Finite Group Databasebeta

← C53×#7084 #27214596

Subgroups — C53×#7084 #27214596

Order 13568.

316 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 50 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×#1999C211yesmaximal, characteristic
2C53×#1314C211yesmaximal, characteristic, complemented
3C53×#1293C211yesmaximal, characteristic, complemented
4C22×D8×C53C411yescharacteristic
5C2×#121×C53—21no—
6C2×#102×C53—21no—
7C53×#407—21no—
8C2×C22⋊C8×C53C2211yescharacteristic
9C53×C22⋊C16—21no—
10C53×#340C411yescharacteristic
11C22×D4×C53—21no—
12C22×D4×C53—41no—
13C23×C4×C53D411yescharacteristic
14C2×D8×C53—42no—
15C2×D8×C53—41no—
16C2×M16×C53—41no—
17C22×C8×C53C2×C411yescharacteristic
18C22×C8×C53D411yescharacteristic
19#121×C53—42no—
20#121×C53—42no—
21C2×C4⋊C4×C53—21no—
22C2×C22⋊C4×C53—41no—
23C2×C16×C53—41no—
24#102×C53—42no—
25C22⋊C8×C53—22no—
26C22⋊C8×C53—41no—
27C22⋊C8×C53—41no—
28C24×C53—41no—
29C24×C53SD1611yescharacteristic
30C2×D4×C53—22no—
31C2×D4×C53—41no—
32C2×D4×C53—41no—
33C2×D4×C53—41no—
34C2×D4×C53—44no—
35C2×D4×C53—41no—
36C2×D4×C53—41no—
37C2×D4×C53—82no—
38C2×D4×C53—82no—
39C22×C4×C53—21no—
40C22×C4×C53—21no—
41C22×C4×C53—41no—
42C22×C4×C53—41no—
43C22×C4×C53—41no—
44C22×C4×C53C22⋊C411yescharacteristic
45C22×C4×C53D811yescharacteristic
46D8×C53—42no—
47D8×C53—42no—
48M16×C53—81no—
49M16×C53—81no—
50C2×C8×C53—21no—
51C2×C8×C53—21no—
52C2×C8×C53—41no—
53C2×C8×C53—41no—
54C2×C8×C53—42no—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—41no—
57C22⋊C4×C53—44no—
58C16×C53—81no—
59C23×C53—21no—
60C23×C53—21no—
61C23×C53—21no—
62C23×C53—41no—
63C23×C53—41no—
64C23×C53—42no—
65C23×C53—81no—
66C23×C53—81no—
67C23×C53—81no—
68C23×C53—81no—
69C23×C53—81no—
70C23×C53—81no—
71C23×C53#10211yescharacteristic
72D4×C53—42no—
73D4×C53—42no—
74D4×C53—81no—
75D4×C53—88no—
76D4×C53—81no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—41no—
81C2×C4×C53—41no—
82C2×C4×C53—41no—
83C2×C4×C53—42no—
84C2×C4×C53—81no—
85C2×C4×C53—81no—
86C2×C4×C53—81no—
87C2×C4×C53#9911yescharacteristic
88C2×C4×C53#10411yescharacteristic
89C8×C53—41no—
90C8×C53—41no—
91C8×C53—82no—
92#7084C5311yesmaximal, characteristic, direct product factor, Sylow
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—42no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53—81no—
100C22×C53—82no—
101C22×C53—82no—
102C22×C53—82no—
103C22×C53—161no—
104C22×C53—161no—
105C22×C53—161no—
106C22×C53—161no—
107C22×C53#32711yescharacteristic
108C22×C53#35911yescharacteristic
109C22×C53#36111yescharacteristic
110C4×C53—21no—
111C4×C53—41no—
112C4×C53—82no—
113C4×C53AΣL(1,𝔽16)11yescharacteristic
114C4×C53#35311yescharacteristic
115#1999C2×C5311yescharacteristic
116#1314C2×C5311yescharacteristic, complemented
117#1293C2×C5311yescharacteristic, complemented
118C2×C53—21no—
119C2×C53—21no—
120C2×C53—81no—
121C2×C53—162no—
122C2×C53#134111yescharacteristic
123C2×C53#134311yescharacteristic
124C2×C53#134911yescharacteristic
125C22×D8C4×C5311yescharacteristic
126C2×#121—21no—
127C2×#102—21no—
128#407—21no—
129C2×C22⋊C8C22×C5311yescharacteristic
130C22⋊C16—21no—
131#340C4×C5311yescharacteristic
132C53#708411yescharacteristic, direct product factor, Sylow
133C22×D4—21no—
134C22×D4—41no—
135C23×C4D4×C5311yescharacteristic
136C2×D8—41no—
137C2×D8—42no—
138C2×M16—41no—
139C22×C8C2×C4×C5311yescharacteristic
140C22×C8D4×C5311yescharacteristic
141#121—42no—
142#121—42no—
143C2×C4⋊C4—21no—
144C2×C22⋊C4—41no—
145C2×C16—41no—
146#102—42no—
147C22⋊C8—22no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C24—41no—
151C24SD16×C5311yescharacteristic
152C2×D4—22no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—44no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—82no—
160C2×D4—82no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4C22⋊C4×C5311yescharacteristic
167C22×C4D8×C5311yescharacteristic
168D8—42no—
169D8—42no—
170M16—81no—
171M16—81no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—42no—
177C4⋊C4—22no—
178C4⋊C4—41no—
179C22⋊C4—44no—
180C16—81no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—41no—
185C23—41no—
186C23—42no—
187C23—81no—
188C23—81no—
189C23—81no—
190C23—81no—
191C23—81no—
192C23—81no—
193C23#102×C5311yescharacteristic
194D4—42no—
195D4—42no—
196D4—81no—
197D4—88no—
198D4—81no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4#99×C5311yescharacteristic
210C2×C4#104×C5311yescharacteristic
211C8—41no—
212C8—41no—
213C8—82no—
214C22—21no—
215C22—21no—
216C22—42no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—82no—
222C22—82no—
223C22—82no—
224C22—161no—
225C22—161no—
226C22—161no—
227C22—161no—
228C22C53×#32711yescharacteristic
229C22C53×#35911yescharacteristic
230C22C53×#36111yescharacteristic
231C4—21no—
232C4—41no—
233C4—82no—
234C4C53×AΣL(1,𝔽16)11yescharacteristic
235C4C53×#35311yescharacteristic
236C2—21no—
237C2—21no—
238C2—81no—
239C2—162no—
240C2C53×#134111yescharacteristic
241C2C53×#134311yescharacteristic
242C2C53×#134911yescharacteristic
subgroupG/H
center, socleC22×C53#327
commutatorC22×C8C2×C4×C53
Frattini, cosocleC2×C22⋊C8C22×C53
prime radicalC53×#1999C2
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×C22⋊C8×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1314C2semidirect
3. C53×#1293C2semidirect
92. #7084C53direct (N × H)
116. #1314C2×C53semidirect
117. #1293C2×C53semidirect
132. C53#7084direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#19992
C53×#13142
C53×#12932
#708453