Finite Group Databasebeta

← C53×#12247 #27219759

Subgroups — C53×#12247 #27219759

Order 13568.

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

normal subgroups only

This group has 394 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×#2582C211yesmaximal, characteristic
2C53×#2060C211yesmaximal, characteristic
3C53×#1611C212yesmaximal
4C53×#1346C212yesmaximal
5C53×#1325C211yesmaximal, characteristic
6C53×#544C2211yescharacteristic
7C2×C4×Q8×C53—21no—
8C53×#483—22no—
9C53×#428—21no—
10C53×#398—21no—
11C53×#398C2212yes—
12C53×#391—22no—
13C53×#389—22no—
14C53×#384C2211yescharacteristic
15C53×#343C2211yescharacteristic
16C53×#339—21no—
17C53×#328—22no—
18C53×#324C2212yes—
19C22×Q8×C53—21no—
20C2×M16×C53—21no—
21C4⋊Q8×C53—41no—
22#125×C53—41no—
23C22⋊Q8×C53—22no—
24C4×Q8×C53—42no—
25C4×Q8×C53—42no—
26#117×C53—21no—
27C2×C4⋊C4×C53—21no—
28C2×C4⋊C4×C53—21no—
29C2×C4⋊C4×C53—22no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53D411yescharacteristic
32C2×C4⋊C4×C53D412yes—
33C2×C4⋊C4×C53C2311yescharacteristic
34C2×C42×C53—22no—
35C2×C42×C53D411yescharacteristic
36#107×C53—41no—
37#106×C53—41no—
38#106×C53—42no—
39#103×C53—42no—
40C22⋊C8×C53—22no—
41#95×C53—22no—
42#95×C53—21no—
43#95×C53—22no—
44#95×C53D412yes—
45C2×Q8×C53—24no—
46C2×Q8×C53—22no—
47C2×Q8×C53—42no—
48C2×Q8×C53—42no—
49C22×C4×C53—21no—
50C22×C4×C53—22no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53C2×D411yescharacteristic
54C22×C4×C53C2×D412yes—
55M16×C53—22no—
56M16×C53—41no—
57C2×C8×C53—41no—
58C2×C8×C53—42no—
59C4⋊C4×C53—21no—
60C4⋊C4×C53—22no—
61C4⋊C4×C53—21no—
62C4⋊C4×C53—41no—
63C4⋊C4×C53—44no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—42no—
66C4⋊C4×C53—42no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—81no—
71C22⋊C4×C53—41no—
72C42×C53—21no—
73C42×C53—44no—
74C42×C53—41no—
75C42×C53SD1612yes—
76C23×C53#12011yescharacteristic
77Q8×C53—48no—
78Q8×C53—82no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—24no—
85C2×C4×C53—22no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53C2×SD1611yescharacteristic
97C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
98C8×C53—42no—
99C8×C53—82no—
100#12247C5311yesmaximal, characteristic, direct product factor, Sylow
101C22×C53—21no—
102C22×C53—21no—
103C22×C53#45011yescharacteristic
104C22×C53#45611yescharacteristic
105C22×C53B(4,𝔽2)11yescharacteristic
106C4×C53—24no—
107C4×C53—21no—
108C4×C53—21no—
109C4×C53—44no—
110C4×C53—42no—
111C4×C53—82no—
112C4×C53—81no—
113C4×C53—81no—
114C4×C53—81no—
115#2582C2×C5311yescharacteristic
116#2060C2×C5311yescharacteristic
117#1611C2×C5312yes—
118#1346C2×C5312yes—
119#1325C2×C5311yescharacteristic
120C2×C53—21no—
121C2×C53—21no—
122C2×C53#160411yescharacteristic
123C2×C53#219811yescharacteristic
124C2×C53#220511yescharacteristic
125#544C22×C5311yescharacteristic
126C2×C4×Q8—21no—
127#483—22no—
128#428—21no—
129#398—21no—
130#398C22×C5312yes—
131#391—22no—
132#389—22no—
133#384C22×C5311yescharacteristic
134#343C22×C5311yescharacteristic
135#339—21no—
136#328—22no—
137#324C22×C5312yes—
138C53#1224711yescharacteristic, direct product factor, Sylow
139C22×Q8—21no—
140C2×M16—21no—
141C4⋊Q8—41no—
142#125—41no—
143C22⋊Q8—22no—
144C4×Q8—42no—
145C4×Q8—42no—
146#117—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4D4×C5311yescharacteristic
152C2×C4⋊C4D4×C5312yes—
153C2×C4⋊C4C23×C5311yescharacteristic
154C2×C42—22no—
155C2×C42D4×C5311yescharacteristic
156#107—41no—
157#106—41no—
158#106—42no—
159#103—42no—
160C22⋊C8—22no—
161#95—22no—
162#95—21no—
163#95—22no—
164#95D4×C5312yes—
165C2×Q8—24no—
166C2×Q8—22no—
167C2×Q8—42no—
168C2×Q8—42no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4C2×D4×C5311yescharacteristic
174C22×C4C2×D4×C5312yes—
175M16—22no—
176M16—41no—
177C2×C8—41no—
178C2×C8—42no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—41no—
183C4⋊C4—42no—
184C4⋊C4—41no—
185C4⋊C4—42no—
186C4⋊C4—44no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—81no—
191C22⋊C4—41no—
192C42—21no—
193C42—44no—
194C42—41no—
195C42SD16×C5312yes—
196C23#120×C5311yescharacteristic
197Q8—48no—
198Q8—82no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4C2×SD16×C5311yescharacteristic
217C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
218C8—42no—
219C8—82no—
220C22—21no—
221C22—21no—
222C22C53×#45011yescharacteristic
223C22C53×#45611yescharacteristic
224C22C53×B(4,𝔽2)11yescharacteristic
225C4—24no—
226C4—21no—
227C4—21no—
228C4—44no—
229C4—42no—
230C4—81no—
231C4—81no—
232C4—81no—
233C4—82no—
234C2—21no—
235C2—21no—
236C2C53×#160411yescharacteristic
237C2C53×#219811yescharacteristic
238C2C53×#220511yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C53
prime radicalC23×C53#120
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
100. #12247C53direct (N × H)
138. C53#12247direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#25822
C53×#20602
C53×#16112
C53×#13462
C53×#13252
#1224753