Finite Group Databasebeta

← C53×#12365 #27219877

Subgroups — C53×#12365 #27219877

Order 13568.

346 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 78 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×#3291C211yesmaximal, characteristic
2C53×#2230C212yesmaximal, complemented
3C53×#1708C211yesmaximal, characteristic, complemented
4C53×#1359C211yesmaximal, characteristic, complemented
5C53×#1335C212yesmaximal, complemented
6C53×#577—22no—
7C53×#498C2211yescharacteristic, complemented
8C53×#492C2211yescharacteristic
9C53×#487—21no—
10C53×#486—21no—
11C53×#471—22no—
12C53×#456—21no—
13C53×#453—21no—
14C53×#443C2212yes—
15C53×#365—22no—
16C53×#364C2212yescomplemented
17C53×#360—22no—
18C82×C53C2211yescharacteristic, complemented
192−1+4×C53—21no—
20H(5,𝔽2)×C53—21no—
21#137×C53—22no—
22#137×C53—42no—
23AGL(1,ℤ/8)×C53—22no—
24AGL(1,ℤ/8)×C53—42no—
25#135×C53—42no—
26#135×C53D412yes—
27C2×SD16×C53—22no—
28C2×SD16×C53—42no—
29#131×C53—22no—
30C4⋊Q8×C53—21no—
31C4⋊Q8×C53D411yescharacteristic
32#127×C53D411yescharacteristic
33M32×C53—22no—
34#108×C53D412yes—
35#106×C53—22no—
36#104×C53—22no—
37#103×C53—42no—
38#102×C53—42no—
39#101×C53—21no—
40#100×C53—21no—
41C4×C8×C53—21no—
42C4×C8×C53C2311yescharacteristic
43#41×C53—22no—
44#41×C53—42no—
45#41×C53—42no—
46#41×C53—81no—
47C2×Q8×C53—21no—
48C2×Q8×C53—41no—
49C2×Q8×C53—42no—
50C2×D4×C53—21no—
51C2×D4×C53—41no—
52C2×D4×C53—42no—
53C2×D4×C53—81no—
54Q16×C53—42no—
55Q16×C53SD1612yes—
56SD16×C53—22no—
57SD16×C53—42no—
58SD16×C53—42no—
59SD16×C53—42no—
60SD16×C53—42no—
61D8×C53—42no—
62D8×C53SD1612yes—
63M16×C53—22no—
64M16×C53—42no—
65C2×C8×C53—22no—
66C2×C8×C53—42no—
67C2×C8×C53C2×D412yes—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—41no—
71C42×C53C2×D411yescharacteristic
72C16×C53—42no—
73C23×C53—41no—
74C23×C53—81no—
75Q8×C53—22no—
76Q8×C53—42no—
77Q8×C53—82no—
78Q8×C53—81no—
79D4×C53—22no—
80D4×C53—42no—
81D4×C53—42no—
82D4×C53—82no—
83D4×C53—82no—
84D4×C53—81no—
85C2×C4×C53—21no—
86C2×C4×C53—42no—
87C2×C4×C53—41no—
88C2×C4×C53—81no—
89C2×C4×C53—82no—
90C2×C4×C53—81no—
91C2×C4×C53#12011yescharacteristic
92C8×C53—42no—
93C8×C53—42no—
94C8×C53C2×SD1612yes—
95C8×C53AGL(1,ℤ/8)12yes—
96#12365C5311yesmaximal, characteristic, direct product factor, Sylow
97C22×C53—42no—
98C22×C53—81no—
99C22×C53—161no—
100C22×C53—321no—
101C22×C53#45311yescharacteristic
102C4×C53—22no—
103C4×C53—42no—
104C4×C53—81no—
105C4×C53—161no—
106C4×C53#45012yes—
107#3291C2×C5311yescharacteristic
108#2230C2×C5312yescomplemented
109#1708C2×C5311yescharacteristic, complemented
110#1359C2×C5311yescharacteristic, complemented
111#1335C2×C5312yescomplemented
112C2×C53—21no—
113C2×C53—82no—
114C2×C53—161no—
115C2×C53#162911yescharacteristic
116#577—22no—
117#498C22×C5311yescharacteristic, complemented
118#492C22×C5311yescharacteristic
119#487—21no—
120#486—21no—
121#471—22no—
122#456—21no—
123#453—21no—
124#443C22×C5312yes—
125#365—22no—
126#364C22×C5312yescomplemented
127#360—22no—
128C82C22×C5311yescharacteristic, complemented
129C53#1236511yescharacteristic, direct product factor, Sylow
1302−1+4—21no—
131H(5,𝔽2)—21no—
132#137—22no—
133#137—42no—
134AGL(1,ℤ/8)—22no—
135AGL(1,ℤ/8)—42no—
136#135—42no—
137#135D4×C5312yes—
138C2×SD16—22no—
139C2×SD16—42no—
140#131—22no—
141C4⋊Q8—21no—
142C4⋊Q8D4×C5311yescharacteristic
143#127D4×C5311yescharacteristic
144M32—22no—
145#108D4×C5312yes—
146#106—22no—
147#104—22no—
148#103—42no—
149#102—42no—
150#101—21no—
151#100—21no—
152C4×C8—21no—
153C4×C8C23×C5311yescharacteristic
154#41—22no—
155#41—42no—
156#41—42no—
157#41—81no—
158C2×Q8—21no—
159C2×Q8—41no—
160C2×Q8—42no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4—42no—
164C2×D4—81no—
165Q16—42no—
166Q16SD16×C5312yes—
167SD16—22no—
168SD16—42no—
169SD16—42no—
170SD16—42no—
171SD16—42no—
172D8—42no—
173D8SD16×C5312yes—
174M16—22no—
175M16—42no—
176C2×C8—22no—
177C2×C8—42no—
178C2×C8C2×D4×C5312yes—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C42C2×D4×C5311yescharacteristic
183C16—42no—
184C23—41no—
185C23—81no—
186Q8—22no—
187Q8—42no—
188Q8—82no—
189Q8—81no—
190D4—22no—
191D4—42no—
192D4—42no—
193D4—82no—
194D4—82no—
195D4—81no—
196C2×C4—21no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—81no—
200C2×C4—82no—
201C2×C4—81no—
202C2×C4#120×C5311yescharacteristic
203C8—42no—
204C8—42no—
205C8C2×SD16×C5312yes—
206C8AGL(1,ℤ/8)×C5312yes—
207C22—42no—
208C22—81no—
209C22—161no—
210C22—321no—
211C22C53×#45311yescharacteristic
212C4—22no—
213C4—42no—
214C4—81no—
215C4—161no—
216C4C53×#45012yes—
217C2—21no—
218C2—82no—
219C2—161no—
220C2C53×#162911yescharacteristic
subgroupG/H
center, socleC2×C53#1629
commutator, Frattini, cosocleC4×C8C23×C53
prime radicalC53×#3291C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C4×C8×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2230C2semidirect
3. C53×#1708C2semidirect
4. C53×#1359C2semidirect
5. C53×#1335C2semidirect
7. C53×#498C22semidirect
16. C53×#364C22semidirect
18. C82×C53C22semidirect
96. #12365C53direct (N × H)
108. #2230C2×C53semidirect
109. #1708C2×C53semidirect
110. #1359C2×C53semidirect
111. #1335C2×C53semidirect
117. #498C22×C53semidirect
126. #364C22×C53semidirect
128. C82C22×C53semidirect
129. C53#12365direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32912
C53×#22302
C53×#17082
C53×#13592
C53×#13352
#1236553