Finite Group Databasebeta

← C53×#12152 #27219664

Subgroups — C53×#12152 #27219664

Order 13568.

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

normal subgroups only

This group has 588 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×#2537C211yesmaximal, characteristic, complemented
2C53×#2039C211yesmaximal, characteristic, complemented
3C53×#1598C212yesmaximal, complemented
4C53×#1345C212yesmaximal, complemented
5C53×#1325C211yesmaximal, characteristic, complemented
6C53×#537C2211yescharacteristic
7C2×#121×C53—22no—
8C2×C4×D4×C53—21no—
9C53×#447—22no—
10C2×#104×C53—21no—
11C53×#417—21no—
12C53×#394—21no—
13C53×#394C2212yes—
14C53×#384C2211yescharacteristic
15C53×#380—22no—
16C53×#343C2211yescharacteristic
17C53×#327—22no—
18C53×#324C2212yes—
19C2×#41×C53—21no—
20C22×D4×C53—21no—
21C22×D4×C53—21no—
22C22×D4×C53D412yes—
23C23×C4×C53—22no—
24C2×D8×C53—42no—
25C2×M16×C53—21no—
26C22⋊Q8×C53—41no—
27#121×C53—22no—
28#121×C53—41no—
29#121×C53—48no—
30C4×D4×C53—42no—
31C4×D4×C53—42no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53D411yescharacteristic
35C2×C4⋊C4×C53C2311yescharacteristic
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53—22no—
38C2×C22⋊C4×C53—22no—
39C2×C42×C53D411yescharacteristic
40#104×C53—42no—
41#103×C53—41no—
42#102×C53—41no—
43#102×C53—42no—
44C22⋊C8×C53—22no—
45#95×C53—21no—
46#95×C53D412yes—
47C24×C53—22no—
48#41×C53—41no—
49#41×C53—41no—
50#41×C53—41no—
51#41×C53—41no—
52C2×Q8×C53—21no—
53C2×D4×C53—24no—
54C2×D4×C53—21no—
55C2×D4×C53—22no—
56C2×D4×C53—42no—
57C2×D4×C53—42no—
58C2×D4×C53—42no—
59C2×D4×C53—44no—
60C2×D4×C53—44no—
61C2×D4×C53—41no—
62C2×D4×C53—81no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—21no—
66C22×C4×C53—44no—
67C22×C4×C53—44no—
68C22×C4×C53—41no—
69C22×C4×C53—44no—
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D412yes—
72D8×C53—84no—
73M16×C53—41no—
74M16×C53—41no—
75C2×C8×C53—42no—
76C2×C8×C53—41no—
77C4⋊C4×C53—22no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—44no—
81C4⋊C4×C53—41no—
82C22⋊C4×C53—42no—
83C22⋊C4×C53—44no—
84C22⋊C4×C53—44no—
85C22⋊C4×C53—42no—
86C22⋊C4×C53—42no—
87C42×C53—21no—
88C42×C53D812yescomplemented
89C23×C53—24no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—42no—
96C23×C53—41no—
97C23×C53#12011yescharacteristic
98Q8×C53—41no—
99Q8×C53—41no—
100D4×C53—48no—
101D4×C53—41no—
102D4×C53—41no—
103D4×C53—44no—
104D4×C53—82no—
105D4×C53—82no—
106D4×C53—84no—
107D4×C53—81no—
108D4×C53—81no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—22no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—88no—
123C2×C4×C53—81no—
124C2×C4×C53—81no—
125C2×C4×C53C2×D811yescharacteristic
126C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
127C8×C53—81no—
128C8×C53—82no—
129#12152C5311yesmaximal, characteristic, direct product factor, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—44no—
133C22×C53—44no—
134C22×C53—44no—
135C22×C53—82no—
136C22×C53—82no—
137C22×C53—82no—
138C22×C53—82no—
139C22×C53—81no—
140C22×C53—81no—
141C22×C53—81no—
142C22×C53#44711yescharacteristic
143C22×C53#45411yescharacteristic
144C22×C53B(4,𝔽2)11yescharacteristic
145C4×C53—24no—
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—42no—
149C4×C53—82no—
150C4×C53—81no—
151C4×C53—81no—
152#2537C2×C5311yescharacteristic, complemented
153#2039C2×C5311yescharacteristic, complemented
154#1598C2×C5312yescomplemented
155#1345C2×C5312yescomplemented
156#1325C2×C5311yescharacteristic, complemented
157C2×C53—21no—
158C2×C53—21no—
159C2×C53—84no—
160C2×C53—161no—
161C2×C53#159811yescharacteristic
162C2×C53#219611yescharacteristic
163C2×C53#220411yescharacteristic
164#537C22×C5311yescharacteristic
165C2×#121—22no—
166C2×C4×D4—21no—
167#447—22no—
168C2×#104—21no—
169#417—21no—
170#394—21no—
171#394C22×C5312yes—
172#384C22×C5311yescharacteristic
173#380—22no—
174#343C22×C5311yescharacteristic
175#327—22no—
176#324C22×C5312yes—
177C53#1215211yescharacteristic, direct product factor, Sylow
178C2×#41—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4D4×C5312yes—
182C23×C4—22no—
183C2×D8—42no—
184C2×M16—21no—
185C22⋊Q8—41no—
186#121—22no—
187#121—41no—
188#121—48no—
189C4×D4—42no—
190C4×D4—42no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4D4×C5311yescharacteristic
194C2×C4⋊C4C23×C5311yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C42D4×C5311yescharacteristic
199#104—42no—
200#103—41no—
201#102—42no—
202#102—41no—
203C22⋊C8—22no—
204#95—21no—
205#95D4×C5312yes—
206C24—22no—
207#41—41no—
208#41—41no—
209#41—41no—
210#41—41no—
211C2×Q8—21no—
212C2×D4—24no—
213C2×D4—21no—
214C2×D4—22no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—41no—
221C2×D4—81no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—41no—
228C22×C4—44no—
229C22×C4C2×D4×C5311yescharacteristic
230C22×C4C2×D4×C5312yes—
231D8—84no—
232M16—41no—
233M16—41no—
234C2×C8—42no—
235C2×C8—41no—
236C4⋊C4—22no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C22⋊C4—42no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C42—21no—
247C42D8×C5312yescomplemented
248C23—24no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—41no—
256C23#120×C5311yescharacteristic
257Q8—41no—
258Q8—41no—
259D4—48no—
260D4—41no—
261D4—41no—
262D4—44no—
263D4—82no—
264D4—82no—
265D4—84no—
266D4—81no—
267D4—81no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—88no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4C2×D8×C5311yescharacteristic
285C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
286C8—81no—
287C8—82no—
288C22—21no—
289C22—21no—
290C22—44no—
291C22—44no—
292C22—44no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22C53×#44711yescharacteristic
301C22C53×#45411yescharacteristic
302C22C53×B(4,𝔽2)11yescharacteristic
303C4—24no—
304C4—21no—
305C4—21no—
306C4—42no—
307C4—82no—
308C4—81no—
309C4—81no—
310C2—21no—
311C2—21no—
312C2—84no—
313C2—161no—
314C2C53×#159811yescharacteristic
315C2C53×#219611yescharacteristic
316C2C53×#220411yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C53
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C4⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#2537C2semidirect
2. C53×#2039C2semidirect
3. C53×#1598C2semidirect
4. C53×#1345C2semidirect
5. C53×#1325C2semidirect
88. C42×C53D8semidirect
129. #12152C53direct (N × H)
152. #2537C2×C53semidirect
153. #2039C2×C53semidirect
154. #1598C2×C53semidirect
155. #1345C2×C53semidirect
156. #1325C2×C53semidirect
177. C53#12152direct (N × H)
247. C42D8×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#25372
C53×#20392
C53×#15982
C53×#13452
C53×#13252
#1215253