Finite Group Databasebeta

← C53×#9028 #27216540

Subgroups — C53×#9028 #27216540

Order 13568.

696 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×C53×#397C211yesmaximal, characteristic
2C53×#2312C211yesmaximal, characteristic
3C53×#1823C212yesmaximal, complemented
4C2×C53×#323C211yesmaximal, characteristic
5C53×#1294C212yesmaximal, complemented
6C22×C4⋊C4×C53C2211yescharacteristic
7C22×C22⋊C4×C53—21no—
8C22×C22⋊C4×C53C2211yescharacteristic
9C2×C22⋊C8×C53C2212yes—
10C53×#397—22no—
11C53×#397C414yescomplemented
12C53×#388—24no—
13C53×#380C2212yes—
14C53×#379—21no—
15C4×C22⋊C4×C53—21no—
16C2×#95×C53—21no—
17C2×#95×C53C2211yescharacteristic
18C53×#340—24no—
19C53×#323—22no—
20C25×C53D411yescharacteristic
21C23×C4×C53—21no—
22C23×C4×C53—21no—
23C23×C4×C53D411yescharacteristic
24C23×C4×C53C2311yescharacteristic
25C22×C8×C53—22no—
26C2×C4⋊C4×C53—21no—
27C2×C4⋊C4×C53—22no—
28C2×C4⋊C4×C53—22no—
29C2×C4⋊C4×C53C2×C412yes—
30C2×C4⋊C4×C53D414yes—
31C2×C22⋊C4×C53—22no—
32C2×C22⋊C4×C53—22no—
33C2×C22⋊C4×C53—24no—
34C2×C22⋊C4×C53—22no—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53—44no—
38C2×C22⋊C4×C53C2×C412yes—
39C2×C22⋊C4×C53D412yes—
40C2×C42×C53—22no—
41C22⋊C8×C53—22no—
42C22⋊C8×C53—22no—
43#95×C53—22no—
44#95×C53—24no—
45#95×C53—21no—
46#95×C53—22no—
47#95×C53C2×C412yes—
48C24×C53—21no—
49C24×C53—22no—
50C24×C53—21no—
51C24×C53—22no—
52C24×C53—42no—
53C24×C53—42no—
54C24×C53C22⋊C412yes—
55C24×C53C2×D411yescharacteristic
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—24no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53C22⋊C412yes—
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D412yes—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53#4112yes—
75C2×C8×C53—44no—
76C2×C8×C53—42no—
77C4⋊C4×C53—24no—
78C4⋊C4×C53—44no—
79C4⋊C4×C53—42no—
80C22⋊C4×C53—24no—
81C22⋊C4×C53—24no—
82C22⋊C4×C53—48no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C22⋊C4×C53—82no—
86C22⋊C4×C53—82no—
87C42×C53—82no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—48no—
96C23×C53—42no—
97C23×C53—42no—
98C23×C53—42no—
99C23×C53—42no—
100C23×C53—81no—
101C23×C53—81no—
102C23×C53—84no—
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53#12011yescharacteristic
105C23×C53#12311yescharacteristic
106C23×C53#13512yes—
107C23×C53AGL(1,ℤ/8)12yes—
108C2×C4×C53—24no—
109C2×C4×C53—22no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—44no—
122C2×C4×C53—44no—
123C2×C4×C53—44no—
124C2×C4×C53—81no—
125C2×C4×C53—84no—
126C2×C4×C53—82no—
127C2×C4×C53—82no—
128C2×C4×C53C4×D412yes—
129C2×C4×C53#12112yes—
130C8×C53—82no—
131#9028C5311yesmaximal, characteristic, direct product factor, Sylow
132C22×C53—24no—
133C22×C53—24no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—44no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—42no—
142C22×C53—82no—
143C22×C53—84no—
144C22×C53—84no—
145C22×C53#38611yescharacteristic
146C22×C53#41611yescharacteristic
147C22×C53#41811yescharacteristic
148C22×C53#44912yes—
149C22×C53#45512yes—
150C22×C53#45812yes—
151C22×C53#48212yes—
152C4×C53—22no—
153C4×C53—22no—
154C4×C53—42no—
155C4×C53—82no—
156C4×C53—84no—
157C4×C53—82no—
158C2×#397C2×C5311yescharacteristic
159#2312C2×C5311yescharacteristic
160#1823C2×C5312yescomplemented
161C2×#323C2×C5311yescharacteristic
162#1294C2×C5312yescomplemented
163C2×C53—21no—
164C2×C53—21no—
165C2×C53—22no—
166C2×C53—82no—
167C2×C53#161214yes—
168C2×C53#187511yescharacteristic
169C2×C53#188511yescharacteristic
170C2×C53#189011yescharacteristic
171C22×C4⋊C4C22×C5311yescharacteristic
172C22×C22⋊C4—21no—
173C22×C22⋊C4C22×C5311yescharacteristic
174C2×C22⋊C8C22×C5312yes—
175#397—22no—
176#397C4×C5314yescomplemented
177#388—24no—
178#380C22×C5312yes—
179#379—21no—
180C4×C22⋊C4—21no—
181C2×#95—21no—
182C2×#95C22×C5311yescharacteristic
183#340—24no—
184#323—22no—
185C53#902811yescharacteristic, direct product factor, Sylow
186C25D4×C5311yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4D4×C5311yescharacteristic
190C23×C4C23×C5311yescharacteristic
191C22×C8—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C2×C4×C5312yes—
196C2×C4⋊C4D4×C5314yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4C2×C4×C5312yes—
205C2×C22⋊C4D4×C5312yes—
206C2×C42—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209#95—22no—
210#95—21no—
211#95—22no—
212#95—24no—
213#95C2×C4×C5312yes—
214C24—21no—
215C24—22no—
216C24—21no—
217C24—22no—
218C24—42no—
219C24—42no—
220C24C22⋊C4×C5312yes—
221C24C2×D4×C5311yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4C22⋊C4×C5312yes—
237C22×C4C22×C4×C5311yescharacteristic
238C22×C4C2×D4×C5312yes—
239C22×C4C2×D4×C5311yescharacteristic
240C22×C4#41×C5312yes—
241C2×C8—44no—
242C2×C8—42no—
243C4⋊C4—24no—
244C4⋊C4—44no—
245C4⋊C4—42no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—48no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—82no—
252C22⋊C4—82no—
253C42—82no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—42no—
260C23—42no—
261C23—48no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—81no—
267C23—81no—
268C23—84no—
269C23C2×C22⋊C4×C5311yescharacteristic
270C23#120×C5311yescharacteristic
271C23#123×C5311yescharacteristic
272C23#135×C5312yes—
273C23AGL(1,ℤ/8)×C5312yes—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—81no—
291C2×C4—84no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4C4×D4×C5312yes—
295C2×C4#121×C5312yes—
296C8—82no—
297C22—24no—
298C22—24no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—44no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—82no—
308C22—84no—
309C22—84no—
310C22C53×#38611yescharacteristic
311C22C53×#41611yescharacteristic
312C22C53×#41811yescharacteristic
313C22C53×#44912yes—
314C22C53×#45512yes—
315C22C53×#45812yes—
316C22C53×#48212yes—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—82no—
321C4—84no—
322C4—82no—
323C2—21no—
324C2—21no—
325C2—22no—
326C2—82no—
327C2C53×#161214yes—
328C2C53×#187511yescharacteristic
329C2C53×#188511yescharacteristic
330C2C53×#189011yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC25×C53D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C53×#1823C2semidirect
5. C53×#1294C2semidirect
11. C53×#397C4semidirect
131. #9028C53direct (N × H)
160. #1823C2×C53semidirect
162. #1294C2×C53semidirect
176. #397C4×C53semidirect
185. C53#9028direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#3972
C53×#23122
C53×#18232
C2×C53×#3232
C53×#12942
#902853