Finite Group Databasebeta

← C53×#12973 #27220485

Subgroups — C53×#12973 #27220485

Order 13568.

460 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2369C211yesmaximal, characteristic
2C2×C53×#356C211yesmaximal, characteristic
3C53×#1886C211yesmaximal, characteristic
4C53×#1857C212yesmaximal
5C53×#1394C212yesmaximal
6C2×C4⋊Q8×C53C2211yescharacteristic
7C53×#519C2211yescharacteristic
8C53×#428—22no—
9C53×#419—21no—
10C2×#101×C53C2211yescharacteristic
11C2×#101×C53C2212yes—
12C53×#391—21no—
13C53×#391C2212yes—
14C53×#389—22no—
15C4×C4⋊C4×C53—22no—
16C4×C4⋊C4×C53—21no—
17C53×#356C414yescomplemented
18C53×#339—22no—
19C53×#339—21no—
20C22×Q8×C53D411yescharacteristic
21C22×Q8×C53C2311yescharacteristic
22C2×M16×C53—22no—
23C2×M16×C53—21no—
24C4⋊Q8×C53—22no—
25C4⋊Q8×C53C2×C412yes—
26C4⋊Q8×C53Q812yes—
27C4×Q8×C53—41no—
28C4×Q8×C53—41no—
29#117×C53—21no—
30#117×C53—22no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53D412yes—
36C2×C42×C53—22no—
37C2×C42×C53—21no—
38C2×C42×C53D411yescharacteristic
39C2×C42×C53D412yes—
40#107×C53—42no—
41#106×C53—42no—
42#103×C53—41no—
43#103×C53—41no—
44#101×C53—22no—
45#101×C53—22no—
46#101×C53C2×C414yes—
47#95×C53—22no—
48#95×C53—22no—
49#95×C53—22no—
50C2×Q8×C53—21no—
51C2×Q8×C53—41no—
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53—41no—
55C2×Q8×C53C4⋊C412yes—
56C2×Q8×C53C22×C411yescharacteristic
57C2×Q8×C53C2×Q811yescharacteristic
58C2×Q8×C53#4112yes—
59C22×C4×C53—21no—
60C22×C4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—22no—
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D412yes—
65M16×C53—24no—
66M16×C53—42no—
67M16×C53—41no—
68M16×C53—41no—
69C2×C8×C53—42no—
70C2×C8×C53—41no—
71C4⋊C4×C53—24no—
72C4⋊C4×C53—24no—
73C4⋊C4×C53—24no—
74C4⋊C4×C53—21no—
75C4⋊C4×C53—21no—
76C4⋊C4×C53—22no—
77C4⋊C4×C53—22no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—81no—
81C22⋊C4×C53—41no—
82C42×C53—21no—
83C42×C53—44no—
84C42×C53—41no—
85C42×C53—42no—
86C42×C53—42no—
87C42×C53—81no—
88C42×C53C4⋊C412yes—
89C23×C53#12011yescharacteristic
90Q8×C53—41no—
91Q8×C53—41no—
92Q8×C53—42no—
93Q8×C53—42no—
94Q8×C53—81no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—24no—
103C2×C4×C53—22no—
104C2×C4×C53—42no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53C2×C4⋊C411yescharacteristic
114C2×C4×C53C4×D412yes—
115C2×C4×C53C22⋊Q812yes—
116C2×C4×C53#12311yescharacteristic
117C8×C53—44no—
118C8×C53—81no—
119#12973C5311yesmaximal, characteristic, direct product factor, Sylow
120C22×C53—21no—
121C22×C53—21no—
122C22×C53#38511yescharacteristic
123C22×C53B(4,𝔽2)11yescharacteristic
124C22×C53#45811yescharacteristic
125C4×C53—24no—
126C4×C53—22no—
127C4×C53—21no—
128C4×C53—21no—
129C4×C53—22no—
130C4×C53—44no—
131C4×C53—42no—
132C4×C53—81no—
133C4×C53—82no—
134C4×C53—81no—
135#2369C2×C5311yescharacteristic
136C2×#356C2×C5311yescharacteristic
137#1886C2×C5311yescharacteristic
138#1857C2×C5312yes—
139#1394C2×C5312yes—
140C2×C53—21no—
141C2×C53—21no—
142C2×C53#186711yescharacteristic
143C2×C53#220411yescharacteristic
144C2×C53#220511yescharacteristic
145C2×C4⋊Q8C22×C5311yescharacteristic
146#519C22×C5311yescharacteristic
147#428—22no—
148#419—21no—
149C2×#101C22×C5311yescharacteristic
150C2×#101C22×C5312yes—
151#391—21no—
152#391C22×C5312yes—
153#389—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—21no—
156#356C4×C5314yescomplemented
157#339—22no—
158#339—21no—
159C53#1297311yescharacteristic, direct product factor, Sylow
160C22×Q8D4×C5311yescharacteristic
161C22×Q8C23×C5311yescharacteristic
162C2×M16—22no—
163C2×M16—21no—
164C4⋊Q8—22no—
165C4⋊Q8C2×C4×C5312yes—
166C4⋊Q8Q8×C5312yes—
167C4×Q8—41no—
168C4×Q8—41no—
169#117—21no—
170#117—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4D4×C5312yes—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42D4×C5311yescharacteristic
179C2×C42D4×C5312yes—
180#107—42no—
181#106—42no—
182#103—41no—
183#103—41no—
184#101—22no—
185#101—22no—
186#101C2×C4×C5314yes—
187#95—22no—
188#95—22no—
189#95—22no—
190C2×Q8—21no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8C4⋊C4×C5312yes—
196C2×Q8C22×C4×C5311yescharacteristic
197C2×Q8C2×Q8×C5311yescharacteristic
198C2×Q8#41×C5312yes—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4C2×D4×C5311yescharacteristic
204C22×C4C2×D4×C5312yes—
205M16—24no—
206M16—42no—
207M16—41no—
208M16—41no—
209C2×C8—42no—
210C2×C8—41no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—81no—
221C22⋊C4—41no—
222C42—21no—
223C42—42no—
224C42—44no—
225C42—42no—
226C42—41no—
227C42—81no—
228C42C4⋊C4×C5312yes—
229C23#120×C5311yescharacteristic
230Q8—41no—
231Q8—41no—
232Q8—42no—
233Q8—42no—
234Q8—81no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4C2×C4⋊C4×C5311yescharacteristic
254C2×C4C4×D4×C5312yes—
255C2×C4C22⋊Q8×C5312yes—
256C2×C4#123×C5311yescharacteristic
257C8—44no—
258C8—81no—
259C22—21no—
260C22—21no—
261C22C53×#38511yescharacteristic
262C22C53×B(4,𝔽2)11yescharacteristic
263C22C53×#45811yescharacteristic
264C4—24no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—22no—
269C4—44no—
270C4—42no—
271C4—82no—
272C4—81no—
273C4—81no—
274C2—21no—
275C2—21no—
276C2C53×#186711yescharacteristic
277C2C53×#220411yescharacteristic
278C2C53×#220511yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutatorC2×Q8C22×C4×C53
Frattini, cosocleC22×Q8C23×C53
prime radicalC23×C53#120
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C22×Q8×C53 ◁ G

Complements

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

NH ≅ G/Ntype
17. C53×#356C4semidirect
119. #12973C53direct (N × H)
156. #356C4×C53semidirect
159. C53#12973direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#23692
C2×C53×#3562
C53×#18862
C53×#18572
C53×#13942
#1297353