Finite Group Databasebeta

← C53×#9157 #27216669

Subgroups — C53×#9157 #27216669

Order 13568.

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

normal subgroups only

This group has 648 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
1C53×#2854C211yesmaximal, characteristic
2C2×C53×#383C211yesmaximal, characteristic, complemented
3C53×#1817C211yesmaximal, characteristic, complemented
4C2×C53×#327C212yesmaximal
5C53×#1294C212yesmaximal, complemented
6C2×#123×C53—21no—
7C2×#121×C53C2211yescharacteristic
8C22×C42×C53C2211yescharacteristic
9C2×C22⋊C8×C53C2212yes—
10C53×#390C412yescomplemented
11C53×#389C412yescomplemented
12C53×#388—22no—
13C53×#386—21no—
14C53×#383—24no—
15C53×#380C2211yescharacteristic
16C2×#95×C53—21no—
17C2×#95×C53—21no—
18C2×#95×C53C2212yes—
19C53×#336—22no—
20C53×#327—24no—
21C22×D4×C53—21no—
22C22×D4×C53C2×C411yescharacteristic
23C23×C4×C53—22no—
24C23×C4×C53D411yescharacteristic
25C23×C4×C53D411yescharacteristic
26C23×C4×C53C2311yescharacteristic
27C22×C8×C53—22no—
28#123×C53—44no—
29#121×C53—22no—
30#121×C53—22no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53C2×C411yescharacteristic
33C2×C4⋊C4×C53C2×C412yes—
34C2×C22⋊C4×C53—21no—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53—21no—
38C2×C42×C53—22no—
39C2×C42×C53—22no—
40C2×C42×C53C2×C412yes—
41C2×C42×C53D412yes—
42C22⋊C8×C53—22no—
43C22⋊C8×C53—22no—
44#95×C53—24no—
45#95×C53—24no—
46#95×C53—22no—
47#95×C53—22no—
48#95×C53—21no—
49#95×C53D414yes—
50C24×C53—21no—
51C24×C53C2×D411yescharacteristic
52C2×D4×C53—22no—
53C2×D4×C53—42no—
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C2×D4×C53—42no—
57C2×D4×C53—42no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53—22no—
65C22×C4×C53—21no—
66C22×C4×C53—24no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53—44no—
70C22×C4×C53—44no—
71C22×C4×C53—42no—
72C22×C4×C53—42no—
73C22×C4×C53C22⋊C412yes—
74C22×C4×C53C22⋊C412yes—
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D412yes—
78C22×C4×C53#4112yes—
79C2×C8×C53—44no—
80C2×C8×C53—42no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—44no—
83C4⋊C4×C53—42no—
84C22⋊C4×C53—44no—
85C22⋊C4×C53—44no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—42no—
88C42×C53—24no—
89C42×C53—24no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—41no—
93C23×C53—42no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53—42no—
97C23×C53C2×C22⋊C411yescharacteristic
98C23×C53#12011yescharacteristic
99C23×C53#12311yescharacteristic
100C23×C53#13512yes—
101C23×C53AGL(1,ℤ/8)12yes—
102D4×C53—82no—
103D4×C53—82no—
104D4×C53—82no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
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—44no—
120C2×C4×C53—44no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—44no—
128C2×C4×C53—84no—
129C2×C4×C53—84no—
130C2×C4×C53C4×D412yes—
131C2×C4×C53#12112yes—
132C8×C53—82no—
133#9157C5311yesmaximal, characteristic, direct product factor, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—24no—
139C22×C53—24no—
140C22×C53—81no—
141C22×C53—81no—
142C22×C53—81no—
143C22×C53—82no—
144C22×C53—82no—
145C22×C53#38611yescharacteristic
146C22×C53#41012yes—
147C22×C53#41611yescharacteristic
148C22×C53#41811yescharacteristic
149C22×C53#42112yes—
150C22×C53#44912yes—
151C22×C53#48212yes—
152C4×C53—22no—
153C4×C53—22no—
154C4×C53—42no—
155C4×C53—42no—
156C4×C53—84no—
157C4×C53—81no—
158C4×C53—82no—
159#2854C2×C5311yescharacteristic
160C2×#383C2×C5311yescharacteristic, complemented
161#1817C2×C5311yescharacteristic, complemented
162C2×#327C2×C5312yes—
163#1294C2×C5312yescomplemented
164C2×C53—22no—
165C2×C53—22no—
166C2×C53—161no—
167C2×C53#151412yes—
168C2×C53#152112yes—
169C2×C53#187511yescharacteristic
170C2×C53#189711yescharacteristic
171C2×C53#189911yescharacteristic
172C2×#123—21no—
173C2×#121C22×C5311yescharacteristic
174C22×C42C22×C5311yescharacteristic
175C2×C22⋊C8C22×C5312yes—
176#390C4×C5312yescomplemented
177#389C4×C5312yescomplemented
178#388—22no—
179#386—21no—
180#383—24no—
181#380C22×C5311yescharacteristic
182C2×#95—21no—
183C2×#95—21no—
184C2×#95C22×C5312yes—
185#336—22no—
186#327—24no—
187C53#915711yescharacteristic, direct product factor, Sylow
188C22×D4—21no—
189C22×D4C2×C4×C5311yescharacteristic
190C23×C4—22no—
191C23×C4D4×C5311yescharacteristic
192C23×C4D4×C5311yescharacteristic
193C23×C4C23×C5311yescharacteristic
194C22×C8—22no—
195#123—44no—
196#121—22no—
197#121—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4C2×C4×C5311yescharacteristic
200C2×C4⋊C4C2×C4×C5312yes—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C2×C4×C5312yes—
208C2×C42D4×C5312yes—
209C22⋊C8—22no—
210C22⋊C8—22no—
211#95—24no—
212#95—24no—
213#95—22no—
214#95—21no—
215#95—22no—
216#95D4×C5314yes—
217C24—21no—
218C24C2×D4×C5311yescharacteristic
219C2×D4—22no—
220C2×D4—42no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—42no—
224C2×D4—42no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4C22⋊C4×C5312yes—
241C22×C4C22⋊C4×C5312yes—
242C22×C4C22×C4×C5311yescharacteristic
243C22×C4C2×D4×C5311yescharacteristic
244C22×C4C2×D4×C5312yes—
245C22×C4#41×C5312yes—
246C2×C8—44no—
247C2×C8—42no—
248C4⋊C4—22no—
249C4⋊C4—42no—
250C4⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C42—24no—
256C42—24no—
257C23—22no—
258C23—22no—
259C23—41no—
260C23—42no—
261C23—41no—
262C23—41no—
263C23—42no—
264C23C2×C22⋊C4×C5311yescharacteristic
265C23#120×C5311yescharacteristic
266C23#123×C5311yescharacteristic
267C23#135×C5312yes—
268C23AGL(1,ℤ/8)×C5312yes—
269D4—82no—
270D4—82no—
271D4—82no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—84no—
296C2×C4—84no—
297C2×C4C4×D4×C5312yes—
298C2×C4#121×C5312yes—
299C8—82no—
300C22—24no—
301C22—24no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—82no—
307C22—82no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22C53×#38611yescharacteristic
312C22C53×#41012yes—
313C22C53×#41611yescharacteristic
314C22C53×#41811yescharacteristic
315C22C53×#42112yes—
316C22C53×#44912yes—
317C22C53×#48212yes—
318C4—22no—
319C4—22no—
320C4—42no—
321C4—42no—
322C4—84no—
323C4—82no—
324C4—81no—
325C2—22no—
326C2—22no—
327C2—161no—
328C2C53×#151412yes—
329C2C53×#152112yes—
330C2C53×#187511yescharacteristic
331C2C53×#189711yescharacteristic
332C2C53×#189911yescharacteristic
subgroupG/H
center, socleC23×C53C2×C22⋊C4
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC2×#121×C53C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C22×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C53×#383C2semidirect
3. C53×#1817C2semidirect
5. C53×#1294C2semidirect
10. C53×#390C4semidirect
11. C53×#389C4semidirect
133. #9157C53direct (N × H)
160. C2×#383C2×C53semidirect
161. #1817C2×C53semidirect
163. #1294C2×C53semidirect
176. #390C4×C53semidirect
177. #389C4×C53semidirect
187. C53#9157direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#28542
C2×C53×#3832
C53×#18172
C2×C53×#3272
C53×#12942
#915753