Finite Group Databasebeta

← C53×#7758 #27215270

Subgroups — C53×#7758 #27215270

Order 13568.

536 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2733C211yesmaximal, characteristic
2C53×#2023C212yesmaximal, complemented
3C53×#2016C211yesmaximal, characteristic, complemented
4C53×#1272C211yesmaximal, characteristic, complemented
5C53×#1270C212yesmaximal, complemented
6C22×Q16×C53—21no—
7C2×C4⋊Q8×C53C2211yescharacteristic
8C2×#124×C53—21no—
9C2×#103×C53—21no—
10C2×#102×C53—22no—
11C53×#394—21no—
12C53×#392—22no—
13C53×#391C2211yescharacteristic
14C53×#390C2212yes—
15C53×#388—22no—
16C53×#340—22no—
17C53×#336C2211yescharacteristic
18C53×#336C2212yes—
19C22×Q8×C53—21no—
20C22×Q8×C53D411yescharacteristic
21C22×D4×C53—21no—
22C22×D4×C53—21no—
23C22×D4×C53D412yes—
24C2×Q16×C53—42no—
25C2×Q16×C53—42no—
26C2×Q16×C53—42no—
27C22×C8×C53—21no—
28C22×C8×C53—22no—
29C4⋊Q8×C53—42no—
30#124×C53—42no—
31#124×C53—42no—
32C2×C4⋊C4×C53—22no—
33C2×C4⋊C4×C53D412yes—
34C2×C4⋊C4×C53D411yescharacteristic
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53—22no—
38C2×C22⋊C4×C53—22no—
39C2×C42×C53—21no—
40C2×C42×C53C2311yescharacteristic
41#103×C53—42no—
42#102×C53—44no—
43#95×C53—21no—
44#95×C53—22no—
45C24×C53—22no—
46C2×Q8×C53—21no—
47C2×Q8×C53—22no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53—41no—
55C2×Q8×C53D812yes—
56C2×D4×C53—22no—
57C2×D4×C53—22no—
58C2×D4×C53—42no—
59C2×D4×C53—42no—
60C2×D4×C53—42no—
61C2×D4×C53—81no—
62C2×D4×C53—81no—
63C2×D4×C53—81no—
64C2×D4×C53—82no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—22no—
68C22×C4×C53—41no—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D412yes—
71Q16×C53—48no—
72C2×C8×C53—22no—
73C2×C8×C53—41no—
74C2×C8×C53—41no—
75C2×C8×C53—42no—
76C2×C8×C53—42no—
77C2×C8×C53—42no—
78C4⋊C4×C53—24no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—42no—
82C22⋊C4×C53—44no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C22⋊C4×C53—42no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—82no—
88C42×C53—41no—
89C42×C53—42no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—42no—
96C23×C53—42no—
97C23×C53#12011yescharacteristic
98Q8×C53—42no—
99Q8×C53—42no—
100Q8×C53—42no—
101Q8×C53—81no—
102Q8×C53—81no—
103Q8×C53—81no—
104D4×C53—44no—
105D4×C53—82no—
106D4×C53—84no—
107D4×C53—84no—
108C2×C4×C53—21no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—21no—
112C2×C4×C53—22no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—41no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—81no—
122C2×C4×C53—81no—
123C2×C4×C53C2×D811yescharacteristic
124C2×C4×C53#13512yes—
125C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C53#13712yes—
127C8×C53—42no—
128C8×C53—82no—
129#7758C5311yesmaximal, characteristic, direct product factor, Sylow
130C22×C53—82no—
131C22×C53—82no—
132C22×C53—82no—
133C22×C53—82no—
134C22×C53—82no—
135C22×C53—82no—
136C22×C53—82no—
137C22×C53#44711yescharacteristic
138C22×C53#45212yes—
139C22×C53#45311yescharacteristic
140C22×C53#45512yes—
141C22×C53B(4,𝔽2)11yescharacteristic
142C4×C53—21no—
143C4×C53—21no—
144C4×C53—22no—
145C4×C53—22no—
146C4×C53—42no—
147C4×C53—82no—
148C4×C53—82no—
149C4×C53—81no—
150#2733C2×C5311yescharacteristic
151#2023C2×C5312yescomplemented
152#2016C2×C5311yescharacteristic, complemented
153#1272C2×C5311yescharacteristic, complemented
154#1270C2×C5312yescomplemented
155C2×C53—162no—
156C2×C53#159511yescharacteristic
157C2×C53#160612yes—
158C2×C53#163411yescharacteristic
159C2×C53#164112yes—
160C2×C53#165911yescharacteristic
161C22×Q16—21no—
162C2×C4⋊Q8C22×C5311yescharacteristic
163C2×#124—21no—
164C2×#103—21no—
165C2×#102—22no—
166#394—21no—
167#392—22no—
168#391C22×C5311yescharacteristic
169#390C22×C5312yes—
170#388—22no—
171#340—22no—
172#336C22×C5311yescharacteristic
173#336C22×C5312yes—
174C53#775811yescharacteristic, direct product factor, Sylow
175C22×Q8—21no—
176C22×Q8D4×C5311yescharacteristic
177C22×D4—21no—
178C22×D4—21no—
179C22×D4D4×C5312yes—
180C2×Q16—42no—
181C2×Q16—42no—
182C2×Q16—42no—
183C22×C8—21no—
184C22×C8—22no—
185C4⋊Q8—42no—
186#124—42no—
187#124—42no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4D4×C5312yes—
190C2×C4⋊C4D4×C5311yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C42—21no—
196C2×C42C23×C5311yescharacteristic
197#103—42no—
198#102—44no—
199#95—21no—
200#95—22no—
201C24—22no—
202C2×Q8—21no—
203C2×Q8—22no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8D8×C5312yes—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—81no—
218C2×D4—81no—
219C2×D4—81no—
220C2×D4—82no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4C2×D4×C5311yescharacteristic
226C22×C4C2×D4×C5312yes—
227Q16—48no—
228C2×C8—22no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C4⋊C4—24no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C22⋊C4—44no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—82no—
244C42—41no—
245C42—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23#120×C5311yescharacteristic
254Q8—42no—
255Q8—42no—
256Q8—42no—
257Q8—81no—
258Q8—81no—
259Q8—81no—
260D4—44no—
261D4—82no—
262D4—84no—
263D4—84no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4C2×D8×C5311yescharacteristic
280C2×C4#135×C5312yes—
281C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
282C2×C4#137×C5312yes—
283C8—42no—
284C8—82no—
285C22—82no—
286C22—82no—
287C22—82no—
288C22—82no—
289C22—82no—
290C22—82no—
291C22—82no—
292C22C53×#44711yescharacteristic
293C22C53×#45212yes—
294C22C53×#45311yescharacteristic
295C22C53×#45512yes—
296C22C53×B(4,𝔽2)11yescharacteristic
297C4—21no—
298C4—21no—
299C4—22no—
300C4—22no—
301C4—42no—
302C4—81no—
303C4—82no—
304C4—82no—
305C2—162no—
306C2C53×#159511yescharacteristic
307C2C53×#160612yes—
308C2C53×#163411yescharacteristic
309C2C53×#164112yes—
310C2C53×#165911yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC53×#2733C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2023C2semidirect
3. C53×#2016C2semidirect
4. C53×#1272C2semidirect
5. C53×#1270C2semidirect
129. #7758C53direct (N × H)
151. #2023C2×C53semidirect
152. #2016C2×C53semidirect
153. #1272C2×C53semidirect
154. #1270C2×C53semidirect
174. C53#7758direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#27332
C53×#20232
C53×#20162
C53×#12722
C53×#12702
#775853