Finite Group Databasebeta

← C53×#7564 #27215076

Subgroups — C53×#7564 #27215076

Order 13568.

550 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 174 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×#2296C211yesmaximal, characteristic
2C4×C22⋊C8×C53C212yesmaximal
3C53×#1271C214yesmaximal, complemented
4C2×#117×C53C411yescharacteristic
5C2×#117×C53C411yescharacteristic
6C22×C42×C53C2211yescharacteristic
7C2×C22⋊C8×C53—22no—
8C2×C4×C8×C53—22no—
9C53×#389C412yes—
10C53×#389C2212yes—
11C53×#385—22no—
12C4×C4⋊C4×C53—22no—
13C53×#336C2214yes—
14C23×C4×C53C2×C411yescharacteristic
15C23×C4×C53D412yes—
16C22×C8×C53—24no—
17#117×C53—22no—
18#117×C53—42no—
19#117×C53C814yescomplemented
20C2×C4⋊C4×C53—22no—
21C2×C4⋊C4×C53—22no—
22C2×C4⋊C4×C53C2×C412yes—
23C2×C4⋊C4×C53C2×C412yes—
24C2×C22⋊C4×C53—21no—
25C2×C22⋊C4×C53—21no—
26C2×C42×C53—21no—
27C2×C42×C53—21no—
28C2×C42×C53—42no—
29C2×C42×C53C2×C411yescharacteristic
30C2×C42×C53D412yes—
31C2×C42×C53C2311yescharacteristic
32C22⋊C8×C53—28no—
33C4×C8×C53—44no—
34#95×C53—22no—
35#95×C53—22no—
36C24×C53C22⋊C411yescharacteristic
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—24no—
41C22×C4×C53—24no—
42C22×C4×C53—22no—
43C22×C4×C53—42no—
44C22×C4×C53C22⋊C412yes—
45C22×C4×C53C22⋊C411yescharacteristic
46C22×C4×C53C2×C812yes—
47C22×C4×C53M1612yes—
48C22×C4×C53C22×C411yescharacteristic
49C22×C4×C53C2×D412yes—
50C2×C8×C53—28no—
51C2×C8×C53—44no—
52C2×C8×C53—44no—
53C4⋊C4×C53—22no—
54C4⋊C4×C53—42no—
55C4⋊C4×C53—44no—
56C4⋊C4×C53—44no—
57C4⋊C4×C53C2×C814yes—
58C22⋊C4×C53—24no—
59C22⋊C4×C53—42no—
60C42×C53—24no—
61C42×C53—24no—
62C42×C53—41no—
63C42×C53—41no—
64C42×C53—42no—
65C23×C53—21no—
66C23×C53—21no—
67C23×C53—22no—
68C23×C53—22no—
69C23×C53C22⋊C812yes—
70C23×C53C2×C22⋊C411yescharacteristic
71C2×C4×C53—21no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—24no—
78C2×C4×C53—24no—
79C2×C4×C53—24no—
80C2×C4×C53—48no—
81C2×C4×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53C22⋊C812yes—
87C2×C4×C53#9912yes—
88C2×C4×C53#10112yes—
89C2×C4×C53#10414yes—
90C2×C4×C53C22×C811yescharacteristic
91C2×C4×C53C2×M1611yescharacteristic
92C2×C4×C53#13514yes—
93C8×C53—48no—
94#7564C5311yesmaximal, characteristic, direct product factor, Sylow
95C22×C53—22no—
96C22×C53—24no—
97C22×C53—41no—
98C22×C53—42no—
99C22×C53—41no—
100C22×C53C2×C22⋊C811yescharacteristic
101C22×C53C2×#9911yescharacteristic
102C22×C53C2×#10111yescharacteristic
103C22×C53#41612yes—
104C22×C53C2×#10412yes—
105C4×C53—21no—
106C4×C53—21no—
107C4×C53—24no—
108C4×C53—24no—
109C4×C53—44no—
110C4×C53—82no—
111C4×C53#32414yes—
112#2296C2×C5311yescharacteristic
113C4×C22⋊C8C2×C5312yes—
114#1271C2×C5314yescomplemented
115C2×C53—42no—
116C2×C53C2×#32411yescharacteristic
117C2×C53#147612yes—
118C2×C53#150112yes—
119C2×C53#152611yescharacteristic
120C2×C53#154411yescharacteristic
121C2×#117C4×C5311yescharacteristic
122C2×#117C4×C5311yescharacteristic
123C22×C42C22×C5311yescharacteristic
124C2×C22⋊C8—22no—
125C2×C4×C8—22no—
126#389C4×C5312yes—
127#389C22×C5312yes—
128#385—22no—
129C4×C4⋊C4—22no—
130#336C22×C5314yes—
131C53#756411yescharacteristic, direct product factor, Sylow
132C23×C4C2×C4×C5311yescharacteristic
133C23×C4D4×C5312yes—
134C22×C8—24no—
135#117—22no—
136#117—42no—
137#117C8×C5314yescomplemented
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4C2×C4×C5312yes—
141C2×C4⋊C4C2×C4×C5312yes—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—42no—
147C2×C42C2×C4×C5311yescharacteristic
148C2×C42D4×C5312yes—
149C2×C42C23×C5311yescharacteristic
150C22⋊C8—28no—
151C4×C8—44no—
152#95—22no—
153#95—22no—
154C24C22⋊C4×C5311yescharacteristic
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—42no—
162C22×C4C22⋊C4×C5312yes—
163C22×C4C22⋊C4×C5311yescharacteristic
164C22×C4C2×C8×C5312yes—
165C22×C4M16×C5312yes—
166C22×C4C22×C4×C5311yescharacteristic
167C22×C4C2×D4×C5312yes—
168C2×C8—28no—
169C2×C8—44no—
170C2×C8—44no—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4C2×C8×C5314yes—
176C22⋊C4—24no—
177C22⋊C4—42no—
178C42—24no—
179C42—24no—
180C42—41no—
181C42—41no—
182C42—42no—
183C23—21no—
184C23—21no—
185C23—22no—
186C23—22no—
187C23C22⋊C8×C5312yes—
188C23C2×C22⋊C4×C5311yescharacteristic
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—48no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4C22⋊C8×C5312yes—
205C2×C4#99×C5312yes—
206C2×C4#101×C5312yes—
207C2×C4#104×C5314yes—
208C2×C4C22×C8×C5311yescharacteristic
209C2×C4C2×M16×C5311yescharacteristic
210C2×C4#135×C5314yes—
211C8—48no—
212C22—22no—
213C22—24no—
214C22—42no—
215C22—41no—
216C22—41no—
217C22C2×C22⋊C8×C5311yescharacteristic
218C22C2×#99×C5311yescharacteristic
219C22C2×#101×C5311yescharacteristic
220C22C53×#41612yes—
221C22C2×#104×C5312yes—
222C4—21no—
223C4—21no—
224C4—24no—
225C4—24no—
226C4—44no—
227C4—82no—
228C4C53×#32414yes—
229C2—42no—
230C2C2×C53×#32411yescharacteristic
231C2C53×#147612yes—
232C2C53×#150112yes—
233C2C53×#152611yescharacteristic
234C2C53×#154411yescharacteristic
subgroupG/H
centerC22×C4×C53C22⋊C4
commutatorC2×C4C22×C8×C53
Frattini, cosocleC2×C42C23×C53
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C53 ◁ C22×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C53×#1271C2semidirect
19. #117×C53C8semidirect
94. #7564C53direct (N × H)
114. #1271C2×C53semidirect
131. C53#7564direct (N × H)
137. #117C8×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#22962
C4×C22⋊C8×C532
C53×#12712
#756453