Finite Group Databasebeta

← C53×#12207 #27219719

Subgroups — C53×#12207 #27219719

Order 13568.

388 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2817C211yesmaximal, characteristic
2C53×#2030C211yesmaximal, characteristic
3C53×#1608C212yesmaximal
4C53×#1344C212yesmaximal
5C53×#1317C211yesmaximal, characteristic
6C53×#543C2211yescharacteristic
7C2×C22⋊Q8×C53—21no—
8C53×#483—22no—
9C53×#400—22no—
10C53×#397—21no—
11C53×#397C2212yes—
12C53×#395—22no—
13C53×#381—21no—
14C53×#381C2211yescharacteristic
15C53×#342—21no—
16C53×#342C2211yescharacteristic
17C53×#328—22no—
18C53×#323C2212yes—
19C22×Q8×C53—21no—
20C23×C4×C53D411yescharacteristic
21C22⋊Q8×C53—22no—
22C22⋊Q8×C53—42no—
23C22⋊Q8×C53—42no—
24C22⋊Q8×C53—42no—
25C2×C4⋊C4×C53—22no—
26C2×C4⋊C4×C53—21no—
27C2×C4⋊C4×C53—21no—
28C2×C4⋊C4×C53D412yes—
29C2×C22⋊C4×C53—22no—
30C2×C22⋊C4×C53—21no—
31C2×C22⋊C4×C53—21no—
32C2×C22⋊C4×C53D411yescharacteristic
33C2×C22⋊C4×C53C2311yescharacteristic
34#106×C53—42no—
35#103×C53—42no—
36C22⋊C8×C53—22no—
37#95×C53—22no—
38#95×C53—21no—
39#95×C53—22no—
40#95×C53—22no—
41#95×C53—42no—
42#95×C53D412yes—
43C24×C53C2×D411yescharacteristic
44C2×Q8×C53—22no—
45C2×Q8×C53—42no—
46C2×Q8×C53—81no—
47C2×Q8×C53—81no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—41no—
55C22×C4×C53—41no—
56C22×C4×C53—41no—
57C22×C4×C53SD1612yes—
58C22×C4×C53C2×D412yes—
59C2×C8×C53—42no—
60C4⋊C4×C53—22no—
61C4⋊C4×C53—44no—
62C4⋊C4×C53—42no—
63C4⋊C4×C53—42no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—81no—
66C22⋊C4×C53—41no—
67C22⋊C4×C53—41no—
68C22⋊C4×C53—41no—
69C22⋊C4×C53—42no—
70C22⋊C4×C53—42no—
71C22⋊C4×C53—41no—
72C22⋊C4×C53—41no—
73C22⋊C4×C53—41no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53—42no—
77C23×C53#12011yescharacteristic
78C23×C53C2×SD1611yescharacteristic
79C23×C53AGL(1,ℤ/8)11yescharacteristic
80Q8×C53—84no—
81Q8×C53—82no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—41no—
85C2×C4×C53—41no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53—44no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—81no—
97C2×C4×C53—82no—
98C2×C4×C53—81no—
99C2×C4×C53—82no—
100C8×C53—82no—
101#12207C5311yesmaximal, characteristic, direct product factor, Sylow
102C22×C53—22no—
103C22×C53—21no—
104C22×C53—21no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53—42no—
108C22×C53—81no—
109C22×C53#45011yescharacteristic
110C22×C53#45511yescharacteristic
111C22×C53B(4,𝔽2)11yescharacteristic
112C4×C53—42no—
113C4×C53—42no—
114C4×C53—82no—
115C4×C53—81no—
116C4×C53—82no—
117C4×C53—82no—
118C4×C53—161no—
119#2817C2×C5311yescharacteristic
120#2030C2×C5311yescharacteristic
121#1608C2×C5312yes—
122#1344C2×C5312yes—
123#1317C2×C5311yescharacteristic
124C2×C53—21no—
125C2×C53—21no—
126C2×C53—41no—
127C2×C53—41no—
128C2×C53#160111yescharacteristic
129C2×C53#220112yes—
130#543C22×C5311yescharacteristic
131C2×C22⋊Q8—21no—
132#483—22no—
133#400—22no—
134#397—21no—
135#397C22×C5312yes—
136#395—22no—
137#381—21no—
138#381C22×C5311yescharacteristic
139#342—21no—
140#342C22×C5311yescharacteristic
141#328—22no—
142#323C22×C5312yes—
143C53#1220711yescharacteristic, direct product factor, Sylow
144C22×Q8—21no—
145C23×C4D4×C5311yescharacteristic
146C22⋊Q8—22no—
147C22⋊Q8—42no—
148C22⋊Q8—42no—
149C22⋊Q8—42no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4D4×C5312yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4D4×C5311yescharacteristic
158C2×C22⋊C4C23×C5311yescharacteristic
159#106—42no—
160#103—42no—
161C22⋊C8—22no—
162#95—22no—
163#95—21no—
164#95—22no—
165#95—22no—
166#95—42no—
167#95D4×C5312yes—
168C24C2×D4×C5311yescharacteristic
169C2×Q8—22no—
170C2×Q8—42no—
171C2×Q8—81no—
172C2×Q8—81no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4SD16×C5312yes—
183C22×C4C2×D4×C5312yes—
184C2×C8—42no—
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—44no—
189C4⋊C4—42no—
190C4⋊C4—81no—
191C22⋊C4—41no—
192C22⋊C4—41no—
193C22⋊C4—41no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—41no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C23—21no—
200C23—21no—
201C23—42no—
202C23#120×C5311yescharacteristic
203C23C2×SD16×C5311yescharacteristic
204C23AGL(1,ℤ/8)×C5311yescharacteristic
205Q8—82no—
206Q8—84no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—81no—
222C2×C4—82no—
223C2×C4—81no—
224C2×C4—82no—
225C8—82no—
226C22—22no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—81no—
233C22C53×#45011yescharacteristic
234C22C53×#45511yescharacteristic
235C22C53×B(4,𝔽2)11yescharacteristic
236C4—42no—
237C4—42no—
238C4—82no—
239C4—81no—
240C4—82no—
241C4—82no—
242C4—161no—
243C2—21no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2C53×#160111yescharacteristic
248C2C53×#220112yes—
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C53
prime radicalC24×C53C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C22⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
101. #12207C53direct (N × H)
143. C53#12207direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#28172
C53×#20302
C53×#16082
C53×#13442
C53×#13172
#1220753