×C53—42no—
69D4×C53—82no—
70D4×C53—84no—
71D4×C53—82no—
72D4×C53—82no—
73C2×C4×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—42no—
77C2×C4×C53—42no—
78C2×C4×C53—84no—
79C2×C4×C53—164no—
80C2×C4×C53#9911yescharacteristic
81C2×C4×C53C2×C22⋊C411yescharacteristic
82C8×C53—42no—
83#13595C5311yesmaximal, characteristic, direct product factor, Sylow
84C22×C53—41no—
85C22×C53—42no—
86C22×C53—42no—
87C22×C53—82no—
88C22×C53—82no—
89C22×C53—84no—
90C22×C53—162no—
91C22×C53#35412yes—
92C22×C53C2×#9911yescharacteristic
93C4×C53—22no—
94C4×C53—21no—
95C4×C53—21no—
96C4×C53—41no—
97C4×C53—82no—
98C4×C53—164no—
99#3138C2×C5311yescharacteristic
100C2×#353C2×C5312yes—
101#1412C2×C5314yescomplemented
102C2×C53—21no—
103C2×C53—22no—
104C2×C53—82no—
105C2×C53—82no—
106C2×C53C2×#35411yescharacteristic
107#533C4×C5311yescharacteristic, complemented
108C2×#127C22×C5311yescharacteristic
109#490C4×C5312yescomplemented
110#490C22×C5312yes—
111#486—22no—
112#418—22no—
113C2×#99—22no—
114C4×M16C4×C5311yescharacteristic, complemented
115#353—22no—
116#353C22×C5314yes—
117C53#1359511yescharacteristic, direct product factor, Sylow
118C22×D4—41no—
119C22×D4D4×C5312yes—
120C2×M16—21no—
121C4⋊Q8—21no—
122#127—41no—
123#127C2×C4×C5311yescharacteristic
124#127D4×C5312yes—
125#127C23×C5311yescharacteristic
126#125C2×C4×C5312yescomplemented
127C22⋊Q8—41no—
128C4×Q8—41no—
129#117—21no—
130C2×C22⋊C4—42no—
131C2×C42C2×C4×C5311yescharacteristic
132#102—24no—
133#102—24no—
134#99—24no—
135#99—42no—
136#97C2×C4×C5312yescomplemented
137C4×C8—21no—
138C24—22no—
139C2×Q8—41no—
140C2×D4—22no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—41no—
144C2×D4—42no—
145C2×D4—41no—
146C2×D4—84no—
147C2×D4C22⋊C4×C5312yes—
148C2×D4C2×D4×C5312yes—
149C22×C4—21no—
150C22×C4—82no—
151C22×C4C22⋊C4×C5311yescharacteristic
152M16—42no—
153C2×C8—22no—
154C4⋊C4—22no—
155C4⋊C4—42no—
156C4⋊C4—41no—
157C22⋊C4—41no—
158C22⋊C4—44no—
159C22⋊C4—82no—
160C42—21no—
161C42—41no—
162C42C22⋊C4×C5311yescharacteristic
163C42C22×C4×C5311yescharacteristic
164C23—22no—
165C23—22no—
166C23—42no—
167C23—42no—
168C23—84no—
169C23#99×C5311yescharacteristic
170Q8—82no—
171D4—42no—
172D4—42no—
173D4—82no—
174D4—84no—
175D4—82no—
176D4—82no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—84no—
183C2×C4—164no—
184C2×C4#99×C5311yescharacteristic
185C2×C4C2×C22⋊C4×C5311yescharacteristic
186C8—42no—
187C22—41no—
188C22—42no—
189C22—42no—
190C22—82no—
191C22—82no—
192C22—84no—
193C22—162no—
194C22C53×#35412yes—
195C22C2×#99×C5311yescharacteristic
196C4—22no—
197C4—21no—
198C4—21no—
199C4—41no—
200C4—82no—
201C4—164no—
202C2—21no—
203C2—22no—
204C2—82no—
205C2—82no—
206C2C2×C53×#35411yescharacteristic
subgroupG/H
center, socleC2×C53C2×#354
commutatorC42C22×C4×C53
Frattini, cosocle#127C23×C53
prime radicalC2×#127×C53C22
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C23×C53 ◁ C22×C4×C53 ◁ C2×#127×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C53×#1412C2semidirect
4. C53×#533C4semidirect
6. C53×#490C4semidirect
11. C4×M16×C53C4semidirect
22. #125×C53C2×C4semidirect
32. #97×C53C2×C4semidirect
83. #13595C53direct (N × H)
101. #1412C2×C53semidirect
107. #533C4×C53semidirect
109. #490C4×C53semidirect
114. C4×M16C4×C53semidirect
117. C53#13595direct (N × H)
126. #125C2×C4×C53semidirect
136. #97C2×C4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#31382
C2×C53×#3532
C53×#14122
#1359553