Finite Group Databasebeta

← C53×#3125 #20272062

Subgroups — C53×#3125 #20272062

Order 6784.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#542C211yesmaximal, characteristic, complemented
2C53×#520C211yesmaximal, characteristic, complemented
3C53×#483C211yesmaximal, characteristic, complemented
4C53×#481C211yesmaximal, characteristic, complemented
5C53×#476C211yesmaximal, characteristic, complemented
6C53×#475C211yesmaximal, characteristic, complemented
7C53×#464C211yesmaximal, characteristic, complemented
8C53×#463C211yesmaximal, characteristic, complemented
9C53×#452C211yesmaximal, characteristic, complemented
10C53×#449C211yesmaximal, characteristic
11C53×#442C211yesmaximal, characteristic, complemented
12C53×#441C211yesmaximal, characteristic, complemented
13C4×SD16×C53C211yesmaximal, characteristic, complemented
14C4×SD16×C53C211yesmaximal, characteristic, complemented
15C53×#433C211yesmaximal, characteristic, complemented
16C2×#41×C53C2211yescharacteristic
17C2×Q16×C53C2211yescharacteristic
18C2×SD16×C53C2211yescharacteristic
19C2×SD16×C53C2211yescharacteristic
20C2×D8×C53C2211yescharacteristic
21C4⋊Q8×C53C2211yescharacteristic
22#126×C53—21no—
23#124×C53—21no—
24#124×C53C2211yescharacteristic
25#123×C53—21no—
26C22⋊Q8×C53—21no—
27C22⋊Q8×C53C2211yescharacteristic
28#121×C53C2211yescharacteristic
29C4×Q8×C53C2211yescharacteristic
30C4×Q8×C53C2211yescharacteristic
31C4×Q8×C53C2211yescharacteristic
32C4×D4×C53—21no—
33C4×D4×C53—21no—
34C4×D4×C53C2211yescharacteristic
35C4×D4×C53C2211yescharacteristic
36C4×D4×C53C2211yescharacteristic
37#117×C53—21no—
38#117×C53C2211yescharacteristic
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2211yescharacteristic
41#106×C53C2211yescharacteristic
42#106×C53C2211yescharacteristic
43#106×C53C2211yescharacteristic
44#106×C53C2211yescharacteristic
45C4⋊C8×C53C2211yescharacteristic
46C4⋊C8×C53C2211yescharacteristic
47#103×C53C2211yescharacteristic
48#103×C53C2211yescharacteristic
49#103×C53C2211yescharacteristic
50#103×C53C2211yescharacteristic
51#102×C53C2211yescharacteristic
52#102×C53C2211yescharacteristic
53#102×C53C2211yescharacteristic
54#102×C53C2211yescharacteristic
55C22⋊C8×C53C2211yescharacteristic
56C22⋊C8×C53C2211yescharacteristic
57#97×C53C2211yescharacteristic
58C4×C8×C53C2211yescharacteristic
59#41×C53—22no—
60#41×C53—22no—
61C2×Q8×C53—21no—
62C2×Q8×C53C2311yescharacteristic
63C2×Q8×C53C2311yescharacteristic
64C2×D4×C53—21no—
65C2×D4×C53—21no—
66C2×D4×C53C2311yescharacteristic
67C2×D4×C53C2311yescharacteristic
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53C2311yescharacteristic
74Q16×C53—22no—
75SD16×C53—22no—
76SD16×C53—22no—
77D8×C53—22no—
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C4⋊C4×C53—21no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53—21no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53D411yescharacteristic
89C4⋊C4×C53D411yescharacteristic
90C4⋊C4×C53C2311yescharacteristic
91C4⋊C4×C53C2311yescharacteristic
92C4⋊C4×C53C2311yescharacteristic
93C4⋊C4×C53C2311yescharacteristic
94C22⋊C4×C53—21no—
95C22⋊C4×C53—21no—
96C22⋊C4×C53—21no—
97C22⋊C4×C53—21no—
98C22⋊C4×C53—21no—
99C22⋊C4×C53—21no—
100C22⋊C4×C53—21no—
101C22⋊C4×C53D411yescharacteristic
102C22⋊C4×C53D411yescharacteristic
103C42×C53—21no—
104C42×C53—21no—
105C42×C53—21no—
106C42×C53C2311yescharacteristic
107C42×C53C2311yescharacteristic
108C23×C53—21no—
109C23×C53—21no—
110C23×C53C2×D411yescharacteristic
111Q8×C53—21no—
112Q8×C53—21no—
113Q8×C53—21no—
114Q8×C53—81no—
115Q8×C53#4112yes—
116D4×C53—24no—
117D4×C53—21no—
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—81no—
121D4×C53#4112yes—
122C2×C4×C53—24no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—81no—
132C2×C4×C53—81no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53C2×D411yescharacteristic
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2×D411yescharacteristic
140C2×C4×C53C2411yescharacteristic
141C8×C53—21no—
142C8×C53—21no—
143C8×C53—21no—
144C8×C53—21no—
145C22×C53—21no—
146C22×C53—22no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53C22×D411yescharacteristic
155C4×C53—21no—
156C4×C53—22no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53C2×#4111yescharacteristic
167C4×C53C2×#4111yescharacteristic
168#3125C5311yesmaximal, characteristic, direct product factor, Sylow
169C2×C53—41no—
170C2×C53—42no—
171C2×C53—81no—
172C2×C53#52511yescharacteristic
173C2×C53#57711yescharacteristic
174C2×C53#57711yescharacteristic
175#542C2×C5311yescharacteristic, complemented
176#520C2×C5311yescharacteristic, complemented
177#483C2×C5311yescharacteristic, complemented
178#481C2×C5311yescharacteristic, complemented
179#476C2×C5311yescharacteristic, complemented
180#475C2×C5311yescharacteristic, complemented
181#464C2×C5311yescharacteristic, complemented
182#463C2×C5311yescharacteristic, complemented
183#452C2×C5311yescharacteristic, complemented
184#449C2×C5311yescharacteristic
185#442C2×C5311yescharacteristic, complemented
186#441C2×C5311yescharacteristic, complemented
187C4×SD16C2×C5311yescharacteristic, complemented
188C4×SD16C2×C5311yescharacteristic, complemented
189#433C2×C5311yescharacteristic, complemented
190C53#312511yescharacteristic, direct product factor, Sylow
191C2×#41C22×C5311yescharacteristic
192C2×Q16C22×C5311yescharacteristic
193C2×SD16C22×C5311yescharacteristic
194C2×SD16C22×C5311yescharacteristic
195C2×D8C22×C5311yescharacteristic
196C4⋊Q8C22×C5311yescharacteristic
197#126—21no—
198#124—21no—
199#124C22×C5311yescharacteristic
200#123—21no—
201C22⋊Q8—21no—
202C22⋊Q8C22×C5311yescharacteristic
203#121C22×C5311yescharacteristic
204C4×Q8C22×C5311yescharacteristic
205C4×Q8C22×C5311yescharacteristic
206C4×Q8C22×C5311yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4C22×C5311yescharacteristic
210C4×D4C22×C5311yescharacteristic
211C4×D4C22×C5311yescharacteristic
212#117—21no—
213#117C22×C5311yescharacteristic
214C2×C4⋊C4—21no—
215C2×C4⋊C4C22×C5311yescharacteristic
216#106C22×C5311yescharacteristic
217#106C22×C5311yescharacteristic
218#106C22×C5311yescharacteristic
219#106C22×C5311yescharacteristic
220C4⋊C8C22×C5311yescharacteristic
221C4⋊C8C22×C5311yescharacteristic
222#103C22×C5311yescharacteristic
223#103C22×C5311yescharacteristic
224#103C22×C5311yescharacteristic
225#103C22×C5311yescharacteristic
226#102C22×C5311yescharacteristic
227#102C22×C5311yescharacteristic
228#102C22×C5311yescharacteristic
229#102C22×C5311yescharacteristic
230C22⋊C8C22×C5311yescharacteristic
231C22⋊C8C22×C5311yescharacteristic
232#97C22×C5311yescharacteristic
233C4×C8C22×C5311yescharacteristic
234#41—22no—
235#41—22no—
236C2×Q8—21no—
237C2×Q8C23×C5311yescharacteristic
238C2×Q8C23×C5311yescharacteristic
239C2×D4—21no—
240C2×D4—21no—
241C2×D4C23×C5311yescharacteristic
242C2×D4C23×C5311yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4C23×C5311yescharacteristic
249Q16—22no—
250SD16—22no—
251SD16—22no—
252D8—22no—
253C2×C8C23×C5311yescharacteristic
254C2×C8C23×C5311yescharacteristic
255C2×C8C23×C5311yescharacteristic
256C2×C8C23×C5311yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4D4×C5311yescharacteristic
264C4⋊C4D4×C5311yescharacteristic
265C4⋊C4C23×C5311yescharacteristic
266C4⋊C4C23×C5311yescharacteristic
267C4⋊C4C23×C5311yescharacteristic
268C4⋊C4C23×C5311yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4D4×C5311yescharacteristic
277C22⋊C4D4×C5311yescharacteristic
278C42—21no—
279C42—21no—
280C42—21no—
281C42C23×C5311yescharacteristic
282C42C23×C5311yescharacteristic
283C23—21no—
284C23—21no—
285C23C2×D4×C5311yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—81no—
290Q8#41×C5312yes—
291D4—24no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—81no—
296D4#41×C5312yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4C2×D4×C5311yescharacteristic
311C2×C4C2×D4×C5311yescharacteristic
312C2×C4C2×D4×C5311yescharacteristic
313C2×C4C2×D4×C5311yescharacteristic
314C2×C4C2×D4×C5311yescharacteristic
315C2×C4C24×C5311yescharacteristic
316C8—21no—
317C8—21no—
318C8—21no—
319C8—21no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22C22×D4×C5311yescharacteristic
330C4—21no—
331C4—22no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4C2×#41×C5311yescharacteristic
342C4C2×#41×C5311yescharacteristic
343C2—41no—
344C2—42no—
345C2—81no—
346C2C53×#52511yescharacteristic
347C2C53×#57711yescharacteristic
348C2C53×#57711yescharacteristic
subgroupG/H
center, socleC22×C53C22×D4
commutator, Frattini, cosocleC2×C4C24×C53
prime radicalC53×#449C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ #117×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#542C2semidirect
2. C53×#520C2semidirect
3. C53×#483C2semidirect
4. C53×#481C2semidirect
5. C53×#476C2semidirect
6. C53×#475C2semidirect
7. C53×#464C2semidirect
8. C53×#463C2semidirect
9. C53×#452C2semidirect
11. C53×#442C2semidirect
12. C53×#441C2semidirect
13. C4×SD16×C53C2semidirect
14. C4×SD16×C53C2semidirect
15. C53×#433C2semidirect
168. #3125C53direct (N × H)
175. #542C2×C53semidirect
176. #520C2×C53semidirect
177. #483C2×C53semidirect
178. #481C2×C53semidirect
179. #476C2×C53semidirect
180. #475C2×C53semidirect
181. #464C2×C53semidirect
182. #463C2×C53semidirect
183. #452C2×C53semidirect
185. #442C2×C53semidirect
186. #441C2×C53semidirect
187. C4×SD16C2×C53semidirect
188. C4×SD16C2×C53semidirect
189. #433C2×C53semidirect
190. C53#3125direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#5422
C53×#5202
C53×#4832
C53×#4812
C53×#4762
C53×#4752
C53×#4642
C53×#4632
C53×#4522
C53×#4492
C53×#4422
C53×#4412
C4×SD16×C532
C53×#4332
#312553