Finite Group Databasebeta

← C53×#7598 #27215110

Subgroups — C53×#7598 #27215110

Order 13568.

514 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 166 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
2C53×#1797C212yesmaximal
3C53×#1271C214yesmaximal, complemented
4C2×#117×C53C411yescharacteristic
5C2×#117×C53C411yescharacteristic
6C22×C42×C53C2211yescharacteristic
7C2×C4⋊C8×C53—22no—
8C2×C22⋊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
32C4⋊C8×C53—44no—
33C22⋊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—22no—
41C22×C4×C53—22no—
42C22×C4×C53—22no—
43C22×C4×C53—42no—
44C22×C4×C53—42no—
45C22×C4×C53C22⋊C412yes—
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C2×C812yes—
48C22×C4×C53M1612yes—
49C22×C4×C53C22×C411yescharacteristic
50C22×C4×C53C2×D412yes—
51C2×C8×C53—44no—
52C2×C8×C53—44no—
53C2×C8×C53—44no—
54C4⋊C4×C53—22no—
55C4⋊C4×C53—44no—
56C4⋊C4×C53—42no—
57C4⋊C4×C53—42no—
58C4⋊C4×C53—42no—
59C4⋊C4×C53C2×C814yes—
60C22⋊C4×C53—24no—
61C22⋊C4×C53—42no—
62C42×C53—24no—
63C42×C53—24no—
64C42×C53—41no—
65C42×C53—41no—
66C42×C53—42no—
67C23×C53—21no—
68C23×C53—21no—
69C23×C53—22no—
70C23×C53—22no—
71C23×C53C22⋊C812yes—
72C23×C53C2×C22⋊C411yescharacteristic
73C2×C4×C53—21no—
74C2×C4×C53—22no—
75C2×C4×C53—21no—
76C2×C4×C53—22no—
77C2×C4×C53—24no—
78C2×C4×C53—24no—
79C2×C4×C53—24no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—44no—
84C2×C4×C53—44no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53C22⋊C812yes—
91C2×C4×C53#9912yes—
92C2×C4×C53#10112yes—
93C2×C4×C53C22×C811yescharacteristic
94C2×C4×C53C2×M1611yescharacteristic
95C2×C4×C53AGL(1,ℤ/8)12yes—
96C2×C4×C53#13712yes—
97C8×C53—84no—
98#7598C5311yesmaximal, characteristic, direct product factor, Sylow
99C22×C53—22no—
100C22×C53—24no—
101C22×C53—41no—
102C22×C53—41no—
103C22×C53—42no—
104C22×C53C2×C22⋊C811yescharacteristic
105C22×C53C2×#9911yescharacteristic
106C22×C53C2×#10111yescharacteristic
107C22×C53#41712yes—
108C22×C53#42112yes—
109C4×C53—21no—
110C4×C53—21no—
111C4×C53—22no—
112C4×C53—22no—
113C4×C53—44no—
114C4×C53—42no—
115C4×C53—82no—
116C4×C53#32414yes—
117#2296C2×C5311yescharacteristic
118#1797C2×C5312yes—
119#1271C2×C5314yescomplemented
120C2×C53—42no—
121C2×C53C2×#32411yescharacteristic
122C2×C53#147712yes—
123C2×C53#151312yes—
124C2×C53#153711yescharacteristic
125C2×C53#156111yescharacteristic
126C2×#117C4×C5311yescharacteristic
127C2×#117C4×C5311yescharacteristic
128C22×C42C22×C5311yescharacteristic
129C2×C4⋊C8—22no—
130C2×C22⋊C8—22no—
131#389C4×C5312yes—
132#389C22×C5312yes—
133#385—22no—
134C4×C4⋊C4—22no—
135#336C22×C5314yes—
136C53#759811yescharacteristic, direct product factor, Sylow
137C23×C4C2×C4×C5311yescharacteristic
138C23×C4D4×C5312yes—
139C22×C8—24no—
140#117—22no—
141#117—42no—
142#117C8×C5314yescomplemented
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4C2×C4×C5312yes—
146C2×C4⋊C4C2×C4×C5312yes—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—42no—
152C2×C42C2×C4×C5311yescharacteristic
153C2×C42D4×C5312yes—
154C2×C42C23×C5311yescharacteristic
155C4⋊C8—44no—
156C22⋊C8—44no—
157#95—22no—
158#95—22no—
159C24C22⋊C4×C5311yescharacteristic
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4C22⋊C4×C5312yes—
169C22×C4C22⋊C4×C5311yescharacteristic
170C22×C4C2×C8×C5312yes—
171C22×C4M16×C5312yes—
172C22×C4C22×C4×C5311yescharacteristic
173C22×C4C2×D4×C5312yes—
174C2×C8—44no—
175C2×C8—44no—
176C2×C8—44no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—44no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4C2×C8×C5314yes—
183C22⋊C4—24no—
184C22⋊C4—42no—
185C42—24no—
186C42—24no—
187C42—41no—
188C42—41no—
189C42—42no—
190C23—21no—
191C23—21no—
192C23—22no—
193C23—22no—
194C23C22⋊C8×C5312yes—
195C23C2×C22⋊C4×C5311yescharacteristic
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4C22⋊C8×C5312yes—
214C2×C4#99×C5312yes—
215C2×C4#101×C5312yes—
216C2×C4C22×C8×C5311yescharacteristic
217C2×C4C2×M16×C5311yescharacteristic
218C2×C4AGL(1,ℤ/8)×C5312yes—
219C2×C4#137×C5312yes—
220C8—84no—
221C22—22no—
222C22—24no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22C2×C22⋊C8×C5311yescharacteristic
227C22C2×#99×C5311yescharacteristic
228C22C2×#101×C5311yescharacteristic
229C22C53×#41712yes—
230C22C53×#42112yes—
231C4—21no—
232C4—21no—
233C4—22no—
234C4—22no—
235C4—44no—
236C4—42no—
237C4—82no—
238C4C53×#32414yes—
239C2—42no—
240C2C2×C53×#32411yescharacteristic
241C2C53×#147712yes—
242C2C53×#151312yes—
243C2C53×#153711yescharacteristic
244C2C53×#156111yescharacteristic
subgroupG/H
center, socleC23×C53C2×C22⋊C4
commutatorC2×C4C22×C8×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC24×C53C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×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
98. #7598C53direct (N × H)
119. #1271C2×C53semidirect
136. C53#7598direct (N × H)
142. #117C8×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#22962
C53×#17972
C53×#12712
#759853