Finite Group Databasebeta

← C53×#7817 #27215329

Subgroups — C53×#7817 #27215329

Order 13568.

410 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2734C211yesmaximal, characteristic
2C53×#2096C211yesmaximal, characteristic
3C53×#2084C212yesmaximal
4C53×#1272C211yesmaximal, characteristic
5C53×#1271C212yesmaximal
6C2×C4⋊Q8×C53—21no—
7C2×#106×C53—22no—
8C2×#106×C53—21no—
9C2×#103×C53—21no—
10C53×#398—21no—
11C53×#395—22no—
12C53×#391C2211yescharacteristic
13C53×#389C2212yes—
14C53×#387—22no—
15C53×#384C2211yescharacteristic
16C53×#340—22no—
17C53×#336C2211yescharacteristic
18C53×#336C2212yes—
19C22×Q8×C53—21no—
20C22×Q8×C53D411yescharacteristic
21C22×C8×C53—22no—
22C22×C8×C53—21no—
23C4⋊Q8×C53—42no—
24C4⋊Q8×C53—42no—
25C2×C4⋊C4×C53—22no—
26C2×C4⋊C4×C53—21no—
27C2×C4⋊C4×C53—22no—
28C2×C4⋊C4×C53—22no—
29C2×C4⋊C4×C53—21no—
30C2×C4⋊C4×C53D412yes—
31C2×C4⋊C4×C53D412yes—
32C2×C4⋊C4×C53D411yescharacteristic
33C2×C42×C53—21no—
34C2×C42×C53C2311yescharacteristic
35#106×C53—42no—
36#106×C53—44no—
37#103×C53—42no—
38#95×C53—21no—
39#95×C53—22no—
40#95×C53—22no—
41#95×C53—22no—
42C2×Q8×C53—21no—
43C2×Q8×C53—42no—
44C2×Q8×C53—41no—
45C2×Q8×C53—41no—
46C2×Q8×C53—81no—
47C2×Q8×C53—81no—
48C2×Q8×C53SD1612yes—
49C22×C4×C53—22no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—41no—
54C22×C4×C53C2×D411yescharacteristic
55C22×C4×C53C2×D412yes—
56C2×C8×C53—22no—
57C2×C8×C53—24no—
58C2×C8×C53—41no—
59C2×C8×C53—42no—
60C2×C8×C53—42no—
61C2×C8×C53—41no—
62C4⋊C4×C53—22no—
63C4⋊C4×C53—44no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—42no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—81no—
73C4⋊C4×C53SD1614yes—
74C42×C53—41no—
75C42×C53—42no—
76C23×C53#12011yescharacteristic
77Q8×C53—42no—
78Q8×C53—81no—
79Q8×C53—84no—
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—24no—
85C2×C4×C53—22no—
86C2×C4×C53—42no—
87C2×C4×C53—41no—
88C2×C4×C53—42no—
89C2×C4×C53—41no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—82no—
96C2×C4×C53—81no—
97C2×C4×C53C2×SD1612yes—
98C2×C4×C53C2×SD1611yescharacteristic
99C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
100C2×C4×C53#13712yes—
101C8×C53—44no—
102C8×C53—42no—
103#7817C5311yesmaximal, characteristic, direct product factor, Sylow
104C22×C53#44812yes—
105C22×C53#45011yescharacteristic
106C22×C53#45311yescharacteristic
107C22×C53#45612yes—
108C22×C53#45811yescharacteristic
109C4×C53—22no—
110C4×C53—22no—
111C4×C53—21no—
112C4×C53—21no—
113C4×C53—42no—
114C4×C53—44no—
115C4×C53—82no—
116C4×C53—81no—
117C4×C53—161no—
118#2734C2×C5311yescharacteristic
119#2096C2×C5311yescharacteristic
120#2084C2×C5312yes—
121#1272C2×C5311yescharacteristic
122#1271C2×C5312yes—
123C2×C53#161112yes—
124C2×C53#161311yescharacteristic
125C2×C53#162311yescharacteristic
126C2×C53#165112yes—
127C2×C53#166011yescharacteristic
128C2×C4⋊Q8—21no—
129C2×#106—22no—
130C2×#106—21no—
131C2×#103—21no—
132#398—21no—
133#395—22no—
134#391C22×C5311yescharacteristic
135#389C22×C5312yes—
136#387—22no—
137#384C22×C5311yescharacteristic
138#340—22no—
139#336C22×C5312yes—
140#336C22×C5311yescharacteristic
141C53#781711yescharacteristic, direct product factor, Sylow
142C22×Q8—21no—
143C22×Q8D4×C5311yescharacteristic
144C22×C8—22no—
145C22×C8—21no—
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4D4×C5312yes—
154C2×C4⋊C4D4×C5312yes—
155C2×C4⋊C4D4×C5311yescharacteristic
156C2×C42—21no—
157C2×C42C23×C5311yescharacteristic
158#106—42no—
159#106—44no—
160#103—42no—
161#95—21no—
162#95—22no—
163#95—22no—
164#95—22no—
165C2×Q8—21no—
166C2×Q8—42no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—81no—
170C2×Q8—81no—
171C2×Q8SD16×C5312yes—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—41no—
177C22×C4C2×D4×C5311yescharacteristic
178C22×C4C2×D4×C5312yes—
179C2×C8—22no—
180C2×C8—24no—
181C2×C8—42no—
182C2×C8—41no—
183C2×C8—42no—
184C2×C8—41no—
185C4⋊C4—22no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C4⋊C4—44no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—81no—
196C4⋊C4SD16×C5314yes—
197C42—41no—
198C42—42no—
199C23#120×C5311yescharacteristic
200Q8—42no—
201Q8—81no—
202Q8—84no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—82no—
219C2×C4—81no—
220C2×C4C2×SD16×C5312yes—
221C2×C4C2×SD16×C5311yescharacteristic
222C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
223C2×C4#137×C5312yes—
224C8—44no—
225C8—42no—
226C22C53×#44812yes—
227C22C53×#45011yescharacteristic
228C22C53×#45311yescharacteristic
229C22C53×#45612yes—
230C22C53×#45811yescharacteristic
231C4—22no—
232C4—22no—
233C4—21no—
234C4—21no—
235C4—42no—
236C4—44no—
237C4—81no—
238C4—82no—
239C4—161no—
240C2C53×#161112yes—
241C2C53×#161311yescharacteristic
242C2C53×#162311yescharacteristic
243C2C53×#165112yes—
244C2C53×#166011yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C53#120
commutator, Frattini, cosocleC2×C42C23×C53
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
103. #7817C53direct (N × H)
141. C53#7817direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#27342
C53×#20962
C53×#20842
C53×#12722
C53×#12712
#781753