Finite Group Databasebeta

← C53×#11537 #27219049

Subgroups — C53×#11537 #27219049

Order 13568.

348 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2929C211yesmaximal, characteristic
2C53×#2166C212yesmaximal, complemented
3C53×#1321C212yesmaximal, complemented
4C53×#1320C212yesmaximal, complemented
5C2×#131×C53C411yescharacteristic
6C23×C8×C53C411yescharacteristic
7C53×#525C411yescharacteristic
8C53×#435—22no—
9C8×D4×C53—22no—
10C53×#424C411yescharacteristic
11C53×#408C2211yescharacteristic
12C53×#407C2211yescharacteristic
13C53×#405C2211yescharacteristic
14C53×#364—22no—
15C53×#349C2214yes—
16C53×#347—22no—
17C2×#41×C53C2×C411yescharacteristic
18C23×C4×C53C2×C411yescharacteristic
19#131×C53—42no—
20C2×M16×C53—21no—
21C2×M16×C53C2×C411yescharacteristic
22C2×M16×C53C2311yescharacteristic
23C22×C8×C53—21no—
24C22×C8×C53—21no—
25C22×C8×C53—21no—
26C22×C8×C53—42no—
27C22×C8×C53C2×C411yescharacteristic
28C22×C8×C53C2×C411yescharacteristic
29#123×C53—21no—
30C22⋊Q8×C53—21no—
31#121×C53—21no—
32#120×C53—21no—
33C4×D4×C53—22no—
34#117×C53C2×C411yescharacteristic
35M32×C53—24no—
36C4⋊C8×C53—22no—
37C22⋊C8×C53—22no—
38C22⋊C8×C53—22no—
39#97×C53D412yes—
40C4×C8×C53D412yes—
41C24×C53M1611yescharacteristic
42#41×C53—41no—
43#41×C53—41no—
44C2×Q8×C53M1611yescharacteristic
45C2×D4×C53—22no—
46C2×D4×C53—21no—
47C2×D4×C53M1611yescharacteristic
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53—41no—
52C22×C4×C53—41no—
53C22×C4×C53M1611yescharacteristic
54C22×C4×C53C22×C411yescharacteristic
55M16×C53—41no—
56M16×C53—42no—
57M16×C53—41no—
58C2×C8×C53—22no—
59C2×C8×C53—21no—
60C2×C8×C53—21no—
61C2×C8×C53—42no—
62C2×C8×C53—42no—
63C2×C8×C53—42no—
64C2×C8×C53—41no—
65C2×C8×C53C22⋊C411yescharacteristic
66C2×C8×C53C22⋊C411yescharacteristic
67C2×C8×C53C2×D412yes—
68C4⋊C4×C53—22no—
69C4⋊C4×C53—21no—
70C22⋊C4×C53—22no—
71C22⋊C4×C53—22no—
72C22⋊C4×C53—21no—
73C42×C53C22⋊C412yes—
74C16×C53—44no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53—41no—
78C23×C53—42no—
79C23×C53C2×M1611yescharacteristic
80Q8×C53—41no—
81D4×C53—41no—
82D4×C53—44no—
83D4×C53—42no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—42no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53C2×C22⋊C411yescharacteristic
96C2×C4×C53C2×M1611yescharacteristic
97C8×C53—22no—
98C8×C53—21no—
99C8×C53—41no—
100C8×C53—41no—
101C8×C53#10012yes—
102#11537C5311yesmaximal, characteristic, direct product factor, Sylow
103C22×C53—22no—
104C22×C53—21no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53—42no—
108C22×C53—42no—
109C22×C53—42no—
110C22×C53—41no—
111C22×C53—41no—
112C22×C53#40711yescharacteristic
113C4×C53—21no—
114C4×C53—42no—
115C4×C53—41no—
116C4×C53—41no—
117C4×C53#41011yescharacteristic
118C4×C53C2×#10011yescharacteristic
119#2929C2×C5311yescharacteristic
120#2166C2×C5312yescomplemented
121#1321C2×C5312yescomplemented
122#1320C2×C5312yescomplemented
123C2×C53—21no—
124C2×C53—42no—
125C2×C53—41no—
126C2×C53—81no—
127C2×C53#146611yescharacteristic
128C2×#131C4×C5311yescharacteristic
129C23×C8C4×C5311yescharacteristic
130#525C4×C5311yescharacteristic
131#435—22no—
132C8×D4—22no—
133#424C4×C5311yescharacteristic
134#408C22×C5311yescharacteristic
135#407C22×C5311yescharacteristic
136#405C22×C5311yescharacteristic
137#364—22no—
138#349C22×C5314yes—
139#347—22no—
140C53#1153711yescharacteristic, direct product factor, Sylow
141C2×#41C2×C4×C5311yescharacteristic
142C23×C4C2×C4×C5311yescharacteristic
143#131—42no—
144C2×M16—21no—
145C2×M16C2×C4×C5311yescharacteristic
146C2×M16C23×C5311yescharacteristic
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—42no—
151C22×C8C2×C4×C5311yescharacteristic
152C22×C8C2×C4×C5311yescharacteristic
153#123—21no—
154C22⋊Q8—21no—
155#121—21no—
156#120—21no—
157C4×D4—22no—
158#117C2×C4×C5311yescharacteristic
159M32—24no—
160C4⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8—22no—
163#97D4×C5312yes—
164C4×C8D4×C5312yes—
165C24M16×C5311yescharacteristic
166#41—41no—
167#41—41no—
168C2×Q8M16×C5311yescharacteristic
169C2×D4—22no—
170C2×D4—21no—
171C2×D4M16×C5311yescharacteristic
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4M16×C5311yescharacteristic
178C22×C4C22×C4×C5311yescharacteristic
179M16—41no—
180M16—42no—
181M16—41no—
182C2×C8—22no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—41no—
189C2×C8C22⋊C4×C5311yescharacteristic
190C2×C8C22⋊C4×C5311yescharacteristic
191C2×C8C2×D4×C5312yes—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—21no—
197C42C22⋊C4×C5312yes—
198C16—44no—
199C23—21no—
200C23—21no—
201C23—41no—
202C23—42no—
203C23C2×M16×C5311yescharacteristic
204Q8—41no—
205D4—41no—
206D4—44no—
207D4—42no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4C2×C22⋊C4×C5311yescharacteristic
220C2×C4C2×M16×C5311yescharacteristic
221C8—22no—
222C8—21no—
223C8—41no—
224C8—41no—
225C8#100×C5312yes—
226C22—22no—
227C22—21no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—41no—
234C22—41no—
235C22C53×#40711yescharacteristic
236C4—21no—
237C4—42no—
238C4—41no—
239C4—41no—
240C4C53×#41011yescharacteristic
241C4C2×#100×C5311yescharacteristic
242C2—21no—
243C2—42no—
244C2—41no—
245C2—81no—
246C2C53×#146611yescharacteristic
subgroupG/H
centerC4×C53#410
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×M16C23×C53
socleC2×C53#1466
prime radicalC53×#525C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C53 ◁ C2×C8×C53 ◁ C53×#405 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2166C2semidirect
3. C53×#1321C2semidirect
4. C53×#1320C2semidirect
102. #11537C53direct (N × H)
120. #2166C2×C53semidirect
121. #1321C2×C53semidirect
122. #1320C2×C53semidirect
140. C53#11537direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#29292
C53×#21662
C53×#13212
C53×#13202
#1153753