Finite Group Databasebeta

← C53×#7326 #27214838

Subgroups — C53×#7326 #27214838

Order 13568.

374 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 78 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×#1463C212yesmaximal
2C53×#1445C211yesmaximal, characteristic
3C22×C4⋊C4×C53C2211yescharacteristic
4C53×#407—22no—
5C2×#95×C53—22no—
6C2×#95×C53C411yescharacteristic
7C2×#95×C53C411yescharacteristic
8C53×#324—22no—
9C53×#324C414yescomplemented
10C23×C4×C53—21no—
11C23×C4×C53—21no—
12C23×C4×C53C2×C411yescharacteristic
13C23×C4×C53D412yes—
14C2×M16×C53—42no—
15C2×C4⋊C4×C53—41no—
16C2×C4⋊C4×C53—41no—
17C2×C4⋊C4×C53C2×C412yes—
18C2×C4⋊C4×C53D411yescharacteristic
19C2×C4⋊C4×C53Q811yescharacteristic
20C22⋊C8×C53—24no—
21C22⋊C8×C53—42no—
22#95×C53—44no—
23#95×C53—41no—
24#95×C53—41no—
25C24×C53C22⋊C411yescharacteristic
26C22×C4×C53—21no—
27C22×C4×C53—21no—
28C22×C4×C53—22no—
29C22×C4×C53—22no—
30C22×C4×C53—22no—
31C22×C4×C53—22no—
32C22×C4×C53—42no—
33C22×C4×C53—42no—
34C22×C4×C53—42no—
35C22×C4×C53—42no—
36C22×C4×C53—42no—
37C22×C4×C53—42no—
38C22×C4×C53—41no—
39C22×C4×C53—41no—
40C22×C4×C53—42no—
41C22×C4×C53—41no—
42C22×C4×C53—41no—
43C22×C4×C53C4211yescharacteristic
44C22×C4×C53C22⋊C412yes—
45C22×C4×C53C4⋊C411yescharacteristic
46C22×C4×C53C4⋊C412yes—
47M16×C53—82no—
48M16×C53—82no—
49C2×C8×C53—44no—
50C4⋊C4×C53—42no—
51C4⋊C4×C53—41no—
52C4⋊C4×C53—41no—
53C23×C53—22no—
54C23×C53—21no—
55C23×C53—21no—
56C23×C53—21no—
57C23×C53—21no—
58C23×C53#9511yescharacteristic
59C23×C53#9911yescharacteristic
60C23×C53#9911yescharacteristic
61C2×C4×C53—21no—
62C2×C4×C53—22no—
63C2×C4×C53—21no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—42no—
68C2×C4×C53—42no—
69C2×C4×C53—42no—
70C2×C4×C53—42no—
71C2×C4×C53—42no—
72C2×C4×C53—42no—
73C2×C4×C53—44no—
74C2×C4×C53—44no—
75C2×C4×C53—42no—
76C2×C4×C53—42no—
77C2×C4×C53—82no—
78C2×C4×C53—82no—
79C2×C4×C53—82no—
80C2×C4×C53—81no—
81C2×C4×C53—81no—
82C2×C4×C53—81no—
83C2×C4×C53—81no—
84C8×C53—84no—
85#7326C5311yesmaximal, characteristic, direct product factor, Sylow
86C22×C53—22no—
87C22×C53—22no—
88C22×C53—22no—
89C22×C53—22no—
90C22×C53—21no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—21no—
96C22×C53#34211yescharacteristic
97C22×C53#34412yes—
98C22×C53#35311yescharacteristic
99C22×C53#35411yescharacteristic
100C22×C53#35511yescharacteristic
101C22×C53#35611yescharacteristic
102C4×C53—41no—
103C4×C53—41no—
104C4×C53—42no—
105C4×C53—42no—
106C4×C53—82no—
107C4×C53—82no—
108#1463C2×C5312yes—
109#1445C2×C5311yescharacteristic
110C2×C53—21no—
111C2×C53—21no—
112C2×C53—22no—
113C2×C53—21no—
114C2×C53—21no—
115C2×C53#130611yescharacteristic
116C2×C53#139311yescharacteristic
117C2×C53#139411yescharacteristic
118C22×C4⋊C4C22×C5311yescharacteristic
119#407—22no—
120C2×#95—22no—
121C2×#95C4×C5311yescharacteristic
122C2×#95C4×C5311yescharacteristic
123#324—22no—
124#324C4×C5314yescomplemented
125C53#732611yescharacteristic, direct product factor, Sylow
126C23×C4—21no—
127C23×C4—21no—
128C23×C4C2×C4×C5311yescharacteristic
129C23×C4D4×C5312yes—
130C2×M16—42no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4C2×C4×C5312yes—
134C2×C4⋊C4D4×C5311yescharacteristic
135C2×C4⋊C4Q8×C5311yescharacteristic
136C22⋊C8—24no—
137C22⋊C8—42no—
138#95—41no—
139#95—41no—
140#95—44no—
141C24C22⋊C4×C5311yescharacteristic
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4C42×C5311yescharacteristic
160C22×C4C22⋊C4×C5312yes—
161C22×C4C4⋊C4×C5311yescharacteristic
162C22×C4C4⋊C4×C5312yes—
163M16—82no—
164M16—82no—
165C2×C8—44no—
166C4⋊C4—42no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C23—22no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23#95×C5311yescharacteristic
175C23#99×C5311yescharacteristic
176C23#99×C5311yescharacteristic
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C8—84no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22C53×#34211yescharacteristic
212C22C53×#34412yes—
213C22C53×#35311yescharacteristic
214C22C53×#35411yescharacteristic
215C22C53×#35511yescharacteristic
216C22C53×#35611yescharacteristic
217C4—41no—
218C4—41no—
219C4—42no—
220C4—42no—
221C4—82no—
222C4—82no—
223C2—21no—
224C2—21no—
225C2—22no—
226C2—21no—
227C2—21no—
228C2C53×#130611yescharacteristic
229C2C53×#139311yescharacteristic
230C2C53×#139411yescharacteristic
subgroupG/H
center, socleC22×C53#342
commutatorC22×C4C42×C53
Frattini, cosocleC22×C4⋊C4C22×C53
prime radicalC24×C53C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C24×C53 ◁ C22×C4⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
9. C53×#324C4semidirect
85. #7326C53direct (N × H)
124. #324C4×C53semidirect
125. C53#7326direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#14632
C53×#14452
#732653