Finite Group Databasebeta

← C53×#12185 #27219697

Subgroups — C53×#12185 #27219697

Order 13568.

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

normal subgroups only

This group has 360 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×#2844C211yesmaximal, characteristic, complemented
2C53×#2040C211yesmaximal, characteristic
3C53×#1616C212yesmaximal
4C53×#1353C212yesmaximal, complemented
5C53×#1324C211yesmaximal, characteristic, complemented
6C53×#537C2211yescharacteristic
7C53×#481—22no—
8C2×#104×C53—21no—
9C53×#417—21no—
10C53×#400—22no—
11C53×#400C2212yes—
12C53×#395—22no—
13C53×#395—21no—
14C53×#382—21no—
15C53×#382C2211yescharacteristic
16C53×#343C2211yescharacteristic
17C53×#327—22no—
18C53×#324C2212yes—
19C2×#41×C53—21no—
20C22×Q8×C53—21no—
21C2×M16×C53—21no—
22C22⋊Q8×C53—41no—
23#121×C53—22no—
24#121×C53—41no—
25C2×C4⋊C4×C53—21no—
26C2×C4⋊C4×C53—21no—
27C2×C4⋊C4×C53—22no—
28C2×C4⋊C4×C53D411yescharacteristic
29C2×C4⋊C4×C53D412yes—
30C2×C4⋊C4×C53C2311yescharacteristic
31C2×C42×C53D411yescharacteristic
32#106×C53—42no—
33#104×C53—42no—
34#103×C53—41no—
35#102×C53—41no—
36#102×C53—42no—
37C22⋊C8×C53—22no—
38#95×C53—22no—
39#95×C53—21no—
40#95×C53—22no—
41#95×C53—22no—
42#95×C53—22no—
43#95×C53D412yes—
44#41×C53—41no—
45#41×C53—41no—
46#41×C53—41no—
47#41×C53—41no—
48C2×Q8×C53—21no—
49C2×Q8×C53—82no—
50C2×Q8×C53—81no—
51C2×D4×C53—21no—
52C2×D4×C53—22no—
53C2×D4×C53—41no—
54C22×C4×C53—21no—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—21no—
58C22×C4×C53—41no—
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D412yes—
61M16×C53—41no—
62M16×C53—41no—
63C2×C8×C53—42no—
64C2×C8×C53—41no—
65C4⋊C4×C53—22no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—81no—
70C4⋊C4×C53—81no—
71C4⋊C4×C53—82no—
72C22⋊C4×C53—42no—
73C42×C53—21no—
74C42×C53SD1612yes—
75C23×C53—41no—
76C23×C53#12011yescharacteristic
77Q8×C53—41no—
78Q8×C53—41no—
79Q8×C53—84no—
80D4×C53—41no—
81D4×C53—41no—
82D4×C53—82no—
83D4×C53—81no—
84D4×C53—81no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—41no—
99C2×C4×C53—42no—
100C2×C4×C53—41no—
101C2×C4×C53—81no—
102C2×C4×C53—81no—
103C2×C4×C53C2×SD1611yescharacteristic
104C2×C4×C53#13711yescharacteristic
105C8×C53—81no—
106C8×C53—82no—
107#12185C5311yesmaximal, characteristic, direct product factor, Sylow
108C22×C53—21no—
109C22×C53—21no—
110C22×C53—81no—
111C22×C53—81no—
112C22×C53—81no—
113C22×C53#44811yescharacteristic
114C22×C53#45411yescharacteristic
115C22×C53B(4,𝔽2)11yescharacteristic
116C4×C53—21no—
117C4×C53—21no—
118C4×C53—42no—
119C4×C53—42no—
120C4×C53—82no—
121C4×C53—81no—
122C4×C53—81no—
123C4×C53—82no—
124#2844C2×C5311yescharacteristic, complemented
125#2040C2×C5311yescharacteristic
126#1616C2×C5312yes—
127#1353C2×C5312yescomplemented
128#1324C2×C5311yescharacteristic, complemented
129C2×C53—21no—
130C2×C53—21no—
131C2×C53—161no—
132C2×C53#159911yescharacteristic
133C2×C53#220111yescharacteristic
134C2×C53#220211yescharacteristic
135#537C22×C5311yescharacteristic
136#481—22no—
137C2×#104—21no—
138#417—21no—
139#400—22no—
140#400C22×C5312yes—
141#395—22no—
142#395—21no—
143#382—21no—
144#382C22×C5311yescharacteristic
145#343C22×C5311yescharacteristic
146#327—22no—
147#324C22×C5312yes—
148C53#1218511yescharacteristic, direct product factor, Sylow
149C2×#41—21no—
150C22×Q8—21no—
151C2×M16—21no—
152C22⋊Q8—41no—
153#121—22no—
154#121—41no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4D4×C5311yescharacteristic
159C2×C4⋊C4D4×C5312yes—
160C2×C4⋊C4C23×C5311yescharacteristic
161C2×C42D4×C5311yescharacteristic
162#106—42no—
163#104—42no—
164#103—41no—
165#102—41no—
166#102—42no—
167C22⋊C8—22no—
168#95—22no—
169#95—21no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95D4×C5312yes—
174#41—41no—
175#41—41no—
176#41—41no—
177#41—41no—
178C2×Q8—21no—
179C2×Q8—82no—
180C2×Q8—81no—
181C2×D4—21no—
182C2×D4—22no—
183C2×D4—41no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—41no—
189C22×C4C2×D4×C5311yescharacteristic
190C22×C4C2×D4×C5312yes—
191M16—41no—
192M16—41no—
193C2×C8—42no—
194C2×C8—41no—
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C4⋊C4—82no—
202C22⋊C4—42no—
203C42—21no—
204C42SD16×C5312yes—
205C23—41no—
206C23#120×C5311yescharacteristic
207Q8—41no—
208Q8—41no—
209Q8—84no—
210D4—41no—
211D4—41no—
212D4—82no—
213D4—81no—
214D4—81no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4C2×SD16×C5311yescharacteristic
234C2×C4#137×C5311yescharacteristic
235C8—81no—
236C8—82no—
237C22—21no—
238C22—21no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22C53×#44811yescharacteristic
243C22C53×#45411yescharacteristic
244C22C53×B(4,𝔽2)11yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—82no—
250C4—81no—
251C4—82no—
252C4—81no—
253C2—21no—
254C2—21no—
255C2—161no—
256C2C53×#159911yescharacteristic
257C2C53×#220111yescharacteristic
258C2C53×#220211yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C53
prime radicalC53×#537C22
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C4⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#2844C2semidirect
4. C53×#1353C2semidirect
5. C53×#1324C2semidirect
107. #12185C53direct (N × H)
124. #2844C2×C53semidirect
127. #1353C2×C53semidirect
128. #1324C2×C53semidirect
148. C53#12185direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#28442
C53×#20402
C53×#16162
C53×#13532
C53×#13242
#1218553