Finite Group Databasebeta

← C53×#7508 #27215020

Subgroups — C53×#7508 #27215020

Order 13568.

290 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C2×C53×#355C211yesmaximal, characteristic
2C2×C53×#354C211yesmaximal, characteristic
3C53×#1959C211yesmaximal, characteristic
4C2×#124×C53C2211yescharacteristic
5C2×#103×C53—21no—
6C2×#102×C53—21no—
7C2×#101×C53—21no—
8C2×#99×C53—21no—
9C2×#97×C53C411yescharacteristic, complemented
10C53×#384C411yescharacteristic, complemented
11C53×#355—21no—
12C53×#355C412yescomplemented
13C53×#354—21no—
14C53×#354C412yescomplemented
15C22×Q8×C53D411yescharacteristic
16C22×D4×C53D411yescharacteristic
17C2×M16×C53—41no—
18C22×C8×C53—21no—
19#124×C53—41no—
20#124×C53C2×C411yescharacteristic
21#124×C53C2×C411yescharacteristic
22#124×C53D411yescharacteristic
23#124×C53Q811yescharacteristic
24C2×C4⋊C4×C53—21no—
25C2×C4⋊C4×C53—41no—
26C2×C22⋊C4×C53—41no—
27C2×C22⋊C4×C53—41no—
28C2×C42×C53C2×C411yescharacteristic
29#103×C53—42no—
30#102×C53—42no—
31#101×C53—22no—
32#101×C53—41no—
33#99×C53—22no—
34#99×C53—41no—
35#97×C53—41no—
36C24×C53—21no—
37C2×Q8×C53—21no—
38C2×Q8×C53—41no—
39C2×Q8×C53—41no—
40C2×Q8×C53C22⋊C411yescharacteristic
41C2×Q8×C53C4⋊C411yescharacteristic
42C2×D4×C53—21no—
43C2×D4×C53—41no—
44C2×D4×C53—41no—
45C2×D4×C53C22⋊C411yescharacteristic
46C2×D4×C53C4⋊C411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—41no—
50C22×C4×C53—81no—
51C22×C4×C53C22⋊C411yescharacteristic
52M16×C53—42no—
53M16×C53—81no—
54C2×C8×C53—22no—
55C2×C8×C53—41no—
56C2×C8×C53—41no—
57C2×C8×C53—81no—
58C4⋊C4×C53—41no—
59C4⋊C4×C53—41no—
60C4⋊C4×C53—82no—
61C22⋊C4×C53—41no—
62C22⋊C4×C53—41no—
63C22⋊C4×C53—41no—
64C22⋊C4×C53—41no—
65C22⋊C4×C53—42no—
66C22⋊C4×C53—81no—
67C42×C53—21no—
68C42×C53C4211yescharacteristic
69C42×C53C4⋊C411yescharacteristic
70C23×C53—21no—
71C23×C53—21no—
72C23×C53—41no—
73C23×C53—41no—
74C23×C53—81no—
75C23×C53—81no—
76C23×C53#9911yescharacteristic
77Q8×C53—41no—
78Q8×C53—41no—
79Q8×C53—81no—
80D4×C53—41no—
81D4×C53—41no—
82D4×C53—81no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—82no—
94C2×C4×C53—82no—
95C2×C4×C53—81no—
96C2×C4×C53—161no—
97C2×C4×C53—161no—
98C2×C4×C53#9511yescharacteristic
99C2×C4×C53#9911yescharacteristic
100C8×C53—42no—
101C8×C53—82no—
102#7508C5311yesmaximal, characteristic, direct product factor, Sylow
103C22×C53—21no—
104C22×C53—21no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53—81no—
108C22×C53—81no—
109C22×C53—81no—
110C22×C53—81no—
111C22×C53—161no—
112C22×C53#34211yescharacteristic
113C22×C53#35311yescharacteristic
114C22×C53#35411yescharacteristic
115C4×C53—21no—
116C4×C53—21no—
117C4×C53—41no—
118C4×C53—41no—
119C4×C53—41no—
120C4×C53—41no—
121C4×C53—162no—
122C4×C53—161no—
123C2×#355C2×C5311yescharacteristic
124C2×#354C2×C5311yescharacteristic
125#1959C2×C5311yescharacteristic
126C2×C53—21no—
127C2×C53—21no—
128C2×C53—81no—
129C2×C53—81no—
130C2×C53#139311yescharacteristic
131C2×C53#140611yescharacteristic
132C2×C53#140711yescharacteristic
133C2×#124C22×C5311yescharacteristic
134C2×#103—21no—
135C2×#102—21no—
136C2×#101—21no—
137C2×#99—21no—
138C2×#97C4×C5311yescharacteristic, complemented
139#384C4×C5311yescharacteristic, complemented
140#355—21no—
141#355C4×C5312yescomplemented
142#354—21no—
143#354C4×C5312yescomplemented
144C53#750811yescharacteristic, direct product factor, Sylow
145C22×Q8D4×C5311yescharacteristic
146C22×D4D4×C5311yescharacteristic
147C2×M16—41no—
148C22×C8—21no—
149#124—41no—
150#124C2×C4×C5311yescharacteristic
151#124C2×C4×C5311yescharacteristic
152#124D4×C5311yescharacteristic
153#124Q8×C5311yescharacteristic
154C2×C4⋊C4—21no—
155C2×C4⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C42C2×C4×C5311yescharacteristic
159#103—42no—
160#102—42no—
161#101—22no—
162#101—41no—
163#99—22no—
164#99—41no—
165#97—41no—
166C24—21no—
167C2×Q8—21no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8C22⋊C4×C5311yescharacteristic
171C2×Q8C4⋊C4×C5311yescharacteristic
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4C22⋊C4×C5311yescharacteristic
176C2×D4C4⋊C4×C5311yescharacteristic
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—41no—
180C22×C4—81no—
181C22×C4C22⋊C4×C5311yescharacteristic
182M16—42no—
183M16—81no—
184C2×C8—22no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—81no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—82no—
191C22⋊C4—41no—
192C22⋊C4—41no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C22⋊C4—42no—
196C22⋊C4—81no—
197C42—21no—
198C42C42×C5311yescharacteristic
199C42C4⋊C4×C5311yescharacteristic
200C23—21no—
201C23—21no—
202C23—41no—
203C23—41no—
204C23—81no—
205C23—81no—
206C23#99×C5311yescharacteristic
207Q8—41no—
208Q8—41no—
209Q8—81no—
210D4—41no—
211D4—41no—
212D4—81no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—81no—
226C2×C4—161no—
227C2×C4—161no—
228C2×C4#95×C5311yescharacteristic
229C2×C4#99×C5311yescharacteristic
230C8—42no—
231C8—82no—
232C22—21no—
233C22—21no—
234C22—41no—
235C22—41no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—161no—
241C22C53×#34211yescharacteristic
242C22C53×#35311yescharacteristic
243C22C53×#35411yescharacteristic
244C4—21no—
245C4—21no—
246C4—41no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—162no—
251C4—161no—
252C2—21no—
253C2—21no—
254C2—81no—
255C2—81no—
256C2C53×#139311yescharacteristic
257C2C53×#140611yescharacteristic
258C2C53×#140711yescharacteristic
subgroupG/H
center, socleC22×C53#353
commutatorC42C42×C53
Frattini, cosocleC2×#124C22×C53
prime radicalC22×D4×C53D4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C22×C4×C53 ◁ C2×#124×C53 ◁ G

Complements

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

NH ≅ G/Ntype
9. C2×#97×C53C4semidirect
10. C53×#384C4semidirect
12. C53×#355C4semidirect
14. C53×#354C4semidirect
102. #7508C53direct (N × H)
138. C2×#97C4×C53semidirect
139. #384C4×C53semidirect
141. #355C4×C53semidirect
143. #354C4×C53semidirect
144. C53#7508direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#3552
C2×C53×#3542
C53×#19592
#750853