Finite Group Databasebeta

← C53×#3121 #20272058

Subgroups — C53×#3121 #20272058

Order 6784.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#542C211yesmaximal, characteristic
2C53×#519C211yesmaximal, characteristic, complemented
3C53×#485C211yesmaximal, characteristic, complemented
4C53×#482C211yesmaximal, characteristic
5C53×#477C211yesmaximal, characteristic, complemented
6C53×#475C211yesmaximal, characteristic, complemented
7C53×#464C211yesmaximal, characteristic, complemented
8C53×#464C211yesmaximal, characteristic, complemented
9C53×#451C211yesmaximal, characteristic, complemented
10C53×#448C211yesmaximal, characteristic
11C53×#441C211yesmaximal, characteristic, complemented
12C53×#440C211yesmaximal, characteristic, complemented
13C4×Q16×C53C211yesmaximal, characteristic, complemented
14C4×SD16×C53C211yesmaximal, characteristic, complemented
15C53×#433C211yesmaximal, characteristic, complemented
16C22×Q8×C53C2211yescharacteristic
17C2×Q16×C53C2211yescharacteristic
18C2×Q16×C53C2211yescharacteristic
19C2×SD16×C53C2211yescharacteristic
20C2×SD16×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×C53—21no—
30C4×Q8×C53—21no—
31C4×Q8×C53C2211yescharacteristic
32C4×Q8×C53C2211yescharacteristic
33C4×Q8×C53C2211yescharacteristic
34C4×Q8×C53C2211yescharacteristic
35C4×Q8×C53C2211yescharacteristic
36C4×D4×C53C2211yescharacteristic
37#117×C53—21no—
38#117×C53—21no—
39#117×C53C2211yescharacteristic
40#117×C53C2211yescharacteristic
41#107×C53C2211yescharacteristic
42#107×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#103×C53C2211yescharacteristic
52#103×C53C2211yescharacteristic
53#102×C53C2211yescharacteristic
54#102×C53C2211yescharacteristic
55C22⋊C8×C53C2211yescharacteristic
56C22⋊C8×C53C2211yescharacteristic
57#97×C53C2211yescharacteristic
58C4×C8×C53C2211yescharacteristic
59C2×Q8×C53—22no—
60C2×Q8×C53—22no—
61C2×Q8×C53—21no—
62C2×Q8×C53—21no—
63C2×Q8×C53C2311yescharacteristic
64C2×Q8×C53C2311yescharacteristic
65C2×Q8×C53C2311yescharacteristic
66C2×D4×C53—21no—
67C2×D4×C53C2311yescharacteristic
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53C2311yescharacteristic
71Q16×C53—22no—
72Q16×C53—22no—
73SD16×C53—22no—
74SD16×C53—22no—
75C2×C8×C53C2311yescharacteristic
76C2×C8×C53C2311yescharacteristic
77C2×C8×C53C2311yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C4⋊C4×C53—21no—
80C4⋊C4×C53—21no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53D411yescharacteristic
87C4⋊C4×C53D411yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C4⋊C4×C53C2311yescharacteristic
90C4⋊C4×C53C2311yescharacteristic
91C4⋊C4×C53C2311yescharacteristic
92C22⋊C4×C53—21no—
93C22⋊C4×C53—21no—
94C22⋊C4×C53—21no—
95C22⋊C4×C53—21no—
96C22⋊C4×C53—21no—
97C22⋊C4×C53—21no—
98C22⋊C4×C53D411yescharacteristic
99C22⋊C4×C53D411yescharacteristic
100C42×C53—21no—
101C42×C53—21no—
102C42×C53—21no—
103C42×C53—21no—
104C42×C53—21no—
105C42×C53—21no—
106C42×C53C2311yescharacteristic
107C42×C53C2311yescharacteristic
108C23×C53—21no—
109C23×C53C2×D411yescharacteristic
110Q8×C53—24no—
111Q8×C53—21no—
112Q8×C53—21no—
113Q8×C53—21no—
114Q8×C53—21no—
115Q8×C53—81no—
116Q8×C53#4112yes—
117Q8×C53#4112yes—
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—81no—
121C2×C4×C53—24no—
122C2×C4×C53—21no—
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×C53C2×D411yescharacteristic
133C2×C4×C53C2×D411yescharacteristic
134C2×C4×C53C2×D411yescharacteristic
135C2×C4×C53C2×D411yescharacteristic
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2411yescharacteristic
138C8×C53—21no—
139C8×C53—21no—
140C8×C53—21no—
141C8×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—41no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53C22×D411yescharacteristic
149C4×C53—22no—
150C4×C53—22no—
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53C2×#4111yescharacteristic
162C4×C53C2×#4111yescharacteristic
163#3121C5311yesmaximal, characteristic, direct product factor, Sylow
164C2×C53—41no—
165C2×C53—81no—
166C2×C53#52511yescharacteristic
167C2×C53#57711yescharacteristic
168C2×C53#57811yescharacteristic
169#542C2×C5311yescharacteristic
170#519C2×C5311yescharacteristic, complemented
171#485C2×C5311yescharacteristic, complemented
172#482C2×C5311yescharacteristic
173#477C2×C5311yescharacteristic, complemented
174#475C2×C5311yescharacteristic, complemented
175#464C2×C5311yescharacteristic, complemented
176#464C2×C5311yescharacteristic, complemented
177#451C2×C5311yescharacteristic, complemented
178#448C2×C5311yescharacteristic
179#441C2×C5311yescharacteristic, complemented
180#440C2×C5311yescharacteristic, complemented
181C4×Q16C2×C5311yescharacteristic, complemented
182C4×SD16C2×C5311yescharacteristic, complemented
183#433C2×C5311yescharacteristic, complemented
184C53#312111yescharacteristic, direct product factor, Sylow
185C22×Q8C22×C5311yescharacteristic
186C2×Q16C22×C5311yescharacteristic
187C2×Q16C22×C5311yescharacteristic
188C2×SD16C22×C5311yescharacteristic
189C2×SD16C22×C5311yescharacteristic
190C4⋊Q8C22×C5311yescharacteristic
191#126—21no—
192#124—21no—
193#124C22×C5311yescharacteristic
194#123—21no—
195C22⋊Q8—21no—
196C22⋊Q8C22×C5311yescharacteristic
197#121C22×C5311yescharacteristic
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8C22×C5311yescharacteristic
201C4×Q8C22×C5311yescharacteristic
202C4×Q8C22×C5311yescharacteristic
203C4×Q8C22×C5311yescharacteristic
204C4×Q8C22×C5311yescharacteristic
205C4×D4C22×C5311yescharacteristic
206#117—21no—
207#117—21no—
208#117C22×C5311yescharacteristic
209#117C22×C5311yescharacteristic
210#107C22×C5311yescharacteristic
211#107C22×C5311yescharacteristic
212#106C22×C5311yescharacteristic
213#106C22×C5311yescharacteristic
214C4⋊C8C22×C5311yescharacteristic
215C4⋊C8C22×C5311yescharacteristic
216#103C22×C5311yescharacteristic
217#103C22×C5311yescharacteristic
218#103C22×C5311yescharacteristic
219#103C22×C5311yescharacteristic
220#103C22×C5311yescharacteristic
221#103C22×C5311yescharacteristic
222#102C22×C5311yescharacteristic
223#102C22×C5311yescharacteristic
224C22⋊C8C22×C5311yescharacteristic
225C22⋊C8C22×C5311yescharacteristic
226#97C22×C5311yescharacteristic
227C4×C8C22×C5311yescharacteristic
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8C23×C5311yescharacteristic
233C2×Q8C23×C5311yescharacteristic
234C2×Q8C23×C5311yescharacteristic
235C2×D4—21no—
236C2×D4C23×C5311yescharacteristic
237C22×C4—21no—
238C22×C4—21no—
239C22×C4C23×C5311yescharacteristic
240Q16—22no—
241Q16—22no—
242SD16—22no—
243SD16—22no—
244C2×C8C23×C5311yescharacteristic
245C2×C8C23×C5311yescharacteristic
246C2×C8C23×C5311yescharacteristic
247C2×C8C23×C5311yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4D4×C5311yescharacteristic
256C4⋊C4D4×C5311yescharacteristic
257C4⋊C4C23×C5311yescharacteristic
258C4⋊C4C23×C5311yescharacteristic
259C4⋊C4C23×C5311yescharacteristic
260C4⋊C4C23×C5311yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4D4×C5311yescharacteristic
268C22⋊C4D4×C5311yescharacteristic
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42C23×C5311yescharacteristic
276C42C23×C5311yescharacteristic
277C23—21no—
278C23C2×D4×C5311yescharacteristic
279Q8—24no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—81no—
285Q8#41×C5312yes—
286Q8#41×C5312yes—
287D4—21no—
288D4—21no—
289D4—81no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—81no—
301C2×C4C2×D4×C5311yescharacteristic
302C2×C4C2×D4×C5311yescharacteristic
303C2×C4C2×D4×C5311yescharacteristic
304C2×C4C2×D4×C5311yescharacteristic
305C2×C4C2×D4×C5311yescharacteristic
306C2×C4C24×C5311yescharacteristic
307C8—21no—
308C8—21no—
309C8—21no—
310C8—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22C22×D4×C5311yescharacteristic
318C4—22no—
319C4—22no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4C2×#41×C5311yescharacteristic
331C4C2×#41×C5311yescharacteristic
332C2—41no—
333C2—81no—
334C2C53×#52511yescharacteristic
335C2C53×#57711yescharacteristic
336C2C53×#57811yescharacteristic
subgroupG/H
center, socleC22×C53C22×D4
commutator, Frattini, cosocleC2×C4C24×C53
prime radical#121×C53C22
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
2. C53×#519C2semidirect
3. C53×#485C2semidirect
5. C53×#477C2semidirect
6. C53×#475C2semidirect
7. C53×#464C2semidirect
8. C53×#464C2semidirect
9. C53×#451C2semidirect
11. C53×#441C2semidirect
12. C53×#440C2semidirect
13. C4×Q16×C53C2semidirect
14. C4×SD16×C53C2semidirect
15. C53×#433C2semidirect
163. #3121C53direct (N × H)
170. #519C2×C53semidirect
171. #485C2×C53semidirect
173. #477C2×C53semidirect
174. #475C2×C53semidirect
175. #464C2×C53semidirect
176. #464C2×C53semidirect
177. #451C2×C53semidirect
179. #441C2×C53semidirect
180. #440C2×C53semidirect
181. C4×Q16C2×C53semidirect
182. C4×SD16C2×C53semidirect
183. #433C2×C53semidirect
184. C53#3121direct (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×#5192
C53×#4852
C53×#4822
C53×#4772
C53×#4752
C53×#4642
C53×#4512
C53×#4482
C53×#4412
C53×#4402
C4×Q16×C532
C4×SD16×C532
C53×#4332
#312153